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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 x6 in (2 + 2x)10 is ……….
(a) 10C6
(b) 26
(c) 10C626
(d) 10C6210
Solution:
(d) 10C6210
Hint.
tr + 1 = 210(nCr)
To find coefficient of x6 put r = 6
∴ coefficient of x6 = 210 [10C6]
Question 3.
The coefficient of x8y12 in the expansion of (2x + 3y)20 is …….
(a) 0
(b) 28312
(c) 28312 + 21238
(d) 20C828312
Solution:
(d) 20C828312
Question 4.
If nC10 > nCr for all possible r, then a value of n is ……..
(a) 10
(b) 21
(c) 19
(d) 20
Solution:
(d) 20
Hint.
20C10 > 20Cr 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 + x2)2 (1 + x)n = a0 + a1x + a2x2 + …. + xn + 4 and if a0, a1, a2 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 Sn 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) d2
Solution:
(a) d
Question 11.
The remainder when 3815 is divided by 13 is …….
(a) 12
(b) 1
(c) 11
(d) 5
Solution:
(a) 12
Hint.
(38)15 = (39 – 1)15
= (39)15 – 15C1(39 )14 + 15C2(39)13 + . . . . . . . + 15C14 (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 nth 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 x5 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) \(\frac{3}{2} \log \left(\frac{5}{3}\right)\)