Mathematics - Sequences and Series
Exam Duration: 45 Mins Total Questions : 30
The sides a, b, c of the \(\triangle ABC\) are in G.P., where log a - log 2b, log 2b - log 3c, log 3c - log a are in A.P., then the \(\triangle ABC\) is
- (a)
acute angled
- (b)
obtuse angled
- (c)
right angled
- (d)
NONE OF THESE
Coefficient of x99 in the polynomial (x-1)(x-2)(x-3)....(x-100) is
- (a)
1000
- (b)
1002
- (c)
-5050
- (d)
NONE OF THESE
The sum of integers from 1 to 100 that are divisible by 2 or 5 is
- (a)
2550
- (b)
1050
- (c)
550
- (d)
3050
The sum of the first ten terms of the series \(log3+log\frac { 9 }{ 4 } +log\frac { 27 }{ 16 } +log\frac { 81 }{ 64 } +....\quad is\)
- (a)
45 log 3 - 5 log 4
- (b)
50 log 3 + 20 log 4
- (c)
55 log 3 - 45 log 4
- (d)
NONE OF THESE
If a, b, c, d and p are distinct natural numbers such that \((a^2+b^2+c^2)P^2-(ab+bc+cd)p+(b^2+c^2+d^2)\le 0\) then
- (a)
a, b, c, d are in A.P.
- (b)
a, b, c, d are in G.P.
- (c)
a, b, c, d are in H.P.
- (d)
ab = cd
If \(\alpha ,\quad \beta \) are the roots of the equation x2 - px + q = 0, then the value of \((\alpha +\beta )x-\left( \frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } \right) { x }^{ 2 }+\left( \frac { { \alpha }^{ 3 }+{ \beta }^{ 3 } }{ 3 } \right) { x }^{ 3 }-...\) is
- (a)
log (1-px+qx2)
- (b)
log(1+px-qx2)
- (c)
log(1+px+qx2)
- (d)
NONE OF THESE
If \(y\quad =\quad 1-x+\frac { { x }^{ 2 } }{ 2! } -\frac { { x }^{ 3 } }{ 3! } +...\) and \(z\quad =\quad \left( y+\frac { { y }^{ 2 } }{ 2 } +\frac { { y }^{ 3 } }{ 3 } +... \right) \) then value of x is
- (a)
log (1-e2)
- (b)
\(log\left( \frac { 1 }{ 1-{ e }^{ z } } \right) \)
- (c)
log (1+ez)
- (d)
\(log\left( \frac { 1 }{ 1-{ +e }^{ z } } \right) \)
The minimum value of the expression \({ 3 }^{ X }+{ 3 }^{ 1-X },X\epsilon R,\) is
- (a)
0
- (b)
\(\frac{1}{3}\)
- (c)
3
- (d)
\(2\sqrt{3}\)
The natural number a for which \(\sum _{ k=1 }^{ n }{ f\left( a+k \right) } =16\left( { 2 }^{ n }-1 \right) \), where the function f satisfies f (X + Y) = f(X). f(Y) for all natural numbers X and Y and further f(1) = 2 is
- (a)
3
- (b)
4
- (c)
6
- (d)
5
If 1 + \(\lambda \)+ \({ \lambda }^{ 2 }\)+ ... + \({ \lambda }^{ n }\)= (1 + \({ \lambda }\)) (1 + \({ \lambda }^{ 2 }\)) (1 + \({ \lambda }^{ 4 }\)) (1 + \({ \lambda }^{ 8 }\)) (1 + \({ \lambda }^{ 16 }\)), then the value n is (where n \(\in \) N)
- (a)
32
- (b)
16
- (c)
31
- (d)
15
If a1, a2,a3, .. ,a20 are AMs between 13 and 67, then the maximum value of a1a2a3 ...a20 is
- (a)
(20)20
- (b)
(40)20
- (c)
(60)20
- (d)
(80)20
If a1,a2,...., an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 + ... + an-1 + 2an is
- (a)
n(2c)1/n
- (b)
(n+1)c1/n
- (c)
2nc1/n
- (d)
(n+1)(2c)1/n
The coefficient of x49 in the product (x - 1) (x - 3)....(x - 99) is
- (a)
-992
- (b)
1
- (c)
-2500
- (d)
none of these
A monkey while trying to reach the top of a pole of height 12 m takes every time a jump of 2 m but slips 1 m while holding the pole. The number of jumps required to reach the top of the pole, is
- (a)
6
- (b)
10
- (c)
11
- (d)
12
The solution of log\(\sqrt { 3 } \)x+log\(\sqrt [ 4 ]{ 3 } \)x+log\(\sqrt [ 6 ]{ 3 } \)x+....+log\(\sqrt [ 16 ]{ 3 } \)x = 36 is
- (a)
x = 3
- (b)
x = 4\(\sqrt { 3 } \)
- (c)
x = 9
- (d)
x = \(\sqrt { 3 } \)
The sum to infinity of the series, \(1+2\left( 1-\frac { 1 }{ n } \right) +3{ \left( 1-\frac { 1 }{ n } \right) }^{ 2 }+....\) is
- (a)
n2
- (b)
n ( n+1)
- (c)
\(n{ \left( 1+\frac { 1 }{ n } \right) }^{ 2 }\)
- (d)
none of these
If a, a1, a2, a3, ... , a2n, b are in AP and a, g1, g2, g3......, g2n, b are in GP and h is the HM of a and b, then \(\frac { { a }_{ 1 }+{ a }_{ 2n } }{ { g }_{ 1 }{ g }_{ 2n } } +\frac { { a }_{ 2 }+{ a }_{ 2n-1 } }{ { g }_{ 2 }{ g }_{ 2n-1 } } +....+\frac { { a }_{ n }+{ a }_{ n+1 } }{ { g }_{ n }{ g }_{ n+1 } } \) is equal to
- (a)
\(\frac { 2n }{ h } \)
- (b)
2nh
- (c)
nh
- (d)
\(\frac { n }{ h } \)
the pth term Tp of HP is q (p+q) and qth term Tq is p(p+q) when p>1, q>1, then
- (a)
Tp+q =pq
- (b)
Tpq=p+q
- (c)
Tp+q>Tpq
- (d)
Tpq>Tp+q
If the arithmetic mean of two positive numbers a and b (a>b) is twice their geometric mean, then a : b is
- (a)
2 + \(\sqrt { 3 } \) : 2 - \(\sqrt { 3 } \)
- (b)
7 + 4\(\sqrt { 3 } \) : 1
- (c)
1 : 7 - 4 \(\sqrt { 3 } \)
- (d)
2: \(\sqrt { 3 } \)
Suppose P is the first of n (n > 1) AM's between two positive numbers a and b; q the first of n HM's between the same two numbers.
The value of \(\left( \frac { p }{ q } -1 \right) \)is
- (a)
\(\frac { n }{ { (n-1) }^{ 2 } } { \left( \sqrt { \frac { a }{ b } } -\sqrt { \frac { b }{ a } } \right) }^{ 2 }\)
- (b)
\(\frac { n }{ { (n-1) }^{ 2 } } { \left( \sqrt { \frac { a }{ b } } +\sqrt { \frac { b }{ a } } \right) }^{ 2 }\)
- (c)
\(\frac { n }{ { (n+1) }^{ 2 } } { \left( \sqrt { \frac { a }{ b } } -\sqrt { \frac { b }{ a } } \right) }^{ 2 }\)
- (d)
\(\frac { n }{ { (n+1) }^{ 2 } } { \left( \sqrt { \frac { a }{ b } } +\sqrt { \frac { b }{ a } } \right) }^{ 2 }\)
In an A.P., if mth term is n and the nth term is m, where m\(\neq\)n, then find its pth term.
- (a)
n-m+p
- (b)
n+m+p
- (c)
n+m-p
- (d)
n-m-p
Which term of the sequence 4,9,14,19,....is 124 ?
- (a)
25th
- (b)
20th
- (c)
26th
- (d)
21st
If S1,S2,S3 are the sum of n,2n,3n terms respectively of an A.P., then S3/(S2-S1)=
- (a)
1
- (b)
2
- (c)
3
- (d)
4
The natural numbers are grouped as follows:
S1={1},S2={2,3,4},S3={5,6,7,8,9},........, then the first element of S21 is
- (a)
391
- (b)
399
- (c)
401
- (d)
442
If a1,a2,a3,a4 and b are real numbers such that (\(a_1^2+a_2^2+a_2^3\))b2- 2(a1a2 +a2a3 +a3a4)b+(\(a_2^2+a_3^2+a_4^2\))\(\le 0,\) then a1,a2,a3,a4 are
- (a)
in A.P.
- (b)
in G.P.
- (c)
in A.G.P.
- (d)
such that (a1+a2)(a3-a4)=(a1+a3)(a2-a4)
Calculate sum to n terms of the series 4+44+444+...
- (a)
\({4\over9}[{10\over9}(10^n-1)-n]\)
- (b)
\({4\over9}[{10\over9}(10^n+1)+n]\)
- (c)
\({10\over9}(10^n-1)\)
- (d)
None of these
Let x be the arithmetic mean and y, z be the two geometric mean and y,z be the two geometric means between any two positive numbers. The value of \({y^3+z^3\over xyz}\)is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
The sum of the series 12+(12+22)+(12+22+32)to n terms=...
- (a)
\({n(n+1)^2(n+2)\over12}\)
- (b)
\({n(n+1)^2(n+3)\over12}\)
- (c)
\({n(n+2)^2(n+1)\over12}\)
- (d)
\({n(n+2)^2(n+1)\over14}\)
If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is
- (a)
0
- (b)
22
- (c)
220
- (d)
198
Consider first three terms of a sequence Sn is [x - 1], [x - 3],which are in A.P.
Statement-I : The sixth term of Sn is 7 < third term.
Statement-II : If a, a + d, a + 2d, ... are in A.P. (d \(\neq\) 0), then sixth term is (a + 5d).
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -1.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.