# Free Mathematics Notes for 9th Class

Updated: 13 Aug 2019

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Welcome to our comprehensive mathematics notes for 9th-class of Khyber Pakhtunkhwa (KPK) all boards, including Peshawar board, Mardan, Swat, Malakand, Kohat, and DI Khan board. Here, we will provide detailed information on the topics covered in class 9th mathematics and tips and tricks for preparing and scoring high in your exams.

The notes are based on the latest syllabus and examination pattern prescribed by the KPK boards for class 9 (IX) SSC-I matric.

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These notes are according to the syllabus of KPK text book. Other board notes will also be uploaded time to time.

## Class 9 Mathematics Board Papers

## Class 9 Mathematics Notes

### Mathematics Class 9 Notes (KPK) Chapter # 1

Ex # 1.1 MCQs/Explanation

Introduction to Matrices

Ex # 1.2 MCQs/Explanation

Types of Matrices

Ex # 1.3 MCQs/Explanation

Addition Matrices

Ex # 1.4 MCQs/Explanation

Multiplication of Matrices

Ex # 1.4 MCQs/Explanation/Review Exercise Unit # 1

Determinant of a Matrix

### Mathematics Class 9 Notes (KPK) Chapter # 2

**1. The number that can be expressed in the form of \frac{p}{q} \ where \ p \ and \ q are integers and q \neq 0 is called _______ numbers.**

O Rational

O Irrational

O Imaginary

O None of these

Answer: Rational

Explanation:

Definition of rational number.

**2. A rational in the form of \frac{p}{q} where \ p \ and q are _______**

O Imaginary

O Complex

O Integers

O All of them

Answer: Integers

Explanation:

**3. A rational in the form of \frac{p}{q} where q _______ 0.**

O Equal

O Not equal

O Both a and b

O None of these

Answer: Not equal

Explanation:

if q=0 then it becomes infinite.

**4. The word Rational is derived from _______**

O Ratio

O Not Ratio

O lota

O All of them

Answer: Ratio

Explanation:

**5. The rules of rational number for p \ and \ q are _______.**

O \frac{p}{q}

O Integers

O q \neq 0

O All of them

Answer: All of them

Explanation:

These all are the rules for ratinal number.

**6. Rational number is denoted by _______.**

O Q

O Q^{\prime}

O Both a and b

O None of these

Answer: Q

Explanation:

**7. Irrational number is denoted by _______**

O Q

O Q^{\prime}

O Both a and b

O None of these

Answer: Q^{\prime}

Explanation:

**8. The word irrational means_______**

O Ratio

O Not Ratio

O lota

O None of these

Answer: Not ratio

Explanation:

**9. Irrational numbers consist of all the numbers which are_______**

O Rational

O Not rational

O lota

O None of these

Answer: Not rational

Explanation:

Numbers other than rational numbers are called Irrational numbers.

**10. 1\frac{3}{4} is_______**

O Rational

O Irrational

O Complex

O None of these

Answer: Rational

Explanation:

1\frac{3}{4}=\frac{7}{4}=1.75

Terminating decimals are Rational numbers.

**11. \sqrt{3} is_______**

O Rational

O Irrational

O Complex

O None of these

Answer: Irrational

Explanation:

\sqrt{3}=1.7320508076 \dots

Non-terminating and non-recurring (repeating) decimals are irrational numbers.

**12. \pi is_______**

O Rational

O Irrational

O Complex

O None of these

Answer: Irrational

Explanation:

The approximate value of \pi \ is \ 3.1415926 \dots which is Non-terminating and non-recuring decimal. Thus \pi is an irrational number.

**13. R is the symbol of _______ number.**

O Rational

O Irrational

O Real

O Both a & b

Answer: Real

Explanation:

**14. The union of rational and irrational numbers is the set of _______ numbers.**

O Complex

O lota

O Prime

O Real

Answer: Real

Explanation:

Rational and irrational numbers are collectively called real number.

**15. Q \cup Q^{\prime}= _______**

O i

O Prime

O R

O N

Answer: R

Explanation:

The union of rational and irrational numbers is the set of Real numbers.

**16. All the numbers on the number line are _______ numbers.**

O i

O Complex

O Real

O None of these

Answer: Real

Explanation:

Every number there is a point on the line. The number associated with a point is called the coordinate of that point on the line and the point is called the graph of the number.

**17. All the numbers on the number line are _______ numbers.**

O Rational

O Irrationa|

O Real

O All of them

Answer: All of them

Explanation:

Rational, Irrational and Real all are shown by number line.

**18. -17 is _______ numbers.**

O Whole

O Natural

O Integers

O All of them

Answer: Integers

Explanation:

Natural and whole number is always be positive.

**19. A whole number is a number that does not contain_______**

O Decimal

O Negative

O Fraction

O All of them

Answer: All of them

Explanation:

Whole number does not contain Decimal, Negative and Fraction.

**20. All terminating and repeating decimals are_______**

O Rational

O Irrational

O Complex

O All of them

Answer: Rational

Explanation:

This is the rule for rational number.

**21. _______ decimals are Rational numbers.**

O Terminating

O Repeating

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Terminating and repeating decimals are rational numbers.

2

**2. \frac{3}{8}=0.375 is _______ decimal.**

O Terminating

O Repeating

O Both a & b

O None of these

Answer: Terminating

Explanation:

A decimal number that contains a finit number of digits after the decimal point is called terminting decimal.

**23. \frac{2}{15}=0.133 \ldots is _______ decimal**

O Terminating

O Repeating

O Both a & b

O None of these

Answer: Repeating

Explanation:

When some digits are repeated in same order after decimal point is called repeating decimal.

**24. 0.1 \overline{3} is _______ decimal**

O Terminating

O Repeating

O Both a & b

O None of these

Answer: Repeating

Explanation:

The bar over the digit 3 means that this digit repeat forever.

**25. Bar over the digit means that this digit is_______**

O Terminated

O Repeated

O Both a & b

O None of these

Answer: Repeated

Explanation:

The bar over the digit means that this digit repeat forever.

**26. A decimal which is non-terminating and non-repeating is called _______ numbers.**

O Rational

O Irrational

O Complex

O All of them

Answer: Irrational

Explanation:

Non-terminating and non-recurring (repeating) decimals are irrational numbers.

**27. The number line help in visualizing the set of _______ numbers.**

O Complex

O Imaginary

O Real

O None of these

Answer: Real

Explanation:

We assume that for any point on a line there is a real number.

**28. For every real number there is a _______ on the line.**

O Point

O Line

O Both a & b

O None of these

Answer: Point

Explanation:

For every real number, there must be a point on the line.

**29. The number associated with a point on the line is called the of _______ that point.**

O Coordinate

O Zero

O Both a & b

O None of these

Answer: Coordinate

Explanation:

It is the graphical representation of real number on line.

**30. The point on the number line is called the _______ of the number.**

O Coordinate

O Zero

O Graph

O None of these

Answer: Graph

Explanation:

**31. The numbers to the right of “0” on a number line are called _______ numbers.**

O Positive

O Negative

O Complex

O None of these

Answer: Positve

Explanation:

The numbers greater than “0” are written on the right side of zero.

**32. The numbers to the left of “O” on a number line are called _______ numbers.**

O Positive

O Negative

O Complex

O None of these

Answer: Negative

Explanation:

The numbers less than “0” are written on the left side of zero.

**33. 0 is _______**

O Positive integers

O Negative integers

O Neither positive nor negative

O Not an integer

Answer: Neither positive nor negative

Explanation:

0 is the only number which is neither positive nor negative.

**34. Repeating decimals are called _______ decimals.**

O Recurring

O Non-Recurring

O Both a & b

O None of these

Answer: Recurring

Explanation:

Repeating decimals are also called Recurring decimals.

**35. Non-Repeating decimals are called _______ decimals.**

O Recurring

O Non-Recurring

O Botha \& b

O None of these

Answer: Non-Recurring

Explanation:

The Non-Repeating decimals are also called Non-Recurring

**36. \frac{5}{27}=0.185185185 \dots is called _______ recurring decimals.**

O Terminating

O Non-Terminating

O Both a & b

O None of these

Answer: Non-Terminating

Explanation:

Here the number of digits repeated infinitely.

**37. \frac{5}{7} is _______ number.**

O Rational

O Irrational

O Imaginary

O None of these

Answer: Rational

Explanation:

\frac{5}{7}=0.714285714285 \dots

Non-terminating recurring (repeating) decimals are rational numbers.

**38. \sqrt{36} is _______ .**

O Rational

O Whole

O Natural

O All of them

Answer: All of them

Explanation:

\sqrt{36}=6

Thus 6 shows Natural, Whole and Rational number at a time.

**39. \frac{14}{3} is _______ number.**

O Rational

O Whole

O Both a & b

O Irrational

Answer: Rational

Explanation:

\frac{14}{3}=4.66666 \dots Non-terminating recurring (repeating) decimals are rational numbers.

**1. The set of Real number is the union of two ________ sets.**

O Zero

O New

O Disjoint

O None of these

Answer: Disjoint

Explanation:

**2. Q \cap Q^{\prime}= ________**]

O Q

O Q^{\prime}

O \emptyset

O All of them

Answer: \emptyset

Explanation:

The intersection of Rational and Irrational set is empty set.

**3. The sum of two real number is also a real number is called ________ property w.r.t Addition.**

O Closure

O Commutative

O Associative

O None of these

Answer: Closure

Explanation:

Statement for Closure Property.

**4 . The ________ of two real number is also a real number is called closure property.**

O Product

O Commutative

O Associative

O None of these

Answer: Product

Explanation:

In Closure property the product of two real number is alway be a real number.

**5. Example of closure property:**

O 7+9=16

O 7 \times 9=63

O Both a & b

O None of these

Answer: Both a & b

Explanation:

In Closure property, the sum and product of two real numbers must be the real number. Thus both a & b obey closure property.

**6. Commutative property w.r.t addition is ________**

O a+b=b+c

O a+c=b+c

O a+b+c=a+b

O a+b=b+c

Answer: a+b=b+c

Explanation:

General form of Commutative property w.r.t Addition.

**7. Commutative property w.r.t multiplication is ________**

O a b=b c

O a c=b c

O a b c=a b

O a b=b c

Answer: a b=b c

Explanation:

General form of Commutative property w.r.t Multiplication.

**8. Commutative property is ________**

O a+b=b+a

O a b=b a

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Both a & b Showed the Commutative property of Addition and Multiplication respectively.

**9. Associative property w.rt Addition is ________**

O a(bc)=(ab)c

O a+(b+c)=(a+b)+c

O Both a & b

O None of these

Answer: a+(b+c)=(a+b)+c

Explanation:

General form of Associative property w.r.t Addition.

**10. Zero is called ________**

O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive identity

Explanation:

Zero (0) is called Additive identity because adding “0” to a number does not change that number.

**11. a+0=0+a=a is**

O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive identity

Explanation:

Zero (0) is called Additive identity because adding “0” to a number does not change that number.

**12. The product of real number and zero is________**

O a

O That number

O Imaginary

O Zero

Answer: Zero

Explanation:

Any number multiplied to zero is always be zero.

**13. 1 is called ________ w.r.t multiplication.**

O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative identity

Explanation:

1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

**14. a \times 1=1 \times a=a is ________ property. O Multiplicative identityg>**

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative identity

Explanation:

1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

O 10

O Zero

O That number

O None of these

Answer: That number

Explanation:

1 is called Multiplicative identity because multiplying “1” to any number does not change that number.

O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive inverse

Explanation:

Definition of Additive inverse

O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive inverse

Explanation:

When a real number and its opposite, the result will always be 0.

O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive inverse

Explanation:

When a real number and its opposite, the result will always be 0.

O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative inverse

Explanation:

When a real number is multiplied by its inverse or reciprocal, the result will always be 1.

O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative inverse

Explanation:

When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative inverse

Explanation:

When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

O a(b+c)=ab+ac

O (b+c)a=ba+ca

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Both a & b showed the Distributive Property of Multiplication over Addition

O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Reflexive

Explanation:

Every number is equal to itself is known as Reflexive property.

O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Symmetric

Explanation:

By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Transitive

Explanation:

If a equal to b under a rule and b equal to c under the same rules then a equal to c is known as transitive property.

O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Symmetric

Explanation:

By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Transitive

Explanation:

If a equal to b under a rule and b equal to c under the same rules then a equal to c is known as transitive property.

O Symmetric

O Reflexive

O None of these

Answer: Reflexive

Explanation:

Every number is equal to itself is known as Reflexive property.

O Multiplicative

O Both a & b

O None of these

Answer: Additive

Explanation:

If we add the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

O Additive

O Multiplicative

O Both a & b

O None of these

Answer: Multiplicative

Explanation:

If we Multiply the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

O Addition

O Multiplication

O Both a & b

O None of these

Answer: Addition

Explanation:

In this, cancelled the non-zero common factor from both side of the equation by Adding or Subtraction.

O Addition

O Multiplication

O Both a & b

O None of these

Answer: Multiplication

Explanation:

In this, cancelled the non-zero common factor from both side of the equation by Multiplication or Divison.

O Increasing

O Decreasing

O Comparing

O Equating

Answer: Comparing

Explanation:

See MCQs No. 34

O a=b

O a>b

O a < b

O All of them

Answer: All of them

Explanation:

Trichotomy property is used for compare two numbers.

O 5=5

O 3 < 5

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Trichotomy property is used for compare two numbers.

O Additive

O Multiplicative

O Transitive

O All of them

Answer: Transitive

Explanation:

If a greater than b under a rule and b greater than c under the same rule then a greater than c is known as transitive property of inequlity.

O Multiplicative

O Transitive

O All of them

Answer: Transitive

Explanation:

If a less than b under a rule and b less than c under the same rule then a less than c is known as transitive property of inequlity.

O Additive

O Multiplicative

O Transitive

O All of them

Answer: Additive

Explanation:

If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.

O Additive

O Multiplicative

O Transitive

O All of them

Answer: Additive

Explanation:

If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is less than right side.

O x \times 2 > 5 \times 2

O x \times 2 < 5 \times 2

O Both a & b

O None of these

Answer: x \times 2 > 5 \times 2

Explanation:

If we multiply the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.

Note:

The number should be positive.

O x \times -2 > 5 \times -2

O x \times -2 < 5 \times -2

O Both a & b

O None of these

Answer: x \times -2 < 5 \times -2

Explanation:

If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

O ac < bc

O ac > bc

O Both a & b

O None of these

Answer: ac < bc

Explanation:

If we multiply the same Positive number to both sides of an inequality, the result will remain same. i.e. left side is less than right side.

O ac < bc

O ac > bc

O Both a & b

O None of these

Answer: ac > bc

Explanation:

If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes greaer than right side.

O ac < bc

O ac > bc

O Both a & b

O None of these

Answer: ac < bc

Explanation:

If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative identity

Explanation:

1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

**15. The product of 1 and a number is________**O 10

O Zero

O That number

O None of these

Answer: That number

Explanation:

1 is called Multiplicative identity because multiplying “1” to any number does not change that number.

**16. The sum of two numbers is zero (0) is called________**O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive inverse

Explanation:

Definition of Additive inverse

**17. If a+a^{\prime}=a^{\prime}+a=0 \ then \ a^{\prime} is called ________ of a .**O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive inverse

Explanation:

When a real number and its opposite, the result will always be 0.

**18. If a+(-a)=-a+a=0 \ then \ -a is called of ________ a .**O Additive identity

O Additive inverse

O Both a & b

O None of these

Answer: Additive inverse

Explanation:

When a real number and its opposite, the result will always be 0.

**19. The product of two numbers is 1 is called________**O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative inverse

Explanation:

When a real number is multiplied by its inverse or reciprocal, the result will always be 1.

**20. If a \cdot a^{-1}=a^{-1} \cdot a=1 \ then \ a^{-1} is called ________ of a .**O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative inverse

Explanation:

When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

**21 If a \cdot \frac{1}{a}=\frac{1}{a}, a=1 \ then \ \frac{1}{a} is called of ________ a .**O Multiplicative identity

O Imaginary

O Multiplicative inverse

O None of these

Answer: Multiplicative inverse

Explanation:

When the Product of two numbers is “1” then it is said to be Multiplicative inverse.

**22. Distributive Property of Multiplication over Addition is ________**O a(b+c)=ab+ac

O (b+c)a=ba+ca

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Both a & b showed the Distributive Property of Multiplication over Addition

**23. If a=a , then it is ________ property.**O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Reflexive

Explanation:

Every number is equal to itself is known as Reflexive property.

**24. If a=b , then also b=a , it is ________ property.**O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Symmetric

Explanation:

By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

**25. If a=b \ and \ b=c then a=c , it is ________ property.**O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Transitive

Explanation:

If a equal to b under a rule and b equal to c under the same rules then a equal to c is known as transitive property.

**26. If y=x^2 \ then \ also \ x^2=y , it is ________ property.**O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Symmetric

Explanation:

By interchanging the sides of an equation doesn’t effect the result is known as symmetric Prperty.

**27. I x+y=z \ and \ z=a+b then x+y=a+b , it is ________ property.**O Transitive

O Symmetric

O Reflexive

O None of these

Answer: Transitive

Explanation:

If a equal to b under a rule and b equal to c under the same rules then a equal to c is known as transitive property.

**28. If 3=3 , then it is ________ property.**O Transitive>O Symmetric

O Reflexive

O None of these

Answer: Reflexive

Explanation:

Every number is equal to itself is known as Reflexive property.

**29. If a=b , then also a+c=b+c , it is ________ property of equality.**O Ad>ditiveO Multiplicative

O Both a & b

O None of these

Answer: Additive

Explanation:

If we add the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

**30. If a=b then also ac=bc , it is ________ property of equality.**O Additive

O Multiplicative

O Both a & b

O None of these

Answer: Multiplicative

Explanation:

If we Multiply the same number or expression on both sides of an equation, the equation does not change which means both the sides remain equal.

**31. If a+c=b+c then a=b , it is Cancellation property w.r.t ________**O Addition

O Multiplication

O Both a & b

O None of these

Answer: Addition

Explanation:

In this, cancelled the non-zero common factor from both side of the equation by Adding or Subtraction.

**32. If ac=bc then a=b , it is Cancellation property w.r.t ________**O Addition

O Multiplication

O Both a & b

O None of these

Answer: Multiplication

Explanation:

In this, cancelled the non-zero common factor from both side of the equation by Multiplication or Divison.

**33. Trichotomy property is used for ________ two numbers.**O Increasing

O Decreasing

O Comparing

O Equating

Answer: Comparing

Explanation:

See MCQs No. 34

**34. Trichotomy property must be true for ________**O a=b

O a>b

O a < b

O All of them

Answer: All of them

Explanation:

Trichotomy property is used for compare two numbers.

**35. Trichotomy property must be true for________**O 5=5

O 3 < 5

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Trichotomy property is used for compare two numbers.

**36. If a > b and b>c then a > c , it is ________ property of inequality.**O Additive

O Multiplicative

O Transitive

O All of them

Answer: Transitive

Explanation:

If a greater than b under a rule and b greater than c under the same rule then a greater than c is known as transitive property of inequlity.

**37. If a < b & b < c then a < c, it is ________ property.**O Additive>O Multiplicative

O Transitive

O All of them

Answer: Transitive

Explanation:

If a less than b under a rule and b less than c under the same rule then a less than c is known as transitive property of inequlity.

**38. If a > b then a+c > b+c , it is ________ property of inequlity.**O Additive

O Multiplicative

O Transitive

O All of them

Answer: Additive

Explanation:

If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.

**39. If a < b then a+c < b+c , it is ________ property.**O Additive

O Multiplicative

O Transitive

O All of them

Answer: Additive

Explanation:

If we add the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is less than right side.

**40. If x > 5 then ________**O x \times 2 > 5 \times 2

O x \times 2 < 5 \times 2

O Both a & b

O None of these

Answer: x \times 2 > 5 \times 2

Explanation:

If we multiply the same number or expression on both sides of an inequality, but the result will remain the same. i.e. left side is greater than right side.

Note:

The number should be positive.

**41. If x>5 then ________**O x \times -2 > 5 \times -2

O x \times -2 < 5 \times -2

O Both a & b

O None of these

Answer: x \times -2 < 5 \times -2

Explanation:

If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

**42. For c > 0 and a < b then Multiplicative property ________**O ac < bc

O ac > bc

O Both a & b

O None of these

Answer: ac < bc

Explanation:

If we multiply the same Positive number to both sides of an inequality, the result will remain same. i.e. left side is less than right side.

**43. For c < 0 and a < b then Multiplicative property ________**O ac < bc

O ac > bc

O Both a & b

O None of these

Answer: ac > bc

Explanation:

If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes greaer than right side.

**44. For c < 0 and a > b then Multiplicative property ________**O ac < bc

O ac > bc

O Both a & b

O None of these

Answer: ac < bc

Explanation:

If we multiply the same Negative number to both sides of an inequality, the result will changed. i.e. left side becomes less than right side.

**
**

O Index

O Radical

O Radicand

O All of them

Answer: Radical

Explanation:

O Index

O Radical

O Radicand

O All of them

Answer: Radicand

Explanation:

3.

O Index

O Radical

O Radicand

O All of them

Answer: Index

Explanation:

O a^n

O a^2

O a^{\frac{2}{n}}

O a^{\frac{1}{n}}

Answer: a^{\frac{1}{n}}

Explanation:

It is the General Exponential form of any radical form

O 0

O 1

O 2

O All of them

Answer: 2

Explanation:

If the index is not given means it is 2 which we cannot write.

O 1

O 0

O -1

O 2

Answer: 1

Explanation:

Square root of 1 will always be 1.

O 1

O 0

O -1

O 2

Answer: 1

Explanation:

Cube root of 1 will always be 1

O 1

O 0

O -1

O 2

Answer: 1

Explanation:

Forth root of 1 will always be 1.

O 6

O 0

O -6

O 4

Answer: 6

Explanation:

\sqrt{36}=\sqrt{6^2}

\sqrt{36}=(6^2)^\frac{1}{2}

\sqrt{36}=6

O 4

O 5

O 6

O 7

Answer: 6

Explanation:

\sqrt[3]{216}=\sqrt[3]{6^3}

\sqrt[3]{216}=(6^3)^\frac{1}{3}

\sqrt[3]{216}=6

O 4

O 5

O 6

O 7

Answer: 4

Explanation:

\sqrt[4]{256}=\sqrt[4]{4^4}

\sqrt[4]{256}=(4^4)^\frac{1}{4}

\sqrt[4]{256}=4

O 4

O 5

O 6

O 7

Answer: 5

Explanation:

\sqrt[4]{625}=\sqrt[4]{5^4}

\sqrt[4]{625}=(5^4)^\frac{1}{4}

\sqrt[4]{625}=5

O 4

O 5

O 6

O 7

Answer: 6

Explanation:

\sqrt[4]{1296}=\sqrt[4]{6^4}

\sqrt[4]{1296}=(6^4)^\frac{1}{4}

\sqrt[4]{1296}=6

O 4

O -4

O Both a & b

O None of these

Answer: Both a & b

Explanation:

This means what numbers squared becomes 16. Thus x \ can \ be \ 4 \ or \ -4 \ like \ (4)^2=16 \ and \ also \ (-4)^2=16.

Hence the value of x=\pm 4 .

O 4

O -4

O Both a & b

O None of these

Answer: 4

Explanation:

Here x is the principal square root of 16, which has always a positive value such is x=4 .

O 4^{th}

O 5^{th}

O 2^{nd}

O Square

Answer: 4^{th}

Explanation:

Here is the index of 4, thus it is called 4^{th} root.

O 4

O -4

O Imaginary

O None of these

Answer: 4

Explanation:

\sqrt[3]{64}=\sqrt[3]{4^3}

\sqrt[3]{64}=(4^3)^\frac{1}{3}

\sqrt[3]{64}=4

O 4

O -4

O Imaginary

O None of these

Answer: -4

Explanation:

If a is negative, then n must be odd for the nth root of a to be a real number.

\sqrt[3]{-64}=\sqrt[3]{(-4)^3}

\sqrt[3]{-64}=\left[(-4)^3\right]^\frac{1}{3}

\sqrt[3]{-64}=-4

O 4

O -4

O Imaginary

O None of these

Answer: Imaginary

Explanation:

If radicand is negative, then index must be odd, here the index is 2 which is even.

Hence, \sqrt{-64}= imaginary

O 1

O 0

O n

O -0

Answer: 0

Explanation:

If a is zero, then

\sqrt[n]{0}=0

O \sqrt[n]{a} \cdot \sqrt[n]{b}

O \sqrt[n]{a}+\sqrt[n]{b}

O \sqrt[n]{a} \cdot \sqrt{b}

O \sqrt{a} \cdot \sqrt[n]{b}

Answer: \sqrt[n]{a} \cdot \sqrt[n]{b}

Explanation:

It is the product rule of Radical.

O \sqrt[n]{a} \cdot \sqrt[n]{b}

O \sqrt[n]{a}+\sqrt[n]{b}

O \sqrt[n]{a}-\sqrt[n]{b}

O \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Answer: \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Explanation:

It is the Quotient rule of Radicand.

O 2 \sqrt{y}

O 10 \sqrt{y}

O 2 \sqrt{2 y}

O 10 \sqrt{2 y}

Answer: 10 \sqrt{2 y}

Explanation:

2 \sqrt{\frac{150 x y}{3 x}}=2 \sqrt{50y}

2 \sqrt{\frac{150 x y}{3 x}}=2 \sqrt{25 \times 2y}

2 \sqrt{\frac{150 x y}{3 x}}=2 \times 5 \sqrt{2y}

2 \sqrt{\frac{150 x y}{3 x}}=10 \sqrt{2y}

O a^{\frac{m}{n}}

O a^{\frac{1}{n}}

O a^{\frac{n}{m}}

O All of them

Answer: a^{\frac{1}{n}}

Explanation:

It is the exponential form of radical.

O \sqrt[n]{a}

O \sqrt[n]{a^m}

O Both a & b

O None of these

Answer: \sqrt[n]{a^m}

Explanation:

a^{\frac{m}{n}}=(a^m)^\frac{1}{n}

a^{\frac{m}{n}}=\sqrt[n]{a^m}

O Exponential

O Radical

O Quadratic

O Cubic

Answer: Radical

Explanation:

This is the way to represent the radical form.

O Exponential

O Radical

O Quadratic

O Cubic

Answer: Exponential

Explanation:

This is the way to represent the Exponential form.

O 16

O -16

O Both a & b

O None of these

Answer: 16

Explanation:

2^4= 2 \times 2 \times 2 \times 2

2^4= 16

O 16

O -16

O Both a & b

O None of these

Answer: -16

Explanation:

-2^4= -(2 \times 2 \times 2 \times 2)

-2^4= -16

O 16

O -16

O Both a & b

O None of these

Answer: 16

Explanation:

(-2)^4= -2 \times -2 \times -2 \times -2

(-2)^4= 16

O a^{\frac{m}{n}}

O a^{\frac{1}{n}}

O a^{\frac{n}{m}}

O All of them

Answer: a^{\frac{m}{n}}

Explanation:

\sqrt[n]{a^m}=(a^m)^\frac{1}{n}

\sqrt[n]{a^m}= a^{\frac{m}{n}}

1. a^m \cdot a^n=

O a^{\frac{m}{n}}

O a^{m-n}

O a^{\frac{n}{m}}

O a^{m+n}

Answer: a^{m+n}

Explanation:

To multiply powers of the same base, keep the base same and add the exponents.

2. \frac{a^m}{a^n}=

O a^{\frac{m}{n}}

O a^{m-n}

O a^{\frac{n}{m}}

O a^{m+n}

Answer: a^{m-n}

Explanation:

To divide two expressions with the same bases and different exponents, keep the base same and subtract the exponents.

3. \left(a^m\right)^n=

O a^{\frac{m}{n}}

O a^{m-n}

O a^{m n}

O a^{m+n}

Answer: a^{m n}

Explanation:

To raise an exponential expression to a power, keep the base same and multiply the exponents.

4. (a b)^n=

O a b^n

O a^n b

O a b

O a^n b^n

Answer: a^n b^n

Explanation:

To multiply different bases with same exponent, keep the exponent same and multiply the expressions with the same exponent.

5. \left(\frac{a}{b}\right)^n=

O \frac{a^n}{b^n}

O \frac{a^n}{b}

O \frac{a}{b^n}

O All of these

Answer: \frac{a^n}{b^n}

Explanation:

To divide the two expressions with the same exponent, keep the exponent same and divide the expressions.

6. a^0=

O a

O 1

O 0

O None of theses

Answer: 0

Explanation:

Any non-zero number raised to the zero power equals one.

7. 400^{\circ}=

O 400

O 1

O 0

O None of these

Answer: 1

Explanation:

Any non-zero number raised to the zero power equals one.

8. a^{-n}=

O \frac{1}{a^n}

O \frac{1}{a^{-n}}

O a^n

O None of these

Answer: \frac{1}{a^n}

Explanation:

If we convert the Numerator having a Negative exponent, moves it to Denominator and the exponent becomes positive.

9. \frac{1}{a^{-n}}=

O \frac{1}{a^n}

O \frac{1}{a^{-n}}

O a^n

O None of these

Answer: a^n

Explanation:

If we convert the Denominator having a Negative exponent, moves it to Numerator and the exponent becomes positive.

10. (-a)^3 \times(-a)^4=

O a

O a^7

O a^4

O -a^7

Answer: -a^7

Explanation:

(-a)^3 \times(-a)^4=(-a)^{3+4}

(-a)^3 \times(-a)^4=(-a)^7

(-a)^3 \times(-a)^4=-a^7

11. \left(-2 a^2 b^3\right)^3=

O 8 a^6 b^9

O 2 a^2 b^3

O 2 a b

O -8 a^6 b^9

Answer: -8 a^6 b^9

Explanation:

\left(-2 a^2 b^3\right)^3=\left(-2 \right)^3 a^{2 \times 3} b^{3 \times 3}

\left(-2 a^2 b^3\right)^3=-8 a^6 b^9

12. \frac{a^0 \cdot b^0}{2}=

O 1

O 2

O \frac{1}{2}

O 0

Answer: \frac{1}{2}

Explanation:

\frac{a^0 \cdot b^0}{2}=\frac{1 \times 1}{2}

\frac{a^0 \cdot b^0}{2}=\frac{1}{2}

13. \left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=

O \frac{a^2}{b^4}

O \frac{a}{b}

O \frac{a^3}{b^6}

O None of these

Answer: \frac{a^3}{b^6}

Explanation:

\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^{2 \times \frac{3}{2}}}{b^{4 \times \frac{3}{2}}}

\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^{1 \times 3}}{b^{2 \times 3}}

\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^3}{b^6}

14. (\sqrt[3]{a})^{\frac{1}{2}}=

O a^{\frac{1}{6}}

O a^{\frac{1}{3}}

O a^{\frac{3}{2}}

O a^{\frac{2}{3}}

Answer: a^{\frac{1}{6}}

Explanation:

See Ex # 2.4 Q No. 3 Part (iv)

15. \sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=

O \sqrt[8]{x^8}

O x^8

O x

O x^2

Answer: x^2

Explanation:

\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=\left(x^8 \right)^{\frac{1}{8}} \left(x^8 \right)^{\frac{1}{8}}

\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=x \cdot x

\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=x^2

O Whole

O Natural

O Real

O Complex

Answer: Complex

Explanation:

Definintion of Complex Number.

O Real

O Imaginary

O Conjugate

O Transpose

Answer: Real

Explanation:

O Real

O Imaginary

O Conjugate

O Transpose

Answer: Imaginary

Explanation:

O -a-b i

O -a+b i

O a-b i

O None of these

Answer: a-b i

Explanation:

A conjugate of a complex number is obtained by changing the sign of imaginary part.

O -a-b i

O -a+b i

O a-b i

O None of these

Answer: a-b i

Explanation:

The conjugate is denoted by \overline{a+bi}

O a=c

O b=d

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Z_1=Z_2 if real parts are equal i.e. a=c and imaginary parts are equal i.e. b=d .

O 1

O i

O -1

O All of them

Answer: -1

Explanation:

i^2=-1

O 1

O i

O -1

O \sqrt{-1}

Answer: \sqrt{-1}

Explanation:

i=\sqrt{-1}

O 8-5i

O 10-8i

O 10+8i

O None of these

Answer:

Explanation:

2i(4-5i)=8i-10i^2

2i(4-5i)=8i-10(-1)

2i(4-5i)=8i+10

2i(4-5i)=10+8i

O 2+3i

O 3-2i

O -13

O 13

Answer: 13

Explanation:

(3-2i)(3+2i)=3^2-(2i)^2

(3-2i)(3+2i)=9-4i^2

(3-2i)(3+2i)=9-4(-1)

(3-2i)(3+2i)=9+4

(3-2i)(3+2i)=13

a. -\sqrt{5}

b. \frac{1}{\sqrt{5}}

c. \sqrt{-3}

d. -5

Answer: -\sqrt{5}

Explanation:

When the sum of two numbers is zero (0)

OR

Additive inverse is obtained by changing the sign. So, the additive inverse of

\sqrt{5} \ is \ -\sqrt{5}

a. Commutative

b. Associative

c. Distributive

d. Closure

Answer: Distributive

Explanation:

Distributive Property is:

a(b+c)=ab+ac

a. 1

b. i

c. -1

d. 0

Answer: -1

Explanation:

\sqrt{-1} \times \sqrt{-1}

\sqrt{-1} \times \sqrt{-1}=i \times i \qquad \because \sqrt{-1}=i

\sqrt{-1} \times \sqrt{-1}=i^2

\sqrt{-1} \times \sqrt{-1}=-1

4. Which of the following represents numbers greater than -3 but less than 6 ?

Answer: {x: -3 < x < 6}

Explanation:

a. -4

b. -2

c. 0

d. 8

Answer: -4

Explanation:

16 \times 2^m=4^{n-8}

Put \ n=8

16 \times 2^m=4^{8-8}

2^4 \times 2^m=4^0

2^4 \times 2^m=1

2^m=\frac{1}{2^4}

2^m=2^{-4}

Thus \ m=-4

a. 1

b. -1

c. -i

d. i

Answer: 1

Explanation:

(i).(-i)=-i^2

(i).(-i)=-(-1)

(i).(-i)=-1

a. 0

b. 1

c. -1

d. R

Answer: 1

Explanation:

1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

a. a positive integer

b. a negative integer

c. neither positive nor negative

d. not an integer

Answer: neither positive nor negative

Explanation:

0 is the only number which is neither positive nor negative.

a. 5

b. -15

c. 5i

d. 15i

Answer: -15

Explanation:

As we have

3i(2+5i)=x+6i

6i+15i^2=x+6i

15(-1)=x \qquad \because \sqrt{-1}=i

-15=x

a. 0

b. 1

c. -1

d. Not defined

Answer: 0

a. 9

b. 9+i

c. 9-i

d. 9i

Answer: 9i

Explanation:

\sqrt{-(-9)^2}=\sqrt{-81}

\sqrt{-(-9)^2}=\sqrt{-1 \times 81}

\sqrt{-(-9)^2}=\sqrt{-1} \times \sqrt{81}

\sqrt{-(-9)^2}=i \times 9 \qquad \because \sqrt{-1}=i

\sqrt{-(-9)^2}=9i

**1. In \sqrt[n]{a},\ then \ \sqrt{ \quad } is called**

O Index

O Radical

O Radicand

O All of them

Answer: Radical

Explanation:

**2. In \sqrt[n]{a}, \ then \ a is called**

O Index

O Radical

O Radicand

O All of them

Answer: Radicand

Explanation:

3.

**In \sqrt[n]{a}, \ then \ n is called**

O Index

O Radical

O Radicand

O All of them

Answer: Index

Explanation:

**4. The exponential form of \sqrt[n]{a} is**

O a^n

O a^2

O a^{\frac{2}{n}}

O a^{\frac{1}{n}}

Answer: a^{\frac{1}{n}}

Explanation:

It is the General Exponential form of any radical form

**5. \sqrt{2} , the index is**

O 0

O 1

O 2

O All of them

Answer: 2

Explanation:

If the index is not given means it is 2 which we cannot write.

**6. \sqrt{1}=**

O 1

O 0

O -1

O 2

Answer: 1

Explanation:

Square root of 1 will always be 1.

**7. \sqrt[3]{1}=**

O 1

O 0

O -1

O 2

Answer: 1

Explanation:

Cube root of 1 will always be 1

**8. \sqrt[3]{1}=**

O 1

O 0

O -1

O 2

Answer: 1

Explanation:

Forth root of 1 will always be 1.

**9. \sqrt{36}=**

O 6

O 0

O -6

O 4

Answer: 6

Explanation:

\sqrt{36}=\sqrt{6^2}

\sqrt{36}=(6^2)^\frac{1}{2}

\sqrt{36}=6

**10. \sqrt[3]{216}=**

O 4

O 5

O 6

O 7

Answer: 6

Explanation:

\sqrt[3]{216}=\sqrt[3]{6^3}

\sqrt[3]{216}=(6^3)^\frac{1}{3}

\sqrt[3]{216}=6

**11. \sqrt[4]{256}=**

O 4

O 5

O 6

O 7

Answer: 4

Explanation:

\sqrt[4]{256}=\sqrt[4]{4^4}

\sqrt[4]{256}=(4^4)^\frac{1}{4}

\sqrt[4]{256}=4

**12. \sqrt[4]{625}=**

O 4

O 5

O 6

O 7

Answer: 5

Explanation:

\sqrt[4]{625}=\sqrt[4]{5^4}

\sqrt[4]{625}=(5^4)^\frac{1}{4}

\sqrt[4]{625}=5

**13. \sqrt[4]{1296}=**

O 4

O 5

O 6

O 7

Answer: 6

Explanation:

\sqrt[4]{1296}=\sqrt[4]{6^4}

\sqrt[4]{1296}=(6^4)^\frac{1}{4}

\sqrt[4]{1296}=6

**14. If x^2=16 , then x=**

O 4

O -4

O Both a & b

O None of these

Answer: Both a & b

Explanation:

This means what numbers squared becomes 16. Thus x \ can \ be \ 4 \ or \ -4 \ like \ (4)^2=16 \ and \ also \ (-4)^2=16.

Hence the value of x=\pm 4 .

**15. If x=\sqrt{16} , then x=**

O 4

O -4

O Both a & b

O None of these

Answer: 4

Explanation:

Here x is the principal square root of 16, which has always a positive value such is x=4 .

**16. \sqrt[4]{1296} is called root**

O 4^{th}

O 5^{th}

O 2^{nd}

O Square

Answer: 4^{th}

Explanation:

Here is the index of 4, thus it is called 4^{th} root.

**17. \sqrt[3]{64}=**

O 4

O -4

O Imaginary

O None of these

Answer: 4

Explanation:

\sqrt[3]{64}=\sqrt[3]{4^3}

\sqrt[3]{64}=(4^3)^\frac{1}{3}

\sqrt[3]{64}=4

**18. \sqrt[3]{-64}=**

O 4

O -4

O Imaginary

O None of these

Answer: -4

Explanation:

If a is negative, then n must be odd for the nth root of a to be a real number.

\sqrt[3]{-64}=\sqrt[3]{(-4)^3}

\sqrt[3]{-64}=\left[(-4)^3\right]^\frac{1}{3}

\sqrt[3]{-64}=-4

**19. \sqrt{-64}=**

O 4

O -4

O Imaginary

O None of these

Answer: Imaginary

Explanation:

If radicand is negative, then index must be odd, here the index is 2 which is even.

Hence, \sqrt{-64}= imaginary

**20. \sqrt[n]{0}=**

O 1

O 0

O n

O -0

Answer: 0

Explanation:

If a is zero, then

\sqrt[n]{0}=0

**21. \sqrt[n]{ab}=**

O \sqrt[n]{a} \cdot \sqrt[n]{b}

O \sqrt[n]{a}+\sqrt[n]{b}

O \sqrt[n]{a} \cdot \sqrt{b}

O \sqrt{a} \cdot \sqrt[n]{b}

Answer: \sqrt[n]{a} \cdot \sqrt[n]{b}

Explanation:

It is the product rule of Radical.

**22. \sqrt[n]{\frac{a}{b}}=**

O \sqrt[n]{a} \cdot \sqrt[n]{b}

O \sqrt[n]{a}+\sqrt[n]{b}

O \sqrt[n]{a}-\sqrt[n]{b}

O \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Answer: \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Explanation:

It is the Quotient rule of Radicand.

**23. 2 \sqrt{\frac{150 x y}{3 x}}=**

O 2 \sqrt{y}

O 10 \sqrt{y}

O 2 \sqrt{2 y}

O 10 \sqrt{2 y}

Answer: 10 \sqrt{2 y}

Explanation:

2 \sqrt{\frac{150 x y}{3 x}}=2 \sqrt{50y}

2 \sqrt{\frac{150 x y}{3 x}}=2 \sqrt{25 \times 2y}

2 \sqrt{\frac{150 x y}{3 x}}=2 \times 5 \sqrt{2y}

2 \sqrt{\frac{150 x y}{3 x}}=10 \sqrt{2y}

**24. \sqrt[n]{a}=**

O a^{\frac{m}{n}}

O a^{\frac{1}{n}}

O a^{\frac{n}{m}}

O All of them

Answer: a^{\frac{1}{n}}

Explanation:

It is the exponential form of radical.

**25. a^{\frac{m}{n}}=**

O \sqrt[n]{a}

O \sqrt[n]{a^m}

O Both a & b

O None of these

Answer: \sqrt[n]{a^m}

Explanation:

a^{\frac{m}{n}}=(a^m)^\frac{1}{n}

a^{\frac{m}{n}}=\sqrt[n]{a^m}

**26. \sqrt{13} is ________ form**

O Exponential

O Radical

O Quadratic

O Cubic

Answer: Radical

Explanation:

This is the way to represent the radical form.

**27. 13^2 is ________ form.**

O Exponential

O Radical

O Quadratic

O Cubic

Answer: Exponential

Explanation:

This is the way to represent the Exponential form.

**28. 2^4=**

O 16

O -16

O Both a & b

O None of these

Answer: 16

Explanation:

2^4= 2 \times 2 \times 2 \times 2

2^4= 16

**29. -2^4=**

O 16

O -16

O Both a & b

O None of these

Answer: -16

Explanation:

-2^4= -(2 \times 2 \times 2 \times 2)

-2^4= -16

**30. (-2)^4=**

O 16

O -16

O Both a & b

O None of these

Answer: 16

Explanation:

(-2)^4= -2 \times -2 \times -2 \times -2

(-2)^4= 16

**31. \sqrt[n]{a^m}=**

O a^{\frac{m}{n}}

O a^{\frac{1}{n}}

O a^{\frac{n}{m}}

O All of them

Answer: a^{\frac{m}{n}}

Explanation:

\sqrt[n]{a^m}=(a^m)^\frac{1}{n}

\sqrt[n]{a^m}= a^{\frac{m}{n}}

O a^{\frac{m}{n}}

O a^{m-n}

O a^{\frac{n}{m}}

O a^{m+n}

Answer: a^{m+n}

Explanation:

To multiply powers of the same base, keep the base same and add the exponents.

2. \frac{a^m}{a^n}=

O a^{\frac{m}{n}}

O a^{m-n}

O a^{\frac{n}{m}}

O a^{m+n}

Answer: a^{m-n}

Explanation:

To divide two expressions with the same bases and different exponents, keep the base same and subtract the exponents.

3. \left(a^m\right)^n=

O a^{\frac{m}{n}}

O a^{m-n}

O a^{m n}

O a^{m+n}

Answer: a^{m n}

Explanation:

To raise an exponential expression to a power, keep the base same and multiply the exponents.

4. (a b)^n=

O a b^n

O a^n b

O a b

O a^n b^n

Answer: a^n b^n

Explanation:

To multiply different bases with same exponent, keep the exponent same and multiply the expressions with the same exponent.

5. \left(\frac{a}{b}\right)^n=

O \frac{a^n}{b^n}

O \frac{a^n}{b}

O \frac{a}{b^n}

O All of these

Answer: \frac{a^n}{b^n}

Explanation:

To divide the two expressions with the same exponent, keep the exponent same and divide the expressions.

6. a^0=

O a

O 1

O 0

O None of theses

Answer: 0

Explanation:

Any non-zero number raised to the zero power equals one.

7. 400^{\circ}=

O 400

O 1

O 0

O None of these

Answer: 1

Explanation:

Any non-zero number raised to the zero power equals one.

8. a^{-n}=

O \frac{1}{a^n}

O \frac{1}{a^{-n}}

O a^n

O None of these

Answer: \frac{1}{a^n}

Explanation:

If we convert the Numerator having a Negative exponent, moves it to Denominator and the exponent becomes positive.

9. \frac{1}{a^{-n}}=

O \frac{1}{a^n}

O \frac{1}{a^{-n}}

O a^n

O None of these

Answer: a^n

Explanation:

If we convert the Denominator having a Negative exponent, moves it to Numerator and the exponent becomes positive.

10. (-a)^3 \times(-a)^4=

O a

O a^7

O a^4

O -a^7

Answer: -a^7

Explanation:

(-a)^3 \times(-a)^4=(-a)^{3+4}

(-a)^3 \times(-a)^4=(-a)^7

(-a)^3 \times(-a)^4=-a^7

11. \left(-2 a^2 b^3\right)^3=

O 8 a^6 b^9

O 2 a^2 b^3

O 2 a b

O -8 a^6 b^9

Answer: -8 a^6 b^9

Explanation:

\left(-2 a^2 b^3\right)^3=\left(-2 \right)^3 a^{2 \times 3} b^{3 \times 3}

\left(-2 a^2 b^3\right)^3=-8 a^6 b^9

12. \frac{a^0 \cdot b^0}{2}=

O 1

O 2

O \frac{1}{2}

O 0

Answer: \frac{1}{2}

Explanation:

\frac{a^0 \cdot b^0}{2}=\frac{1 \times 1}{2}

\frac{a^0 \cdot b^0}{2}=\frac{1}{2}

13. \left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=

O \frac{a^2}{b^4}

O \frac{a}{b}

O \frac{a^3}{b^6}

O None of these

Answer: \frac{a^3}{b^6}

Explanation:

\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^{2 \times \frac{3}{2}}}{b^{4 \times \frac{3}{2}}}

\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^{1 \times 3}}{b^{2 \times 3}}

\left(\frac{a^2}{b^4}\right)^{\frac{3}{2}}=\frac{a^3}{b^6}

14. (\sqrt[3]{a})^{\frac{1}{2}}=

O a^{\frac{1}{6}}

O a^{\frac{1}{3}}

O a^{\frac{3}{2}}

O a^{\frac{2}{3}}

Answer: a^{\frac{1}{6}}

Explanation:

See Ex # 2.4 Q No. 3 Part (iv)

15. \sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=

O \sqrt[8]{x^8}

O x^8

O x

O x^2

Answer: x^2

Explanation:

\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=\left(x^8 \right)^{\frac{1}{8}} \left(x^8 \right)^{\frac{1}{8}}

\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=x \cdot x

\sqrt[8]{x^8} \cdot \sqrt[8]{x^8}=x^2

**1. A number in the form of a+bi where a \ and \ b are real number is called_______ number.**

O Whole

O Natural

O Real

O Complex

Answer: Complex

Explanation:

Definintion of Complex Number.

**2. In complex number a+bi, \ "a" is called part.**

O Real

O Imaginary

O Conjugate

O Transpose

Answer: Real

Explanation:

**3. In complex number a+bi, \ "b" is called part.**

O Real

O Imaginary

O Conjugate

O Transpose

Answer: Imaginary

Explanation:

**4. The conjugate of a+bi is**

O -a-b i

O -a+b i

O a-b i

O None of these

Answer: a-b i

Explanation:

A conjugate of a complex number is obtained by changing the sign of imaginary part.

**5. \overline{a+bi}=**

O -a-b i

O -a+b i

O a-b i

O None of these

Answer: a-b i

Explanation:

The conjugate is denoted by \overline{a+bi}

**6. Let Z_1=a+b i \ and \ Z_2=c+di \ then \ Z_1=Z_2 if**

O a=c

O b=d

O Both a & b

O None of these

Answer: Both a & b

Explanation:

Z_1=Z_2 if real parts are equal i.e. a=c and imaginary parts are equal i.e. b=d .

**7. i^2=**

O 1

O i

O -1

O All of them

Answer: -1

Explanation:

i^2=-1

**8. i=**

O 1

O i

O -1

O \sqrt{-1}

Answer: \sqrt{-1}

Explanation:

i=\sqrt{-1}

**9. 2i(4-5i)**

O 8-5i

O 10-8i

O 10+8i

O None of these

Answer:

Explanation:

2i(4-5i)=8i-10i^2

2i(4-5i)=8i-10(-1)

2i(4-5i)=8i+10

2i(4-5i)=10+8i

**10. (3-2i)(3+2i)=**

O 2+3i

O 3-2i

O -13

O 13

Answer: 13

Explanation:

(3-2i)(3+2i)=3^2-(2i)^2

(3-2i)(3+2i)=9-4i^2

(3-2i)(3+2i)=9-4(-1)

(3-2i)(3+2i)=9+4

(3-2i)(3+2i)=13

**1. The additive inverse of \sqrt{5} is**

a. -\sqrt{5}

b. \frac{1}{\sqrt{5}}

c. \sqrt{-3}

d. -5

Answer: -\sqrt{5}

Explanation:

When the sum of two numbers is zero (0)

OR

Additive inverse is obtained by changing the sign. So, the additive inverse of

\sqrt{5} \ is \ -\sqrt{5}

**2. 2(3+4)=2 \times 3+2 \times 4 , here the property used is.**

a. Commutative

b. Associative

c. Distributive

d. Closure

Answer: Distributive

Explanation:

Distributive Property is:

a(b+c)=ab+ac

**3. \sqrt{-1} \times \sqrt{-1}=**

a. 1

b. i

c. -1

d. 0

Answer: -1

Explanation:

\sqrt{-1} \times \sqrt{-1}

\sqrt{-1} \times \sqrt{-1}=i \times i \qquad \because \sqrt{-1}=i

\sqrt{-1} \times \sqrt{-1}=i^2

\sqrt{-1} \times \sqrt{-1}=-1

4. Which of the following represents numbers greater than -3 but less than 6 ?

Answer: {x: -3 < x < 6}

Explanation:

**5. If n=8 \ and \ 16 \times 2^m=4^{n-8} , then \ m= ?**

a. -4

b. -2

c. 0

d. 8

Answer: -4

Explanation:

16 \times 2^m=4^{n-8}

Put \ n=8

16 \times 2^m=4^{8-8}

2^4 \times 2^m=4^0

2^4 \times 2^m=1

2^m=\frac{1}{2^4}

2^m=2^{-4}

Thus \ m=-4

**6. (i).(-i)=**

a. 1

b. -1

c. -i

d. i

Answer: 1

Explanation:

(i).(-i)=-i^2

(i).(-i)=-(-1)

(i).(-i)=-1

**7. The multiplicative identity of real number is**

a. 0

b. 1

c. -1

d. R

Answer: 1

Explanation:

1 is called Multiplicative identity because multiplying “1” to a number does not change that number.

**8. 0 is**

a. a positive integer

b. a negative integer

c. neither positive nor negative

d. not an integer

Answer: neither positive nor negative

Explanation:

0 is the only number which is neither positive nor negative.

**9. For i=\sqrt{-1} , if \ 3i(2+5i)=x+6i , then \ x= ?**

a. 5

b. -15

c. 5i

d. 15i

Answer: -15

Explanation:

As we have

3i(2+5i)=x+6i

6i+15i^2=x+6i

15(-1)=x \qquad \because \sqrt{-1}=i

-15=x

**10. \sqrt{0}=**

a. 0

b. 1

c. -1

d. Not defined

Answer: 0

**11. \sqrt{-(-9)^2}= ?**

a. 9

b. 9+i

c. 9-i

d. 9i

Answer: 9i

Explanation:

\sqrt{-(-9)^2}=\sqrt{-81}

\sqrt{-(-9)^2}=\sqrt{-1 \times 81}

\sqrt{-(-9)^2}=\sqrt{-1} \times \sqrt{81}

\sqrt{-(-9)^2}=i \times 9 \qquad \because \sqrt{-1}=i

\sqrt{-(-9)^2}=9i

O Standard notation

O Binary notation

O Scientific notation

O All of them

Answer: Scientific notation

Explanation:

O Standard

O Binary

O Decimal

O All of them

Answer: Decimal

Explanation:

O x=a \times 10^n

O x=a \times 10^a

O x=a \times 10^{10}

O None of them

Answer: x=a \times 10^n

Explanation:

Here "a" is real number greater than or equal to 1 but less than 10 and integer power n of 10. i.e.

x=a \times 10^n

O Zero

O First

O Second

O All of them

Answer: First

Explanation:

Rules to convert standard to scientific notation.

O Positive

O Negative

O Both of them

O None of them

Answer: Positive

Explanation:

Rules to convert standard to scientific notation.

O Positive

O Negative

O Both of them

O None of them

Answer: Negative

Explanation:

Rules to convert standard to scientific notation.

O Left

O Right

O Both of them

O None of them

Answer: Left

Explanation:

Rules to convert Scientific to Standard notation.

O Left

O Right

O Both of them

O None of them

Answer: Right

Explanation:

Rules to convert Scientific to Standard notation.

O 1.67 \times 10^7

O 1.67 \times 10^8

O 167.00 \times 10^7

O All of them

Answer: 1.67 \times 10^7

Explanation:

Place the Decimal after first non-zero digit which is 1.

Count the digits towards left which are 7

Here the decimal moves towards left, So power of 10 is positive. Thus,

1.67 \times 10^7

O 3.9 \times 10^7

O 3.9 \times 10^{-7}

O 39 \times 10^8

O 39 \times 10^{-8}

Answer: 3.9 \times 10^{-7}

Explanation:

Place the Decimal after first non-zero digit which is 3.

Count the digits towards right which are 7

Here the decimal moves towards Right, So power of 10 is Negative. Thus,

3.9 \times 10^{-7}

O 5 \times 10^{-5}

O 5.0 \times 10^{-5}

O 5 \times 10^5

O Both a & b

Answer: 5.0 \times 10^{-5}

Explanation:

0.05 \times 10^{-3}=5.0 \times 10^{-2} \times 10^{-3}

0.05 \times 10^{-3}=5.0 \times 10^{-2-3}

0.05 \times 10^{-3}=5.0 \times 10^{-5}

O 0.0000003

O 0.00000315

O 315

O 3150000

Answer: 0.00000315

Explanation:

If the Power of 10 is Negative, So Move the decimal towards Left.

Here the power is -6 then move the decimal upto 6 Digits towards left. Thus,

0.00000315

O 0.0000003

O 0.00000315

O 315

O 3150000

Answer: 3150000

Explanation:

If the Power of 10 is Positive, So Move the decimal towards Right.

Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,

3150000

O -2.6

O 0.0000026

O 2600000

O -2600000

Answer: -2600000

Explanation:

If the Power of 10 is Positive, So Move the decimal towards Right.

Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,

-2600000

O a^y=x

O y=x

O a^x=y

O a^x=y

Answer: a^x=y

Explanation:

We called \log _a y=x \ like \ log \ of \ y \ to \ the \ base \ a \ equal \ to \ x.

O \log _a y=x

O \log _a x=y

O \log x

O \log y

Answer: \log _a y=x

Explanation:

If a^x=y then the index x is called the logarithm of y to the base a and wirtthe as:

\log _a y=x

O \log _{-6} \frac{1}{64}=2

O \log _{64} 2=\frac{1}{64}

O \log _2 \frac{1}{64}=-6

O \log _a y=x

Answer: \log _2 \frac{1}{64}=-6

Explanation:

General form of Conversion is:

a^x=y \longleftrightarrow \log _a y=x

O \log _{10} 1=1

O \log _{10} 1=0

O \log _0 1=1

O \log _a y=x

Answer: \log _{10} 1=0

Explanation:

a^x=y \longleftrightarrow \log _a y=x

O \log _y x=\frac{3}{4}

O \log _{\frac{3}{4}} y=x

O \log _x y=\frac{3}{4}

O All of them

Answer: \log _x y=\frac{3}{4}

Explanation:

a^x=y \longleftrightarrow \log _a y=x

O 2^{-7}=\frac{1}{128}

O 2^{-7}=128

O -7^2=\frac{1}{128}

O a^x=y

Answer: 2^{-7}=\frac{1}{128}

Explanation:

\log _a y=x \longleftrightarrow a^x=y

O 1^a=1

O 1=1

O a^1=1

O None of them

Answer: a^1=1

Explanation:

\log _a y=x \longleftrightarrow a^x=y

O 1^a=0

O 1=1

O a^0=1

O a=1

Answer: a^0=1

Explanation:

\log _a y=x \longleftrightarrow a^x=y

O 4^{\frac{-3}{2}}=\frac{1}{8}

O 4^{\frac{1}{3}}=\frac{-3}{2}

O Both a & b

O None of them

Answer: 4^{\frac{-3}{2}}=\frac{1}{8}

Explanation:

\log _a y=x \longleftrightarrow a^x=y

O (\sqrt{5})^x=125

O \left(5^{\frac{1}{2}}\right)^x=125

O 5^{\frac{x}{2}}=125

O All of them

Answer: (\sqrt{5})^x=125

Explanation:

\log _a y=x \longleftrightarrow a^x=y

O 3^x=5

O 3^2=5 x+1

O 3^2=5 x

O None of them

Answer: 3^2=5 x+1

Explanation:

\log _a y=x \longleftrightarrow a^x=y

O 5

O 125

O 6

O None of them

Answer: 6

Explanation:

See Ex # 3.2

Q No. 3

Part (i)

O \frac{1}{64}

O 64

O 4

O -3

Answer: \frac{1}{64}

Explanation:

See Ex # 3.2

Q No. 3

Part (ii)

O \frac{1}{2}

O 81

O 9

O -3

Answer: \frac{1}{2}

Explanation:

See Ex # 3.2

Q No. 3

Part (iii)

O \frac{5}{8}

O \frac{8}{5}

O 5

O 2

Answer: \frac{8}{5}

Explanation:

See Ex # 3.2

Q No. 3

Part (iv)

O 2

O 7

O 0

O 128

Answer: 128

Explanation:

See Ex # 3.2

Q No. 3

Part (v)

O \frac{5}{10}

O \frac{1}{2}

O 0.5

O All of them

Answer: \frac{1}{2}

Explanation:

See Ex # 3.2

Q No. 3

Part (vi)

O 1

O 10

O 0

O All of them

Answer: 10

Explanation:

See Ex # 3.2

Q No. 3

Part (vii)

O 64

O 2

O 8

O All of them

Answer: 8

Explanation:

See Ex # 3.2

Q No. 3

Part (viii)

O 6561

O 4

O 3

O None of these

Answer: 6561

Explanation:

See Ex # 3.2

Q No. 3

Part (ix)

O Natural

O Common

O Briggs

O Both b & c

Answer: Common

Explanation:

O Natural

O Briggs

O Both a & b

O None of these

Answer: Briggs

Explanation:

O Characteristics

O Mantissa

O Both a & b

O None of these

Answer: Characteristics

Explanation:

In 1.5377 Characteristics is 1.

O Characteristics

O Mantissa

O Both a & b

O None of these

Answer: Mantissa

Explanation:

In 1.5377 Mantissa \ is \ .5377 .

O 1

O .5377

O 1.5377

O None of these

Answer: 1

Explanation:

The digit before the decimal point or Integral part is called characteristics.

O 1

O .5377

O 1.5377

O None of these

Answer: .5377

Explanation:

The decimal fraction part is Mantissa.

O Characteristics

O Mantissa

O Both a & b

O None of these

Answer: Mantissa

Explanation:

The mean difference is the third part to find the mantissa and it is added to mantissa.

O .8825

O .8828

O 2

O 76

Answer: .8828

Explanation:

(i). First ignore the decimal point

(ii). Take first two digits e.g. 76 and proceed along this row until we come to column headed by third digit 3 of the number which is 8825

(iii). Now take fourth digit i.e. 5 and proceed along this row in mean difference column which is 5.

(iv). Now add 8825+3=8828

Thus Mantissa of 763.5 \ is \ .8828

O 0

O 2

O 3

O 4

Answer: 2

Explanation:

First convert 982.5 to Scientific form:

9.825 \times 10^2

Thus Characteristics is 2

O 0

O 1

O 2

O 3

Answer: 3

Explanation:

First convert 7824 to Scientific form:

7.824 \times 10^3

Thus Characteristics is 3

O 0

O 1

O 2

O 3

Answer: 1

Explanation:

First convert 56.3 to Scientific form:

5.63 \times 10^1

Thus Characteristics is 1

O 0

O 1

O 2

O 3

Answer: 0

Explanation:

First convert 7.43 to Scientific form:

7.43 \times 10^0

Thus Characteristics is 0

O 1

O -1

O 2

O -2

Answer: -1

Explanation:

First convert 0.71 to Scientific form:

7.1 \times 10^{-1}

Thus Characteristics is -1

O 0

O 2

O 3

O 4

Answer: 4

Explanation:

First convert 37300 to Scientific form:

3.73 \times 10^4

Thus Characteristics is 4

O 1

O -1

O -3

O -2

Answer:

Explanation:

First convert 0.00159 to Scientific form:

0.00159 \times 10^{-3}

Thus Characteristics is -3

O .3927

O .3938

O 3

O None of these

Answer:

Explanation:

(i). First ignore the decimal point

(ii). Take first two digits e.g. 24 and proceed along this row until we come to column headed by third digit 7 of the number which is 3927

(iii). Now take fourth digit i.e. 6 and proceed along this row in mean difference column which is 11.

(iv). Now add 3927+11=3938

Thus Mantissa of 2476 \ is \ .3938

O 24

O 0.3802

O 2.3802

O None of these

Answer: 0.3802

Explanation:

See Ex # 3.3

Q No. 3

Part No. (ii)

O .6836

O 2.6836

O 2.6830

O None of these

Answer: 2.6836

Explanation:

See Ex # 3.3

Q No. 3

Part No. (iv)

O .8938

O \overline{1} .8938

O 1.8938

O None of these

Answer: \overline{1} .8938

Explanation:

See Ex # 3.3

Q No. 3

Part No. (v)

O \overline{2} .9805

O \overline{2} .9808

O 2.9808

O None of these

Answer: \overline{2} .9808

Explanation:

See Ex # 3.3

Q No. 3

Part No. (vi)

O .8451

O 1.8451

O 2.8451

O None of these

Answer: 2.8451

Explanation:

See Ex # 3.3

Q No. 3

Part No. (viii)

O 1.781

O 17.81

O 1781

O None of these

Answer: 17.81

Explanation:

See Ex # 3.4

Q No. 1

Part No. (i)

O 6.920

O 69.20

O 6920

O None of these

Answer: 6.920

Explanation:

See Ex # 3.4

Q No. 1

Part No. (ii)

O 1.781

O 17.81

O 1781

O 0.01781

Answer: 0.01781

Explanation:

See Ex # 3.4

Q No. 1

Part No. (iv)

O \log _a m+\log _a n

O \log _a m-\log _a n

O n \log _a m

O All of them

Answer: \log _a m+\log _a n

Explanation:

Let \log _a m=x \ and \ \log _a n=y

Write them in Exponential form:

a^x=m \ and \ a^y=n

Now multiply these:

a^x \times a^y=mn

Or

mn=a^x \times a^y

mn=a^{x+y}

Taking \log _a on B.S

\log _a m n=\log _a a^{x+y}

\log _a m n=(x+y) \log _a a

\log _a m n=(x+y)(1) \qquad \log _a a=1

\log _a m n=x+y

\log _a m n=\log _a m+\log _a n

O \log _a m+\log _a n

O \log _a m-\log _a n

O n \log _a m

O All of them

Answer: \log _a m-\log _a n

Explanation:

Let \log _a m=x \ and \ \log _a n=y

Write them in Exponential form:

a^x=m and a^y=n

Now Divide these:

\frac{a^x}{a^y}=\frac{m}{n}

Or

\frac{m}{n}=\frac{a^x}{a^y}

\frac{m}{n}=a^{x-y}

Taking \log _a on B.S

\log _a \frac{m}{n}=\log _a a^{x-y}

\log _a \frac{m}{n}=(x-y) \log _a a

\log _a \frac{m}{n}=(x-y)(1) \qquad \log _a a=1

\log _a \frac{m}{n}=x-y

Hence \ \log _a \frac{m}{n}=\log _a m-\log _a n

O \log _a m+\log _a n

O \log _a m-\log _a n

O n \log _a m

O All of them

Answer: n \log _a m

Explanation:

Let \log _a m=x

In Exponential form:

a^x=m

Or

m=a^x

Taking power ‘ n ‘ on B.S

m^n=\left(a^x\right)^n

m^n=a^{n x}

Taking \log _a on B.S

\log _a m^n=\log _a a^{n x}

\log _a m^n=n x \log _a a

\log _a m^n=n x(1) \qquad \log _a a=1

\log _a m^n=n x

\log _a m^n=n \log _a m

O \log _a \frac{m}{n}

O \log _a m n

O n \log _a m

O All of them

Answer: \log _a m n

Explanation:

\log _a m n=\log _a m+\log _a n

O \log _a \frac{m}{n}

O \log _a m n

O n \log _a m

O All of them

Answer: \log _a \frac{m}{n}

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

O \log _a \frac{m}{n}

O \log _a m n

O \log _a m^n

O All of them

Answer:

Explanation:

\log _a m^n=n \log _a m

O \log m+\log n

O \log m-\log n

O n \log m

O All of them

Answer: \log m+\log n

Explanation:

\log _a m n=\log _a m+\log _a n

O \log m+\log n

O \log m-\log n

O n \log m

O All of them

Answer: \log m-\log n

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

O \log m+\log n

O \log m-\log n

O n \log m

O All of them

Answer: n \log m

Explanation:

\log _a m^n=n \log _a m

O \log \frac{m}{n}

O \log m n

O \log m^n

O All of them

Answer: \log m n

Explanation:

\log _a m n=\log _a m+\log _a n

O \log \frac{m}{n}

O \log m n

O \log m^n

O All of them

Answer: \log \frac{m}{n}

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

O \log \frac{m}{n}

O \log m n

O \log m^n

O All of them

Answer: \log m^n

Explanation:

\log _a m^n=n \log _a m

O \log 2+\log 3

O \log 2-\log 3

O 2 \log 3

O All of them

Answer: \log 2+\log 3

Explanation:

\log _a m n=\log _a m+\log _a n

O \log 2+\log 3

O \log 2-\log 3

O 2 \log 3

O All of them

Answer: \log 2-\log 3

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

O \log 2+1

O \log 2-1

O 2 \log 3

O All of them

Answer: 2 \log 3

Explanation:

\log _a m^n=n \log _a m

O \log 2 \times 3

O \log 6

O \log 2

O Both a & b

Answer: \log 6

Explanation:

\log 2+\log 3 =\log 2\times 3

\log 2+\log 3 =\log 6

O \log \frac{2}{3}

O \log 2 \times 3

O \log 3^2

O All of them

Answer: \log \frac{2}{3}

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

O \log \frac{2}{3}

O \log 2 \times 3

O \log 3^2

O All of them

Answer: \log 3^2

Explanation:

\log _a m^n=n \log _a m

2 \log 3= \log 3^2

2 \log 3= \log 9

O 6

O 7

O 24

O 42

Answer: 42

Explanation:

\log _2 6+\log _2 7=\log _2 a

As \log _a m n=\log _a m+\log _a n

\log _2 6 \times 7=\log _2 a

\log _2 42=\log _2 a

Thus \ a=42

O \log _a n

O \log _a m

O Both a & b

O None of these

Answer: \log _a n

Explanation:

Let \log _a m=x and \log _m n=y

Write them in Exponential form:

a^x=m \ and \ m^y=n

Now multiply these:

As a^{x y}=\left(a^x\right)^y

But \left(a^x\right)^y=m

So a^{x y}=(m)^y=n

Then a^{x y}=n

Taking \log _a on B.S

\log _a a^{x y}=\log _a n

(x y) \log _a a=\log _a n

x y(1)=\log _a n \qquad As \ \log _a a=1

Now

\log _a m \log _m n=\log _a n

O \log _5 2

O \log _2 5

O Both a & b

O None of these

Answer: \log _2 5

Explanation:

\log _a m \log _m n=\log _a n

O \log _5 2

O \log _2 5

O Both a & b

O None of these

Answer: \log _2 5

Explanation:

\log _a m \log _m n=\log _a n

O Logarithm

O Change of Base

O Change of Logarithm

O None of these

Answer: Change of Base

Explanation:

O \log _m n

O \log _t r

O \log 10

O None of these

Answer: \log _m n

Explanation:

O \log _m n

O \log _t r

O \log 10

O None of these

Answer: \log _t r

Explanation:

O 0

O 1

O 10

O None of these

Answer: 1

Explanation:

O 0

O 1

O 10

O None of these

Answer: 1

Explanation:

O 0

O 1

O 10

O None of these

Answer: 1

Explanation:

O 0

O 1

O 10

O None of these

Answer: 0

Explanation:

O 0

O 1

O 10

O None of these

Answer:

Explanation:

O -1

O -2

O 2

O Does not exist

Answer: -2

Explanation:

\log _9 \frac{1}{81}=\log _9 \frac{1}{9^2}

\log _9 \frac{1}{81}=\log _9 9^{-2}

\log _9 \frac{1}{81}=-2 \log _9 9

\log _9 \frac{1}{81}=-2(1)

\log _9 \frac{1}{81}=-2

O 64

O 3^2

O 3

O 2^8

Answer: 3

Explanation:

\log _2 8=x

\log _2 2^3=x

3 \log _2 2=x

3(1)=x

3=x

O 10

O e

O \pi

O 5

Answer: 10

Explanation:

O -1

O -\frac{1}{2}

O \frac{1}{2}

O 2

Answer: \frac{1}{2}

Explanation:

\log \sqrt{10} =\log (10)^{\frac{1}{2}}

\log \sqrt{10} =\frac{1}{2} \log 10

\log \sqrt{10} =\frac{1}{2}(1)

\log \sqrt{10} =\frac{1}{2}

O 2

O 1

O 0

O 10

Answer: 1

Explanation:

O t=m^b

O b^m=t

O m=b^t

O m^t=b

Answer: m=b^t

Explanation:

O 2

O 3

O 0

O 1

Answer: 1

Explanation:

O -5

O 5

O -4

O 4

Answer: -5

Explanation:

O -1

O -\frac{1}{2}

O \frac{1}{2}

O 2

Answer: -\frac{1}{2}

Explanation:

\log _7 \frac{1}{\sqrt{7}} =\log _7 \frac{1}{(7)^{\frac{1}{2}}}

\log _7 \frac{1}{\sqrt{7}} =\log _7 7^{-\frac{1}{2}}

\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2} \log _7 7

\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}(1)

\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}

O 10

O e

O \pi

O 1

Answer:

Explanation:

O \log m\log n

O \log m-\log n

O \log mn

O \log \frac{m}{n}

Answer: \log mn

Explanation:

O 6.9 \times 10^3

O 6.9 \times 10^{-2}

O 0.69 \times 10^3

O 69 \times 10^2

Answer: 6.9 \times 10^{-2}

Explanation:

O \ln \frac{x}{y}

O \ln x y^2

O \ln \frac{x^2}{y}

O \ln \frac{x}{y^2}

Answer: \ln \frac{x}{y^2}

Explanation:

1. (a+b)^2=

Answer:

a^2+b^2+2 a b

2. (a-b)^2=

Answer:

a^2+b^2-2 a b

3. \left(x+\frac{1}{x}\right)^2=

Answer:

x^2+\frac{1}{x^2}+2(x)\left(\frac{1}{x}\right)

4. \left(x-\frac{1}{x}\right)^2=

Answer:

x^2+\frac{1}{x^2}-2(x)\left(\frac{1}{x}\right)

5. \left(x+\frac{1}{x}\right)^2=

Answer:

x^2+\frac{1}{x^2}+2

6. \left(x-\frac{1}{x}\right)^2=

Answer:

x^2+\frac{1}{x^2}-2

7. (a+b)^2+(a-b)^2=

Answer:

2\left(a^2+b^2\right)

8. (a+b)^2-(a-b)^2=

Answer:

4 a b

9. (a+b)(a-b)=

Answer:

a^2-b^2

10. (a+b+c)^2=

Answer:

a^2+b^2+c^2+2(a b+b c+c a)

11. (a+b)^3=

Answer:

a^3+b^3+3 a b(a+b)

12. (a-b)^3=

Answer:

a^3-b^3-3 a b(a-b)

13. \left(x+\frac{1}{x}\right)^3=

Answer:

x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right)

14. \left(x-\frac{1}{x}\right)^3=

Answer:

x^3-\frac{1}{x^3}-3\left(x-\frac{1}{x}\right)

15. a^3+b^3=

Answer:

(a+b)\left(a^2-a b+b^2\right)

16. a^3-b^3=

Answer:

(a-b)\left(a^2+a b+b^2\right)

17. x^3+\frac{1}{x^3}=

Answer:

\left(x+\frac{1}{x}\right)\left(x^2-(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\right)

18. x^3-\frac{1}{x^3}=

Answer:

\left(x-\frac{1}{x}\right)\left(x^2+(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\right)

19. x^3+\frac{1}{x^3}=

Answer:

\left(x+\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}-1\right)

20. x^3-\frac{1}{x^3}=

Answer:

\left(x-\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}+1\right)

21. a^2+2 a b+b^2=

Answer:

(a+b)^2

22. a^2-2 a b+b^2=

Answer:

(a-b)^2

23) x^2+2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}=

Answer:

\left(x+\frac{1}{x}\right)^2

24) x^2-2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}=

Answer:

\left(x-\frac{1}{x}\right)^2

25) x^2+2+\frac{1}{x^2}=

Answer:

\left(x+\frac{1}{x}\right)^2

26) x^2-2+\frac{1}{x^2}=

Answer:

\left(x-\frac{1}{x}\right)^2

27) a^2-b^2=

Answer:

(a+b)(a-b)

28) (a+b)^2-c^2=

Answer:

(a+b+c)(c+b-c)

29) (a+b)^3=

Answer:

a^3+3 a^2 b+3 a b^2+b^3

30) (a-b)^3=

Answer:

a^3-3 a^2 b+3 a b^2-b^3

31) (a+b)\left(a^2-a b+b^2\right)=

Answer:

a^3+b^3

32) (a-b)\left(a^2+a b+b^2\right)=

Answer:

a^3-b^3

1. A number of the form of \sqrt[n]{a} is called ______, where a is a positive rational number.

O Conjugate

O Surd

O Rational

O None of these

Answer:

Surd

2. The rules for surd is/are _________

O It is irrational

O It is a root

O A root of a rational number.

O All of these

Answer:

All of these

3. In a surd, \sqrt[n]{a}, n is called ______ of the surd.

O type

O order

O conjugate

O None of these

Answer:

Order

4. \sqrt{3} is

O a surd

O not a surd

O conjugate of a surd

O None of these

Answer: a surd

Explanation: Here is the root of rational number, so it is a surd

5. \sqrt{5+\sqrt{3}} is

O a surd

O not a surd

O conjugate of a surd

O None of these

Answer: not a surd

Explanation: Here the root of irrational number, so it is not a surd.

6. \sqrt[3]{8} is

O a surd

O not a surd

O conjugate of a surd

O None of these

Answer: not a surd

Explanation:

\sqrt[3]{8} is not a surd because its value is 2 which is rational.

7. \sqrt{3} is called _____ surd.

O Quadratic

O Cubic

O Biquadratic

O None of these

Answer: Quadratic

Explanation: Here the surd index order is 2, so it is called quadratic.

8. \sqrt[3]{3} is called _____ surd.

O Quadratic

O Cubic

O Biquadratic

O None of these

Answer: Cubic

Explanation:

Here the surd index order is 3, so it is called cubic.

9. \sqrt[4]{3} is called _____ surd.

O Quadratic

O Cubic

O Biquadratic

O None of these

Answer: Biquadratic

Explanation:

Here the surd index order is 4, so it is called biquadratic.

10. \sqrt[3]{3} \ and \ \sqrt[3]{5} is called _____ surds.

O Quadratic

O equiradical

O Biquadratic

O None of these

Answer: equiradical

Explanation:

Surds of the same order are called equiradical surds which is 3.

11. \sqrt{3} is called _____ surd.

O Monomial

O Binomial

O Trinomial

O None of these

Answer: Monomial

Explanation:

Surd consists of one term is called monomial.

12. \sqrt{3}-2 is called _____ surd.

O Monomial

O Binomial

O Trinomial

O None of these

Answer: Binomial

Explanation:

Surd consists of Two terms is called binomial.

13. 2-\sqrt{3}+\sqrt{5} is called _____ surd.

O Monomial

O Binomial

O Trinomial

O None of these

Answer: Trinomial

Explanation:

Surd consists of three terms is called trinomial.

14. Expressions containing two or more surds are called _____ surds.

O Monomial

O Binomial

O Compound

O None of these

Answer:

Compound

15. \sqrt[3]{5} is called _____ surd.

O an entire

O mixed

O Trinomial

O None of these

Answer: an entire

Explanation:

If the coefficient of the surd is \pm1 is called an entire surd.

16. -4 \sqrt[3]{5} is called _____ surd.

O an entire

O mixed

O Trinomial

O None of these

Answer: mixed

Explanation:

If the coefficient of the surd is other than \pm1 is called mixed surd.

17. \sqrt{8} and \sqrt{18} are _____ surds.

O Similar

O Like

O Both a & b

O None of these

Answer: Both a & b

Explanation:

\sqrt{8}=\sqrt{4 \times 2}

\sqrt{8}=2\sqrt{2}

AND

\sqrt{18}=\sqrt{9 \times 2}

\sqrt{18}=3\sqrt{2}

18. Conjugate of 3-\sqrt{5} is

O -3-\sqrt{5}

O -3+\sqrt{5}

O 3+\sqrt{5}

O \sqrt{3}+\sqrt{5}

Answer: 3+\sqrt{5}

Explanation:

The conjugate of

a\sqrt{x}+b\sqrt{y} is

a\sqrt{x}-b\sqrt{y}

1. m r^2+3 m r^2-5 m r^2=

O -m r^2

O -m r

O -m r

O m r^2

Answer: -m r^2

Explanation:

m r^2+3 m r^2-5 m r^2

=4 m r^2-5 m r^2

=-m r^2

2. \left(x^3 y^2\right)\left(x^2 y^3\right)=

O x^5 y^5

O x^5 y^4

O x^4 y^5

O x^4 y^4

Answer: x^5 y^5

Explanation:

\left(x^3 y^2\right)\left(x^2 y^3\right)

=x^{3+2} y^{2+3}

=x^5 y^5

3. \left(4 x y^4\right)^3=

O 64 x^3 y^8

O 64 x^3 y^{10}

O 64 x^3 y^{12}

O 64 x^3 y^7

Answer: 64 x^3 y^{12}

Explanation:

\left(4 x y^4\right)^3

=4^3 x^3 y^{4 \times 3}

=64 x^3 y^{12}

4. (7x+4 y)-(3 x-6 y)=

O 3 x

O 2 x+10 y

O 4 x+y

O 4 x+10 y

Answer: 4 x+10 y

Explanation:

(7 x+4 y)-(3 x-6 y)

=7 x+4 y-3 x+6 y

=7 x-3 x+4 y+6 y

=4 x+10 y

5. (a+b)^2-(a-b)^2=

O 4 a b

O 2\left(a^2+b^2\right)

O a^2-4 a b+2 b^2

O a^4-b^4

Answer:

4 a b

6. (a+b+c)^2=

O a^2+b^2+c^2

O a^2+b^2+c^2+2(a+b+c)

O a^2+b^2+c^2+2(a b+b c+c a)

O a+b+c+2(a b+b c+c a)

Answer:

a^2+b^2+c^2+2(a b+b c+c a)

7. a^3+b^3=

O (a+b)^3-2 a b(a+b)

O (a+b)\left(a^2+a b+b^2\right)

O (a-b)\left(a^2-a b+b^2\right)

O (a+b)\left(a^2-a b+b^2\right)

Answer:

(a+b)\left(a^2-a b+b^2\right)

8. Conjugate of 3-\sqrt{5} is

O -3-\sqrt{5}

O -3+\sqrt{5}

O 3+\sqrt{5}

O \sqrt{3}+\sqrt{5}

Answer: 3+\sqrt{5}

Explanation:

The conjugate of

a\sqrt{x}+b\sqrt{y} is

a\sqrt{x}-b\sqrt{y}

9. Which of the following expression is equivalent to the expression \left(m^2+4\right)^{-\frac{1}{2}}

O \frac{m^2+4}{2}

O -\sqrt{m^2+4}

O \frac{1}{\sqrt{m^2+4}}

O \frac{1}{m+2}

Answer: \frac{1}{\sqrt{m^2+4}}

Explanation:

\left(m^2+4\right)^{-\frac{1}{2}}

=\frac{1}{(m 2+4)^{\frac{1}{2}}}

=\frac{1}{\sqrt{m^2+4}}

10. For which of the following expression a+b is not a factor?

O a^2-b^2

O a^2+b^2

O a^3+b^3

O a^4+b^4

Answer: a^2+b^2

Explanation:

a^2+b^2 cannot be factorize further, so

a+b is not its factor.

11. \frac{p}{q} \div \frac{r}{q} \cdot \frac{p}{q}=

O \frac{p}{q r}

O \frac{p^2}{q^2 r}

O \frac{p}{q^2 r^2}

O \frac{p^2}{q r}

Answer: \frac{p^2}{q r}

Explanation:

\frac{p}{q} \div \frac{r}{q} \cdot \frac{p}{q}

=\frac{p}{q} \cdot \frac{q}{r} \cdot \frac{p}{q}

=\frac{p}{q} \cdot \frac{1}{r} \cdot \frac{p}{1}

=\frac{p^2}{q r}

12. Evaluate k^2(2 l-3 m)

when k=-2, l=3, m=4

O 24

O -24

O 6

O 4

Answer:

Explanation:

k^2(2 l-3 m)

=(-2)^2[2(3)-3(4)]

=4(6-12)

=4(-6)

=-24

13. Evaluate 5 m \sqrt{k^2+l^2}

when k=-2, l=3, m=4.

O 10 \sqrt{13}

O 20

O 20 \sqrt{13}

O \sqrt{13}

Answer:

Explanation:

5 m \sqrt{k^2+l^2}

=5(4) \sqrt{(-2)^2+(3)^2}

=20 \sqrt{4+9}

=20 \sqrt{13}

### Mathematics Class 9 Notes (KPK) Chapter # 3

**1. __________ is a way of writing numbers that are too big or too small to be easily written in decimal form.**

O Standard notation

O Binary notation

O Scientific notation

O All of them

Answer: Scientific notation

Explanation:

**2. Scientific notation is a way of writing numbers that are too big or too small to be easily written in _________ form.**

O Standard

O Binary

O Decimal

O All of them

Answer: Decimal

Explanation:

**3. General form of scientific notation is**

O x=a \times 10^n

O x=a \times 10^a

O x=a \times 10^{10}

O None of them

Answer: x=a \times 10^n

Explanation:

Here "a" is real number greater than or equal to 1 but less than 10 and integer power n of 10. i.e.

x=a \times 10^n

**4. In standard to scientific notation, the decimal should place after ____________ non – zero digit.**

O Zero

O First

O Second

O All of them

Answer: First

Explanation:

Rules to convert standard to scientific notation.

**5. From standard to scientific notation, if decimal moved towards left, then power of 10 will be ____________**

O Positive

O Negative

O Both of them

O None of them

Answer: Positive

Explanation:

Rules to convert standard to scientific notation.

**6. From standard to scientific notation, if decimal moved towards right, then power of 10 will be ____________**

O Positive

O Negative

O Both of them

O None of them

Answer: Negative

Explanation:

Rules to convert standard to scientific notation.

**7. From scientific to standard notation, it the exponent of 10 is negative, the decimal will move towards ___________**

O Left

O Right

O Both of them

O None of them

Answer: Left

Explanation:

Rules to convert Scientific to Standard notation.

**8. From scientific to standard notation, it the exponent of 10 is positive, the decimal will move towards ___________**

O Left

O Right

O Both of them

O None of them

Answer: Right

Explanation:

Rules to convert Scientific to Standard notation.

**9. The scientific form of 16700000 is**

O 1.67 \times 10^7

O 1.67 \times 10^8

O 167.00 \times 10^7

O All of them

Answer: 1.67 \times 10^7

Explanation:

Place the Decimal after first non-zero digit which is 1.

Count the digits towards left which are 7

Here the decimal moves towards left, So power of 10 is positive. Thus,

1.67 \times 10^7

**10. The scientific form of 0.00000039 is**

O 3.9 \times 10^7

O 3.9 \times 10^{-7}

O 39 \times 10^8

O 39 \times 10^{-8}

Answer: 3.9 \times 10^{-7}

Explanation:

Place the Decimal after first non-zero digit which is 3.

Count the digits towards right which are 7

Here the decimal moves towards Right, So power of 10 is Negative. Thus,

3.9 \times 10^{-7}

**11. The scientific form of 0.05 \times 10^{-3} is**

O 5 \times 10^{-5}

O 5.0 \times 10^{-5}

O 5 \times 10^5

O Both a & b

Answer: 5.0 \times 10^{-5}

Explanation:

0.05 \times 10^{-3}=5.0 \times 10^{-2} \times 10^{-3}

0.05 \times 10^{-3}=5.0 \times 10^{-2-3}

0.05 \times 10^{-3}=5.0 \times 10^{-5}

**12. The standard form of 3.15 \times 10^{-6} is**

O 0.0000003

O 0.00000315

O 315

O 3150000

Answer: 0.00000315

Explanation:

If the Power of 10 is Negative, So Move the decimal towards Left.

Here the power is -6 then move the decimal upto 6 Digits towards left. Thus,

0.00000315

**13. The standard form of 3.15 \times 10^{6} is**

O 0.0000003

O 0.00000315

O 315

O 3150000

Answer: 3150000

Explanation:

If the Power of 10 is Positive, So Move the decimal towards Right.

Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,

3150000

**14. The standard form of -2.6 \times 10^{6} is**

O -2.6

O 0.0000026

O 2600000

O -2600000

Answer: -2600000

Explanation:

If the Power of 10 is Positive, So Move the decimal towards Right.

Here the power is 6 then move the decimal upto 6 Digits towards Right. Thus,

-2600000

**1. The exponential form of \log _a y=x is**

O a^y=x

O y=x

O a^x=y

O a^x=y

Answer: a^x=y

Explanation:

We called \log _a y=x \ like \ log \ of \ y \ to \ the \ base \ a \ equal \ to \ x.

**2. The logarithm form of a^x=y is**

O \log _a y=x

O \log _a x=y

O \log x

O \log y

Answer: \log _a y=x

Explanation:

If a^x=y then the index x is called the logarithm of y to the base a and wirtthe as:

\log _a y=x

**3. The logarithm form of 2^{-6}=\frac{1}{64} is**

O \log _{-6} \frac{1}{64}=2

O \log _{64} 2=\frac{1}{64}

O \log _2 \frac{1}{64}=-6

O \log _a y=x

Answer: \log _2 \frac{1}{64}=-6

Explanation:

General form of Conversion is:

a^x=y \longleftrightarrow \log _a y=x

**4. The logarithm form of 10^{\circ}=1 is**

O \log _{10} 1=1

O \log _{10} 1=0

O \log _0 1=1

O \log _a y=x

Answer: \log _{10} 1=0

Explanation:

a^x=y \longleftrightarrow \log _a y=x

**5. The logarithm form of x^{\frac{3}{4}}=y is**

O \log _y x=\frac{3}{4}

O \log _{\frac{3}{4}} y=x

O \log _x y=\frac{3}{4}

O All of them

Answer: \log _x y=\frac{3}{4}

Explanation:

a^x=y \longleftrightarrow \log _a y=x

**6. The exponential form of \log _2 \frac{1}{128}=-7 is**

O 2^{-7}=\frac{1}{128}

O 2^{-7}=128

O -7^2=\frac{1}{128}

O a^x=y

Answer: 2^{-7}=\frac{1}{128}

Explanation:

\log _a y=x \longleftrightarrow a^x=y

**7. The exponential form of \log _a a=1 is**

O 1^a=1

O 1=1

O a^1=1

O None of them

Answer: a^1=1

Explanation:

\log _a y=x \longleftrightarrow a^x=y

**8. The exponential form of \log_a 1=0 is**

O 1^a=0

O 1=1

O a^0=1

O a=1

Answer: a^0=1

Explanation:

\log _a y=x \longleftrightarrow a^x=y

**9. The exponential form of \log_4 \frac{1}{8}=\frac{-3}{2} is**

O 4^{\frac{-3}{2}}=\frac{1}{8}

O 4^{\frac{1}{3}}=\frac{-3}{2}

O Both a & b

O None of them

Answer: 4^{\frac{-3}{2}}=\frac{1}{8}

Explanation:

\log _a y=x \longleftrightarrow a^x=y

**10. The exponential form of \log _{\sqrt{5}} 125=x is**

O (\sqrt{5})^x=125

O \left(5^{\frac{1}{2}}\right)^x=125

O 5^{\frac{x}{2}}=125

O All of them

Answer: (\sqrt{5})^x=125

Explanation:

\log _a y=x \longleftrightarrow a^x=y

**11. The exponential form of \log _3(5 x+1)=2 is**

O 3^x=5

O 3^2=5 x+1

O 3^2=5 x

O None of them

Answer: 3^2=5 x+1

Explanation:

\log _a y=x \longleftrightarrow a^x=y

**12. In \log _{\sqrt{5}} 125=x , the Value of x is**

O 5

O 125

O 6

O None of them

Answer: 6

Explanation:

See Ex # 3.2

Q No. 3

Part (i)

**13. In log x=-3 , the Value of x is**

O \frac{1}{64}

O 64

O 4

O -3

Answer: \frac{1}{64}

Explanation:

See Ex # 3.2

Q No. 3

Part (ii)

**14. In \log _{81} 9=x , the Value of x is**

O \frac{1}{2}

O 81

O 9

O -3

Answer: \frac{1}{2}

Explanation:

See Ex # 3.2

Q No. 3

Part (iii)

**15. In \log _3(5 x+1)=2 , the Value of x is**

O \frac{5}{8}

O \frac{8}{5}

O 5

O 2

Answer: \frac{8}{5}

Explanation:

See Ex # 3.2

Q No. 3

Part (iv)

**16. In \log _2 x=7 , the Value of x is**

O 2

O 7

O 0

O 128

Answer: 128

Explanation:

See Ex # 3.2

Q No. 3

Part (v)

**17. In \log _x 0.25=2 , the Value of x is**

O \frac{5}{10}

O \frac{1}{2}

O 0.5

O All of them

Answer: \frac{1}{2}

Explanation:

See Ex # 3.2

Q No. 3

Part (vi)

**18. In \log _x(0.001)=-3 , the Value of x is**

O 1

O 10

O 0

O All of them

Answer: 10

Explanation:

See Ex # 3.2

Q No. 3

Part (vii)

**19. In \log _x \frac{1}{64}=-2 , the Value of x is**

O 64

O 2

O 8

O All of them

Answer: 8

Explanation:

See Ex # 3.2

Q No. 3

Part (viii)

**20. In \log _{\sqrt{3}} x=16 , the Value of x is**

O 6561

O 4

O 3

O None of these

Answer: 6561

Explanation:

See Ex # 3.2

Q No. 3

Part (ix)

**1. Logarithms having base 10 are called________ Logarithms**

O Natural

O Common

O Briggs

O Both b & c

Answer: Common

Explanation:

**2. Common logarithm is also called __________ logarithm**

O Natural

O Briggs

O Both a & b

O None of these

Answer: Briggs

Explanation:

**3. The digit before the decimal point or integral part is called _____________**

O Characteristics

O Mantissa

O Both a & b

O None of these

Answer: Characteristics

Explanation:

In 1.5377 Characteristics is 1.

**4. The decimal fraction part is called ________**

O Characteristics

O Mantissa

O Both a & b

O None of these

Answer: Mantissa

Explanation:

In 1.5377 Mantissa \ is \ .5377 .

**5. In 1.5377 , characteristics is**

O 1

O .5377

O 1.5377

O None of these

Answer: 1

Explanation:

The digit before the decimal point or Integral part is called characteristics.

**6. In 1.5377 , Mantissa is**

O 1

O .5377

O 1.5377

O None of these

Answer: .5377

Explanation:

The decimal fraction part is Mantissa.

**7. The mean difference digits are added to ______________**

O Characteristics

O Mantissa

O Both a & b

O None of these

Answer: Mantissa

Explanation:

The mean difference is the third part to find the mantissa and it is added to mantissa.

**8 The mantissa of 763.5 is**

O .8825

O .8828

O 2

O 76

Answer: .8828

Explanation:

(i). First ignore the decimal point

(ii). Take first two digits e.g. 76 and proceed along this row until we come to column headed by third digit 3 of the number which is 8825

(iii). Now take fourth digit i.e. 5 and proceed along this row in mean difference column which is 5.

(iv). Now add 8825+3=8828

Thus Mantissa of 763.5 \ is \ .8828

**9. The characteristics of 982.5 is**

O 0

O 2

O 3

O 4

Answer: 2

Explanation:

First convert 982.5 to Scientific form:

9.825 \times 10^2

Thus Characteristics is 2

**10. The characteristics of 7824 is**

O 0

O 1

O 2

O 3

Answer: 3

Explanation:

First convert 7824 to Scientific form:

7.824 \times 10^3

Thus Characteristics is 3

**11. The characteristics of 56.3 is**

O 0

O 1

O 2

O 3

Answer: 1

Explanation:

First convert 56.3 to Scientific form:

5.63 \times 10^1

Thus Characteristics is 1

**12. The characteristics of 7.43 is**

O 0

O 1

O 2

O 3

Answer: 0

Explanation:

First convert 7.43 to Scientific form:

7.43 \times 10^0

Thus Characteristics is 0

**13. The characteristics of 0.71 is**

O 1

O -1

O 2

O -2

Answer: -1

Explanation:

First convert 0.71 to Scientific form:

7.1 \times 10^{-1}

Thus Characteristics is -1

**14. The characteristics of 37300 is**

O 0

O 2

O 3

O 4

Answer: 4

Explanation:

First convert 37300 to Scientific form:

3.73 \times 10^4

Thus Characteristics is 4

**15. The characteristics of 0.00159 is**

O 1

O -1

O -3

O -2

Answer:

Explanation:

First convert 0.00159 to Scientific form:

0.00159 \times 10^{-3}

Thus Characteristics is -3

**16. The mantissa of 2476 is**

O .3927

O .3938

O 3

O None of these

Answer:

Explanation:

(i). First ignore the decimal point

(ii). Take first two digits e.g. 24 and proceed along this row until we come to column headed by third digit 7 of the number which is 3927

(iii). Now take fourth digit i.e. 6 and proceed along this row in mean difference column which is 11.

(iv). Now add 3927+11=3938

Thus Mantissa of 2476 \ is \ .3938

**17. The log of 2.4 is**

O 24

O 0.3802

O 2.3802

O None of these

Answer: 0.3802

Explanation:

See Ex # 3.3

Q No. 3

Part No. (ii)

**18. The log of 482.7 is**

O .6836

O 2.6836

O 2.6830

O None of these

Answer: 2.6836

Explanation:

See Ex # 3.3

Q No. 3

Part No. (iv)

**19. The log of 0.783 is**

O .8938

O \overline{1} .8938

O 1.8938

O None of these

Answer: \overline{1} .8938

Explanation:

See Ex # 3.3

Q No. 3

Part No. (v)

**20. The log of 0.09566 is**

O \overline{2} .9805

O \overline{2} .9808

O 2.9808

O None of these

Answer: \overline{2} .9808

Explanation:

See Ex # 3.3

Q No. 3

Part No. (vi)

**21. The log of 700 is**

O .8451

O 1.8451

O 2.8451

O None of these

Answer: 2.8451

Explanation:

See Ex # 3.3

Q No. 3

Part No. (viii)

**22. The anti-\log 1.2508 is**

O 1.781

O 17.81

O 1781

O None of these

Answer: 17.81

Explanation:

See Ex # 3.4

Q No. 1

Part No. (i)

**23. The anti -\log 0.8401 is**

O 6.920

O 69.20

O 6920

O None of these

Answer: 6.920

Explanation:

See Ex # 3.4

Q No. 1

Part No. (ii)

**24. The anti-\log \overline{2} .2508 is**

O 1.781

O 17.81

O 1781

O 0.01781

Answer: 0.01781

Explanation:

See Ex # 3.4

Q No. 1

Part No. (iv)

**1. \log _a m n=**

O \log _a m+\log _a n

O \log _a m-\log _a n

O n \log _a m

O All of them

Answer: \log _a m+\log _a n

Explanation:

Let \log _a m=x \ and \ \log _a n=y

Write them in Exponential form:

a^x=m \ and \ a^y=n

Now multiply these:

a^x \times a^y=mn

Or

mn=a^x \times a^y

mn=a^{x+y}

Taking \log _a on B.S

\log _a m n=\log _a a^{x+y}

\log _a m n=(x+y) \log _a a

\log _a m n=(x+y)(1) \qquad \log _a a=1

\log _a m n=x+y

\log _a m n=\log _a m+\log _a n

**2. \log _a \frac{m}{n}=**

O \log _a m+\log _a n

O \log _a m-\log _a n

O n \log _a m

O All of them

Answer: \log _a m-\log _a n

Explanation:

Let \log _a m=x \ and \ \log _a n=y

Write them in Exponential form:

a^x=m and a^y=n

Now Divide these:

\frac{a^x}{a^y}=\frac{m}{n}

Or

\frac{m}{n}=\frac{a^x}{a^y}

\frac{m}{n}=a^{x-y}

Taking \log _a on B.S

\log _a \frac{m}{n}=\log _a a^{x-y}

\log _a \frac{m}{n}=(x-y) \log _a a

\log _a \frac{m}{n}=(x-y)(1) \qquad \log _a a=1

\log _a \frac{m}{n}=x-y

Hence \ \log _a \frac{m}{n}=\log _a m-\log _a n

**3. \log _a m^n=**

O \log _a m+\log _a n

O \log _a m-\log _a n

O n \log _a m

O All of them

Answer: n \log _a m

Explanation:

Let \log _a m=x

In Exponential form:

a^x=m

Or

m=a^x

Taking power ‘ n ‘ on B.S

m^n=\left(a^x\right)^n

m^n=a^{n x}

Taking \log _a on B.S

\log _a m^n=\log _a a^{n x}

\log _a m^n=n x \log _a a

\log _a m^n=n x(1) \qquad \log _a a=1

\log _a m^n=n x

\log _a m^n=n \log _a m

**4. \quad \log _a m+\log _a n**

O \log _a \frac{m}{n}

O \log _a m n

O n \log _a m

O All of them

Answer: \log _a m n

Explanation:

\log _a m n=\log _a m+\log _a n

**5. \log _a m-\log _a n**

O \log _a \frac{m}{n}

O \log _a m n

O n \log _a m

O All of them

Answer: \log _a \frac{m}{n}

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

**6. n \log _a m**

O \log _a \frac{m}{n}

O \log _a m n

O \log _a m^n

O All of them

Answer:

Explanation:

\log _a m^n=n \log _a m

**7. \log m n=**

O \log m+\log n

O \log m-\log n

O n \log m

O All of them

Answer: \log m+\log n

Explanation:

\log _a m n=\log _a m+\log _a n

**8. \log \frac{m}{n}=**

O \log m+\log n

O \log m-\log n

O n \log m

O All of them

Answer: \log m-\log n

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

**9. \log m^n=**

O \log m+\log n

O \log m-\log n

O n \log m

O All of them

Answer: n \log m

Explanation:

\log _a m^n=n \log _a m

**10. \log m+\log n**

O \log \frac{m}{n}

O \log m n

O \log m^n

O All of them

Answer: \log m n

Explanation:

\log _a m n=\log _a m+\log _a n

**11. \log m-\log n**

O \log \frac{m}{n}

O \log m n

O \log m^n

O All of them

Answer: \log \frac{m}{n}

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

**12. n \log m**

O \log \frac{m}{n}

O \log m n

O \log m^n

O All of them

Answer: \log m^n

Explanation:

\log _a m^n=n \log _a m

**13. \log 2 \times 3=**

O \log 2+\log 3

O \log 2-\log 3

O 2 \log 3

O All of them

Answer: \log 2+\log 3

Explanation:

\log _a m n=\log _a m+\log _a n

**14. \log \frac{2}{3}=**

O \log 2+\log 3

O \log 2-\log 3

O 2 \log 3

O All of them

Answer: \log 2-\log 3

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

**15. \log 3^2=**

O \log 2+1

O \log 2-1

O 2 \log 3

O All of them

Answer: 2 \log 3

Explanation:

\log _a m^n=n \log _a m

**16. \log 2+\log 3**

O \log 2 \times 3

O \log 6

O \log 2

O Both a & b

Answer: \log 6

Explanation:

\log 2+\log 3 =\log 2\times 3

\log 2+\log 3 =\log 6

**17. \log 2-\log 3**

O \log \frac{2}{3}

O \log 2 \times 3

O \log 3^2

O All of them

Answer: \log \frac{2}{3}

Explanation:

\log _a \frac{m}{n}=\log _a m-\log _a n

**18. 2 \log 3=**

O \log \frac{2}{3}

O \log 2 \times 3

O \log 3^2

O All of them

Answer: \log 3^2

Explanation:

\log _a m^n=n \log _a m

2 \log 3= \log 3^2

2 \log 3= \log 9

**19. If \log _2 6+\log _2 7=\log _2 a \ then \ a=**

O 6

O 7

O 24

O 42

Answer: 42

Explanation:

\log _2 6+\log _2 7=\log _2 a

As \log _a m n=\log _a m+\log _a n

\log _2 6 \times 7=\log _2 a

\log _2 42=\log _2 a

Thus \ a=42

**20. \log _a m \log _m n=**

O \log _a n

O \log _a m

O Both a & b

O None of these

Answer: \log _a n

Explanation:

Let \log _a m=x and \log _m n=y

Write them in Exponential form:

a^x=m \ and \ m^y=n

Now multiply these:

As a^{x y}=\left(a^x\right)^y

But \left(a^x\right)^y=m

So a^{x y}=(m)^y=n

Then a^{x y}=n

Taking \log _a on B.S

\log _a a^{x y}=\log _a n

(x y) \log _a a=\log _a n

x y(1)=\log _a n \qquad As \ \log _a a=1

Now

\log _a m \log _m n=\log _a n

**21. \log _2 3 \log _3 5=**

O \log _5 2

O \log _2 5

O Both a & b

O None of these

Answer: \log _2 5

Explanation:

\log _a m \log _m n=\log _a n

**22. \log _2 3 \log _3 4 \log _4 5=**

O \log _5 2

O \log _2 5

O Both a & b

O None of these

Answer: \log _2 5

Explanation:

\log _a m \log _m n=\log _a n

**23. \log _m n=\frac{\log _a n}{\log _a m} is called ____________ law**

O Logarithm

O Change of Base

O Change of Logarithm

O None of these

Answer: Change of Base

Explanation:

**24. \frac{\log _a n}{\log _a m}=**

O \log _m n

O \log _t r

O \log 10

O None of these

Answer: \log _m n

Explanation:

**25. \frac{\log _7 r}{\log _7 t}=**

O \log _m n

O \log _t r

O \log 10

O None of these

Answer: \log _t r

Explanation:

**26. \log _a a=**

O 0

O 1

O 10

O None of these

Answer: 1

Explanation:

**27. \log _{10} 10=**

O 0

O 1

O 10

O None of these

Answer: 1

Explanation:

**28. log10= __________**

O 0

O 1

O 10

O None of these

Answer: 1

Explanation:

**29. log_a 1= __________**

O 0

O 1

O 10

O None of these

Answer: 0

Explanation:

**29. log1= __________**

O 0

O 1

O 10

O None of these

Answer:

Explanation:

**1. \log _9 \frac{1}{81}=**

O -1

O -2

O 2

O Does not exist

Answer: -2

Explanation:

\log _9 \frac{1}{81}=\log _9 \frac{1}{9^2}

\log _9 \frac{1}{81}=\log _9 9^{-2}

\log _9 \frac{1}{81}=-2 \log _9 9

\log _9 \frac{1}{81}=-2(1)

\log _9 \frac{1}{81}=-2

**2. If \log _2 8=x \ then \ x=**

O 64

O 3^2

O 3

O 2^8

Answer: 3

Explanation:

\log _2 8=x

\log _2 2^3=x

3 \log _2 2=x

3(1)=x

3=x

**3. Base of common log is:**

O 10

O e

O \pi

O 5

Answer: 10

Explanation:

**4. \log \sqrt{10}=**

O -1

O -\frac{1}{2}

O \frac{1}{2}

O 2

Answer: \frac{1}{2}

Explanation:

\log \sqrt{10} =\log (10)^{\frac{1}{2}}

\log \sqrt{10} =\frac{1}{2} \log 10

\log \sqrt{10} =\frac{1}{2}(1)

\log \sqrt{10} =\frac{1}{2}

**5. For any non-zero value of x \cdot x^0=**

O 2

O 1

O 0

O 10

Answer: 1

Explanation:

**6. Rewrite t=\log _b m as an exponent equation**

O t=m^b

O b^m=t

O m=b^t

O m^t=b

Answer: m=b^t

Explanation:

**7. \log _{10} 10=**

O 2

O 3

O 0

O 1

Answer: 1

Explanation:

**8. Characteristics of \log 0.000059 is**

O -5

O 5

O -4

O 4

Answer: -5

Explanation:

**9. Evaluate \log _7 \frac{1}{\sqrt{7}}**

O -1

O -\frac{1}{2}

O \frac{1}{2}

O 2

Answer: -\frac{1}{2}

Explanation:

\log _7 \frac{1}{\sqrt{7}} =\log _7 \frac{1}{(7)^{\frac{1}{2}}}

\log _7 \frac{1}{\sqrt{7}} =\log _7 7^{-\frac{1}{2}}

\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2} \log _7 7

\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}(1)

\log _7 \frac{1}{\sqrt{7}} =-\frac{1}{2}

**10. Base of natural log is**

O 10

O e

O \pi

O 1

Answer:

Explanation:

**11. \log m+\log n=**

O \log m\log n

O \log m-\log n

O \log mn

O \log \frac{m}{n}

Answer: \log mn

Explanation:

**12. 0.069 can be written in scientific notation as**

O 6.9 \times 10^3

O 6.9 \times 10^{-2}

O 0.69 \times 10^3

O 69 \times 10^2

Answer: 6.9 \times 10^{-2}

Explanation:

**13. \ln x-2 \ln y**

O \ln \frac{x}{y}

O \ln x y^2

O \ln \frac{x^2}{y}

O \ln \frac{x}{y^2}

Answer: \ln \frac{x}{y^2}

Explanation:

### Mathematics Class 9 Notes (KPK) Chapter # 4

a^2+b^2+2 a b

2. (a-b)^2=

a^2+b^2-2 a b

3. \left(x+\frac{1}{x}\right)^2=

x^2+\frac{1}{x^2}+2(x)\left(\frac{1}{x}\right)

4. \left(x-\frac{1}{x}\right)^2=

x^2+\frac{1}{x^2}-2(x)\left(\frac{1}{x}\right)

5. \left(x+\frac{1}{x}\right)^2=

x^2+\frac{1}{x^2}+2

6. \left(x-\frac{1}{x}\right)^2=

x^2+\frac{1}{x^2}-2

7. (a+b)^2+(a-b)^2=

2\left(a^2+b^2\right)

8. (a+b)^2-(a-b)^2=

4 a b

9. (a+b)(a-b)=

a^2-b^2

10. (a+b+c)^2=

a^2+b^2+c^2+2(a b+b c+c a)

11. (a+b)^3=

a^3+b^3+3 a b(a+b)

12. (a-b)^3=

a^3-b^3-3 a b(a-b)

13. \left(x+\frac{1}{x}\right)^3=

x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right)

14. \left(x-\frac{1}{x}\right)^3=

x^3-\frac{1}{x^3}-3\left(x-\frac{1}{x}\right)

15. a^3+b^3=

(a+b)\left(a^2-a b+b^2\right)

16. a^3-b^3=

Answer:

(a-b)\left(a^2+a b+b^2\right)

17. x^3+\frac{1}{x^3}=

\left(x+\frac{1}{x}\right)\left(x^2-(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\right)

18. x^3-\frac{1}{x^3}=

\left(x-\frac{1}{x}\right)\left(x^2+(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}\right)

19. x^3+\frac{1}{x^3}=

\left(x+\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}-1\right)

20. x^3-\frac{1}{x^3}=

\left(x-\frac{1}{x}\right)\left(x^2+\frac{1}{x^2}+1\right)

21. a^2+2 a b+b^2=

(a+b)^2

22. a^2-2 a b+b^2=

(a-b)^2

23) x^2+2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}=

\left(x+\frac{1}{x}\right)^2

24) x^2-2(x)\left(\frac{1}{x}\right)+\frac{1}{x^2}=

\left(x-\frac{1}{x}\right)^2

25) x^2+2+\frac{1}{x^2}=

\left(x+\frac{1}{x}\right)^2

26) x^2-2+\frac{1}{x^2}=

\left(x-\frac{1}{x}\right)^2

27) a^2-b^2=

(a+b)(a-b)

28) (a+b)^2-c^2=

(a+b+c)(c+b-c)

29) (a+b)^3=

a^3+3 a^2 b+3 a b^2+b^3

30) (a-b)^3=

a^3-3 a^2 b+3 a b^2-b^3

31) (a+b)\left(a^2-a b+b^2\right)=

a^3+b^3

32) (a-b)\left(a^2+a b+b^2\right)=

a^3-b^3

O Conjugate

O Surd

O Rational

O None of these

Surd

2. The rules for surd is/are _________

O It is irrational

O It is a root

O A root of a rational number.

O All of these

All of these

3. In a surd, \sqrt[n]{a}, n is called ______ of the surd.

O type

O order

O conjugate

O None of these

Order

4. \sqrt{3} is

O a surd

O not a surd

O conjugate of a surd

O None of these

Explanation: Here is the root of rational number, so it is a surd

5. \sqrt{5+\sqrt{3}} is

O a surd

O not a surd

O conjugate of a surd

O None of these

Explanation: Here the root of irrational number, so it is not a surd.

6. \sqrt[3]{8} is

O a surd

O not a surd

O conjugate of a surd

O None of these

Explanation:

\sqrt[3]{8} is not a surd because its value is 2 which is rational.

7. \sqrt{3} is called _____ surd.

O Quadratic

O Cubic

O Biquadratic

O None of these

Explanation: Here the surd index order is 2, so it is called quadratic.

8. \sqrt[3]{3} is called _____ surd.

O Quadratic

O Cubic

O Biquadratic

O None of these

Explanation:

Here the surd index order is 3, so it is called cubic.

9. \sqrt[4]{3} is called _____ surd.

O Quadratic

O Cubic

O Biquadratic

O None of these

Explanation:

Here the surd index order is 4, so it is called biquadratic.

10. \sqrt[3]{3} \ and \ \sqrt[3]{5} is called _____ surds.

O Quadratic

O equiradical

O Biquadratic

O None of these

Explanation:

Surds of the same order are called equiradical surds which is 3.

11. \sqrt{3} is called _____ surd.

O Monomial

O Binomial

O Trinomial

O None of these

Explanation:

Surd consists of one term is called monomial.

12. \sqrt{3}-2 is called _____ surd.

O Monomial

O Binomial

O Trinomial

O None of these

Explanation:

Surd consists of Two terms is called binomial.

13. 2-\sqrt{3}+\sqrt{5} is called _____ surd.

O Monomial

O Binomial

O Trinomial

O None of these

Explanation:

Surd consists of three terms is called trinomial.

14. Expressions containing two or more surds are called _____ surds.

O Monomial

O Binomial

O Compound

O None of these

Compound

15. \sqrt[3]{5} is called _____ surd.

O an entire

O mixed

O Trinomial

O None of these

Explanation:

If the coefficient of the surd is \pm1 is called an entire surd.

16. -4 \sqrt[3]{5} is called _____ surd.

O an entire

O mixed

O Trinomial

O None of these

Explanation:

If the coefficient of the surd is other than \pm1 is called mixed surd.

17. \sqrt{8} and \sqrt{18} are _____ surds.

O Similar

O Like

O Both a & b

O None of these

Explanation:

\sqrt{8}=\sqrt{4 \times 2}

\sqrt{8}=2\sqrt{2}

AND

\sqrt{18}=\sqrt{9 \times 2}

\sqrt{18}=3\sqrt{2}

18. Conjugate of 3-\sqrt{5} is

O -3-\sqrt{5}

O -3+\sqrt{5}

O 3+\sqrt{5}

O \sqrt{3}+\sqrt{5}

Explanation:

The conjugate of

a\sqrt{x}+b\sqrt{y} is

a\sqrt{x}-b\sqrt{y}

O -m r^2

O -m r

O -m r

O m r^2

Explanation:

m r^2+3 m r^2-5 m r^2

=4 m r^2-5 m r^2

=-m r^2

2. \left(x^3 y^2\right)\left(x^2 y^3\right)=

O x^5 y^5

O x^5 y^4

O x^4 y^5

O x^4 y^4

Explanation:

\left(x^3 y^2\right)\left(x^2 y^3\right)

=x^{3+2} y^{2+3}

=x^5 y^5

3. \left(4 x y^4\right)^3=

O 64 x^3 y^8

O 64 x^3 y^{10}

O 64 x^3 y^{12}

O 64 x^3 y^7

Explanation:

\left(4 x y^4\right)^3

=4^3 x^3 y^{4 \times 3}

=64 x^3 y^{12}

4. (7x+4 y)-(3 x-6 y)=

O 3 x

O 2 x+10 y

O 4 x+y

O 4 x+10 y

Explanation:

(7 x+4 y)-(3 x-6 y)

=7 x+4 y-3 x+6 y

=7 x-3 x+4 y+6 y

=4 x+10 y

5. (a+b)^2-(a-b)^2=

O 4 a b

O 2\left(a^2+b^2\right)

O a^2-4 a b+2 b^2

O a^4-b^4

4 a b

6. (a+b+c)^2=

O a^2+b^2+c^2

O a^2+b^2+c^2+2(a+b+c)

O a^2+b^2+c^2+2(a b+b c+c a)

O a+b+c+2(a b+b c+c a)

a^2+b^2+c^2+2(a b+b c+c a)

7. a^3+b^3=

O (a+b)^3-2 a b(a+b)

O (a+b)\left(a^2+a b+b^2\right)

O (a-b)\left(a^2-a b+b^2\right)

O (a+b)\left(a^2-a b+b^2\right)

(a+b)\left(a^2-a b+b^2\right)

8. Conjugate of 3-\sqrt{5} is

O -3-\sqrt{5}

O -3+\sqrt{5}

O 3+\sqrt{5}

O \sqrt{3}+\sqrt{5}

Explanation:

The conjugate of

a\sqrt{x}+b\sqrt{y} is

a\sqrt{x}-b\sqrt{y}

9. Which of the following expression is equivalent to the expression \left(m^2+4\right)^{-\frac{1}{2}}

O \frac{m^2+4}{2}

O -\sqrt{m^2+4}

O \frac{1}{\sqrt{m^2+4}}

O \frac{1}{m+2}

Explanation:

\left(m^2+4\right)^{-\frac{1}{2}}

=\frac{1}{(m 2+4)^{\frac{1}{2}}}

=\frac{1}{\sqrt{m^2+4}}

10. For which of the following expression a+b is not a factor?

O a^2-b^2

O a^2+b^2

O a^3+b^3

O a^4+b^4

Explanation:

a^2+b^2 cannot be factorize further, so

a+b is not its factor.

11. \frac{p}{q} \div \frac{r}{q} \cdot \frac{p}{q}=

O \frac{p}{q r}

O \frac{p^2}{q^2 r}

O \frac{p}{q^2 r^2}

O \frac{p^2}{q r}

Explanation:

\frac{p}{q} \div \frac{r}{q} \cdot \frac{p}{q}

=\frac{p}{q} \cdot \frac{q}{r} \cdot \frac{p}{q}

=\frac{p}{q} \cdot \frac{1}{r} \cdot \frac{p}{1}

=\frac{p^2}{q r}

12. Evaluate k^2(2 l-3 m)

when k=-2, l=3, m=4

O 24

O -24

O 6

O 4

Explanation:

k^2(2 l-3 m)

=(-2)^2[2(3)-3(4)]

=4(6-12)

=4(-6)

=-24

13. Evaluate 5 m \sqrt{k^2+l^2}

when k=-2, l=3, m=4.

O 10 \sqrt{13}

O 20

O 20 \sqrt{13}

O \sqrt{13}

Explanation:

5 m \sqrt{k^2+l^2}

=5(4) \sqrt{(-2)^2+(3)^2}

=20 \sqrt{4+9}

=20 \sqrt{13}

### Mathematics Class 9 Notes (KPK) Chapter # 5

### Mathematics Class 9 Notes (KPK) Chapter # 6

### Mathematics Class 9 Notes (KPK) Chapter # 7

### Mathematics Class 9 Notes (KPK) Chapter # 9

### Mathematics Class 9 Notes (KPK) Chapter # 17

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