JEE Mathematics - Mathematical Induction and its Application Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
\(p^{n+1}+(p+1)^{2n-1}\ \ \ (n\epsilon N)\) ,is divisible by
- (a)
p
- (b)
p2
- (c)
p2+p+1
- (d)
p2+p
n(n2-1) is divisible by 24, when n is
- (a)
even
- (b)
odd
- (c)
any integer
- (d)
None of these
The number 2.7n+3.5n-5(n \(\epsilon\) N) is divisible by
- (a)
24
- (b)
36
- (c)
48
- (d)
None of these
The number np - n ( n?\(\epsilon\)? N, P is a prime number ) is a divisible by
- (a)
n
- (b)
n2
- (c)
p
- (d)
p2
x(xn-1-nan-1) + an(n-1) is a divisible by (x-a)k for all n>1,then k equals
- (a)
4
- (b)
3
- (c)
2
- (d)
None of these
Which one of the following is divisible by 133 for every natural number n?
- (a)
11n+2+122n
- (b)
11n+2+122n+1
- (c)
11n+2+12n+1
- (d)
11n+2+12n-1
If m and n are two odd positive integers with n
- (a)
4
- (b)
6
- (c)
8
- (d)
9
For all n ϵ E, 2.42n+1 + 33n+1 is divisible by
- (a)
7
- (b)
5
- (c)
209
- (d)
11
If 10+3.4+k is divisible by 9 for all n ϵ N, then the least positive integral value of k is
- (a)
5
- (b)
7
- (c)
3
- (d)
10
For each n ϵ N, 102n-1+1 is divisible by
- (a)
11
- (b)
13
- (c)
9
- (d)
None of the above
For all n ϵ N, 3.52n+1+23n+1 is divisible by
- (a)
19
- (b)
17
- (c)
23
- (d)
25
The smallest positive integer n for which n!<\(\left( \frac { n+1 }{ 2 } \right) ^{ n }\) holds, is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
Which of the following sum of \(\frac { 1 }{ 2.5 } +\frac { 1 }{ 5.8 } +\frac { 1 }{ 8.11 } +...+\frac { 1 }{ (3n-1)(3n+2) } \) for all n ϵ N?
- (a)
\(\frac { n }{ 6n+3 } \)
- (b)
\(\frac { n }{ 6n+4 } \)
- (c)
\(\frac { n+4 }{ 6n } \)
- (d)
\(\frac { n+5 }{ 6n+4 } \)
If 1+5+12+22+35+......to n terms =\(\frac { { n }^{ 2 }(n+1) }{ 2 } \), then nth term of series is
- (a)
\(\frac { { n }(4n-1) }{ 3 } \)
- (b)
\(\frac { { n }(3n-1) }{ 2 } \)
- (c)
\(\frac { { n }(3n+1) }{ 2 } \)
- (d)
\(\frac { { n }(4n+1) }{ 3 } \)
For all n ϵ N, which one of the following is true?
- (a)
\(cos\theta .cos2\theta .cos4\theta .\quad .....\quad .\quad cos(2^{ n-1 }\theta )=\frac { sin{ 2 }^{ n }\quad \theta }{ 2^{ n }sin\quad \theta } \)
- (b)
\(sin\theta .sin2\theta .sin4\theta .\quad .....\quad .\quad sin(2^{ n-1 }\theta )=\frac { sin{ 2 }^{ n }\quad \theta }{ 2^{ n }sin\quad \theta } \)
- (c)
\(sin\theta .cos2\theta .sin4\theta .\quad .....\quad .\quad cos(2^{ n-1 }\theta )=\frac { sin{ 2 }^{ n }\quad \theta }{ 2^{ n }sin\quad \theta } \)
- (d)
None of the above
For all m ϵ N, \({ \int _{ 0 }^{ \pi }{ \frac { sin(2mx) }{ sin\quad x } } }dx\) is equal to
- (a)
0
- (b)
\(\pi \)
- (c)
\(\frac { \pi }{ 2 } \)
- (d)
\(-\frac { \pi }{ 2 } \)
For all positive integers n > 1, 23n-7n-1 is divisible by
- (a)
49
- (b)
47
- (c)
13
- (d)
None of these
If P(n) :\(1+\frac { 1 }{ 4 } +\frac { 1 }{ 9 } +\frac { 1 }{ 16 } +.....+\frac { 1 }{ { n }^{ 2 } } <2-\frac { 1 }{ n } \) is true and n ϵ N, then
- (a)
\(n\ge 1\)
- (b)
n > 1
- (c)
\(n\ge 2\)
- (d)
n < 2
Which of the following is sum of the series \(\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 } }{ 1+3+5 } +....\)up to n terms?
- (a)
\(\frac { n({ 2n }^{ 2 }+9n+13) }{ 24 } \)
- (b)
\(\frac { n({ 2n }^{ 3 }+3n+13) }{ 24 } \)
- (c)
\(\frac { 2n({ 2n }^{ 2 }+19n+13) }{ 24 } \)
- (d)
\(\frac { { n }^{ 3 } }{ 13 } +\frac { { 3n }^{ 2 } }{ 8 } +\frac { 13n }{ 24 } \)
n(n+1)(n+5) is a multiple of k for all \(n\epsilon N\), then which of the following is a value of K?
- (a)
6
- (b)
5
- (c)
7
- (d)
11
If \(A=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\quad and\quad I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), then which one of the following holds for all \(n\ge 1\), by the principle of mathematical induction?
- (a)
An = 2n-1 A + (n-1) I
- (b)
An = n A + (n-1) I
- (c)
An = 2n-1 A - (n-1) I
- (d)
An = n A - (n-1) I
Let S(k) = 1+3+5+....+(2k-1) = 3+k2. Then, which of the following is true?
- (a)
S(1) is correct
- (b)
S(1) \(\Rightarrow \) S(k+1)
- (c)
S(k) \(\nRightarrow \) S(k+1)
- (d)
Principle of mathematical induction can be used to prove the formula.
A student was asked to prove a statement P(n) by induction. He proved that(P(k+1) is true whenever p(k) is true for all K>5\(\in \) N and also that P(5) is true. On the basis of this he could conclude that P(n) is true for all
- (a)
n\(\in \)N
- (b)
n>5
- (c)
n\(\ge \)5
- (d)
n<5
If P(n): '49n+16n+k is divisible by 64 for n ϵ N" is true, then the least negative integral value of k is
- (a)
-1
- (b)
-2
- (c)
-3
- (d)
-4
For all n≥1,12+22+32+42+ ...+n2=
- (a)
\(\frac { n(n+1) }{ 6 } \)
- (b)
n(n+1)(2n-1)
- (c)
\(\frac { n(n-1)(2n+) }{ 2 } \)
- (d)
\(\frac { n(n+1)(2n+) }{ 6 } \)
For every positive integer n, 7n-3n is divisible by
- (a)
7
- (b)
3
- (c)
4
- (d)
5
P(n):2.7n+3.5n-5 is divisible by
- (a)
24, ∀n∈N
- (b)
21, ∀n∈N
- (c)
35, ∀n∈N
- (d)
50, ∀n∈N
If x and y are distinct integers, then (xn-yn) is divisible by
- (a)
x-y ∀n∈N
- (b)
x2 ∀n∈N
- (c)
y ∀n∈N
- (d)
2(x2+y2) ∀n∈N
Use principle of mathematical induction to find k, where (102n-1 + 1) is divisible by k.
- (a)
11
- (b)
12
- (c)
13
- (d)
9
By the principle of induction ∀n∈N, 32n when divided by 8, leaves remainder
- (a)
2
- (b)
3
- (c)
7
- (d)
1
By mathematical induction, cubes of three consecutive natural numbers is divisible by
- (a)
13
- (b)
5
- (c)
7
- (d)
9
Using principle of mathematical induction,
- (a)
9 ∀n∈N
- (b)
11 ∀n∈N
- (c)
13 ∀n∈N
- (d)
15 ∀n∈N
The greatest positive integer which divides (n+1)(n+2)(n+3)...(n+k) for all n∈W, is
- (a)
r
- (b)
r!
- (c)
n+r
- (d)
(r+1)!
The statement P(n): "1x1!+2x2!+3x3!+......+nx!=(n+1)!-1' is
- (a)
true for all n> 1
- (b)
not true for any n
- (c)
true for all n ∈ N
- (d)
None of these
The smallest postive integer for which the statement 3n+1 <4n holds is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If >-1, then the statement (1+x)n>1+nx is true for
- (a)
all n∊N
- (b)
all n>1, n∊N
- (c)
all n>1, n∊N provided x≠0
- (d)
None of these
If n is apositive integer, then 52n+24n-25 is divisible by
- (a)
574
- (b)
575
- (c)
674
- (d)
576
If P(n) is a statement (n∈N) such that, if P(k) is true, P(k+1) is true for k ∈N, then P(n) is true
- (a)
for all n
- (b)
for all n>1
- (c)
for all n>2
- (d)
nothin can be said
For all n ⋳N, 2.42n+1+33n+1 is divisible by
- (a)
2
- (b)
9
- (c)
3
- (d)
11
For each n∊N, 102n-1+1 is divisible by
- (a)
11
- (b)
13
- (c)
9
- (d)
14
For each n∈N, 32n-1 is divisible by
- (a)
8
- (b)
16
- (c)
32
- (d)
5
The value of \(\left( 1+\frac { 3 }{ 2 } \right) \left( 1+\frac { 5 }{ 4 } \right) \left( 1+\frac { 7 }{ 9 } \right) ...\left( 1+\frac { 2n+1 }{ { n }^{ 2 } } \right) \) is
- (a)
(n+1)
- (b)
(n+1)2
- (c)
2(n+1)2
- (d)
none of these
For all positive integral values of n, 32n-2n +1 is divisible by
- (a)
2
- (b)
4
- (c)
8
- (d)
12
If n ∈N, then 11n+2+12n+1 is divisible by
- (a)
113
- (b)
123
- (c)
133
- (d)
3
For natural number n,2n(n-1)!<nn, if
- (a)
n<2
- (b)
n>2
- (c)
n≥2
- (d)
n>3
For a positive integer n, let a (n) =1+\(\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +\frac { 1 }{ 4 } +..\frac { 1 }{ (2^{ n })-1 } .\)Then
- (a)
a(1000) ≤100
- (b)
a(100)>100
- (c)
1(200)≤100
- (d)
none of these
The remainder when 54n is divided by 13, is
- (a)
1
- (b)
8
- (c)
9
- (d)
10
If 10n+3.4n+2+k is divisible by 9 for all n∊N, then the least positive integral value of k is
- (a)
5
- (b)
3
- (c)
7
- (d)
1
statement-I: If P(n) is defined as2n<! then P(n) is true for all n>3.
Statement-II: In P(n), n may be negative.
- (a)
If both statement-I and statement-II are true and statement-II is the correct explanation of statement-I
- (b)
If both statement-I and statement-II are true but statement-II is not the correct explanation of statement-I
- (c)
If statement-I is true but statement-II is false
- (d)
If statement-I is false but statement-II is true
Statement-I For every natural number n≥2, \(\frac { 1 }{ \sqrt { 1 } } +\frac { 1 }{ \sqrt { 2 } } +...+\frac { 1 }{ \sqrt { n } } >\sqrt { n } .\)
Statement-II: For every natural number n≥2, \(\sqrt { n(n+1) } <n+1.\)
- (a)
If both statement-I and statement-II are true and statement-II is the correct explanation of statement-I
- (b)
If both statement-I and statement-II are true but statement-II is not the correct explanation of statement-I
- (c)
If statement-I is true but statement-II is false
- (d)
If statement-I is false but statement-II is true