1
MCQ (Single Correct Answer)

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 :
A
9
B
7
C
11
D
5

Explanation

An = $$\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} - .... + {\left( { - 1} \right)^{n - 1}}{\left( {{3 \over 4}} \right)^n}$$

Which in a G.P. with a = $${{3 \over 4}}$$, r = $${{{ - 3} \over 4}}$$ and number of terms = n

$$\therefore\,\,\,$$ An = $${{{3 \over 4}\left( {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right)} \over {1 - \left( {{{ - 3} \over 4}} \right)}} = {{{3 \over 4} \times \left( {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right)} \over {{7 \over 4}}}$$

$$ \Rightarrow $$$$\,\,\,$$An = $${{3 \over 7}}$$$$\left[ {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right]$$ $$\,\,\,\,\,\,\,\,\,$$ . . . . . . . . . .(1)

As, Bn = 1 $$-$$ An

For least odd natural number p, such that Bn > An

$$ \Rightarrow $$$$\,\,\,$$ 1 $$-$$ An > An $$ \Rightarrow $$ 1 > 2 $$ \times $$ An $$ \Rightarrow $$ An < $${{1 \over 2}}$$

From eqn. (1), we get

$${{3 \over 7}}$$$$ \times $$ $$\left[ {1 - {{\left( {{{ - 3} \over 4}} \right)}^n}} \right] < {1 \over 2}$$ $$ \Rightarrow $$ 1 $$-$$ $${\left( {{{ - 3} \over 4}} \right)^n} < {7 \over 6}$$

$$ \Rightarrow $$$$\,\,\,$$ 1 $$-$$ $${7 \over 6}$$ < $${\left( {{{ - 3} \over 4}} \right)^n}$$   $$ \Rightarrow $$   $${{ - 1} \over 6} < {\left( {{{ - 3} \over 4}} \right)^n}$$

As n is odd, then $${\left( {{{ - 3} \over 4}} \right)^n}$$ = $$-$$ $${{{{3^n}} \over 4}}$$

So $${{ - 1} \over 6}$$ < $$-$$ $${\left( {{3 \over 4}} \right)^n}$$ $$ \Rightarrow $$ $${1 \over 6}$$ > $${\left( {{3 \over 4}} \right)^n}$$

log$$\left( {{1 \over 6}} \right)$$ = n log$$\left( {{3 \over 4}} \right)$$ $$ \Rightarrow $$ 6.228 < n

Hence, n should be 7.
2
MCQ (Single Correct Answer)

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 :
A
$$38 + {1 \over {{2^{19}}}}$$
B
$$38 + {1 \over {{2^{20}}}}$$
C
$$39 + {1 \over {{2^{20}}}}$$
D
$$39 + {1 \over {{2^{19}}}}$$

Explanation

1 + $${3 \over 2}$$ + $${7 \over 4}$$ + $${15 \over 8}$$ + $${31 \over 16}$$ + . . . .

=  (2 $$-$$ 1) + (2 $$-$$ $${1 \over 2}$$ ) + (2 $$-$$ $${1 \over 4}$$) + (2 $$-$$ $${1 \over 8}$$) + . . . . .+ 20 terms

=   (2 + 2 + . . . . . 20 terms) $$-$$ (1 + $${1 \over 2}$$ + $${1 \over 4}$$ + . . . . . 20 terms)

=   2 $$ \times $$ 20 $$-$$ $$\left( {{{1 - {{\left( {{1 \over 2}} \right)}^{20}}} \over {1 - {1 \over 2}}}} \right)$$

=   40 $$-$$ 2 + 2 $${\left( {{1 \over 2}} \right)^{20}}$$

=  38 + $${1 \over {{2^{19}}}}$$
3
MCQ (Single Correct Answer)

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
$${{13} \over 4}$$

Explanation

$$ \because $$$$\,\,\,$$ $${1 \over {{x_1}}},{1 \over {{x_2}}},{1 \over {{x_3}}},.....,{1 \over {{x_n}}}$$ are in A.P.

x1 = 4 and x21 = 20

Let 'd' be the common difference of this A.P.

$$\therefore\,\,\,$$ its 21st term = $${1 \over {{x_{21}}}} = {1 \over {{x_1}}} + \left[ {\left( {21 - 1} \right) \times d} \right]$$

$$ \Rightarrow $$$$\,\,\,$$ d = $${1 \over {20}}$$ $$ \times $$ $$\left( {{1 \over {20}} - {1 \over 4}} \right)$$ $$ \Rightarrow $$ d = $$-$$ $${1 \over {100}}$$

Also xn > 50(given).

$$\therefore\,\,\,$$ $${1 \over {{x_n}}} = {1 \over {{x_1}}} + \left[ {\left( {n - 1} \right) \times d} \right]$$

$$ \Rightarrow $$$$\,\,\,$$ xn = $${{{x_1}} \over {1 + \left( {n - 1} \right) \times d \times {x_1}}}$$

$$\therefore\,\,\,$$ $${{{x_1}} \over {1 + \left( {n - 1} \right) \times d \times {x_1}}} > 50$$

$$ \Rightarrow $$$$\,\,\,$$ $${4 \over {1 + \left( {n - 1} \right) \times \left( { - {1 \over {100}}} \right) \times 4}} > 50$$

$$ \Rightarrow $$$$\,\,\,$$ 1 + (n $$-$$ 1) $$ \times $$ ($$-$$ $${1 \over {100}}$$) $$ \times $$ 4 < $${4 \over {50}}$$

$$ \Rightarrow $$$$\,\,\,$$ $$-$$ $${1 \over {100}}$$(n $$-$$ 1) < $$-$$ $${{23} \over {100}}$$

$$ \Rightarrow $$$$\,\,\,$$ n $$-$$ > 23   $$ \Rightarrow $$  n > 24

Therefore$$\,\,\,$$ n = 25.

$$ \Rightarrow $$$$\,\,\,$$$$\sum\limits_{i = 1}^{25} {{1 \over {{x_i}}}} $$ = $${{25} \over 2}\left[ {\left( {2 \times {1 \over 4}} \right) + \left( {25 - 1} \right) \times \left( { - {1 \over {100}}} \right)} \right]$$ = $${{13} \over 4}$$
4
MCQ (Single Correct Answer)

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

Explanation

a, b, c are in G.P.

So, b = ar

and c = ar2

given   a + b + c = xb

$$ \Rightarrow $$  a + br + ar2 = x(ar)

$$ \Rightarrow $$  1 + r + r2 = xr

$$ \Rightarrow $$  x = 1 + r + $${1 \over r}$$

let sum of r + $${1 \over r}$$ = M

$$ \therefore $$  r2 + 1 = Mr

$$ \Rightarrow $$  r2 $$-$$ Mr + 1 = 0

this quadratic equation will have

real solution when discriminant is $$ \ge $$ 0

$$ \therefore $$  b2 $$-$$ 4ac $$ \ge $$ 0

M2 $$-$$ 4.1.1 $$ \ge $$ 0

$$ \Rightarrow $$  M2 $$ \ge $$ 4

M $$ \ge $$ 2 or M $$ \le $$ $$-$$ 2

$$ \therefore $$  M $$ \in $$ ($$-$$ $$ \propto $$, $$-$$ 2] $$ \cup $$ [2, $$ \propto $$)

As   x = 1 + r + $${1 \over r}$$

= 1 + M

$$ \therefore $$  x $$ \in $$ ($$-$$ $$ \propto $$, $$-$$ 1] $$ \cup $$ [3, $$ \propto $$)

$$ \therefore $$  x can't be 0, 1, 2.

Questions Asked from Sequences and Series

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Name Indicates No of Questions
AIEEE 2002 (6)
keyboard_arrow_right
AIEEE 2003 (1)
keyboard_arrow_right
AIEEE 2004 (3)
keyboard_arrow_right
AIEEE 2005 (2)
keyboard_arrow_right
AIEEE 2006 (2)
keyboard_arrow_right
AIEEE 2007 (2)
keyboard_arrow_right
AIEEE 2008 (1)
keyboard_arrow_right
AIEEE 2009 (1)
keyboard_arrow_right
AIEEE 2010 (1)
keyboard_arrow_right
AIEEE 2011 (1)
keyboard_arrow_right
AIEEE 2012 (1)
keyboard_arrow_right
JEE Main 2013 (Offline) (1)
keyboard_arrow_right
JEE Main 2014 (Offline) (2)
keyboard_arrow_right
JEE Main 2015 (Offline) (2)
keyboard_arrow_right
JEE Main 2016 (Offline) (2)
keyboard_arrow_right
JEE Main 2016 (Online) 9th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2016 (Online) 10th April Morning Slot (3)
keyboard_arrow_right
JEE Main 2017 (Offline) (2)
keyboard_arrow_right
JEE Main 2017 (Online) 8th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2017 (Online) 9th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2018 (Offline) (2)
keyboard_arrow_right
JEE Main 2018 (Online) 15th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2018 (Online) 15th April Evening Slot (2)
keyboard_arrow_right
JEE Main 2018 (Online) 16th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 9th January Morning Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 9th January Evening Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 10th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 10th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 11th January Morning Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 11th January Evening Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 12th January Morning Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 12th January Evening Slot (3)
keyboard_arrow_right
JEE Main 2019 (Online) 8th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 8th April Evening Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 9th April Morning Slot (1)
keyboard_arrow_right
JEE Main 2019 (Online) 9th April Evening Slot (3)
keyboard_arrow_right
JEE Main 2019 (Online) 10th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 10th April Evening Slot (3)
keyboard_arrow_right
JEE Main 2019 (Online) 12th April Morning Slot (2)
keyboard_arrow_right
JEE Main 2019 (Online) 12th April Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 7th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 7th January Evening Slot (2)
keyboard_arrow_right
JEE Main 2020 (Online) 8th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 8th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 9th January Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 9th January Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 2nd September Morning Slot (2)
keyboard_arrow_right
JEE Main 2020 (Online) 2nd September Evening Slot (2)
keyboard_arrow_right
JEE Main 2020 (Online) 3rd September Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 3rd September Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 4th September Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 4th September Evening Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 5th September Morning Slot (2)
keyboard_arrow_right
JEE Main 2020 (Online) 5th September Evening Slot (2)
keyboard_arrow_right
JEE Main 2020 (Online) 6th September Morning Slot (1)
keyboard_arrow_right
JEE Main 2020 (Online) 6th September Evening Slot (1)
keyboard_arrow_right
JEE Main 2021 (Online) 25th February Morning Slot (1)
keyboard_arrow_right
JEE Main 2021 (Online) 26th February Morning Shift (2)
keyboard_arrow_right
JEE Main 2021 (Online) 26th February Evening Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 18th March Morning Shift (2)
keyboard_arrow_right
JEE Main 2021 (Online) 18th March Evening Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 20th July Evening Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 22th July Evening Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 25th July Morning Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 26th August Morning Shift (2)
keyboard_arrow_right
JEE Main 2021 (Online) 27th August Morning Shift (2)
keyboard_arrow_right
JEE Main 2021 (Online) 27th August Evening Shift (1)
keyboard_arrow_right
JEE Main 2021 (Online) 31st August Morning Shift (2)
keyboard_arrow_right
JEE Main 2021 (Online) 31st August Evening Shift (1)
keyboard_arrow_right

EXAM MAP

Joint Entrance Examination

JEE Advanced JEE Main

Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE ME GATE PI GATE EE GATE CE GATE IN

Medical

NEET