Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

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Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

Plus One Maths Binomial Theorem Three Mark Questions and Answers

Question 1.
Write the middle term in the expansion of the following; (3 score each)
Plus One Maths Binomial Theorem Three Mark Questions and Answers 1
Answer:
i) Here 7 is an odd number. Therefore there are two middle terms \(\left(\frac{7+1}{2}=4\right)^{t h}\) and \(\left(\frac{7+1}{2}+1=5\right)^{t h}\), ie; 4th and 5th terms in the above expansion.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 2

ii) Here 10 is an even number. Therefore middle terms \(\left(\frac{10}{2}+1=6\right)^{t h}\) term in the above expansion.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 3

iii) Here 17 is an odd number. Therefore there are two middle terms \(\left(\frac{17+1}{2}=9\right)^{t h}\), ie; 9th and 10th terms in the above expansion. \(\left(x+\frac{2}{\sqrt{x}}\right)^{17}\).
Plus One Maths Binomial Theorem Three Mark Questions and Answers 4

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

Question 2.
Find the term independent of x in the following expansion. (3 score each)
Plus One Maths Binomial Theorem Three Mark Questions and Answers 5
Answer:
i) General term = tr+1 = (-1)r12Cr(x)12-r \(\left(\frac{1}{x}\right)^{r}\)
=(-1)r12Cr(x)12-r-r = (-1)r12Cr(x)12-2r
Term independent of x in the expansion will be the term in which the power of x is zero, ie; 12 – 2r = 0 ⇒ 12 = 2r
⇒ r = 6
t7 = (-1)612C6x12-2(6)
= \(\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{1 \times 2 \times 3 \times 4 \times 5 \times 6}\) = 924.

ii) General term = tr+1 =(-1)r9Cr(x2)9-r\(\left(\frac{1}{x}\right)^{r}\)
= (-1)r9Cr(x)18-2r-r = (-1)r9Cr(x)18-3r
Term independent of x in the expansion will be the term in which the power of x is zero.
ie; 18 – 3r = 0 ⇒ 18 = 3r ⇒ r = 6
t7 = (-1)69C6x18-3(6)
= \(\frac{9 \times 8 \times 7}{1 \times 2 \times 3}\) = 84.

iii) General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 6
Plus One Maths Binomial Theorem Three Mark Questions and Answers 7
Term independent of x in the expansion will be the term in which the power of x is zero.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 8

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

iv) General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 9
Plus One Maths Binomial Theorem Three Mark Questions and Answers 10
Term independent of x in the expansion will be the term in which the power of x is zero.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 11

Question 3.
Find the coefficient of x10 in the expansion of \(\left(2 x^{2}-\frac{3}{x}\right)^{11}\).
Answer:
General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 12
Given; 22 – 3r = 10 ⇒ 12 = 3r ⇒ r = 4
t5 = (-1)411C4211-4 x22-3(4) 34
= 11C42734x10
Therefore the coefficient of x10 is 11C42734.

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

Question 4.
Find the coefficient of a5b7 in the expansion of (a – 2b)12.
Answer:
General term = tr+1 = (-1)r12Cr(a)12-r(2b)r
= (-1)r12Cr(a)12-r2rbr
The term containing a5b7 is obtained by putting r = 7
⇒ t8 = (-1)712C7(a)12-727b7
Therefore the coefficient of a5b7 is
(-1)712C727 = -12C727.

Question 5.
Find the coefficient of (3 score each)

  1. x11 in the expansion of \(\left(x-\frac{2}{x^{2}}\right)^{17}\)
  2. x9 in the expansion of \(\left(3 x^{2}+\frac{5}{x^{3}}\right)^{12}\)
  3. x20 in the expansion of \(\left(3 x^{3}-\frac{2}{x^{2}}\right)^{40}\)

Answer:
1. General term
Plus One Maths Binomial Theorem Three Mark Questions and Answers 13
The term containing x11 is obtained by
17 – 3r = 11 ⇒ 6 = 3r ⇒ r = 2
⇒ t3 = (-1)217C2 (x)17-3(2) 22 = 17C2(x)11 × 4
Therefore the coefficient of x11 is 17C2 × 4
= \(\frac{17 \times 16}{1 \times 2}\) × 4 = 544

2. General term
Plus One Maths Binomial Theorem Three Mark Questions and Answers 14
The term containing x9 is obtained by
24 – 5r = 9 ⇒ 15 = 5r ⇒ r = 3
⇒ t4 = 12C3(3)12-3(x)24-5(3)53
= 12C3(3)9(x)953
Therefore the coefficient of x9 is 12C3(3)953.

3. General term = tr+1
= (-1)r40Cr(3x3)40-r (\(\frac{2}{x^{2}}\))r
= (-1)r40Cr(3)40-rx120-3r (2)r x-2r
= (-1)r40Cr(3)40-rx120-5r(2)r
The term containing x20 is obtained by
120 – 5r = 20 ⇒ 100 = 5r ⇒ r = 20
⇒ t21 = (-1)2040C20(3)40-20(x) 120-5(20) 220
= 40C20(3)20(x)20220
Therefore the coefficient of x20 is 40C20(3) 20 220.

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

Question 6.

  1. Find the term independent of x in the expansion of \(\left(x^{2}+\frac{2}{x}\right)^{6}\) (3)
  2. If the middle term in the expansion of \(\left(x^{m}+\frac{2}{x}\right)^{6}\) is independent of x, find the value of m.

Answer:
1. tr+1 = nCran-rbr = 6Cr(x2)6-r\(\left(\frac{2}{x}\right)^{r}\)
= 6Crx12-2rx-r(2)r = 6Crx12-3r(2)r
For term independent of x;
12 – 3r = 0 r = 4
t5 = 6C4(2)4 = 6C2 × 16 = \(\frac{6 \times 5}{1 \times 2}\) × 16 = 240

2. m = 1

Plus One Maths Binomial Theorem Four Mark Questions and Answers

Question 1.

  1. Write the general term in the expansion \(\left(\frac{3 x^{2}}{2}-\frac{1}{3 x}\right)^{6}\) (2)
  2. Find the term independent of x in the above expansion. (2)

Answer:
1. General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 15

2. Term independent of x in the expansion will be the term in which the power of x is zero.
ie; 12 – 3r = 0 ⇒ 12 = 3r ⇒ r = 4
Plus One Maths Binomial Theorem Three Mark Questions and Answers 16

Plus One Maths Binomial Theorem Practice Problems Questions and Answers

Question 1.
Expand the following. (2 score each)

  1. (3a2 – 2b)4
  2. (3 – 4x2)5
  3. \(\left(\frac{x}{2}-2 y\right)^{6}\)
  4. \(\left(\frac{x}{2}-2 y\right)^{6}\)

Answer:
1.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 17

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

2.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 18

3.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 19

4.
Plus One Maths Binomial Theorem Three Mark Questions and Answers 20

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

Question 2.
Write the general term in the expansion of the following; (2 score each)
Plus One Maths Binomial Theorem Three Mark Questions and Answers 21
Answer:
i) General term = tr+1
= (-1)r6Cr(x2)6-r(y)r
= (-1)r6Crx12-ryr.

ii) General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 22

iii) General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 23

iv) General term = tr+1
Plus One Maths Binomial Theorem Three Mark Questions and Answers 24

Plus One Maths Chapter Wise Questions and Answers Chapter 8 Binomial Theorem

Question 3.
If the coefficient of x2 in the expansion of (1 + x)n is 6 then the positive value of n.
Answer:
tr+1 = nCrxr, the term containing x2 is obtained by putting r= 2.
nC2 = 6 ⇒ \(\frac{n(n-1)}{2}\) = 6 ⇒ n(n -1) = 12
⇒ n(n -1) = 4 × 3 ⇒ n = 3.

Question 4.
Find the 13th term in the expansion of \(\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}\).
Answer:
Plus One Maths Binomial Theorem Three Mark Questions and Answers 25
= 18C6(3)12-12 = 18C6 = 18564

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