Matrices & Determinants — JEE Maths Practice Questions & Solutions

hard

If

f(x) = \begin{vmatrix} x^5 & |\sin x| & 2x^4 \\ \tan^3 x & 1 & \sec 2x \\ \sin^3 x & x^4 & 5 \end{vmatrix}

, then

\displaystyle\int_{-\pi/2}^{\pi/2} f(x)\,dx

is equal to:

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hard

If

U_n = \begin{vmatrix} 1 & k & k \\ 2n & k^2+k+1 & k^2+k \\ 2n-1 & k^2 & k^2+k+1 \end{vmatrix}

and

\displaystyle\sum_{n=1}^k U_n = 110

, then

k

equals:

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hard

The value of the determinant

\begin{vmatrix} 2 & \tan A\cot B + \cot A\tan B & \tan A\cot C + \cot A\tan C \\ \tan B\cot A + \cot B\tan A & 2 & \tan B\cot C + \cot B\tan C \\ \tan C\cot A + \cot C\tan A & \tan C\cot B + \cot C\tan B & 2 \end{vmatrix}

is:

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hard

If

f(x) = \begin{vmatrix} 4x-4 & (x-2)^2 & x^3 \\ 8x-4\sqrt{2} & (x-2\sqrt{2})^2 & (x+1)^3 \\ 12x-4\sqrt{3} & (x-2\sqrt{3})^2 & (x-1)^3 \end{vmatrix}

, then the coefficient of

x

in

f(x)

is:

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hard

Let

\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}

where

a_{pq} = (i^p)^q

and

i = \sqrt{-1}

. The value of

\Delta

is:

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hard

If

pqr \neq 0

and the system of equations

(p+a)x+by+cz=0

,

ax+(q+b)y+cz=0

,

ax+by+(r+c)z=0

has a non-trivial solution, then

\dfrac{a}{p}+\dfrac{b}{q}+\dfrac{c}{r}

equals:

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hard

Let

S = \left\{(a, b)\;\middle|\; A^3 = A,\ \text{where}\ A = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ a & b \end{pmatrix}\right\}

. Then

n(S)

is:

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hard

Let

D_k

be the

k\times k

matrix with zeros on the main diagonal, unity as the element of the 1st row and

(f(k))^{\text{th}}

column, and

k

for all other off-diagonal entries. If

f(x) = x - \{x\}

where

\{x\}

denotes the fractional part function, then the value of

\det(D_2) + \det(D_3)

equals:

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hard

Let

A = \begin{pmatrix}\ell & m & n\\ p & q & r\\ 1 & 1 & 1\end{pmatrix}

and

B = A^2

. If

(\ell-m)^2+(p-q)^2 = 9

,

(m-n)^2+(q-r)^2 = 16

,

(n-\ell)^2+(r-p)^2 = 25

, then the value of

\det(B)

equals:

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hard

If

f(x) = \begin{vmatrix} x^2-4x+6 & 2x^2+4x+10 & 3x^2-2x+16 \\ x-2 & 2x+2 & 3x-1 \\ 1 & 2 & 3 \end{vmatrix}

, then

f(x)

is:

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hard

If

\Delta_2 = \begin{vmatrix} y^5z^6(z^3-y^3) & x^4z^6(x^3-z^3) & y^4z^5(y^3-z^3) \\ y^5z^3(y^6-z^6) & xz^3(z^6-x^6) & xy^2(x^6-y^6) \\ y^5z^3(z^3-y^3) & xz^3(x^3-z^3) & xy^2(y^3-x^3) \end{vmatrix}

and

\Delta_1 = \begin{vmatrix} x & y & z^3 \\ x^4 & y^5 & z^6 \\ x^7 & y^8 & z^9 \end{vmatrix}

, then

\Delta_1\Delta_2

equals:

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hard

If

\Delta_r = \begin{vmatrix} 2^{r-1} & x & 2^n-1 \\ 2\cdot3^{r-1} & y & 3^n-1 \\ 4\cdot5^{r-1} & z & 5^n-1 \end{vmatrix}

, then

\displaystyle\sum_{r=1}^n \Delta_r

is equal to:

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hard

If

x, y, z

are positive integers and

\begin{vmatrix} x^3+1 & x^2y & x^2z \\ xy^2 & y^3+1 & y^2z \\ xz^2 & yz^2 & z^3+1 \end{vmatrix} = 30

then the sum of all possible values of

x

is:

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hard

The determinant

\begin{vmatrix} 0 & (a-b)^2 & (a-c)^2 \\ (b-a)^2 & 0 & (b-c)^2 \\ (c-a)^2 & (c-b)^2 & 0 \end{vmatrix}

is equal to:

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hard

If

0 < \theta < \dfrac{\pi}{2}

and

\begin{vmatrix} 1+\sin^2\theta & \cos^2\theta & 4\sin4\theta \\ \sin^2\theta & 1+\cos^2\theta & 4\sin4\theta \\ \sin^2\theta & \cos^2\theta & 1+4\sin4\theta \end{vmatrix} = 0

, then

\theta

equals:

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hard

The determinant

\Delta = \begin{vmatrix} -1+x\sin\alpha & \cos(x+\alpha) & \sin(x+\alpha) \\ 13+x\sin\beta & \cos(x+\beta) & \sin(x+\beta) \\ -12+x\sin\gamma & \cos(x+\gamma) & \sin(x+\gamma) \end{vmatrix}

is independent of:

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medium

Let

A

and

B

be two non-singular matrices of order 2 such that

9A^2B - 6AB + B = 9\left\{B - \begin{pmatrix} 2 & -3 \\ 2 & 1 \end{pmatrix}\right\}

If

B = \begin{pmatrix} 3 & 0 \\ 1 & 1 \end{pmatrix}

and

\text{Tr}(A) = \dfrac{10}{3}

, then the value of

\det(A)

can be:

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medium

If

A

and

B

are square matrices of order 3 such that

\det(A) = -2

and

\det(B) = 1

, then

\det\big(A^{-1}\,\text{adj}(B^{-1})\,\text{adj}(2A^{-1})\big)

is equal to:

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medium

If

M = \begin{pmatrix} 1 & 1 & 2 \\ 0 & 2 & 1 \\ 1 & 0 & 2 \end{pmatrix}

and

M^3 = (\alpha M - I)(\beta M - I)

, where

\alpha, \beta

are integers and

I

is the

3\times 3

identity matrix, then

(\alpha + \beta)

equals:

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medium

If

A

and

B

are two non-singular matrices of order 3 such that

AA^T = 2I

and

A^{-1} = A^T - A\,\text{adj}(2B^{-1})

, then

\det(B)

equals:

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medium

For

\theta = \dfrac{3\pi}{5}

, let

B = [b_{ij}]

be a square matrix of order 2 such that

b_{ij} = \begin{cases} \cos\theta, & i = j \\ \cos\left(\frac{j\pi}{2}+\theta\right), & i > j \\ \sin\left(\frac{j\pi}{2}-\theta\right), & i < j \end{cases}

. Then the trace of

B^5

is:

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medium

If

\alpha, \beta, \gamma

are the roots of

x^3+ax^2+bx+c=0

and if the system

\alpha u+\beta v+\gamma w = 0

,

\beta u+\gamma v+\alpha w = 0

,

\gamma u+\alpha v+\beta w = 0

possesses non-zero solutions, and

a^3-6c=0

, then

2c

equals:

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medium

Let

M = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}

, where

i^2 = -1

, and let

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

. Then

I + M + M^2 + M^3 + \cdots + M^{2010}

is equal to:

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medium

The total number of roots of

\begin{vmatrix} x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1 \end{vmatrix} = 0

is:

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medium

Let

A = \begin{pmatrix} 2 & 2 & 1 \\ 2 & 5 & 2 \\ 1 & 2 & 2 \end{pmatrix}

and

B = \begin{pmatrix} -x & -y & z \\ 0 & y & 2z \\ x & -y & z \end{pmatrix}

where

x, y, z \in \mathbb{R}

. If

B^T A B = \begin{pmatrix} 8 & 0 & 0 \\ 0 & 27 & 0 \\ 0 & 0 & 42 \end{pmatrix}

, then the number of ordered triplets

(x, y, z)

is:

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medium

The value of the determinant

\begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix}

is:

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medium

Matrix

A

is given by

A = \begin{pmatrix} 3 & 11 \\ 2 & 8 \end{pmatrix}

. Then the determinant of

A^{2011} - 5A^{2010}

is:

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medium

A

is a

2\times 2

matrix such that

A\begin{pmatrix}1\\-1\end{pmatrix} = \begin{pmatrix}-1\\2\end{pmatrix}

and

A^2\begin{pmatrix}1\\-1\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix}

. The sum of the elements of

A

is:

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medium

In a square matrix

A

of order 3, the diagonal elements

a_{ii}

are the sum of the roots of the equation

x^2 - (a+b)x + ab = 0

;

a_{i,i+1}

are the product of the roots of the same equation;

a_{i+1,i} = 1

for all valid

i

; and the remaining elements are zero. The value of

\det(A)

is:

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medium

For a matrix

A = \begin{pmatrix}1 & 2r-1\\ 0 & 1\end{pmatrix}

, the value of

\displaystyle\prod_{r=1}^{50}\begin{pmatrix}1 & 2r-1\\ 0 & 1\end{pmatrix}

is equal to:

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medium

Consider a matrix

A = \begin{pmatrix}3 & 1\\ -6 & -2\end{pmatrix}

, then

(1+A)^{99}

equals, where

I

is a unit matrix of order 2:

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medium

Let

A = \begin{pmatrix}2 & 0 & 7\\ 0 & 1 & 0\\ 1 & -2 & 1\end{pmatrix}

and

B = \begin{pmatrix}-\alpha & 14\alpha & 7\alpha\\ 0 & 1 & 0\\ \alpha & -4\alpha & -2\alpha\end{pmatrix}

. If

AB = I

, where

I

is the identity matrix of order 3, then the trace of

B

equals:

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medium

If the product of

n

matrices

\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}\begin{pmatrix}1 & 2\\ 0 & 1\end{pmatrix}\begin{pmatrix}1 & 3\\ 0 & 1\end{pmatrix}\cdots\begin{pmatrix}1 & n\\ 0 & 1\end{pmatrix}

equals the matrix

\begin{pmatrix}1 & 378\\ 0 & 1\end{pmatrix}

, then

n

is equal to:

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medium

If

A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}

satisfies the equation

x^2 - (a+d)x + k = 0

, then:

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medium

Let

A

and

B

be square matrices such that

AB = 0

and

B

is non-singular. Then:

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medium

The number of solutions of the matrix equation

X^2 = I

other than

I

is:

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medium

Number of real values of

x

for which the matrix

A = \begin{pmatrix}3-x & 2 & 2\\ 2 & 4-x & 1\\ -2 & -4 & -1-x\end{pmatrix}

is singular, is:

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medium

If

A

is a diagonal matrix of order

3\times 3

that is commutative with every square matrix of order

3\times 3

under multiplication and

\text{tr}(A) = 12

, then:

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medium

If

A = \begin{pmatrix}-1 & \dfrac{3}{2}\\[4pt] -\dfrac{1}{2} & \dfrac{1}{2}\end{pmatrix}

, then

I + A + A^2 + \cdots\infty

is equal to:

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medium

For positive numbers

x, y, z

, the value of the determinant

\begin{vmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{vmatrix}

is:

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medium

If

x, y, z

are in A.P., then the value of the determinant

\begin{vmatrix} 4 & 5 & 6 & x \\ 5 & 6 & 7 & y \\ 6 & 7 & 8 & z \\ x & y & z & 0 \end{vmatrix}

is:

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medium

If

f(\theta) = \begin{vmatrix} \cos^2\theta & \cos\theta\sin\theta & -\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta & \cos\theta \\ \sin\theta & -\cos\theta & 0 \end{vmatrix}

, then:

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medium

Let

f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}

, where

p

is a constant. Then

\dfrac{d^3}{dx^3}[f(x)]\big|_{x=0}

is:

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medium

If

\begin{vmatrix} 3^2+k & 4^2 & 3^2+3+k \\ 4^2+k & 5^2 & 4^2+4+k \\ 5^2+k & 6^2 & 5^2+5+k \end{vmatrix} = 0

, then the value of

k

is:

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medium

If

\Delta_1 = \begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix}

and

\Delta_2 = \begin{vmatrix} x & b \\ a & x \end{vmatrix}

, then:

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medium

The value of the determinant (for

n \in \mathbb{N}

)

\Delta = \begin{vmatrix} n! & (n+1)! & (n+2)! \\ (n+1)! & (n+2)! & (n+3)! \\ (n+2)! & (n+3)! & (n+4)! \end{vmatrix}

is:

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medium

If

\begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = Px - 12

, then:

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medium

The number of distinct real roots of the equation

\begin{vmatrix} x^2 & (x-1)^2 & (x-2)^2 \\ (x-1)^2 & (x-2)^2 & (x-3)^2 \\ (x-2)^2 & (x-3)^2 & (x-4)^2 \end{vmatrix} = 0

is:

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easy

Let

A

be a matrix of order 3 defined by

A = [a_{ij}]_{3\times 3}

, where

a_{ij} = \displaystyle\lim_{x \to 0}\frac{\sin(ix)}{\tan(jx)}

for all

1 \le i, j \le 3

. Then

A^2

is:

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easy

If

A

is a

3 \times 3

matrix and

\det A = 5

, then

\det\big(\text{Adj}(\text{Adj}\,A)\big)

equals:

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easy

The equation

\begin{pmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}

has a solution

(x, y, z)

besides

(0, 0, 0)

. The value of

k

equals:

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easy

If

M

is a square matrix of order 2, then

(\text{tr}(M))^2 - \text{tr}(M^2)

is equal to:

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easy

A

is an involutory matrix given by

A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}

, then the inverse of

\dfrac{A}{2}

will be:

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easy

A square matrix

P

satisfies

P^2 = I - P

, where

I

is the identity matrix. If

P^n = 5I - 8P

, then

n

is:

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easy

Find the roots of the equation

\begin{vmatrix} x & \alpha & 1 \\ \beta & x & 1 \\ \beta & \gamma & 1 \end{vmatrix} = 0

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easy

If

f(x) = \begin{vmatrix} 1 & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) \end{vmatrix}

, then

f(100)

is equal to:

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Matrices & Determinants — JEE Maths Practice Questions & Solutions

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