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 2.7n+3.5n-5(n \(\epsilon\) N) is divisible by
- (a)
24
- (b)
36
- (c)
48
- (d)
None of these
Which one of the following is true for all n \(\epsilon \) N?
- (a)
1+3+5+....+(2n-1) = n2
- (b)
1+3+5+...+(2n-1) = n
- (c)
1+3+5+....+(2n-1) = n3
- (d)
None of the above
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 natural number. which one of the following is divisible by 6?
- (a)
22n-1
- (b)
2.2n+1-1
- (c)
22n+1-2
- (d)
None of above
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
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 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 n is a natural number, then \(\left( \frac { n+1 }{ 2 } \right) ^{ n }\ge n!\) is true, then
- (a)
n > 1
- (b)
\(n\ge 1\)
- (c)
n > 2
- (d)
\(n\ge 2\)
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)
\(\frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \)
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)
\(-\frac { \pi }{ 2 } \)
Which one of the following is true for all n ϵ N?
- (a)
\(\sqrt { a+\sqrt { a+\sqrt { a+...+\sqrt { a } } } } =\frac { 1+\sqrt { \left( 4a+1 \right) } }{ 2 } \)
- (b)
\(\sqrt { a+\sqrt { a+\sqrt { a+...+\sqrt { a } } } } <\frac { 1+\sqrt { \left( 4a+1 \right) } }{ 2 } \)
- (c)
\(\sqrt { a+\sqrt { a+\sqrt { a+...+\sqrt { a } } } } >\frac { 1+\sqrt { \left( 4a+1 \right) } }{ 2 } \)
- (d)
None of the above
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)
\(\frac { { n }^{ 3 } }{ 13 } +\frac { { 3n }^{ 2 } }{ 8 } +\frac { 13n }{ 24 } \)
If P(m) : m3+m is 3n such that n is a positive integer, then which among the following is true?
- (a)
p(1)
- (b)
p(2)
- (c)
p(4)
- (d)
p(3)
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.
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
The statement 2n> 3n is true for all
- (a)
n∈N
- (b)
n≥ 3, n∈N
- (c)
n>3, n∈N
- (d)
n>2, n∈N
The smallest postive integer for which the statement 3n+1 <4n holds is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
By using principle of mathematical induction for every natural number, (ab)n=
- (a)
anbn
- (b)
anb
- (c)
abn
- (d)
1
If \(\frac{4^n}{n+1}<\frac{(2n)!}{(n!)^2}\), then P(n) is true for
- (a)
n≥1, n∈N
- (b)
n>0,n∊N
- (c)
n<0,n∊N
- (d)
n≥2, n∊N
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, 41n-14n is a multiple of
- (a)
26
- (b)
27
- (c)
25
- (d)
53
For natural number n,2n(n-1)!<nn, if
- (a)
n<2
- (b)
n>2
- (c)
n≥2
- (d)
n>3
Let P(n) denotes the statement that n2+n is odd. It is seen that P(n)⇒P(n+1),P(n) is true for all
- (a)
n>1, n∈N
- (b)
n∈N
- (c)
n>2, n∈N
- (d)
none of these
For all n∈n, 3.52n+1 +23n+1 is divisible by
- (a)
19
- (b)
17
- (c)
23
- (d)
25
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