JEE Main Matrices and Determinants Practice Questions With Solutions

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JEE Main Matrices and Determinants Practice Questions with Solutions

The Mathematics section is a very important section of the JEE Main entrance exam, and Matrices and Determinants are an essential part of Mathematics. The JEE Main Matrices and Determinants Practice Test is a valuable resource for all JEE Main aspirants to learn about the main concepts of matrices and determinants and enhance their problem-solving abilities. The JEE Main Matrices and Determinants Practice Questions with Solutions have a variety of questions from the Matrices and Determinants chapter for students to practice and strengthen their mathematics section for the JEE Main entrance test.

About 2 to 3 questions may appear from Matrices and Determinants on the JEE Main question paper. Usually, the question will be in the form of multiple choice questions as well as integer type questions, testing both your conceptual knowledge and numerical abilities. JEE Main is a highly competitive entrance exam for all engineering aspirants. Therefore, it is essential for candidates to attempt these practice tests at least once a week to develop knowledge and precision of the topic.

The important topics of this chapter, from which questions may appear in the exam, are Types of Matrices, Matrix Operations, Rank of a Matrix, Transpose and Inverse, System of Linear Equations, Determinants, Eigenvalues and Eigenvectors, and Area of Triangle using Determinants. The practice tests, containing questions on all these topics along with their detailed solutions, will allow aspirants to understand the basic concepts well by going through the detailed explanations provided.

Thoroughly practising the JEE Main Matrices and Determinants Practice Questions with Solutions on a regular basis will help students to improve their accuracy and manage their time better during the examination. It will also allow them to memorise formulas, understand concepts, and gain confidence in solving questions from this topic with ease in the JEE Main exam.

JEE Main Mathematics Matrices and Determinants Practice Questions

Chemistry Physics

Question 1.

Let $B = [\begin{array}{ll} 1 & 3 \\ 1 & 5 \end{array}]$ and $A$ be a $2 \times 2$ matrix such that $A B^{- 1} = A^{- 1}$ . If $B C B^{- 1} = A$ and $C^{4} + α C^{2} + β I = O$ , then $2 β - α$ is equal to

Question 2.

Let $λ, μ \in R$ . If the system of equations

$\begin{aligned} 3 x + 5 y + λ z = 3 \\ 7 x + 11 y - 9 z = 2 \\ 97 x + 155 y - 189 z = μ \end{aligned}$

has infinitely many solutions, then $μ + 2 λ$ is equal to :

Question 3.

If $α \neq a, β \neq b, γ \neq c$ and $| \begin{array}{lll} α & b & c \\ a & β & c \\ a & b & γ \end{array} | = 0$ , then $\frac{a}{α - a} + \frac{b}{β - b} + \frac{γ}{γ - c}$ is equal to :

Question 4.

If the system of equations $x + 4 y - z = λ, 7 x + 9 y + μ z = - 3, 5 x + y + 2 z = - 1$ has infinitely many solutions, then $(2 μ + 3 λ)$ is equal to :

- 2

- 3

Question 5.

Let $A = [\begin{array}{lll} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array}]$ . If $A^{3} = 4 A^{2} - A - 21 I$ , where $I$ is the identity matrix of order $3 \times 3$ , then $2 a + 3 b$ is equal to

- 10

- 12

- 13

- 9

Question 6.

If $A$ is a square matrix of order 3 such that $\det (A) = 3$ and $\det (adj (- 4 adj (- 3 adj (3 adj ((2 A)^{- 1}))))) = 2^{m} 3^{n}$ , then $m + 2 n$ is equal to :

Question 7.

For $α, β \in R$ and a natural number $n$ , let $A_{r} = | \begin{array}{ccc} r & 1 & \frac{n^{2}}{2} + α \\ 2 r & 2 & n^{2} - β \\ 3 r - 2 & 3 & \frac{n (3 n - 1)}{2} \end{array} |$ . Then $2 A_{10} - A_{8}$ is

4 α + 2 β

2 n

2 α + 4 β

Question 8.

The values of $m, n$ , for which the system of equations

$\begin{aligned} x + y + z = 4, \\ 2 x + 5 y + 5 z = 17, \\ x + 2 y + m z = n \end{aligned}$

has infinitely many solutions, satisfy the equation :

m^{2} + n^{2} - m - n = 46

m^{2} + n^{2} + mn = 68

m^{2} + n^{2} - mn = 39

m^{2} + n^{2} + m + n = 64

Question 9.

Let $α β \neq 0$ and $A = [\begin{array}{rrr} β & α & 3 \\ α & α & β \\ - β & α & 2 α \end{array}]$ . If $B = [\begin{array}{rrr} 3 α & - 9 & 3 α \\ - α & 7 & - 2 α \\ - 2 α & 5 & - 2 β \end{array}]$ is the matrix of cofactors of the elements of $A$ , then $\det (A B)$ is equal to :

343

125

216

Question 10.

Let A and B be two square matrices of order 3 such that $| A | = 3$ and $| B | = 2$ . Then $| A^{T} A (adj (2 A))^{- 1} (adj (4 B)) (adj (AB))^{- 1} {AA}^{T} |$ is equal to :

108

Question 1.

If the system of equations

$\begin{array}{r} 11 x + y + λ z = - 5 \\ 2 x + 3 y + 5 z = 3 \\ 8 x - 19 y - 39 z = μ \end{array}$

has infinitely many solutions, then $λ^{4} - μ$ is equal to :

Question 2.

Let $A = [\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}]$ and $B = I + adj (A) + (adj A)^{2} + \dots + (adj A)^{10}$ .Then, the sum of all the elements of the matrix $B$ is:

-

110

-

124

-

Question 3.

Let $α \in (0, \infty)$ and $A = [\begin{array}{lll} 1 & 2 & α \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{array}]$ . If $\det (adj (2 A - A^{T}) \cdot adj (A - 2 A^{T})) = 2^{8}$ , then $(\det (A))^{2}$ is equal to:

Question 4.

If the system of equations

$\begin{aligned} x + (\sqrt{2} \sin α) y + (\sqrt{2} \cos α) z = 0 \\ x + (\cos α) y + (\sin α) z = 0 \\ x + (\sin α) y - (\cos α) z = 0 \end{aligned}$

has a non-trivial solution, then $α \in (0, \frac{π}{2})$ is equal to :

\frac{5 π}{24}

\frac{11 π}{24}

\frac{7 π}{24}

\frac{3 π}{4}

Question 5.

Let the system of equations

x + 2 y + 3 z = 5, 2 x + 3 y + z = 9, 4 x + 3 y + λ z = μ

have infinite number of solutions. Then

λ + 2 μ

is equal to :

Question 6.

A = [\begin{array}{cc} \sqrt{2} & 1 \\ - 1 & \sqrt{2} \end{array}], B = [\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}], C = {ABA}^{T}

and

X = A^{T} C^{2} A

, then

\det X

is equal to :

243

729

891

Question 7.

If the system of equations

\begin{aligned} 2 x + 3 y - z = 5 \\ x + α y + 3 z = - 4 \\ 3 x - y + β z = 7 \end{aligned}

has infinitely many solutions, then

13 α β

is equal to :

1110

1120

1210

1220

Question 8.

Let $A$ be a $3 \times 3$ real matrix such that

$A (\begin{array}{l} 1 \\ 0 \\ 1 \end{array}) = 2 (\begin{array}{l} 1 \\ 0 \\ 1 \end{array}), A (\begin{array}{l} - 1 \\ 0 \\ 1 \end{array}) = 4 (\begin{array}{l} - 1 \\ 0 \\ 1 \end{array}), A (\begin{array}{l} 0 \\ 1 \\ 0 \end{array}) = 2 (\begin{array}{l} 0 \\ 1 \\ 0 \end{array}) .$

Then, the system $(A - 3 I) (\begin{array}{l} x \\ y \\ z \end{array}) = (\begin{array}{l} 1 \\ 2 \\ 3 \end{array})$ has :

exactly two solutions

infinitely many solutions

unique solution

no solution

Question 9.

If the system of linear equations

$\begin{aligned} x - 2 y + z = - 4 \\ 2 x + α y + 3 z = 5 \\ 3 x - y + β z = 3 \end{aligned}$

has infinitely many solutions, then $12 α + 13 β$ is equal to

Question 10.

Let $R = (\begin{array}{ccc} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{array})$ be a non-zero $3 \times 3$ matrix, where $x \sin θ = y \sin (θ + \frac{2 π}{3}) = z \sin (θ + \frac{4 π}{3}) \neq 0, θ \in (0, 2 π)$ . For a square matrix $M$ , let trace $(M)$ denote the sum of all the diagonal entries of $M$ . Then, among the statements:

(I) Trace $(R) = 0$

(II) If trace $(adj (adj (R)) = 0$ , then $R$ has exactly one non-zero entry.

Only (I) is true

Only (II) is true

Both (I) and (II) are true

Neither (I) nor (II) is true

JEE Main Matrices and Determinants Practice Questions With Solutions

JEE Main Matrices and Determinants Practice Questions with Solutions

JEE Main Mathematics Matrices and Determinants Practice Questions

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

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