# Practice

x^2 – \frac{1}{4}=0
x^2 = \frac{1}{4}
\sqrt {x^2} =\pm \sqrt{ \frac{1}{4}}
x =\pm \frac{1}{2}

1. Logarithms having base 10 are called________ Logarithms
O Natural
O Common
O Briggs
O Both b & c

Explanation:

2. Common logarithm is also called __________ logarithm
O Natural
O Briggs
O Both a & b
O None of these

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

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

Explanation:
In 1.5377 Mantissa \ is \ .5377 .

5. In 1.5377 , characteristics is
O 1
O .5377
O 1.5377
O None of these

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

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

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

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.
Thus Mantissa of 763.5 \ is \ .8828

9. The characteristics of 982.5 is
O 0
O 2
O 3
O 4

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

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

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

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

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

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

Explanation:
First convert 0.00159 to Scientific form:
0.00159 </span> \times 10^{-3}
Thus Characteristics is -3

16. The mantissa of 2476 is
O .3927
O .3938
O 3
O None of these

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.
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 \ As \qquad \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

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

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

Explanation:

24. \quad \frac{\log _a n}{\log _a m}=
O \log _m n
O \log _t r
O \log 10
O None of these

Explanation:

25. \frac{\log _7 r}{\log _7 t}=
O \log _m n
O \log _t r
O \log 10
O None of these

Explanation:

26. \log _a a=
O 0
O 1
O 10
O None of these

Explanation:

27. \log _{10} 10=
O 0
O 1
O 10
O None of these

Explanation:

28. log⁡10= __________
O 0
O 1
O 10
O None of these

Explanation:

29. log_a⁡ 1= __________
O 0
O 1
O 10
O None of these

Explanation:

29. log⁡1= __________
O 0
O 1
O 10
O None of these

Explanation:

1. \log _9 \frac{1}{81}=
O -1
O -2
O 2
O Does not exist

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

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

Explanation:

4. \log \sqrt{10}=
O -1
O -\frac{1}{2}
O \frac{1}{2}
O 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

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

Explanation:

7. \log _{10} 10=
O 2
O 3
O 0
O 1

Explanation:

8. Characteristics of \log 0.000059 is
O -5
O 5
O -4
O 4

Explanation:

9. Evaluate \log _7 \frac{1}{\sqrt{7}}
O -1
O -\frac{1}{2}
O \frac{1}{2}
O 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

Explanation:

11. \log m+\log n=
O \log m\log n
O \log m-\log n
O \logmn
O \log \frac{m}{n}

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

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}

Explanation:

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

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

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

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

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

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

Explanation:

7. Irrational number is denoted by _______
O Q
O Q^{\prime}
O Both a and b
O None of these

Explanation:

8. The word irrational means_______
O Ratio
O Not Ratio
O lota
O None of these

Explanation:

9. Irrational numbers consist of all the numbers which are_______
O Rational
O Not rational
O lota
O None of these

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

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

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

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

Explanation:

14. The union of rational and irrational numbers is the set of _______ numbers.
O Complex
O lota
O Prime
O Real

Explanation:
Rational and irrational numbers are collectively called real number.

15. Q \cup Q^{\prime}= _______
O i
O Prime
O R
O N

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

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

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

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

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

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

Explanation:
Terminating and repeating decimals are rational numbers.

22. \frac{3}{8}=0.375 is _______ decimal.
O Terminating
O Repeating
O Both a & b
O None of these

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Explanation:
Here the number of digits repeated infinitely.

37. \frac{5}{7} is _______ number.
O Rational
O Irrational
O Imaginary
O None of these

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

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

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

Explanation:

2. Q \cap Q^{\prime}= ________ ]
O Q
O Q^{\prime}
O \emptyset
O All of them

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

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

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

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

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

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

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

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

10. Zero is called ________
O Both a & b
O None of these

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 Both a & b
O None of these

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

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

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

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

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 Both a & b
O None of these

Explanation:

17. If a+a^{\prime}=a^{\prime}+a=0 \ then \ a^{\prime} is called ________ of a .
O Both a & b
O None of these

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 Both a & b
O None of these

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

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

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

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

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

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

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

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

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

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

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>ditive
O Multiplicative
O Both a & b
O None of these

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 Multiplicative
O Both a & b
O None of these

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 Multiplication
O Both a & b
O None of these

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 Multiplication
O Both a & b
O None of these

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

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

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

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 Multiplicative
O Transitive
O All of them

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

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 Multiplicative
O Transitive
O All of them

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 Multiplicative
O Transitive
O All of them

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

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

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

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.

1. In \sqrt[n]{a},\ then \ \sqrt{ \quad } is called
O Index
O All of them

Explanation:

2. In \sqrt[n]{a}, \ then \ a is called
O Index
O All of them

Explanation:

3. In \sqrt[n]{a}, \ then \ n is called
O Index
O All of them

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}}

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

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

Explanation:
Square root of 1 will always be 1.

7. \sqrt[3]{1}=
O 1
O 0
O -1
O 2

Explanation:
Cube root of 1 will always be 1

8. \sqrt[3]{1}=
O 1
O 0
O -1
O 2

Explanation:
Forth root of 1 will always be 1.

9. \sqrt{36}=
O 6
O 0
O -6
O 4

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

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

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

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

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

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

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

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

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

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

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

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}

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}}

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}

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

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

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 Cubic

Explanation:
This is the way to represent the radical form.

27. 13^2 is ________ form.
O Exponential
O Cubic

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

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

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

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

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}

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}

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}

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

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

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

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

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

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

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

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

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

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

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}}

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

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 \ & \ b are real number is called_______ number.
O Whole
O Natural
O Real
O Complex

Explanation:
Definintion of Complex Number.

2. In complex number a+bi, \ "a" is called part.
O Real
O Imaginary
O Conjugate
O Transpose

Explanation:

3. In complex number a+bi, \ "b" is called part.
O Real
O Imaginary
O Conjugate
O Transpose

Explanation:

4. The conjugate of a+bi is
O -a-b i
O -a+b i
O a-b i
O None of these

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

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

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= [/katex]
O 1
O i
O -1
O All of them

Explanation:
i^2=-1

8. i=
O 1
O i
O -1
O \sqrt{-1}

Explanation:
i=\sqrt{-1}

9. 2i(4-5i)
O 8-5i
O 10-8i
O 10+8i
O None of these

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

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

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

Explanation:
Distributive Property is:
a(b+c)=ab+ac

3. \sqrt{-1} \times \sqrt{-1}=
a. 1
b. i
c. -1
d. 0

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\} [/katex] </div> <strong>5. If n=8 \ and \ 16 \times 2^m=4^{n-8} , then \ m= ?

a. -4
b. -2
c. 0
d. 8

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

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

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

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

Explanation:
As we have
3i(2+5i)=x+6i
6i+15i^2=x+6i
-15=x

10. \sqrt{0}=
a. 0
b. 1
c. -1
d. Not defined

11. \sqrt{-(-9)^2}= ?
a. 9
b. 9+i
c. 9-i
d. 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

## Formula Section

</p> <p>f_k = f(x_k),: x_k = x^*+kh,: k=-frac{N-1}{2},dots,frac{N-1}{2}

binom{n}{k} = frac{n!}{k!(n-k)!}

where h is some step.
Then we interpolate points ((x_k,f_k)) by polynomial

P_{N-1}(x)=sum_{j=0}^{N-1}{a_jx^j}

Its coefficients {a_j} are found as a solution of system of linear equations:

This is (e=lim_{ntoinfty} left(1+frac{1}{n}right)^nlim_{ntoinfty}frac{n}{sqrt[n]{n!}} )

frac{1+frac{a}{b}}{1+frac{1}{1+frac{1}{a}}}

## Introduction to Real Numbers

###### Set of Natural Numbers

katex] N=\{1,2,3[,4,\dots\} [/katex]

###### Set of Whole Numbers

W=\{0,1,2,3,4,\dots\}
Set of Integers Z=\{0, \pm1, \pm2, \pm3, \pm4, \dots\}
OR
W=\{\dots,-4,-3,-2,-1,0,1,2,3,4, \dots\}

## Rational Numbers

The word Rational means “Ratio”. A rational number is a number that can be expressed in the form of \frac{p}{q} where p and q are integers and  Rational numbers is denoted by Q

### Set of Rational Numbers

Q=\left\{ \frac{p}{q}|p,q \in Z,q \neq0 \right\}

### Irrational Numbers

The word Irrational means “Not Ratio”. Irrational number consists of all those numbers which are not rational. Irrational numbers are denoted by Q^/ .

## Real numbers

The set of rational and irrational numbers is called Real Numbers. Real numbers is denoted by R Thus QUQ^/=R
Note:
All the numbers on the number line are real numbers.

### Terminating Decimal Fraction:

A decimal number that contains a finite number of digits after the decimal point.

### Non-Terminating Decimal Fraction:

A decimal number that has no end after the decimal point.

#### Non-Terminating Repeating Decimal Fraction

In non-terminating decimal fraction, some digits are repeated in same order after decimal point.

#### Non-Terminating Non-Repeating Decimal Fraction.

In non-terminating decimal fraction, the digits are not repeated in same order after decimal point.
Decimal Representation of Rational and Irrational Numbers.

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

Note:

• Repeating decimals are called recurring decimals.
• Non-repeating decimals are called non-recurring decimals

Properties of Real Number
The set R of real number is the union of two disjoint sets. Thus  QUQ^/=R
Note:
Q \cap Q^/=\emptyset
Real Number System

The sum of real number is also a real number. If a, b \in R then a+b \in R
Example:
7+9=16 Where 16 is a real number.
Closure Property w.r.t Multiplication
The Product of real number is also a real number. If a, b \in R then a.b \in R
Example:
7×9=16 Where 63 is a real number.
If a, b \in R then a+b=b+a
Example:
7+9=9+7
16=16
Commutative Property w.r.t Multiplication
If a, b \in R then a.b=b.a
Example:
7.9=9.7
63=63
If a, b, c \in R then a+(b+c)=(a+b)+c
Example:
2+(3+5)=(2+3)+5
2+8=5+5
10=10

Associatve Property w.r.t Multiplication
If a, b, c \in R then a(bc)=(ab)c
Example:
2(3×5)=(2×3)5
2(15)=(6)5
30=30

Zero (0) is called Additive identity because adding “0” to a number does not change that number. If we add 0 to a real number, the sum will be the real number itself.
If a \in R there exists 0 \in R then a+0=0+a=a
Example:

• 3+0=0+3=3
• -5+0=-5
• 9+0=9
• \frac{2}{3}+0=\frac{2}{3}
• 9.5+0=9.5

Multiplicative Identity
1 is called Multiplicative identity because multiplying “1” to a number does not change that number. If we add 1 to a real number, the product will be the real number itself. If a \in R there exists 1 \in R then a.1=1.a=a

Example:

• 3×1=1×3=3
• -5×1=-5
• 9×1=9
• \frac{2}{3} \times 1=\frac{2}{3}
• 9.5×1=9.5

When the sum of two numbers is zero (0). If we add a real number to its opposite real number, the result will always be zero (0). If a in R there exists an element a^/ then a+a^/=a^/+a=0 then a^/ is called additive inverse of a
OR
a+(-a)=-a+a=0
10+(-10)=-10+10=0

Example:

• 3+(-3)=0
• -5+5=5-5=0
• -20+20=0
• 10-10=0
• -\frac{2}{3}+\frac{2}{3}
• \frac{2}{3}+\left ( -\frac{2}{3} \right) =0
• \sqrt{2}+\left(- \sqrt{2} \right) =0
• 9.5-9.5=0

Multiplicative Inverse
When the product of two numbers is 1.
If we multiply 1 to a real number, then the product will be the real number itself. If a in R there exists an element a^{-1} then a.a^{-1}=a^{-1}.a=1 then a^{-1} is called multiplicative inverse of a.
OR
a.\frac{1}{a}= \frac{1}{a}.a=1 10. \frac{1}{10}=\frac{1}{10}.10=1

Examples:

• 5. \frac{1}{5}=1
• -3 \times \frac{1}{-3}=1
• -3 \left ( \frac{1}{-3} \right)=1
• \frac{1}{3} \times 3 =1
• \frac{5}{3} \times \frac{3}{5} =1
• \left (\frac{5}{3} \right) \left (\frac{3}{5} \right) =1
• \left (-\frac{5}{3} \right) \left (-\frac{3}{5} \right) =1
• \left (\frac{-5}{3} \right) \left (\frac{-3}{5} \right) =1
• \sqrt{2} \left ( \frac{1}{\sqrt 2} \right) =1
• 9.5 \left ( \frac{1}{9.5} \right) =1

Distributive Property of Multiplication over Additon

### Propeties of Equality of Real Numbers

Reflexive Property
Every real number or value is equal to itself. e.g. a=a which means that a itself equal to a
Example

• 5=5
• \frac{1}{5}= \frac{1}{5}
• -3 =-3
• -3.8 =-3.8
• \sqrt{2} = \sqrt{2}
• 5.9+\sqrt{2} = 5.9+\sqrt{2}
• x+y=x+y

Symmetric Property

By interchanging the sides of an equation doesn’t effect the result. e.g. a=b then b=a does not effect the result.
In other words,

• Left side equal to right side of an equation
• Right side equal to left side of an equation

Example
9+7=16 then 16=9+7

• x=16 or 16=x
• x+y=z or z=x+y
• x+2=z or z=x+2
• a-5=b or b=a-5
• 5.9+\sqrt{2} =x or x = 5.9+\sqrt{2}

Note

If x=y then x may be replaced by y or y may be replaced by x in any equation or expression

Symmetric Property may not worked in some cases such as Subtraction or Division

Trasnsitive Property

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. e.g.  a=b and  b=c then  a=c

Example

x+y=z and z=a+b then x+y=a+b

x=5+y and 5+y=a+b then x=a+b

If we add the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then a+c=b+c

Example

x=5 then x+2=5+2

x-3=7

x-3+3=7+3

x=10

Subtraction Property

If we Subtract the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then a-c=b-c

Example

x=5 then x-2=5-2

x+3=7

Subtract 3 from Both sides

x+3-3=7-3

x=4

Multiplication Property

If we Multiply the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then a \times c=b \times c

Example

x=5 then x \times 2=5 \times 2

\frac{x}{3}=7

Mutiply 3 on Both sides

\frac{x}{3} \times 3=7 \times 3

x-=21

Division Property

If we Divide the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then \frac{a}{c} = \frac{b}{c}

Example

x=5 then \frac{x}{3} = \frac{5}{3}

2x=24

Divide Both sides by 2

\frac{2x}{2}=\frac{24}{2}

x-=12

Second law of motion