IISER Mathematics - Mathematical Induction and Its Application
Exam Duration: 45 Mins Total Questions : 30
n(n2-1) is divisible by 24, when n is
- (a)
even
- (b)
odd
- (c)
any integer
- (d)
None of these
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
For every positive integer values of n, 32n-2n+1 is divisible by
- (a)
12
- (b)
4
- (c)
8
- (d)
2
The greatest positive integer, which divides (n+2)(n+3)(n+4)(n+5)(n+6) for all n ϵ N is
- (a)
4
- (b)
120
- (c)
24
- (d)
240
For all n ϵ E, 2.42n+1 + 33n+1 is divisible by
- (a)
7
- (b)
5
- (c)
209
- (d)
11
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
If P(m) denotes the statement that m+m is odd. It is seen that P(m)\(\Rightarrow \)P(m+1), P(m) is true for all
- (a)
m > 1
- (b)
m > 2
- (c)
m
- (d)
None of these
Match the columns:
| Column I | Column II |
| A. \({ 10 }^{ 2n-1 }+1,\forall n\ \epsilon \ N\) is divisible by | P. 24 |
| B. For all odd positive integer n, the number, n (n-1) is divisible by 2-1) is divisible by | Q. -1 |
| C. If 49+16n + \(\lambda \) is divisible by 64 for all n ϵ N, then the least negative integral value of \(\lambda \) is | R. 11 |
- (a)
A B C R P Q - (b)
A B C Q R P - (c)
A B C P Q R - (d)
None of the above
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)
\(-\frac { \pi }{ 2 } \)
If P(n): 49n+16n+\(\lambda \) is divisible by 64 for n ϵ N, then the least negative integral value of \(\lambda \) is
- (a)
-1
- (b)
-2
- (c)
-3
- (d)
-4
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
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
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
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 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
By using principle of mathematical induction for every natural number, (ab)n=
- (a)
anbn
- (b)
anb
- (c)
abn
- (d)
1
If n is apositive integer, then 52n+24n-25 is divisible by
- (a)
574
- (b)
575
- (c)
674
- (d)
576
Let P(n):"n2-n+41 is a prime number", then
- (a)
P(1) is not true
- (b)
P(3) is not true
- (c)
P(5) is not true
- (d)
P(41) is not true
If P(n) is a statement such that P(3) is true. Assuming P(k) is true ⇒P(k+1) is true for all k> 3, then P(n) is true.
- (a)
for all n∈N
- (b)
for all n≥3, n∈N
- (c)
for all n>4, n∈N
- (d)
none of these
If m,n are any two odd positive integers with n<m, then the largest positive integers which divides all the numbers of the type m2-n2 is
- (a)
4
- (b)
6
- (c)
8
- (d)
9
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
23n-7n-1 is divisible by
- (a)
64
- (b)
36
- (c)
49
- (d)
25
The remainder when 54n is divided by 13, is
- (a)
1
- (b)
8
- (c)
9
- (d)
10
If xn-1 is divisible by x-k, then the least positive integral value of k is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
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