2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem

Students can Download Basic Maths Question Bank Chapter 4 Binomial Theorem Questions and Answers, Notes Pdf, 2nd PUC Basic Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

Karnataka 2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem

2nd PUC Basic Maths Binomial Theorem Two Marks Questions and Answers

Question 1.
Find the 5th term in \(\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{8}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 1

Question 2.
Find the 6th term in \((\sqrt{x}-\sqrt{y})^{17}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 2

Question 3.
Find the 8th term in \(\left(\frac{a}{2}-\frac{3}{b}\right)^{10}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 3

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 4.
Find the 7th term \(\left(3 x^{2}-\frac{y}{3}\right)^{9}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 4
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 5

Question 5.
Find the 10th term in the Expansion of \(\left(\frac{a}{b}-\frac{2 b}{a^{2}}\right)^{12}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 6

2nd PUC Basic Maths Binomial Theorem Three or Four Marks Questions and Answers

A Binomial Expansion Calculator that doesn’t require any scripting in your browser. Just enter your values and compute.

Question 1.
Expand (1 – 2x)s using binomial theorem. x = 1, a = – 2x, n = 5.
Answer:
(1 + (-2x))5 = 15 + 5C1 . I4 (-2x) + 5C2. I3 (-2x)2 + 5C3. I2(-2x)3 + 5C4 . 1(-2x)4 + 5C5(-2x)5
= 1+5 (-2x) + 10(-2x)2 + 10 (-2x)3+ 5 (-2x)4 + (-2x)5
= 1 – 10x + 10 C4x2 + 10(-8x3) + 5 (16x4) + (-32x5)
= 1 – 10x + 40x2– 80x3 + 80x4– 32x5

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 2.
Expand
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 7

Question 3.
Simplify : \((2+\sqrt{5})^{4}+(2-\sqrt{5})^{4}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 8

Question 4.
Simplify : \((3+\sqrt{2})^{4}-(3-\sqrt{2})^{4}\) using Binomial theorem.
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 9

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 5.
Simplify  \((\sqrt{2}+1)^{6}-(\sqrt{2}-1)^{6}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 10

2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 11

Question 6.
Expand (0.99)5 by Binomial theorem correctly decimal places.
Answer:
(0.99)5 = (1- 0.01)5=15+5C1.14.(- 0.01) + 5C2.13(-0.01)2+5C3.12(-0.01)3+5C4.11(-0.01)4+  5C5 ( – 0:01 )5

2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 12

Question 7.
Expand (0.98)s by Binomial theorem.
Answer:
(0.98)5 = (1 – 0.02)5
= 15 + 5Cr 14 (- 0.02) + 5Cr 13( – 0.02)2 + 5Cr 12( – 0.02)3 + 5C4. 1(- 0.02)4 + 5C5(- 0.02)5 = 1 – 5(0.02) + 10(0.0004) – 10 (0.000008) + 5(0.00000016) – 0.0000000032
= 1-0.1 + 0.004 – 0.00008 + 0.00000080 – 0.0000000032
= 1.0040008 – 0.1000800032 = 0.9039.

Question 8.
Find the value of (1.02)4 using the binomial theorem.
Answer:
(1.02)4 =(1 +0.02)4
= 14 + 4C1.13(0.02) + 4C2.12(0.02)2 + 4Cr 1(0.02)3 + 4C4(0.02)4
= 1 +4 (0.02) + 6 (0.0004) + 4 (0.000008) + 1 .(0.00000016)
= 1 + 0.08 + 0.0024 + 0.000032 + 0.00000016
= 1.08243216.

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 9.
Write down the middle term in the expansion of \(\left(2 x^{2}-\frac{1}{x}\right)^{12}\)
Answer:
Here n = 12,
∴ There are 13 terms. The middle terms is \(\frac{12}{2}+1\) = 7th terms
To find 7th term put r = 6
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 13

Question 10.
Find the middle term of the Expansion of \(\left(\frac{1}{x^{2}}+\frac{x}{2}\right)^{10}\)
Answer:
Here n = 10, There are 11 terms.
∴  Middle term = \(\frac{10}{2}\)+1 = 6 terms
To find 6th term put r = 5
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 14

Question 11.
Find the 5th term in the expansion of \(\left(2-\frac{1}{x}\right)^{3}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 15

Question 12.
Find the 6th term in the expansion of \(\left(3+\frac{2}{x}\right)^{10}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 16
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 17

Question 13.
Find the value of (10.1)5 up to 4 decimal places, using binomial theorem
Answer:
(10 + 0.1)5= 105 + 5Cr104.(0.1) + 5C2.103. (0.1)2 + 5Cr102.(0.1)3 + 5C4.10.(0.1)4+5C5(0.1)5

2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 18

Question 14.
S.T there is no term containing x3 in the expression \(\left(3 x-\frac{1}{2 x}\right)^{8}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 19
To find the term containg x3 equate the power of x to 3
∴ 8 – 2r = 3 ⇒ 5 = 2r = s.r = r = 5 /2
Since r is a fraction there is no term containing x3.

Question 15.
S.T. There is no term independent of x in the expansion of \(\left(2 x-\frac{1}{x^{2}}\right)^{10}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 20
To find the terms independent of x equate the power of x to O, we get 10 – 3r = 0
⇒ 10 = 3r ⇒ r = \(\frac{10}{3}\) is a fraction. Hence there is no terms independent of x.

Question 16.
Find the co-efficient of x1 in the expression of \(\left(x^{2}+\frac{2}{x}\right)^{11}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 21
To find the coeffecint of x1 equate the power of x to 7
∴ 22 – 3r = 7 ⇒ 22 -1 = 3r ⇒ 15 – 3r  ⇒ r = 5.
∴ Co-efficient of x7 is 11C5.25.

Question 17.
\(\left(\frac{a}{x}+b x\right)^{12}\)
Answer:
∵  n = 12, we have 13 terms,
∴ 7th term is the middle tem
To find 7th term put r = 6
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 22
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 23

Question 18.
\(\left(\frac{a}{3}+\frac{b}{2}\right)^{8}\)
Answer:
Here n = 8,
∵  we have only one middle term, i.e. 5th terms.
To find 5th term put r = 4
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 24

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 19.
\((3+\sqrt{2})^{6}-(3-\sqrt{2})^{6}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 25

Question 20.
Find x if the 17th and 18th terms of the expansion (2 + x)50 are equal.
Answer:
17th term = T16 + 1 = 50C17 . 250 – 16. x16 = 50C16 . 234. x16
18th term = T17 + 1 = 50C17 . 550 – 17. x17 = 50C17 . 233 . x17

2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 26

Question 21.
If 2nd 3rd and 4th terms of (x + y)n are 108,54,12 respectively. Find the value of x,y and n.
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 27
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 28

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 22.
If the coefficient of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal then find r.
Answer:
Tr+1 = 18Cr . (1)18 – r . xr
= 18Cr. xr
Here The coefficient of (r + 1 )th term is 18Cr
Therefore the coefficient of (2r + 4))th term is 18C2r+J
Therefore the coefficient of (r – 2))th term is 18Cr – 3

2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 29

Question 23.
Find the middle terms in the expansion of
(i) \(\left(3 x-\frac{2}{x^{2}}\right)^{15}\)
Answer:
∵ n = 15, we have 2 middle terms i.e., 8th and 9th terms,
(a) To find 8th term put r = 7
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 30
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 31

2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 32

Question 24.
\(\left(\frac{x}{2}+\frac{3}{x^{2}}\right)^{19}\)
Answer:
n = 19, (odd no.)
We have two middle terms i.e., 10th and 11th terms,
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 32

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 25.
\(\left(2 x^{2}+\frac{1}{\sqrt{x}}\right)^{11}\)
Answer:
n = 19, (odd no.). We have two middle terms i.e., 10th and 11th terms,
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 33

Question 26.
Find the coefficients of
(a) \(x^{11} \text { in }\left(x+\frac{2}{x^{2}}\right)^{17}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 34

Question 27.
Find the coefficient of y2 in \(\left(7 y^{2}-\frac{2}{y}\right)^{12}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 35

Question 28.
Find the coefficient of y11 in \(\left(\sqrt{x}-\frac{2}{x}\right)^{17}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 36

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 29.
To find coefficient of x-11, equate the power of x, to -11
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 37

Question 30.
Find the term independent of x in.
(i) \(\left(\frac{4 x^{2}}{3}+\frac{3}{2 x}\right)^{9}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 38

To find the term independent of x, equal the power of x to 0
18 – 3r = 0 ⇒ r = 6
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 39

Question 31.
\(\left(x^{3}-\frac{3}{x^{2}}\right)^{15}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 40
To find the term independent of x, equale the power of x to 0
45 – 5r = 0 ⇒ r = 9
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 41

Question 32.
\(\left(\sqrt{x}+\frac{1}{3 x^{2}}\right)^{10}\)
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 42
To find the terms independent of x , Equale the power of x to 0
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 43

2nd PUC Maths Basic Question Bank Chapter 4 Binomial Theorem

Question 33.
The first 3 terms in (1 + ax)n where n is a positive integer are 1,6x, 6×2. Find the values of a and n.
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 44

Question 34.
In the expansion of (3 + Kx)9, the coefficients of x2 and x3 are Equal. Find K.
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 45

Question 35.
Find the ratio of the coefficients of x4 in the two expansion (1 + x)7 and (1 + x)10.
Answer:
2nd PUC Basic Maths Question Bank Chapter 4 Binomial Theorem 46

Leave a Comment

shares