JEE Mathematics - Sequences and Series Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
Let T be the rth term of an A.P. for r = 1, 2, 3 .... If for some positive integers m, n \({ T }_{ m }\quad =\quad \frac { 1 }{ n } ,\quad { T }_{ n }\quad =\quad \frac { 1 }{ m } \), then Tmn equals
- (a)
\(1\over mn\)
- (b)
\({1\over m}+{1\over n}\)
- (c)
1
- (d)
0
If a, b , c are positive numbers, the least value of \(\frac { a+b }{ c } +\frac { b+c }{ a } +\frac { c+a }{ b } \), is
- (a)
6
- (b)
4
- (c)
3
- (d)
NONE OF THESE
The sum of n terms of an A.P. is 3n2+5. The number of the term which equals 159 is
- (a)
13
- (b)
21
- (c)
27
- (d)
NONE OF THESE
The nth terms of the two series 3+10+17+.... and 63+65+67+.... are equal; the value of n is
- (a)
9
- (b)
13
- (c)
19
- (d)
NONE OF THESE
The arithmetic mean between two numbers is A and the geometric means is G. Then these numbers are
- (a)
\(A\pm \sqrt { { A }^{ 2 }-{ G }^{ 2 } } \)
- (b)
\(\sqrt { { A }^{ 2 }-{ G }^{ 2 } } \pm A\)
- (c)
\(\frac { 1 }{ 2 } \left( \sqrt { { A }^{ 2 }-{ +G }^{ 2 } } +A \right) \)
- (d)
NONE OF THESE
The geometric mean between -8 and -18 is
- (a)
12
- (b)
-12
- (c)
-13
- (d)
NONE OF THESE
Let a1, a2, a3,....a10 be in A.P. and h1,h2,h3....h12 be in H.P. If a1 = h1 = 2 and a10 = h10 = 3, then a4h7 equals
- (a)
2
- (b)
3
- (c)
5
- (d)
6
If \(\frac { { e }^{ x } }{ 1-x } \quad =\quad { B }_{ 0 }+{ B }_{ 1 }x+{ B }_{ 2 }{ x }^{ 2 }+...+{ B }_{ n }{ x }^{ n }+...,\) then value of \(B_n -B_{n-1}\) is
- (a)
1
- (b)
\(1\over n\)
- (c)
\(1\over n!\)
- (d)
NONE OF THESE
The sum of the series, \(\frac { 1 }{ \frac { 2 }{ 3! } +\frac { 4 }{ 5! } +\frac { 6 }{ 7! } +...\infty } \), is
- (a)
\(1\over e\)
- (b)
\(1\over {e^2}\)
- (c)
e
- (d)
NONE OF THESE
The sum of the series, \(\frac { 1 }{ { n }^{ 2 } } +\frac { 1 }{ { 2n }^{ 4 } } +\frac { 1 }{ { 3n }^{ 6 } } +...\) is
- (a)
\(log\left( \frac { { n }^{ 2 } }{ { n }^{ 2 }+1 } \right) \)
- (b)
\(log\left( \frac { { n }^{ 2 }+1 }{ { n }^{ 2 } } \right) \)
- (c)
\(log\left( \frac { { n }^{ 2 } }{ { n }^{ 2 }-1 } \right) \)
- (d)
NONE OF THESE
Let Sn denotes the sum of n terms of an AP, whose first term is a. If the common difference \(d={ S }_{ n }-k{ S }_{ n-1 }+{ S }_{ n-2 },\) then k is equal to
- (a)
3
- (b)
2
- (c)
5
- (d)
7
In the series 3, 7, 11, 15, .... and 2, 5, 8, .... each continued to 100 terms. The number of terms which are identical, is
- (a)
22
- (b)
23
- (c)
25
- (d)
20
If x, Y and z are positive integers, then value of expression ( X + Y) (Y + Z) (Z + X) is
- (a)
= 8 xyz
- (b)
> 8 xyz
- (c)
< 8 xyz
- (d)
= 4 xyz
Match the columns
Column I | Column II |
A. If \(\sum { n } =210\), then \(\sum { { n }^{ 2 } } \) is divisible by the greatest prime number which is greater than | P. 16 |
B. Between 4 and 2916 is inserted odd number (2n+1) GM's. Then, the (n+1)th GM is divisible by greatest odd integer which is less than | Q. 10 |
C. In a certain progression, four consecutive terms are 40, 30, 24, 20. Then, the integral part of the next term of the progression is more than | R. 34 |
D. \(1+\frac { 4 }{ 5 } +\frac { 7 }{ { 5 }^{ 2 } } +\frac { 10 }{ { 5 }^{ 3 } } +....to\quad \infty =\frac { a }{ b } \) where HCF (a,b) = 1, then a - b is less than | S. 30 |
- (a)
A B C D (P, Q, R, S) (R, S) (P, Q) (R, S) - (b)
A B C D R, S) (P, Q) (P, S) (Q, S) - (c)
A B C D (Q, S) (P, S) (P, R) (Q, R) - (d)
None of the above
If first and (2n - 1)th terms of an AP, GP and HP are equal and their nth terms are a, b and c respectively, then
- (a)
\(a=b\neq c\)
- (b)
2a + c = 2b
- (c)
a < b < c
- (d)
\(ac-{ b }^{ 2 }=0\)
If X, Y and Z are in AP and tan-1X, tan-1Y and tan-1Z are also in AP, then
- (a)
x = y = z
- (b)
2x = 3y = 6z
- (c)
6x = 3y = 2z
- (d)
6x = 4y = 3z
If \(\sum _{ i=1 }^{ 21 }{ { a }_{ i }=693 } \) , where a1,a2,.....,a21 are in AP, then the value of \(\sum _{ r=0 }^{ 10 }{ { a }_{2r+1 } } \) is
- (a)
361
- (b)
363
- (c)
365
- (d)
398
In the quadratic equation a x2 + bx + c = 0, if \(\Delta ={ b }^{ 2 }-4ac\) and \(\alpha +\beta ,{ a }^{ 2 }+{ \beta }^{ 2 },{ a }^{ 3 }+{ \beta }^{ 3 }\) are in GP, where a, \(\beta\) are the roots of a x2 + bx + c = 0, then
- (a)
\(\Delta \neq 0\)
- (b)
\(b\Delta =0\)
- (c)
\(c\Delta =0\)
- (d)
\(\Delta =0\)
If a, b, c, d, e, f are in AP, then (e - c) is equal to
- (a)
2(c - a)
- (b)
2(d - b)
- (c)
2(f -d)
- (d)
2(d - c)
The sum of the series 1. n + 2 . (n-1) +3.(n-2)+...+n.1 is
- (a)
\(\frac { n(n+1)(n+2) }{ 6 } \)
- (b)
\(\frac { n(n+1)(n+2) }{ 3 } \)
- (c)
\(\frac { n(n+1)(2n+1) }{ 6 } \)
- (d)
\(\frac { n(n+1)(2n+1) }{ 3 } \)
If p, q, r are three positive real numbers are in AP, then the roots of the quadratic equation px2 + qx + r = 0 are all real for
- (a)
\(\left| \frac { r }{ p } -7 \right| \ge 4\sqrt { 3 } \)
- (b)
\(\left| \frac { P }{ r } -7 \right| <4\sqrt { 3 } \)
- (c)
all p and r
- (d)
no p and r
If \(\frac { a+bx }{ a-bx } =\frac { b+cx }{ b-cx } =\frac { c+dx }{ c-dx } (x\neq 0),\) then a, b, c, d are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
Let a, b, c be three positive prime numbers. The progression in which \(\sqrt { a } ,\sqrt { b } ,\sqrt { c } \) can be three terms (not necessarily consecutive) is
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
The consecutive odd integers whose sum is 452 - 212 are
- (a)
43,45,.....,75
- (b)
43,45, ......,79
- (c)
43,45,.......,85
- (d)
43,45, .......,89
If \(\sum { n } ,\frac { \sqrt { 10 } }{ 3 } \sum { { n }^{ 2 },\sum { { n }^{ 3 } } } \) are in GP, then the value of n is
- (a)
3
- (b)
4
- (c)
2
- (d)
non existent
If \(\sum _{ n=1 }^{ k }{ \left[ \frac { 1 }{ 3 } +\frac { n }{ 90 } \right] =21 } \), where [x] denotes the integral part of x, then k is equal to
- (a)
84
- (b)
80
- (c)
85
- (d)
none of these
The coefficient of x15 in the product (1-x)(1-2x)(1-22x)(1-23x)..(1-215x) is
- (a)
2105 - 2121
- (b)
2121-2105
- (c)
2120-2104
- (d)
2105-2104
If tan-1 x, tan-1 y, tan-1 z are in AP and x, y, z are also in AP ( Y being not equal to 0, 1 or - 1), then
- (a)
x,y,z are in GP
- (b)
x,y,z are in HP
- (c)
x=y=z
- (d)
(x-y)2+(y-z)2+(z-x)2=0
\(\sum _{ i=1 }^{ n }{ \sum _{ j=1 }^{ i }{ \sum _{ k=1 }^{ j }{ 1 } } } \) is equal to
- (a)
\(\frac { n(+1)(n+2) }{ 6 } \)
- (b)
\(\sum { { n }^{ 2 } } \)
- (c)
nC 3
- (d)
n+2C3
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 q is
- (a)
\(\frac { ab(n+1) }{ b+an } \)
- (b)
\(\frac { ab(n+1) }{ (a+bn) } \)
- (c)
\(\frac { ab(n-1) }{ b+an } \)
- (d)
\(\frac { ab(n-1) }{ a+bn } \)
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.
Final conclusion is
- (a)
q lies between p and \(\left( \frac { n+1 }{ n-1 } \right) p\)
- (b)
q lies between p and \({ \left( \frac { n+1 }{ n-1 } \right) }^{ 2 }p\)
- (c)
q does not lies between p and \(\left( \frac { n+1 }{ n-1 } \right) p\)
- (d)
q does not lies between p and \({ \left( \frac { n+1 }{ n-1 } \right) }^{ 2 }\)
The sum of the squares of three distinct real numbers which are in strictly increasing GP is S 2. If their sum is \(\alpha\) S.
If we drop the condition that the GP is strictly increasing and take r2 = 1, (where r is common ratio of GP) then the value of \(\alpha\) is
- (a)
0
- (b)
\(\pm \) 1
- (c)
\(\pm \) 2
- (d)
\(\pm \) \(\sqrt { 3 } \)
We are giving the concept of arithmetic mean of mth power, Let a, b > 0 and a \(\neq \) b and let m be a real number. Then
\(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } >{ \left( \frac { a+b }{ 2 } \right) }^{ m },\quad if\quad m\in R\sim [0,1]\)
However if \(m\in (0,1),\) then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } <{ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
Obviously if m \(\in \{ 0,\quad 1\} ,\)then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } ={ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
If a, b be positive and a + b = 1 (a \(\neq \) b) and if A= \(\sqrt [ 3 ]{ a } +\sqrt [ 3 ]{ b } \) then the correct statement is
- (a)
A > 22/3
- (b)
\(A=\frac { { 2 }^{ 2/3 } }{ 3 } \)
- (c)
A < 22/3
- (d)
A = 22/3
We are giving the concept of arithmetic mean of mth power, Let a, b > 0 and a \(\neq \) b and let m be a real number. Then
\(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } >{ \left( \frac { a+b }{ 2 } \right) }^{ m },\quad if\quad m\in R\sim [0,1]\)
However if \(m\in (0,1),\) then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } <{ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
Obviously if m \(\in \{ 0,\quad 1\} ,\)then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } ={ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
If x, y be positive real numbers such that x2 + y2 = 8, then the maximum value of x + y is
- (a)
2
- (b)
4
- (c)
6
- (d)
8
We are giving the concept of arithmetic mean of mth power, Let a, b > 0 and a \(\neq \) b and let m be a real number. Then
\(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } >{ \left( \frac { a+b }{ 2 } \right) }^{ m },\quad if\quad m\in R\sim [0,1]\)
However if \(m\in (0,1),\) then \(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } <{ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
Obviously if m \(\in \{ 0,\quad 1\} ,\)\(\frac { { a }^{ m }+{ b }^{ m } }{ 2 } ={ \left( \frac { a+b }{ 2 } \right) }^{ m }\)
If a, b, c are are positive real numbers but not all equal such that a + b + c = 1, then best option of values \(\frac { { b }^{ 2 }+{ c }^{ 2 } }{ b+c } +\frac { { c }^{ 2 }+{ a }^{ 2 } }{ c+a } +\frac { { a }^{ 2 }+{ b }^{ 2 } }{ a+b } \)lie between
- (a)
\(\left( \frac { 3 }{ 2 } ,\infty \right) \)
- (b)
(1, \(\infty \))
- (c)
(0, \(\infty \))
- (d)
none of these
If ai > 0, i=1,2,3,....,n and m1,m2,m3,...mn be positive rational numbers, then
\(\left( \frac { { m }_{ 1 }{ a }_{ 1 }+{ m }_{ 2 }{ a }_{ 2 }+...+{ m }_{ n }a_{ n } }{ { m }_{ 1 }+{ m }_{ 2 }+....+{ m }_{ n } } \right) \ge ({ a }_{ 1 }^{ { m }_{ 1 } }{ a }_{ 2 }^{ { m }_{ 2 } }....{ a }_{ n }^{ { m }_{ n } })^{ 1/({ m }_{ 1 }+{ m }_{ 2 }+...+{ m }_{ n }) }\)
\(\ge \frac { ({ m }_{ 1 }+{ m }_{ 2 }+....+{ m }_{ n }) }{ \frac { { m }_{ 1 } }{ { a }_{ 1 } } +\frac { { m }_{ 2 } }{ { a }_{ 2 } } +...+\frac { { m }_{ n } }{ { a }_{ n } } } \) is called weighted mean theorem
where \({ A }^{ * }=\frac { { m }_{ 1 }{ a }_{ 1 }+{ m }_{ 2 }{ a }_{ 2 }+....+{ m }_{ n }{ a }_{ n } }{ { m }_{ 1 }+{ m }_{ 2 }+...+{ m }_{ n } } \)
= Weighted arithmetic mean
\({ G }^{ * }=({ a }_{ 1 }^{ { m }_{ 1 } }{ a }_{ 2 }^{ { m }_{ 2 } }....{ a }_{ n }^{ { m }_{ n } })^{ 1/({ m }_{ 1 }+{ m }_{ 2 }+...+{ m }_{ n }) }\)
=Weighted geometric mean
and \({ H }^{ * }=\frac { { m }_{ 1 }+{ m }_{ 2 }+....+{ m }_{ n } }{ \frac { { m }_{ 1 } }{ { a }_{ 1 } } +\frac { { m }_{ 2 } }{ { a }_{ 2 } } +...\frac { { m }^{ n } }{ { a }_{ n } } } \)= Weighted harmonic mean
ie., \(A^{ * }\ge { G }^{ * }\ge { H }^{ * }\)
Now, let a+b + c = 5(a,b,c > 0) and x2y3 = 6(x>0,y>0)
The greatest value of ab3 c is
- (a)
3
- (b)
9
- (c)
27
- (d)
81
If ai > 0, i=1,2,3,....,n and m1,m2,m3,...mn be positive rational numbers, then
\(\left( \frac { { m }_{ 1 }{ a }_{ 1 }+{ m }_{ 2 }{ a }_{ 2 }+...+{ m }_{ n }a_{ n } }{ { m }_{ 1 }+{ m }_{ 2 }+....+{ m }_{ n } } \right) \ge ({ a }_{ 1 }^{ { m }_{ 1 } }{ a }_{ 2 }^{ { m }_{ 2 } }....{ a }_{ n }^{ { m }_{ n } })^{ 1/({ m }_{ 1 }+{ m }_{ 2 }+...+{ m }_{ n }) }\)
\(\ge \frac { ({ m }_{ 1 }+{ m }_{ 2 }+....+{ m }_{ n }) }{ \frac { { m }_{ 1 } }{ { a }_{ 1 } } +\frac { { m }_{ 2 } }{ { a }_{ 2 } } +...+\frac { { m }_{ n } }{ { a }_{ n } } } \) is called weighted mean theorem
where \({ A }^{ * }=\frac { { m }_{ 1 }{ a }_{ 1 }+{ m }_{ 2 }{ a }_{ 2 }+....+{ m }_{ n }{ a }_{ n } }{ { m }_{ 1 }+{ m }_{ 2 }+...+{ m }_{ n } } \)
= Weighted arithmetic mean
\({ G }^{ * }=({ a }_{ 1 }^{ { m }_{ 1 } }{ a }_{ 2 }^{ { m }_{ 2 } }....{ a }_{ n }^{ { m }_{ n } })^{ 1/({ m }_{ 1 }+{ m }_{ 2 }+...+{ m }_{ n }) }\)
=Weighted geometric mean
and \({ H }^{ * }=\frac { { m }_{ 1 }+{ m }_{ 2 }+....+{ m }_{ n } }{ \frac { { m }_{ 1 } }{ { a }_{ 1 } } +\frac { { m }_{ 2 } }{ { a }_{ 2 } } +...\frac { { m }^{ n } }{ { a }_{ n } } } \)= Weighted harmonic mean
ie ., \(A^{ * }\ge { G }^{ * }\ge { H }^{ * }\)
Now, let a+b + c = 5(a,b,c > 0) and x2y3 = 6(x>0,y>0)
The maximum value of \(\lambda \) \(\mu\) \(\nu \) when \(\frac { { \lambda }^{ 2 } }{ { d }_{ 1 }^{ 2 } } +\frac { { { \mu }^{ 2 } } }{ { d }_{ 2 }^{ 2 } } +\frac { { \upsilon }^{ 2 } }{ { d }_{ 3 }^{ 2 } } =1\) is
- (a)
\(\frac { { d }_{ 1 }{ d }_{ 2 }{ d }_{ 3 } }{ \sqrt [ 3 ]{ 3 } } \)
- (b)
\(\frac { { d }_{ 1 }{ d }_{ 2 }{ d }_{ 3 } }{ 3\sqrt { 3 } } \)
- (c)
\(\frac { { d }_{ 1 }d_{ 2 }{ d }_{ 3 } }{ 3 } \)
- (d)
\(\frac { { d }_{ 1 }d_{ 2 }{ d }_{ 3 } }{ \sqrt { 3 } } \)
Write the first three terms of the sequence whose general term is an=2n+5
- (a)
1,3,5
- (b)
2,4,6
- (c)
7,9,11
- (d)
6,8,10
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
Find the sum of first 24 terms of the A.P.a1,a2,a3,...if it is known that a1+a5+a10+a15+a20+a24=225.
- (a)
950
- (b)
900
- (c)
700
- (d)
800
If x, y, z are in A.P., then\({1\over \sqrt{y}+\sqrt{z}},{1\over \sqrt{z}+\sqrt{x}},{1\over \sqrt{x}+\sqrt{y}}\) are in
- (a)
A.P.
- (b)
G.P.
- (c)
A.G.P.
- (d)
no definite sequence
If log2, log(2x-1)and log(2x+3) are in A.P., then the vaule of x is
- (a)
5/2
- (b)
log25
- (c)
log35
- (d)
log53
Let Tr be the rth term of an A.P for r=1,2,3,.....If for some positive intefers m and n, we have Tm=\({1\over n}\) and \(T_n={1\over m},\) then Tmn=
- (a)
\({1\over mn}\)
- (b)
\({1\over m}+{1\over n}\)
- (c)
1
- (d)
0
The sum of two numbers is \({13\over6}\). An ever number of arithmetic means are being inserted between them and their sum exceeds their number by1. Find the number of means inserred.
- (a)
8
- (b)
10
- (c)
12
- (d)
14
Four geometric means are inserted between the numbers 211-1 and 211+1. The product of these geometric means is
- (a)
244-223+4
- (b)
244-222+1
- (c)
244-211+1
- (d)
222-212+1
If arithmetic mean of two distinct positive real numbers a and b(a>b) be twice their geometric mean, then a:b=
- (a)
(2+\(\sqrt{3}\)):(2-\(\sqrt{3}\))
- (b)
(2+\(\sqrt{5}\)):(2-\(\sqrt{5}\))
- (c)
(2+\(\sqrt{2}\)):(2-\(\sqrt{2}\))
- (d)
None of these
If a,b,c are positive numbers in A.P., such that their product is 64, then the minimum value of b is equal to
- (a)
2
- (b)
4
- (c)
1
- (d)
5
The arithmetic mean of two numbers is 3times their geometric mean and the sum of the squares of the two numbers is 34. The two numbers are
- (a)
\(2\sqrt{3}+\sqrt{5},2\sqrt{3}-\sqrt{5}\)
- (b)
\(3+2\sqrt{2},3-2\sqrt{2}\)
- (c)
\(\sqrt{10}+\sqrt{7},\sqrt{10}-\sqrt{7}\)
- (d)
\(\sqrt{5}-2\sqrt{2},\sqrt{5}+2\sqrt{2}\)
If the sum of n terms of an A.P. is given by 5n = 3n + 2n2, then the common difference of the A.P. is
- (a)
3
- (b)
2
- (c)
6
- (d)
4
Statement-I: If a, b, c are in A.P., then 2b = a + c.
Statement-II: If a, b, c are in A.P., then 10a, 10b, 10c are in G.P.
- (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.