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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 5 Binomial Theorem, Sequences and Series Ex 5.5

Choose the correct or the most suitable answer:

Question 1.

The value of 2 + 4 + 6 + … + 2n is …..

Solution:

(d) n(n + 1)

Question 2.

The coefficient of x^{6} in (2 + 2x)^{10} is ……….

(a) ^{10}C_{6}

(b) 2^{6}

(c) ^{10}C_{6}2^{6}

(d) ^{10}C_{6}2^{10}

Solution:

(d) ^{10}C_{6}2^{10}

Hint.

t_{r + 1} = 2^{10}(^{n}C_{r})

To find coefficient of x_{6} put r = 6

∴ coefficient of x_{6} = 210 [^{10}C_{6}]

Question 3.

The coefficient of x^{8}y^{12} in the expansion of (2x + 3y)^{20} is …….

(a) 0

(b) 2^{8}3^{12}

(c) 2^{8}3^{12} + 2^{12}3^{8}

(d) ^{20}C_{8}2^{8}3^{12}

Solution:

(d) ^{20}C_{8}2^{8}3^{12}

Question 4.

If ^{n}C_{10} > ^{n}C_{r} for all possible r, then a value of n is ……..

(a) 10

(b) 21

(c) 19

(d) 20

Solution:

(d) 20

Hint.

20C_{10} > 20C_{r} for all possible value of r.

n = 20

Question 5.

If a is the arithmetic mean and g is the geometric mean of two numbers, then ……..

(a) a ≤ g

(b) a ≥ g

(c) a = g

(d) a > g

Solution:

(b) a ≥ g

Hint. AM ≥ GM

∴ a ≥ g

Question 6.

If (1 + x^{2})^{2} (1 + x)^{n} = a_{0} + a_{1}x + a_{2}x^{2} + …. + x^{n + 4} and if a_{0}, a_{1}, a_{2} are in AP, then n is ……

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

(b or c)n = 2 or 3

Question 7.

If a, 8, b are in A.P, a, 4, b are in G.P, if a, x, b are in HP then x is ……

(a) 2

(b) 1

(c) 4

(d) 16

Solution:

(a) 2

Question 8.

(a) AP

(b) GP

(c) HP

(d) AGP

Solution:

(c) HP

Question 9.

The HM of two positive numbers whose AM and GM are 16, 8 respectively is ………

(a) 10

(b) 6

(c) 5

(d) 4

Solution:

(d) 4

Hint.

Let the two numbers be a and b

Question 10.

If S_{n} denotes the sum of n terms of an AP whose common difference is d, the value of \(\mathrm{S}_{n}-2 \mathrm{S}_{n-1}+S_{n-2}\) is ……

(a) d

(b) 2d

(c) 4d

(d) d^{2}

Solution:

(a) d

Question 11.

The remainder when 38^{15} is divided by 13 is …….

(a) 12

(b) 1

(c) 11

(d) 5

Solution:

(a) 12

Hint.

(38)^{15} = (39 – 1)^{15}

= (39)^{15} – 15C_{1}(39 )^{14} + 15C_{2}(39)^{13} + . . . . . . . + 15C_{14} (39) – 1

In the Binomial expansion , all the terms except the last term (- 1) are divisible by 13

∴ The remainder = 13 – 1 = 12

Question 12.

The n^{th} term of the sequence 1, 2, 4, 7, 11, …… is

Solution:

(d) \(\frac{n^{2}-n+2}{2}\)

Question 13.

Solution:

Solution:

(d) \(\frac{\sqrt{2 n+1}-1}{2}\)

Question 14.

Solution:

Question 15.

Solution:

(c) \(\frac{n(n+1)}{\sqrt{2}}\)

Question 16.

(a) 14

(b) 7

(c) 4

(d) 6

Solution:

(a) 14

Question 17.

The sum of an infinite GP is 18. If the first term is 6, the common ratio is ………

(a) \(\frac{1}{3}\)

(b) \(\frac{2}{3}\)

(c) \(\frac{1}{6}\)

(d) \(\frac{3}{4}\)

Solution:

(b) \(\frac{2}{3}\)

Hint:

18r = 18 – 6 = 12

r = 12/18 = 2/3

Question 18.

The coefficient of x^{5} in the series e^{-2x} is ………

(a) \(\frac{2}{3}\)

(b) \(\frac{3}{2}\)

(c) \(\frac{-4}{15}\)

(d) \(\frac{4}{15}\)

Answer:

(c) \(\frac{-4}{15}\)

Question 19.

Answer:

(c) \(\frac{(e-1)^{2}}{2 e}\)

Question 20.

Solution:

(b) \(\frac{3}{2} \log \left(\frac{5}{3}\right)\)