JEE Mathematics - Determinants and Matrices Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
The value of
\(\left| \begin{matrix} 1+i & 1-i & i \\ 1-i & i & 1+i \\ i & 1+i & 1-i \end{matrix} \right| \) is
- (a)
4-7i
- (b)
4+7i
- (c)
-4+7i
- (d)
none of these
For all values of \(\theta \), the value of the determinant
\(\triangle =\left| \begin{matrix} 1 & sin\theta & 1 \\ -sin\theta & 1 & sin\theta \\ -1 & -sin\theta & 1 \end{matrix} \right| \) lies between
- (a)
1 and 2
- (b)
2 and 3
- (c)
2 and 4
- (d)
none of these
If a,b,c are distinct real numbers and
\(\left| \begin{matrix} a & { a }^{ 2 } & { a }^{ 3 }-1 \\ b & b^{ 2 } & { b }^{ 3 }-1 \\ c & { c }^{ 2 } & { c }^{ 3 }-1 \end{matrix} \right| =0\) then
- (a)
a+b+c=0
- (b)
abc=1
- (c)
a+b+c=1
- (d)
ab+bc+ca=0
If \(\alpha ,\beta \) are thr roots of ax2+bx+c=0 and Sn=1+\(\alpha \)n+\(\beta \)n, then value of
\(\left| \begin{matrix} { s }_{ 0 } & { s }_{ 1 } & { s }_{ 2 } \\ { s }_{ 1 } & { s }_{ 2 } & { s }_{ 3 } \\ { s }_{ 2 } & { s }_{ 3 } & { s }_{ 4 } \end{matrix} \right| \), is
- (a)
\(\frac { { b }^{ 2 }-4ac }{ { a }^{ 4 } } \)
- (b)
\(\frac { (a+b+c)^{ 2 } }{ { a }^{ 4 } } \)
- (c)
\(\frac { { (b }^{ 2 }-4ac)(a+b+c)^{ 2 } }{ { a }^{ 4 } } \)
- (d)
\(\frac { { (b }^{ 2 }+4ac)(a-b+c)^{ 2 } }{ { a }^{ 4 } } \)
If \(\left[ \begin{matrix} x+3 & 2y+x \\ z-1 & 4a-6 \end{matrix} \right] =\left[ \begin{matrix} 0 & -7 \\ 3 & 2a \end{matrix} \right] \) the value of x,y,z and a are respectively
- (a)
-3,-2,4,3
- (b)
3,-4,2,-3
- (c)
4,2,3,-3
- (d)
none of these
If \(A=\left[ \begin{matrix} 1 & 2 \\ -3 & 4 \end{matrix} \right] \), then A2 equals
- (a)
\(\left[ \begin{matrix} 1 & 4 \\ 9 & 16 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 3 & 2 \\ 7 & 4 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 7 & 10 \\ 12 & 24 \end{matrix} \right] \)
- (d)
none of these
The value of x for which the matrix
\(\left[ \begin{matrix} 3-x & 2 \\ 6 & 4-x \end{matrix} \right] \) is singular, are
- (a)
3,7
- (b)
2,-7
- (c)
1,-7
- (d)
none of these
The inverse of a symmetric matrix is
- (a)
symmetric matrix
- (b)
skew-symmetric matrix
- (c)
diagonal matrix
- (d)
none of these
If \(A=\left[ \begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{matrix} \right] and\quad { A }^{ -1 }=\left[ \begin{matrix} \frac { 1 }{ 2 } & -\frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ -4 & 3 & c \\ \frac { 5 }{ 2 } & -\frac { 3 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right] \) then
- (a)
a=2,c=-\(\frac { 1 }{ 2 } \)
- (b)
a=1,c=-1
- (c)
a=-1,c=1
- (d)
a=\(\frac { 1 }{ 2 } \),c=\(\frac { 1 }{ 2 } \)
If \(A=\left[ \begin{matrix} a & b \\ b & a \end{matrix} \right] and\quad { A }^{ 2 }=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \end{matrix} \right] \), then
- (a)
\(\alpha ={ a }^{ 2 }+{ b }^{ 2 },\beta =ab\)
- (b)
\(\alpha ={ a }^{ 2 }+{ b }^{ 2 },\beta =2ab\)
- (c)
\(\alpha =2ab,\beta ={ a }^{ 2 }+{ b }^{ 2 }\)
- (d)
\(\alpha ={ a }^{ 2 }+{ b }^{ 2 },\beta ={ a }^{ 2 }-{ b }^{ 2 }\)
If B is skew-symmetric matrix and A is any other matrix, than ABAt, is a matrix, which is
- (a)
unit
- (b)
symmetric
- (c)
skew-symmetric
- (d)
diagonal
If \(A=\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right] \) and f(x)=x2-2x+3, then f(A) equals
- (a)
5I2
- (b)
3I2
- (c)
I2
- (d)
2I2
If \(A=\begin{bmatrix} \cos ^{ 2 }{ \theta } & sin\theta cos\theta \\ sin\theta cos\theta & \sin ^{ 2 }{ \theta } \end{bmatrix}\), B=\(\begin{bmatrix} \cos ^{ 2 }{ \phi } & sin\phi cos\phi \\ sin\phi cos\phi & \sin ^{ 2 }{ \phi } \end{bmatrix}\) and (\(\theta-\phi\))=\(\pi\over2\), then AB is equal to
- (a)
0
- (b)
I
- (c)
-I
- (d)
A+B
Suppose AB=A and BA=B, where A and B are square matrices, then which of the following option is correct?
- (a)
B=B2 and A2=A
- (b)
A2=B and B2=A
- (c)
AB=BA
- (d)
A2=B2
If A5=0 such that \(A^n\ne I\) for \(1\le n\le 4\) then (I-A)-1 is equal to
- (a)
A6
- (b)
A4
- (c)
I+2A
- (d)
None of these
Let \(\Delta =\begin{vmatrix} AX & x^{ 2 } & 1 \\ By & y^{ 2 } & 1 \\ Cz & z^{ 2 } & 1 \end{vmatrix}\) and \(\Delta _{ 1 }=\begin{vmatrix} A & B & C \\ x & y & z \\ zy & zx & xy \end{vmatrix}\) then
- (a)
\(\Delta_1=-\Delta\)
- (b)
\(\Delta\ne\Delta_1\)
- (c)
\(\Delta-\Delta_1=0\)
- (d)
None of the above
If \(\sqrt{-i}=i\) and \(\omega\) is a non-real root of unity, then the value of \(\begin{vmatrix} 1 & \omega ^{ 2 } & 1+i+\omega ^{ 2 } \\ -i & -1 & -1-i+\omega \\ 1-i & \omega ^{ 2 }-1 & -1 \end{vmatrix}\) is equal to
- (a)
1
- (b)
i
- (c)
\(\omega\)
- (d)
0
Suppose x=-9 is a root of \(\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix}=0\) then the other two roots are
- (a)
7,3
- (b)
2,7
- (c)
9,7
- (d)
4,7
Observe the following columns
Column I | Column II |
---|---|
A. If \(a^2+b^2+c^2=1\) and \(\Delta =\begin{vmatrix} a2+(b2+c2)cos\phi & ab(1-cos\phi ) & ac(1-cos\phi ) \\ ba(1-cos\phi ) & b2+(c2+a2)cos\phi & bc(1-cos\phi ) \\ ca(1-cos\phi ) & cb(1-cos\phi ) & c2+(a2+b2)cos\phi \end{vmatrix}\) then \(\Delta\) is independent of | P. a |
B. If \(\Delta =\begin{vmatrix} sina & cosa & sin(a+\phi ) \\ sinb & cosb & sin(b+\phi ) \\ sinc & cosc & sin(c+\phi ) \end{vmatrix}\)then \(\Delta\) is independent of | q. b |
C. \(\Delta =\begin{vmatrix} 1/c & 1/c & -(a+b)/c^{ 2 } \\ -(b+c)/a^{ 2 } & 1/a & 1/a \\ -\frac { b(b+c) }{ a^{ 2 }c } & \frac { (a+2b+c) }{ ac } & \frac { -(a+b)b }{ ac^{ 2 } } \end{vmatrix}\)\(\Delta\) then is independent of | r. c |
s. \(\phi\) |
|
t.\(\Delta=o\) |
- (a)
A B C (p,q,r) (p,q,r,s,t) (p,q,r,s,t) - (b)
A B C (q,r,t) (p,q,r) (p,q,r,s,t) - (c)
A B C (p,q,r,s,t) (p,q,r,s,t) (p,q,r) - (d)
None of these
If A,B and C are angles of a triangle, then the determinant \(\begin{vmatrix} -1 & cosC & cosB \\ cosC & -1 & cosA \\ cosB & cosA & -1 \end{vmatrix}\) is equal to
- (a)
0
- (b)
-1
- (c)
1
- (d)
None of these
If a>b>c and the system of equations ax+by+cz=0, bx+cy+az=0, cx+ay+bz=0 has a non-trivial solution, then both the roots of the quadratic equation at2+bt+c=0 are
- (a)
non-real
- (b)
of opposite sign
- (c)
positive
- (d)
complex
Let \(f(n)=\begin{vmatrix} n & n+1 & n+2 \\ ^{ n }P_{ n } & ^{ n+1 }P_{ n+1 } & ^{ n+2 }P_{ n+2 } \\ ^{ n }C_{ n } & ^{ n+1 }C_{ n+1 } & ^{ n+2 }C_{ n+2 } \end{vmatrix}\) Where the symbols have their usual meaning. Then f(n) is divisible by
- (a)
n2+n+1
- (b)
2(n+1)!
- (c)
(n+3)!
- (d)
(n+2)!
Let a,b,c be such that (b+c)\(\ne\)0
If \(\begin{vmatrix} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{vmatrix}+\begin{vmatrix} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{ n+2 }a & (-1)^{ n+1 }b & (-1)^{ n }c \end{vmatrix}=0\) then the value of n is
- (a)
0
- (b)
1
- (c)
\(1\over2\)
- (d)
2
If A5 = 0 such that An \(\neq \) I for \(1\le n\le 4\) then (I - A)-1 is equal to
- (a)
A4
- (b)
A3
- (c)
I + A
- (d)
none of these
If \(A=\left[ \begin{matrix} cos\theta & sin\theta \\ -sin\theta & cos\theta \end{matrix} \right] \), then \(\underset { n\rightarrow \infty }{ lim } \frac { { A }^{ n } }{ n } \) is (where \(\theta \in \)R)
- (a)
a zero matrix
- (b)
an identity matrix
- (c)
\(\left[ \begin{matrix} 0 & 1 \\ 0 & -1 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right] \)
the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ -1 & 2 & -3 \end{matrix} \right] \) is
- (a)
idempotent
- (b)
nilpotent
- (c)
involutory
- (d)
orthogonal
If A is non-singular matrix, then det (A-1) is equal to
- (a)
det \(\left( \frac { 1 }{ A^{ 2 } } \right) \)
- (b)
\(\frac { 1 }{ det(A^{ 2 }) } \)
- (c)
\(\frac { 1 }{ det(A^{ 2 }) } \)
- (d)
\(\frac { 1 }{ det(A) } \)
If \(\sqrt { -1 } =i\) and \(\omega\) is a non real cube root of unity, then the value of \(\left| \begin{matrix} 1 & { \omega }^{ 2 } & 1+i+{ \omega }^{ 2 } \\ -i & -1 & -1-i+\omega \\ 1-i & { \omega }^{ 2 }-1 & -1 \end{matrix} \right| \) is equal to
- (a)
1
- (b)
i
- (c)
\(\omega\)
- (d)
0
If a, b, c are sides of a triangle and \(\left| \begin{matrix} { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ { \left( a+1 \right) }^{ 2 } & { \left( b+1 \right) }^{ 2 } & { \left( c+1 \right) }^{ 2 } \\ { \left( a-1 \right) }^{ 2 } & { \left( b-1 \right) }^{ 2 } & { \left( c-1 \right) }^{ 2 } \end{matrix} \right| =0\) then
- (a)
\(\Delta ABC\) is an equilateral triangle
- (b)
\(\Delta ABC\) is a right-angled isosceles triangle
- (c)
\(\Delta ABC\) is an isosceles triangle
- (d)
none of the above
If the system of equations 2x - y + z = 0, x - 2y + z = 0, tx - y + 2z = 0 has infinitely many solutions and f(x) be a continuous function such that f(5 + x) + f(x) = 2, then \(\int _{ 0 }^{ -2t }{ f\left( x \right) dx } \) is equal to
- (a)
0
- (b)
-2t
- (c)
5
- (d)
t
If A + B + C = \(\pi\) then \(\left| \begin{matrix} \sin { \left( A+B+C \right) } & \sin { B } & \cos { C } \\ -\sin { B } & 0 & \tan { A } \\ \cos { \left( A+B \right) } & -\tan { A } & 0 \end{matrix} \right| \) is equal to
- (a)
1
- (b)
0
- (c)
-1
- (d)
2
In a triangle ABC, the value of the determinant \(\left| \begin{matrix} \sin { \frac { A }{ 2 } } & \sin { \frac { B }{ 2 } } & \sin { \frac { C }{ 2 } } \\ \sin { \left( A+B+C \right) } & \sin { \frac { B }{ 2 } } & \cos { \frac { A }{ 2 } } \\ \cos { \left( \frac { A+B+C }{ 2 } \right) } & \tan { \left( A+B+C \right) } & \sin { \frac { C }{ 2 } } \end{matrix} \right| \) is less than or equal to
- (a)
1/2
- (b)
1/4
- (c)
1/8
- (d)
none of these
if \(\left| \begin{matrix} { x }^{ 2 }+x & x+1 & x-2 \\ 2{ x }^{ 2 }+3x-1 & 3x & 3x-3 \\ { x }^{ 2 }+2x+3 & 2x-1 & 2x-1 \end{matrix} \right| =Ax+B\) where A and B are constants, the
- (a)
A + B = 12
- (b)
A - B = 36
- (c)
A2 + B2 = 720
- (d)
A + 2B = 0
[1-x]\(\left[ \begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 2 \\ x \end{matrix} \right] \) = 0,if x =
- (a)
-7
- (b)
-11
- (c)
-2
- (d)
14
If A =\(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \end{matrix} \right] \) then (A-I)(A+I)=0 for
- (a)
a=b=0 only
- (b)
a=0 only
- (c)
b=0 only
- (d)
any a and b
If A = \(\left[ \begin{matrix} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \)and B=\(\left[ \begin{matrix} 1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \) and AB= I3, then x+y equals
- (a)
A
- (b)
r-A
- (c)
I2
- (d)
3A
If A is a square matrix such that A2=A then (I+A)3-7A is equal to
- (a)
A
- (b)
I-A
- (c)
I
- (d)
3A
If A =\(\left[ \begin{matrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{matrix} \right] \) , then AT+A=I2, If
- (a)
θ= nπ, n∈ Z
- (b)
θ= (2n+1)\(\frac { \pi }{ 2 } \) ,n∈Z
- (c)
θ=2nπ+\(\frac { \pi }{ 3 } \),n∈Z
- (d)
None of these
If A =\(\left[ \begin{matrix} 1 \\ -4 \\ 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} -1 & 2 & 1 \end{matrix} \right] \) , then (AB)' is equal to
- (a)
\(\left[ \begin{matrix} -1 & 4 & -3 \\ 2 & -8 & 6 \\ 1 & -4 & 3 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} -1 & 2 & 1 \\ 4 & -8 & -4 \\ -3 & 6 & 3 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 1 & 4 & -3 \\ 2 & -8 & 6 \\ 1 & 4 & 3 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} -1 & 4 & -3 \\ 2 & 8 & 6 \\ 1 & -4 & 3 \end{matrix} \right] \)
If a, b, care cube roots of unity, then \(\left| \begin{matrix} { e }^{ a } & { e }^{ 2a } & { e }^{ 3a }-1 \\ { e }^{ b } & { e }^{ 2b } & { e }^{ 3b }-1 \\ { e }^{ c } & { e }^{ 2c } & { e }^{ 3c }-1 \end{matrix} \right| \)=
- (a)
0
- (b)
e
- (c)
e2
- (d)
e3
If A =\(\left[ \begin{matrix} 6 & 8 & 5 \\ 4 & 2 & 3 \\ 9 & 7 & 1 \end{matrix} \right] \)is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is
- (a)
\(\left[ \begin{matrix} 6 & 6 & 7 \\ 6 & 2 & 5 \\ 7 & 5 & 1 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} 0 & 2 & -2 \\ -2 & 5 & -2 \\ 2 & 2 & 0 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} 6 & 6 & 7 \\ -6 & 2 & -5 \\ -7 & 5 & 1 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} 0 & 6 & -2 \\ 2 & 2 & -2 \\ -2 & -2 & 0 \end{matrix} \right] \)
Find the value of x for which the matrix A=\(\left[ \begin{matrix} 3-x & 2 & 2 \\ 2 & 4-x & 1 \\ -2 & -4 & -1-x \end{matrix} \right] \) is singular.
- (a)
0, 1
- (b)
1, 3
- (c)
0, 3
- (d)
3, 2
If a matrix A is both symmetric and skew-symmetric then
- (a)
A is a diagonal matrix
- (b)
A is a zero matrix
- (c)
A is a scalar matrix
- (d)
A is a square matrix
If x is a positive integer, then \(\left| \begin{matrix} x! & (x+1)! & (x+2)! \\ (x+1)! & (x+2)! & (x+3)! \\ (x+2)! & (x+3)! & (x+4)! \end{matrix} \right| \)
- (a)
2x!(x+1)!
- (b)
2x!(x+1)!(x+2)!
- (c)
2x!(x+3)!
- (d)
2(x+1)!(x+2)!(x+3)!
Find the inverse of each of the following matrices by using elementary row transformation \(\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 3 & 1 & 1 \end{matrix} \right] \)
- (a)
\(\left[ \begin{matrix} -1/2 & 1/2 & 1/2 \\ 3 & -4 & -1 \\ -3/2 & 5/2 & 1/2 \end{matrix} \right] \)
- (b)
\(\left[ \begin{matrix} -1/2 & 1/2 & 1/2 \\ 3 & -5 & -1 \\ -3/2 & 5/2 & 1/2 \end{matrix} \right] \)
- (c)
\(\left[ \begin{matrix} -1/2 & 1/2 & 1/2 \\ 3 & -1 & -2 \\ -3/2 & 5/2 & 1/2 \end{matrix} \right] \)
- (d)
\(\left[ \begin{matrix} -1/2 & 1/2 & 1/2 \\ 3 & -4 & -1 \\ -3/2 & 5/2 & 1/2 \end{matrix} \right] \)
If \(\alpha ,\beta ,\gamma \) are in A.P., then \(\left| \begin{matrix} x-3 & x-4 & x-\alpha \\ x-2 & x-3 & x-\beta \\ x-1 & x-2 & x-\gamma \end{matrix} \right| \) =
- (a)
0
- (b)
(x-2)(x-3)(x-4)
- (c)
(x-\(\alpha \))(x-\(\beta\))(x-\(\gamma \))
- (d)
\(\alpha \beta \gamma (\alpha -\beta )(\beta -\gamma )^{ 2 }\)
The sum of three numbers is 6. If we multiply the third number by 2 and add the first number to the result, we get 7. By adding second and third numbers to three times the first number, we get 12. Find the numbers.
- (a)
3, 1, 2
- (b)
2, 2, 2
- (c)
4, 1, 1
- (d)
1, 1, 4
The system of linear equations x +y + z = 2, 2x +y - z = 3, 3x + 2y + kz = 4 has a unique solution, if k is not equal to
- (a)
4
- (b)
-4
- (c)
0
- (d)
3
If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx - 12y - 14 = 0 has non-trivial solution, then the value of k is
- (a)
\(-2,\frac { 12 }{ 5 } \)
- (b)
\(-1,\frac { 1 }{ 5 } \)
- (c)
\(-6,\frac { 17 }{ 5 } \)
- (d)
\(6,-\frac { 12 }{ 5 } \)
The determinant \(\left| \begin{matrix} { b }^{ 2 }-ab & b-c & bc-ac \\ ab-{ a }^{ 2 } & a-b & { b }^{ 2 }-ab \\ bc-ac & c-a & ab-{ a }^{ 2 } \end{matrix} \right| \) equals
- (a)
abc(b-c)(c-a)(a-b)
- (b)
(b-c)(c-a)(a-b)
- (c)
(a+b+c)(b-c)(c-a)(a-b)
- (d)
none of these