x^2 – \frac{1}{4}=0
x^2 = \frac{1}{4}
\sqrt {x^2} =\pm \sqrt{ \frac{1}{4}}
x =\pm \frac{1}{2}
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 </span> \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^{xy}
Taking \log _a on B.S
\log _a \frac{m}{n}=\log _a a^{xy}
\log _a \frac{m}{n}=(xy) \log _a a
\log _a \frac{m}{n}=(xy)(1) \qquad \log _a a=1
\log _a \frac{m}{n}=xy
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 21
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 \ 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
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. \quad \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 nonzero 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 \logmn
O \log \frac{m}{n}
Answer: \logmn
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 x2 \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:
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
Nonterminating and nonrecurring (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 Nonterminating and nonrecuring 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.
22. \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 nonterminating and nonrepeating is called _______ numbers.
O Rational
O Irrational
O Complex
O All of them
Answer: Irrational
Explanation:
Nonterminating and nonrecurring (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 NonRecurring
O Both a & b
O None of these
Answer: Recurring
Explanation:
Repeating decimals are also called Recurring decimals.
35. NonRepeating decimals are called _______ decimals.
O Recurring
O NonRecurring
O Botha \& b
O None of these
Answer: NonRecurring
Explanation:
The NonRepeating decimals are also called NonRecurring
36. \frac{5}{27}=0.185185185 \dots is called _______ recurring decimals.
O Terminating
O NonTerminating
O Both a & b
O None of these
Answer: NonTerminating
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
Nonterminating 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 Nonterminating recurring (repeating) decimals are rational numbers.
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.
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>ditive
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.
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 nonzero 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 nonzero 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:
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^{mn}
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^{mn}
O a^{\frac{n}{m}}
O a^{m+n}
Answer: a^{mn}
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^{mn}
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 nonzero 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 nonzero 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.
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 ab i
O a+b i
O ab i
O None of these
Answer: ab i
Explanation:
A conjugate of a complex number is obtained by changing the sign of imaginary part.
5. \overline{a+bi}=
O ab i
O a+b i
O ab i
O None of these
Answer: ab 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= [/katex]
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(45i)
O 85i
O 108i
O 10+8i
O None of these
Answer:
Explanation:
2i(45i)=8i10i^2
2i(45i)=8i10(1)
2i(45i)=8i+10
2i(45i)=10+8i
10. (32i)(3+2i)=
O 2+3i
O 32i
O 13
O 13
Answer: 13
Explanation:
(32i)(3+2i)=3^2(2i)^2
(32i)(3+2i)=94i^2
(32i)(3+2i)=94(1)
(32i)(3+2i)=9+4
(32i)(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}
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\} [/katex] </div> <strong>5. If n=8 \ and \ 16 \times 2^m=4^{n8} , then \ m= ?
a. 4
b. 2
c. 0
d. 8
Answer: 4
Explanation:
16 \times 2^m=4^{n8}
Put \ n=8
16 \times 2^m=4^{88}
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. 9i
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
Formula Section
</p> <p>f_k = f(x_k),: x_k = x^*+kh,: k=frac{N1}{2},dots,frac{N1}{2}
binom{n}{k} = frac{n!}{k!(nk)!}
where h is some step.
Then we interpolate points ((x_k,f_k)) by polynomial
P_{N1}(x)=sum_{j=0}^{N1}{a_jx^j}
Its coefficients {a_j} are found as a solution of system of linear equations:
left{ P_{N1}(x_k) = f_kright},quad k=frac{N1}{2},dots,frac{N1}{2}
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.
NonTerminating Decimal Fraction:
A decimal number that has no end after the decimal point.
NonTerminating Repeating Decimal Fraction
In nonterminating decimal fraction, some digits are repeated in same order after decimal point.
NonTerminating NonRepeating Decimal Fraction.
In nonterminating 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.
 Nonterminating recurring (repeating) decimals are rational numbers.
 Nonterminating and nonrecurring (repeating) decimals are irrational numbers.
Note:
 Repeating decimals are called recurring decimals.
 Nonrepeating decimals are called nonrecurring 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
Closure Property w.r.t Addition
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.
Commutative Property w.r.t Addition:
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
Associatve Property w.r.t Addition:
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
Additive Identity:
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
Additive Inverse
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=55=0
 20+20=0
 1010=0
 \frac{2}{3}+\frac{2}{3}
 \frac{2}{3}+\left ( \frac{2}{3} \right) =0
 \sqrt{2}+\left( \sqrt{2} \right) =0
 9.59.5=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
 a5=b or b=a5
 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
Addition Property
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
x3=7
Add 3 on Both sides
x3+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 ac=bc
Example
x=5 then x2=52
x+3=7
Subtract 3 from Both sides
x+33=73
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