JEE Mains Previous Years Questions with Solutions Android App

Download our App

JEE Mains Previous Years Questions with Solutions

4.5 
Star 1 Star 2 Star 3 Star 4
Star 5
  (100k+ )
1

AIEEE 2004

MCQ (Single Correct Answer)
Let $$A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right)$$. The only correct

statement about the matrix $$A$$ is

A
$${A^2} = 1$$
B
$$A=(-1)I,$$ where $$I$$ is a unit matrix
C
$${A^{ - 1}}$$ does not exist
D
$$A$$ is a zero matrix

Explanation

$$A = \left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]$$

clearly $$\,\,\,A \ne 0.\,$$ Also $$\,\,\left| A \right| = - 1 \ne 0$$

$$\therefore$$ $${A^{ - 1}}\,\,$$ exists, further

$$\left( { - 1} \right)I = \left[ {\matrix{ { - 1} & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right] \ne A$$

Also $${A^2} = \left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]\left[ {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] = I$$
2

AIEEE 2004

MCQ (Single Correct Answer)
Let $$A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right).$$ and $$10$$ $$B = \left( {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right)$$. if $$B$$ is

the inverse of matrix $$A$$, then $$\alpha $$ is

A
$$5$$
B
$$-1$$
C
$$2$$
D
$$-2$$

Explanation

Given that $$10B$$ $$\,\,\, = \left[ {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right]$$

$$ \Rightarrow B = {1 \over {10}}\left[ {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right]$$

Also since, $$B = {A^{ - 1}} \Rightarrow AB = I$$

$$ \Rightarrow {1 \over {10}}\left[ {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right]\left[ {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right] = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

$$ \Rightarrow {1 \over {10}}\left[ {\matrix{ {10} & 0 & {5 - 2} \cr 0 & {10} & { - 5 + \alpha } \cr 0 & 0 & {5 + \alpha } \cr } } \right] = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

$$ \Rightarrow {{5 - \alpha } \over {10}} = 0$$

$$ \Rightarrow \alpha = 5$$
3

AIEEE 2003

MCQ (Single Correct Answer)
If the system of linear equations
$$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
has a non - zero solution, then $$a, b, c$$.
A
satisfy $$a+2b+3c=0$$
B
are in A.P
C
are in G.P
D
are in H.P.

Explanation

For homogeneous system of equations to have non zero solution, $$\Delta = 0$$

$$\left| {\matrix{ 1 & {2a} & a \cr 1 & {3b} & b \cr 1 & {4c} & c \cr } } \right| = 0\,{C_2} \to {C_2} - 2{C_3}$$

$$\left| {\matrix{ 1 & 0 & a \cr 1 & b & b \cr 1 & {2c} & c \cr } } \right| = 0\,\,{R_3} \to {R_3} - {R_2},{R_2} \to {R_2} - {R_1}$$

$$\left| {\matrix{ 1 & 0 & a \cr 0 & b & {b - a} \cr 0 & {2c - b} & {c - b} \cr } } \right| = 0$$

$$b\left( {c - b} \right) - \left( {b - a} \right)\left( {2c - b} \right) = 0$$

On simplification, $${2 \over b} = {1 \over a} + {1 \over c}$$

$$\therefore$$ $$a,b,c$$ are in Harmonic Progression.
4

AIEEE 2003

MCQ (Single Correct Answer)
If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then

$$\Delta = \left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr } } \right|$$ is equal to

A
$${\omega ^2}$$
B
$$0$$
C
$$1$$
D
$$\omega $$

Explanation

$$\Delta = \left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr } } \right|$$

$$ = 1\left( {{\omega ^{3n}} - 1} \right) - {\omega ^n}\left( {{\omega ^{2n}} - {\omega ^{2n}}} \right) + $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^{2n}}\left( {{\omega ^n} - {\omega ^{4n}}} \right)$$

$$ = {\omega ^{3n}} - 1 - 0 + {\omega ^{3n}} - {\omega ^{6n}}$$

$$ = 1 - 1 + 1 - 1 = 0$$ $$\left[ {} \right.$$ as $$\,\,\,\,\,$$ $${\omega ^{3n}} = 1$$ $$\left. {} \right]$$

Questions Asked from Matrices and Determinants

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
JEE Main 2021 (Online) 1st September Evening Shift (2)
JEE Main 2021 (Online) 31st August Evening Shift (1)
JEE Main 2021 (Online) 31st August Morning Shift (2)
JEE Main 2021 (Online) 27th August Evening Shift (3)
JEE Main 2021 (Online) 27th August Morning Shift (1)
JEE Main 2021 (Online) 26th August Evening Shift (1)
JEE Main 2021 (Online) 26th August Morning Shift (2)
JEE Main 2021 (Online) 27th July Evening Shift (1)
JEE Main 2021 (Online) 27th July Morning Shift (1)
JEE Main 2021 (Online) 25th July Evening Shift (2)
JEE Main 2021 (Online) 25th July Morning Shift (1)
JEE Main 2021 (Online) 22th July Evening Shift (2)
JEE Main 2021 (Online) 20th July Evening Shift (1)
JEE Main 2021 (Online) 20th July Morning Shift (1)
JEE Main 2021 (Online) 18th March Evening Shift (2)
JEE Main 2021 (Online) 18th March Morning Shift (2)
JEE Main 2021 (Online) 17th March Evening Shift (1)
JEE Main 2021 (Online) 17th March Morning Shift (2)
JEE Main 2021 (Online) 16th March Morning Shift (1)
JEE Main 2021 (Online) 26th February Evening Shift (1)
JEE Main 2021 (Online) 26th February Morning Shift (3)
JEE Main 2021 (Online) 25th February Evening Shift (3)
JEE Main 2021 (Online) 24th February Evening Shift (2)
JEE Main 2021 (Online) 24th February Morning Shift (1)
JEE Main 2020 (Online) 6th September Evening Slot (1)
JEE Main 2020 (Online) 6th September Morning Slot (2)
JEE Main 2020 (Online) 5th September Evening Slot (2)
JEE Main 2020 (Online) 5th September Morning Slot (1)
JEE Main 2020 (Online) 4th September Evening Slot (2)
JEE Main 2020 (Online) 4th September Morning Slot (1)
JEE Main 2020 (Online) 3rd September Evening Slot (1)
JEE Main 2020 (Online) 3rd September Morning Slot (1)
JEE Main 2020 (Online) 2nd September Evening Slot (2)
JEE Main 2020 (Online) 2nd September Morning Slot (2)
JEE Main 2020 (Online) 9th January Evening Slot (1)
JEE Main 2020 (Online) 9th January Morning Slot (2)
JEE Main 2020 (Online) 8th January Evening Slot (2)
JEE Main 2020 (Online) 8th January Morning Slot (1)
JEE Main 2020 (Online) 7th January Evening Slot (1)
JEE Main 2020 (Online) 7th January Morning Slot (2)
JEE Main 2019 (Online) 12th April Evening Slot (1)
JEE Main 2019 (Online) 12th April Morning Slot (2)
JEE Main 2019 (Online) 10th April Evening Slot (2)
JEE Main 2019 (Online) 10th April Morning Slot (2)
JEE Main 2019 (Online) 9th April Evening Slot (2)
JEE Main 2019 (Online) 9th April Morning Slot (2)
JEE Main 2019 (Online) 8th April Evening Slot (1)
JEE Main 2019 (Online) 8th April Morning Slot (2)
JEE Main 2019 (Online) 12th January Evening Slot (2)
JEE Main 2019 (Online) 12th January Morning Slot (2)
JEE Main 2019 (Online) 11th January Evening Slot (2)
JEE Main 2019 (Online) 11th January Morning Slot (2)
JEE Main 2019 (Online) 10th January Evening Slot (2)
JEE Main 2019 (Online) 10th January Morning Slot (2)
JEE Main 2019 (Online) 9th January Evening Slot (2)
JEE Main 2019 (Online) 9th January Morning Slot (2)
JEE Main 2018 (Online) 16th April Morning Slot (2)
JEE Main 2018 (Offline) (2)
JEE Main 2018 (Online) 15th April Evening Slot (2)
JEE Main 2018 (Online) 15th April Morning Slot (2)
JEE Main 2017 (Online) 9th April Morning Slot (1)
JEE Main 2017 (Online) 8th April Morning Slot (3)
JEE Main 2017 (Offline) (2)
JEE Main 2016 (Online) 10th April Morning Slot (2)
JEE Main 2016 (Online) 9th April Morning Slot (2)
JEE Main 2016 (Offline) (3)
JEE Main 2015 (Offline) (2)
JEE Main 2014 (Offline) (2)
JEE Main 2013 (Offline) (1)
AIEEE 2012 (2)
AIEEE 2011 (2)
AIEEE 2010 (3)
AIEEE 2009 (2)
AIEEE 2008 (3)
AIEEE 2007 (2)
AIEEE 2006 (2)
AIEEE 2005 (4)
AIEEE 2004 (3)
AIEEE 2003 (3)
AIEEE 2002 (1)

Joint Entrance Examination

JEE Main JEE Advanced WB JEE

Graduate Aptitude Test in Engineering

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

Medical

NEET

CBSE

Class 12