2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Morning Slot
Let the sum of the first n terms of a non-constant
A.P., a1, a2, a3, ..... be $50n + {{n(n - 7)} \over 2}A$, where
A is a constant. If d is the common difference of
this A.P., then the ordered pair (d, a50) is equal to
A.
(A, 50+45A)
B.
(50, 50+45A)
C.
(A, 50+46A)
D.
(50, 50+46A)
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Evening Slot
If three distinct numbers a, b, c are in G.P. and the
equations ax2
+ 2bx + c = 0 and
dx2
+ 2ex + ƒ = 0 have a common root, then
which one of the following statements is
correct?
A.
$d \over a$, $e \over b$, $f \over c$ are in G.P.
B.
d, e, ƒ are in A.P
C.
d, e, ƒ are in G.P
D.
$d \over a$, $e \over b$, $f \over c$ are in A.P.
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Evening Slot
The sum
$\sum\limits_{k = 1}^{20} {k{1 \over {{2^k}}}} $ is equal to
A.
$2 - {11 \over {{2^{19}}}}$
B.
$2 - {3 \over {{2^{17}}}}$
C.
$1 - {11 \over {{2^{20}}}}$
D.
$2 - {21 \over {{2^{20}}}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Morning Slot
The sum of all natural numbers 'n' such that
100 < n < 200 and H.C.F. (91, n) > 1 is :
A.
3221
B.
3121
C.
3203
D.
3303
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Evening Slot
If sin4$\alpha $ + 4 cos4$\beta $ + 2 = 4$\sqrt 2 $ sin $\alpha $ cos $\beta $; $\alpha $, $\beta $ $ \in $ [0, $\pi $],
then cos($\alpha $ + $\beta $) $-$ cos($\alpha $ $-$ $\beta $) is equal to :
then cos($\alpha $ + $\beta $) $-$ cos($\alpha $ $-$ $\beta $) is equal to :
A.
$ - \sqrt 2 $
B.
0
C.
$-$ 1
D.
$\sqrt 2 $
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Evening Slot
If the sum of the first 15 terms of the series ${\left( {{3 \over 4}} \right)^3} + {\left( {1{1 \over 2}} \right)^3} + {\left( {2{1 \over 4}} \right)^3} + {3^3} + {\left( {3{3 \over 4}} \right)^3} + ....$ is equal to 225 k, then k
is equal to :
A.
9
B.
108
C.
27
D.
54
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Evening Slot
If nC4, nC5 and nC6 are in A.P., then n can be :
A.
11
B.
12
C.
9
D.
14
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Morning Slot
Let Sk = ${{1 + 2 + 3 + .... + k} \over k}.$ If $S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over {12}}$A, then A is equal to :
A.
283
B.
156
C.
301
D.
303
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Morning Slot
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :
A.
36
B.
28
C.
32
D.
24
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression ${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}$ is :
A.
${1 \over 2}$
B.
${1 \over 4}$
C.
${{m + n} \over {6mn}}$
D.
1
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :
A.
2 : 1
B.
4 : 1
C.
1 : 3
D.
3 : 1
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is ${{27} \over {19}}$.Then the common ratio of this series is :
A.
${4 \over 9}$
B.
${1 \over 3}$
C.
${2 \over 3}$
D.
${2 \over 9}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
Let a1, a2, . . . . . ., a10 be a G.P. If ${{{a_3}} \over {{a_1}}} = 25,$ then ${{{a_9}} \over {{a_5}}}$ equals
A.
53
B.
2(52)
C.
4(52)
D.
54
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th January Evening Slot
Let a1, a2, a3, ..... a10 be in G.P. with ai > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k $ \in $ N (the set of natural numbers) for which
$\left| {\matrix{ {{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3}^k} & {{{\log }_e}\,{a_3}^r{a_4}^k} \cr {{{\log }_e}\,{a_4}^r{a_5}^k} & {{{\log }_e}\,{a_5}^r{a_6}^k} & {{{\log }_e}\,{a_6}^r{a_7}^k} \cr {{{\log }_e}\,{a_7}^r{a_8}^k} & {{{\log }_e}\,{a_8}^r{a_9}^k} & {{{\log }_e}\,{a_9}^r{a_{10}}^k} \cr } } \right|$ $=$ 0.
Then the number of elements in S, is -
$\left| {\matrix{ {{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3}^k} & {{{\log }_e}\,{a_3}^r{a_4}^k} \cr {{{\log }_e}\,{a_4}^r{a_5}^k} & {{{\log }_e}\,{a_5}^r{a_6}^k} & {{{\log }_e}\,{a_6}^r{a_7}^k} \cr {{{\log }_e}\,{a_7}^r{a_8}^k} & {{{\log }_e}\,{a_8}^r{a_9}^k} & {{{\log }_e}\,{a_9}^r{a_{10}}^k} \cr } } \right|$ $=$ 0.
Then the number of elements in S, is -
A.
10
B.
4
C.
2
D.
infinitely many
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th January Morning Slot
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -
A.
1356
B.
1256
C.
1365
D.
1465
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Evening Slot
The sum of the following series
$1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2} + {3^2} + {4^2}} \right)} \over 9}$
$ + {{15\left( {{1^2} + {2^2} + ... + {5^2}} \right)} \over {11}} + .....$ up to 15 terms, is :
$1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2} + {3^2} + {4^2}} \right)} \over 9}$
$ + {{15\left( {{1^2} + {2^2} + ... + {5^2}} \right)} \over {11}} + .....$ up to 15 terms, is :
A.
7520
B.
7510
C.
7830
D.
7820
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Evening Slot
Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then ${a \over c}$ equal to :
A.
2
B.
${1 \over 2}$
C.
${7 \over 13}$
D.
4
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Morning Slot
If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be
A.
2
B.
-3
C.
4
D.
-2
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Morning Slot
Let ${a_1},{a_2},.......,{a_{30}}$ be an A.P.,
$S = \sum\limits_{i = 1}^{30} {{a_i}} $ and $T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \right)}}} $.
If $a_5$ = 27 and S - 2T = 75, then $a_{10}$ is equal to :
$S = \sum\limits_{i = 1}^{30} {{a_i}} $ and $T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \right)}}} $.
If $a_5$ = 27 and S - 2T = 75, then $a_{10}$ is equal to :
A.
47
B.
42
C.
52
D.
57
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 16th April Morning Slot
Let ${1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\,$ (xi $ \ne $ 0 for i = 1, 2, ..., n) be in A.P. such that x1=4 and x21 = 20. If n is the least positive integer for which ${x_n} > 50,$ then $\sum\limits_{i = 1}^n {\left( {{1 \over {{x_i}}}} \right)} $ is equal to :
A.
${1 \over 8}$
B.
3
C.
${{13} \over 8}$
D.
${{13} \over 4}$
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 16th April Morning Slot
The sum of the first 20 terms of the series
$1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,$ is :
$1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,$ is :
A.
$38 + {1 \over {{2^{19}}}}$
B.
$38 + {1 \over {{2^{20}}}}$
C.
$39 + {1 \over {{2^{20}}}}$
D.
$39 + {1 \over {{2^{19}}}}$
2018
JEE Mains
MCQ
JEE Main 2018 (Offline)
Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
12 + 2.22 + 32 + 2.42 + 52 + 2.62 ...........
If B - 2A = 100$\lambda $, then $\lambda $ is equal to
12 + 2.22 + 32 + 2.42 + 52 + 2.62 ...........
If B - 2A = 100$\lambda $, then $\lambda $ is equal to
A.
496
B.
232
C.
248
D.
464
2018
JEE Mains
MCQ
JEE Main 2018 (Offline)
Let ${a_1}$, ${a_2}$, ${a_3}$, ......... ,${a_{49}}$ be in A.P. such that
$\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416$ and ${a_9} + {a_{43}} = 66$.
$a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m$, then m is equal to
$\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416$ and ${a_9} + {a_{43}} = 66$.
$a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m$, then m is equal to
A.
33
B.
66
C.
68
D.
34
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Evening Slot
Let An = $\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3}$ $-$. . . . . + ($-$1)n-1 ${\left( {{3 \over 4}} \right)^n}$ and Bn = 1 $-$ An.
Then, the least dd natural numbr p, so that Bn > An , for all n$ \ge $ p, is :
Then, the least dd natural numbr p, so that Bn > An , for all n$ \ge $ p, is :
A.
9
B.
7
C.
11
D.
5
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Evening Slot
If a, b, c are in A.P. and a2, b2, c2 are in G.P. such that
a < b < c and a + b + c = ${3 \over 4},$ then the value of a is :
a < b < c and a + b + c = ${3 \over 4},$ then the value of a is :
A.
${1 \over 4} - {1 \over {4\sqrt 2 }}$
B.
${1 \over 4} - {1 \over {3\sqrt 2 }}$
C.
${1 \over 4} - {1 \over {2\sqrt 2 }}$
D.
${1 \over 4} - {1 \over {\sqrt 2 }}$
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Morning Slot
If x1, x2, . . ., xn and ${1 \over {{h_1}}}$, ${1 \over {{h_2}}}$, . . . , ${1 \over {{h_n}}}$ are two A.P..s such that x3 = h2 = 8 and x8 = h7 = 20, then x5.h10 equals :
A.
2560
B.
2650
C.
3200
D.
1600
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Morning Slot
If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :
A.
($-$ $\infty $, $-$10]
B.
($-$10, 0)
C.
(0, 10)
D.
[10, $\infty $)
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 9th April Morning Slot
Let
Sn = ${1 \over {{1^3}}}$$ + {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n} \over {{1^3} + {2^3} + ...... + {n^3}}}.$
If 100 Sn = n, then n is equal to :
Sn = ${1 \over {{1^3}}}$$ + {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n} \over {{1^3} + {2^3} + ...... + {n^3}}}.$
If 100 Sn = n, then n is equal to :
A.
199
B.
99
C.
200
D.
19
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 9th April Morning Slot
If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :
A.
2
B.
4${^{{1 \over 3}}}$
C.
4${^{{2 \over 3}}}$
D.
4
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
If the sum of the first n terms of the series $\,\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......$ is $435\sqrt 3 ,$ then n equals :
A.
18
B.
15
C.
13
D.
29
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then ${{a + b} \over {a - b}}$ is equal to :
A.
${{\sqrt 6 } \over 2}$
B.
${{3\sqrt 2 } \over 4}$
C.
${{7\sqrt 3 } \over {12}}$
D.
${{5\sqrt 6 } \over {12}}$
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
For any three positive real numbers a, b and c,
9(25${a^2}$ + b2) + 25(c2 - 3$a$c) = 15b(3$a$ + c).
Then
9(25${a^2}$ + b2) + 25(c2 - 3$a$c) = 15b(3$a$ + c).
Then
A.
b, c and $a$ are in G.P.
B.
b, c and $a$ are in A.P.
C.
$a$, b and c are in A.P.
D.
$a$, b and c are in G.P.
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 10th April Morning Slot
If A > 0, B > 0 and A + B = ${\pi \over 6}$,
then the minimum value of tanA + tanB is :
then the minimum value of tanA + tanB is :
A.
$\sqrt 3 - \sqrt 2 $
B.
$2 - \sqrt 3 $
C.
$4 - 2\sqrt 3 $
D.
${2 \over {\sqrt 3 }}$
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 10th April Morning Slot
Let z = 1 + ai be a complex number, a > 0, such that z3 is a real number.
Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :
Then the sum 1 + z + z2 + . . . . .+ z11 is equal to :
A.
$ - 1250\,\sqrt 3 \,i$
B.
$ 1250\,\sqrt 3 \,i$
C.
$1365\,\sqrt 3 i$
D.
$-$ $1365\,\sqrt 3 i$
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 10th April Morning Slot
Let a1, a2, a3, . . . . . . . , an, . . . . . be in A.P.
If a3 + a7 + a11 + a15 = 72,
then the sum of its first 17 terms is equal to :
If a3 + a7 + a11 + a15 = 72,
then the sum of its first 17 terms is equal to :
A.
306
B.
153
C.
612
D.
204
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 9th April Morning Slot
Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0.1) (600)3. Then x3 + y3 + z3is equal to :
A.
270
B.
258
C.
342
D.
216
2016
JEE Mains
MCQ
JEE Main 2016 (Offline)
If the ${2^{nd}},{5^{th}}\,and\,{9^{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :
A.
1
B.
${7 \over 4}$
C.
${8 \over 5}$
D.
${4 \over 3}$
2016
JEE Mains
MCQ
JEE Main 2016 (Offline)
If the sum of the first ten terms of the series ${\left( {1{3 \over 5}} \right)^2} + {\left( {2{2 \over 5}} \right)^2} + {\left( {3{1 \over 5}} \right)^2} + {4^2} + {\left( {4{4 \over 5}} \right)^2} + .......is\,{{16} \over 5}m,$ then m is equal to :
A.
100
B.
99
C.
102
D.
101
2015
JEE Mains
MCQ
JEE Main 2015 (Offline)
The sum of first 9 terms of the series.
${{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......$
${{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......$
A.
142
B.
192
C.
71
D.
96
2015
JEE Mains
MCQ
JEE Main 2015 (Offline)
If m is the A.M. of two distinct real numbers l and n $(l,n > 1)$ and ${G_1},{G_2}$ and ${G_3}$ are three geometric means between $l$ and n, then $G_1^4\, + 2G_2^4\, + G_3^4$ equals:
A.
$4\,lm{n^2}$
B.
$4\,{l^2}{m^2}{n^2}$
C.
$4\,{l^2}m\,n$
D.
$4\,l\,{m^2}n$
2014
JEE Mains
MCQ
JEE Main 2014 (Offline)
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :
A.
$2 - \sqrt 3 $
B.
$2 + \sqrt 3 $
C.
$\sqrt 2 + \sqrt 3 $
D.
$3 + \sqrt 2 $
2014
JEE Mains
MCQ
JEE Main 2014 (Offline)
If ${(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},$, then k is equal to :
A.
100
B.
110
C.
${{121} \over {10}}$
D.
${{441} \over {100}}$
2013
JEE Mains
MCQ
JEE Main 2013 (Offline)
The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is
A.
${7 \over {81}}\left( {179 - {{10}^{ - 20}}} \right)$
B.
$\,{7 \over 9}\left( {99 - {{10}^{ - 20}}} \right)$
C.
${7 \over {81}}\left( {179 + {{10}^{ - 20}}} \right)$
D.
${7 \over 9}\left( {99 + {{10}^{ - 20}}} \right)$
2012
JEE Mains
MCQ
AIEEE 2012
Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.
Statement-2: $\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}$, for any natural number n.
A.
Statement-1 is false, Statement-2 is true.
B.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D.
Statement-1 is true, Statement-2 is false.
2011
JEE Mains
MCQ
AIEEE 2011
A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after
A.
19 months
B.
20 months
C.
21 months
D.
18 months
2010
JEE Mains
MCQ
AIEEE 2010
A person is to count 4500 currency notes. Let ${a_n}$ denote the number of notes he counts in the ${n^{th}}$ minute. If ${a_1}$ = ${a_2}$ = ....= ${a_{10}}$= 150 and ${a_{10}}$, ${a_{11}}$,.... are in an AP with common difference - 2, then the time taken by him to count all notes is
A.
34 minutes
B.
125 minutes
C.
135 minutes
D.
24 minutes
2009
JEE Mains
MCQ
AIEEE 2009
The sum to infinite term of the series $1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....$ is
A.
3
B.
4
C.
6
D.
2
2008
JEE Mains
MCQ
AIEEE 2008
The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is
A.
- 4
B.
- 12
C.
12
D.
4
2007
JEE Mains
MCQ
AIEEE 2007
In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals
A.
${\sqrt 5 }$
B.
$\,{1 \over 2}\left( {\sqrt 5 - 1} \right)$
C.
${1 \over 2}\left( {1 - \sqrt 5 } \right)$
D.
${1 \over 2}\sqrt 5 $.
2007
JEE Mains
MCQ
AIEEE 2007
The sum of series ${1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......$ upto infinity is
A.
${e^{ - {1 \over 2}}}$
B.
${e^{ + {1 \over 2}}}$
C.
${e^{ - 2}}$
D.
${e^{ - 1}}$