NCERT Solutions for class 11 Maths | Chapter 4 - Principles of Mathematical Induction

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Questions
1 Find the number of shots arranged in a complete pyramid the base of which is an equilateral triangle, each side containing n shots.
A n(n+1)(n+2)/3
B n(n+1)(n+2)/6
C n(n+2)/6
D (n+1)(n+2)/6

Answer:n(n+1)(n+2)/6
2 (1 + x)n ≥ ____ for all n ∈ N,where x > -1
A 1 + nx
B 1 – nx
C 1 + nx/2
D 1 – nx/2

Answer: 1 + nx
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3 For any natural number n, 7n – 2n is divisible by
A 3
B 4
C 5
D 7

Answer:5
4 The nth terms of the series 3 + 7 + 13 + 21 +………. is
A 4n – 1
B n² + n + 1
C none of these
D n + 2

Answer:n² + n + 1
5 (n² + n) is ____ for all n ∈ N.
A Even
B odd
C Either even or odd
D None of these

Answer:Even
6 For any natural number n, 7n – 2n is divisible by
A 3
B 4
C 5
D 7

Answer:5
7 The sum of the series 1³ + 2³ + 3³ + ………..n³ is
A {(n + 1)/2}²
B {n/2}²
C n(n + 1)/2
D {n(n + 1)/2}²

Answer:{n(n + 1)/2}²
8 n(n + 1)(n + 5) is a multiple of ____ for all n ∈ N
A 2
B 3
C 5
D 7

Answer:3
9 The sum of the series 1² + 2² + 3² + ………..n² is
A n(n + 1)(2n + 1)
B n(n + 1)(2n + 1)/2
C n(n + 1)(2n + 1)/3
D n(n + 1)(2n + 1)/6

Answer:n(n + 1)(2n + 1)/6
10 For all n ∈ N, 3×52n+1 + 23n+1 is divisible by
A 19
B 17
C 23
D 25

Answer:17
11 102n-1 + 1 is divisible by ____ for all N ∈ N
A 9
B 10
C 11
D 13

Answer:11
12 {1/(3 ∙ 5)} + {1/(5 ∙ 7)} + {1/(7 ∙ 9)} + ……. + 1/{(2n + 1)(2n + 3)} =
A n/(2n + 3)
B n/{2(2n + 3)}
C n/{3(2n + 3)}
D n/{4(2n + 3)}

Answer:n/{3(2n + 3)}
13 If n is an odd positive integer, then an + bn is divisible by :
A a² + b²
B a + b
C a – b
D none of these

Answer:a + b
14 The sum of the series 1² + 2² + 3² + ………..n² is
A n(n + 1)(2n + 1)
B n(n + 1)(2n + 1)/2
C n(n + 1)(2n + 1)/3
D n(n + 1)(2n + 1)/6

Answer:n(n + 1)(2n + 1)/6
15 For all n∈N, 72n − 48n−1 is divisible by :
A 25
B 2304
C 1234
D 26

Answer:2304
16 {1 – (1/2)}{1 – (1/3)}{1 – (1/4)} ……. {1 – 1/(n + 1)} =
A 1/(n + 1) for all n ∈ N.
B 1/(n + 1) for all n ∈ R
C n/(n + 1) for all n ∈ N.
D n/(n + 1) for all n ∈ R

Answer:1/(n + 1) for all n ∈ N.
17 1/(1 ∙ 2) + 1/(2 ∙ 3) + 1/(3 ∙ 4) + ….. + 1/{n(n + 1)}
A n(n + 1)
B n/(n + 1)
C 2n/(n + 1)
D 3n/(n + 1)

Answer:n/(n + 1)
18 {1 – (1/2)}{1 – (1/3)}{1 – (1/4)} ……. {1 – 1/(n + 1)} =
A 1/(n + 1) for all n ∈ N.
B 1/(n + 1) for all n ∈ R
C n/(n + 1) for all n ∈ N.
D n/(n + 1) for all n ∈ R

Answer:1/(n + 1) for all n ∈ N.
19 1/(1 ∙ 2 ∙ 3) + 1/(2 ∙ 3 ∙ 4) + …….. + 1/{n(n + 1)(n + 2)} =
A {n(n + 3)}/{4(n + 1)(n + 2)}
B (n + 3)/{4(n + 1)(n + 2)}
C n/{4(n + 1)(n + 2)}
D None of these

Answer:{n(n + 3)}/{4(n + 1)(n + 2)}
20 Find the number of shots arranged in a complete pyramid the base of which is an equilateral triangle, each side containing n shots.
A n(n+1)(n+2)/3
B n(n+1)(n+2)/6
C n(n+2)/6
D (n+1)(n+2)/6

Answer:n(n+1)(n+2)/6

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