Mathematics - Mathematical Induction and Its Application
Exam Duration: 45 Mins Total Questions : 30
P(n) is the statement "n2-n+41(n \(\epsilon \) N),is a prime".Then which of the following is not true?
- (a)
P(1)
- (b)
P(2)
- (c)
P(3)
- (d)
P(4)
For all n \(\epsilon \) N ,\(\frac { { n }^{ 7 } }{ 7 } +\frac { { n }^{ 5 } }{ 5 } +\frac { 2{ n }^{ 3 } }{ 3 } -\frac { { n }^{ } }{ 105 } \) ,is
- (a)
a positive integer
- (b)
a negative integer
- (c)
0
- (d)
a rational number
The smallest positive integer for which the inequality \(n!<\left( \frac { n+1 }{ 2 } \right) \) is true,is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
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
If m and n are two odd positive integers with n
- (a)
4
- (b)
6
- (c)
8
- (d)
9
If m \(\epsilon \) N, then the highest natural number which divides m(m-1)(m-2) is
- (a)
3
- (b)
6
- (c)
12
- (d)
9
For all n ϵ E, 2.42n+1 + 33n+1 is divisible by
- (a)
7
- (b)
5
- (c)
209
- (d)
11
Let P(n):2n<(1\(\times\)2\(\times\)3\(\times\)......\(\times\)n), then the smallest positive integer for which P(n) is true, is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
\(\sqrt { n } <\frac { 1 }{ \sqrt { n } } +\frac { n }{ \sqrt { 2 } } +......+\frac { 1 }{ \sqrt { n } } \) is true for all
- (a)
natural numbers N
- (b)
integers I
- (c)
real numbers R
- (d)
natural numbers n ≥ 2
Which of the following is sum of the 1.3 + 2.32 + 3.33 +...+ n.3n for all n ϵ N?
- (a)
\(\frac { \left( 2n+1 \right) { 3 }^{ n+1 }+3 }{ 4 } \)
- (b)
\(\frac { \left( 2n-1 \right) { 3 }^{ n+1 }+3 }{ 4 } \)
- (c)
\(\frac { \left( 2n+1 \right) { 3 }^{ n }+3 }{ 4 } \)
- (d)
\(\frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \)
Which of the following is true for all n ϵ N?
- (a)
1+2x+3x2+....+nxn-1 = \(\frac { 1-(n+1)x^{ n }+nx^{ n-1 } }{ (1-x) } \)
- (b)
1+2x+3x2+....+nxn-1 = \(\frac { 1-(n+1)x^{ n }+nx^{ n } }{ (1-x) } \)
- (c)
1+2x+3x2+....+nxn-1 = \(\frac { 1-(n+1)x^{ n }+nx^{ n+1 } }{ (1-x)^{ 2 } } \)
- (d)
None of the above
For all positive integers n > 1, 23n-7n-1 is divisible by
- (a)
49
- (b)
47
- (c)
13
- (d)
None of these
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
Statements I For each natural number N, (n+1)7-n7-1 is divisible by 7.
Statements II For each natural number n, n7 - n is divisible by 7.
- (a)
Statement I is true, Statement II is true; Statement II is the correct explanation II is the correct explanation for Statement I
- (b)
Statement I is true, Statement II is true; Statement II is not the correct explanation for Statement I
- (c)
Statement I is true, Statement II is false
- (d)
Statement I is false, Statement II is true
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):"2.42n+1+16n+33n+1 is divisible by \(\lambda\) for all n\(\in \) N' is true, then the value of \(\lambda\) is
- (a)
1
- (b)
10
- (c)
11
- (d)
15
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
Using principle of mathematical induction,
- (a)
9 ∀n∈N
- (b)
11 ∀n∈N
- (c)
13 ∀n∈N
- (d)
15 ∀n∈N
The smallest postive integer for which the statement 3n+1 <4n holds is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
10n +3(4n+2) +5(n∈N) is divisible by
- (a)
7
- (b)
5
- (c)
9
- (d)
17
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
Let S(k)=1+3+5+....+(2k-1)=3+k2. Then which of the following is true?
- (a)
S(k)⇒S(k-1)
- (b)
S(k)⇒S(k+1)
- (c)
S(1) is correct
- (d)
All of these
Let P(n):n2+n+1 is an even integer.If P(k) is assumed true ⇒P(k+1) is true. Therefore, P(n) is
- (a)
true for n>1, n∈N
- (b)
true for all n∈N
- (c)
true for n>2, n∈N
- (d)
none of these
For all n∈N, 41n-14n is a multiple of
- (a)
26
- (b)
27
- (c)
25
- (d)
53
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 natural number n,2n(n-1)!<nn, if
- (a)
n<2
- (b)
n>2
- (c)
n≥2
- (d)
n>3
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
For all n∈n, 3.52n+1 +23n+1 is divisible by
- (a)
19
- (b)
17
- (c)
23
- (d)
25
statement-I:sin θ+sin2θ+sin3θ+...sinnθ,\(\frac { \frac { sin\quad n\theta }{ 2 } sin\frac { \left( n+1 \right) }{ 2 } \theta }{ sin\left( \frac { \theta }{ 2 } \right) } \) for all n∊N.
statement-II: The number of subsets of a set containing n distinct elements is 2n, for all nєN.
- (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