1st PUC Maths Question Bank Chapter 8 Binomial Theorem

Students can Download Maths Chapter 8 Binomial Theorem Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 1st PUC Maths Question Bank Chapter 8 Binomial Theorem

Question 1.
State and prove Binomial theorem.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 1
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 2

Some observations in a binomial theorem:
(1) The expansion of {a + b)n has (n + 1) terms
(2) The coefficients nCr occurring in the binomial theorem are known as binomial coefficients.
(3) The indices of V go on decreasing and that of ‘a’ go on increasing by 1 at each stage.
i.e., for each term: index of a + index of b-n.
(4) Since nCr=nCn_r we have
nC0 =nCn, nCx =nCn_r and so on.
Thus the coefficients of the terms equidistant from the beginning and the end in a binomial theorem are equal.
(5) General term in (a + b)n: Tr+1 – nCr anrbr
(6) Middle terms in (a + b)n

  • When ‘n’ is even, the middle term
    \(=\left(\frac{n}{2}+1\right)^{t h} \text { term }\)
  • When ‘n’ is odd ,the middle term are
    \(\frac{1}{2}(n+1)^{n} \text { term an } \frac{1}{2}(n+3)^{n} \text { term }\)

(7) Taking a = x and b = -y in the expansion, we get (x-y)n =[x + (-y)]n

1st PUC Maths Question Bank Chapter 8 Binomial Theorem 3

KSEEB Solutions

Question 2.
Expand each of the expression
(i)\(\left(x^{2}+\frac{3}{x}\right)^{4}\)
(ii)\((1-2 x)^{5}\)
(iii)\(\left(\frac{2}{x}-\frac{x}{2}\right)^{5}\)
(iv)\((2 x-3)^{6}\)
(v)\(\left(\frac{x}{3}+\frac{1}{x}\right)^{5}\)
(vi)\(\left(x+\frac{1}{x}\right)^{6}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 4
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 5
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 6

Question 3.
Using binomial theorem, evaluate each of the following:
(i) (98)5
(ii) (96)3
(iii) (102)5
(iv) (101)4
(v) (99)5
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 7
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 8

KSEEB Solutions

Question 4.
Which is longer \((1.01)^{1000000} \text { or } 10,000 ?\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 9

Question 5.
Using Binomial theorem, indicate which number is larger \( (1.1)^{10000} \text { or } 1000 ?\)
Answer:
Spilitting 1.01 and using Binomial theorem write the first few terms, we have.
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 10

Question 6.
Find (a + b)4 – (a -b)4. Hence evaluate
\((\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 11

Question 7.
Find (x +1)6 +(x -1)6. Hence or otherwise evaluate
\((\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 12
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 13

Question 8.
Evaluate:
\((\sqrt{3}+\sqrt{2})^{6}+(\sqrt{3}-\sqrt{2})^{6}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 14

Question 9.
Find the value of
\(\left(a^{2}+\sqrt{a^{2}-1}\right)^{4}+\left(a^{2}-\sqrt{a^{2}-1}\right)^{4}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 15
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 16

KSEEB Solutions

Question 10.
Show that 9n+1 – 8n – 9 is divisible by 64, whenever ‘n’ is a positive integer.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 17

Question 11.
Using binomial theorem, prove that 6n -5n always leaves remainder 1 when
divided by 25.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 18

Question 12.
Prove that
\(\sum_{n=0}^{n} y \cdot c_{n}=4\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 19
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 20

Question 13.
Find the 4th term in the expansion of (x-2 y)12.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 21

Question 14.
Find the 13th term in the expansion of
\(\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x \neq 0 \)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 22

Question 15.
Write the general term in the expression of
(i) (x2 -y)6
(ii) (x2-yx)12,x≠0
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 23

KSEEB Solutions

Question 16.
Find the coefficient of
(i) x5 in (x + 3)8
(ii) a5b7 in (a – 2b)12
(iii) x6y3 in (x + 2y)9
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 24

Question 17.
Find a, if the 17th and 18th terms of the expansion (2 +a)50 are equal.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 25

Question 18.
In the expansion of (1+ a)m+n, prove that coefficients of am and an are equal.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 26

Question 19.
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1+x)2n-1
Answer:
In (1+x)2n we have
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 27
From (1) and (2) we get, the coefficient of xn in (1 + x)2n is twice the coefficient of xn in (1 + x)2n-1.

KSEEB Solutions

Question 20.
Find a positive value of m for which the coefficient of x2 in the expansion (1 + x)m is 6.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 28

Question 21.
Find the middle terms in the expansions of
(i) \(\left(3-\frac{x^{3}}{6}\right)^{7}\)
(ii)\(\left(\frac{x}{3}+9 y\right)^{10}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 29
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 30

Question 22.
Show that the middle term in the expansion of \((1+x)^{2 n} \text { is } \frac{1 \cdot 3 \cdot 5 \ldots(2 n-1)}{\lfloor n} 2^{n} x^{n} \)where ‘n’ is a positive integer.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 31

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Question 23.
The second, third and fourth terms in the binomial expansion (x + a)n are 240, 720 and 1080, respectively. Find x, a and n
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 32
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 33

Question 24.
The coefficients of three consecutive terms in the expansion of (1 + a)n are in the ratio 1:7: 42. Find n
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 34
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 35

KSEEB Solutions

Question 25.
The coefficients (r-1)th, rth,and (r + 1)th, terms in the expansion of (x + 1)th, are in the ratio 1:3:5. Find n and r.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 36
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 37

Question 26.
Find the term independent of x in the expansion of \(\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 38

Question 27.
Find the term independent of x in the expansion of \(\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x>0\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 39

Question 28.
Find a, 6 and n in the expansion of (a + b)n if the first three terms of the expansion are 729,7290 and 30375, respectively.
Answer:
Given: Tx = 729, T2 = 7290 and T3 = 30375
∴ an=729……………….(1)
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 40

KSEEB Solutions

Question 29.
Find a if the coefficients of x2 and x3 in the expansion of (3 +ax)9 are equal.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 41

Question 30.
If the coefficients of (r – 5)th and (2r -1)th terms in the expansion of (1 + x)34 are equal, find r.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 42
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 43

Question 31.
If the coefficients of ar-1, ar and ar+1 in the expansion of (1 + a)n are in arithmetic progression, prove that n2 – n(4r +1) + 4r2 -2 = 0.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 44

Question 32.
Show that the coefficient of the middle term in the expansion of (1 + x)2n is equal to the sum of the coefficients of two middle terms in the expansion of (1 + x)2n-1.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 45
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 46

Question 33.
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of
\(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n} \text { is } \sqrt{6}: 1\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 47

Question 34.
Find the rth term from the end in the expansion of (x + a)n.
Answer:
rth term from the end in (x + a)n
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 48

Question 35.
If a and b are distinct integers, prove that a-b is a factor of an -bn, whenever n is a positive integer.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 49

Question 36.
The sum of the coefficients of the first three terms in the expansion of
\(\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m \)being a natural numbers, is 559. Find the term of the expansion containing x3.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 50

Question 37.
Find the coefficient of x5 in the product (1 + 2x)6(1 – x)7 using binomial theorem.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 51

Question 38.
Find the coefficient of a4 in the product (1 + 2a)4(2-a)s using binomial theorem.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 52

Question 39.
Expand using Binomial theorem
\(\left(1+\frac{x}{2}-\frac{2}{x}\right)^{4}, x \neq 0\)
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 53

KSEEB Solutions

Question 40.
Find the expansion of (3x2 – 1ax + 3a2)3 using binomial theorem.
Answer:
1st PUC Maths Question Bank Chapter 8 Binomial Theorem 54

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