JEE Main Mathematics - Mathematical Induction and its Application
Exam Duration: 60 Mins Total Questions : 30
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 greatest positive integer which divides (n+1)(n+2)(n+3)...(n+k) for all n \(\epsilon\) N. is
- (a)
k
- (b)
k!
- (c)
(k+1)!
- (d)
(k+2)!
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
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
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
For all positive integers n > 1, 23n-7n-1 is divisible by
- (a)
49
- (b)
47
- (c)
13
- (d)
None of these
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): (n!)2 is true for all \(n\epsilon N\) , then
- (a)
\(n\ge 3\)
- (b)
n = 2
- (c)
n = 4
- (d)
n > 4
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)
\(\frac { { n }^{ 3 } }{ 13 } +\frac { { 3n }^{ 2 } }{ 8 } +\frac { 13n }{ 24 } \)
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
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 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
Use principle of mathematical induction to find k, where (102n-1 + 1) is divisible by k.
- (a)
11
- (b)
12
- (c)
13
- (d)
9
10n +3(4n+2) +5(n∈N) is divisible by
- (a)
7
- (b)
5
- (c)
9
- (d)
17
By mathematical induction, \(\frac { 1 }{ 1.2.3 } +\frac { 1 }{ 2.3.4 } +...+\frac { 1 }{ n(n+1)(n+2) } \)is equal to
- (a)
\(\frac { n(n+1) }{ 4(n+2)(n+3) } \)
- (b)
\(\frac { n(n+3) }{ 4(n+1)(n+2) } \)
- (c)
\(\frac { n(n+2) }{ 4(n+1)(n+3) } \)
- (d)
none of these
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
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 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
For all positive integral values of n, 32n-2n +1 is divisible by
- (a)
2
- (b)
4
- (c)
8
- (d)
12
For natural number n,2n(n-1)!<nn, if
- (a)
n<2
- (b)
n>2
- (c)
n≥2
- (d)
n>3
Statement-I: For all n ∊N, 3.52n+1 + 23n+1 is divisible by 21.
Statement -II: For all n∈N, 102n-1+1 is divisible by 11.
- (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: 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