Matrices and Determinants

300 questions · Mathematics · JEE Main

Matrices and Determinants

300 questions · Mathematics · JEE Main

Let $A = [α 6 - 1 β], α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If I denotes $2 \times 2$ identity matrix, then the matrix…2025 · 2 Apr · Shift 1 · Q26 · MCQ
Let $a \in R$ and $A$ be a matrix of order $3 \times 3$ such that $det (A) = - 4$ and $A + I = 12 a a 11 102$ , where $I$ is the identity matrix…2025 · 2 Apr · Shift 1 · Q28 · MCQ
If the system of linear equations $3 x + y + β z = 3 2 x + α y - z = - 3 x + 2 y + z = 4$ has infinitely many solutions, then the value of $22 β - 9 α$ is :2025 · 2 Apr · Shift 1 · Q39 · MCQ
If the system of equations $2 x + λ y + 3 z = 5 3 x + 2 y - z = 7 4 x + 5 y + μ z = 9$ has infinitely many solutions, then $(λ^{2} + μ^{2})$ is equal to :2025 · 2 Apr · Shift 2 · Q30 · MCQ
Let $A$ be a $3 \times 3$ real matrix such that $A^{2} (A - 2 I) - 4 (A - I) = O$ , where $I$ and $O$ are the identity and null matrices, respectively. If $A^{5} = α A^{2} + β A + γ I$ , where $α, β$ , and $γ$ are real constants, then $α + β + γ$ …2025 · 2 Apr · Shift 2 · Q31 · MCQ
Let $A$ be a matrix of order $3 \times 3$ and $∣ A ∣ = 5$ . If $∣2 adj (3 A adj (2 A)) ∣ = 2^{α} \cdot 3^{β} \cdot 5^{γ}, α, β, γ \in N$ , then $α + β + γ$ is equal to2025 · 3 Apr · Shift 1 · Q45 · MCQ
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A = λ 47 25 - 1 362, ∣ A ∣ = - 1$ . Let $B$ be the inverse of the matrix $adj (Aadj (A^{2}))$ …2025 · 3 Apr · Shift 2 · Q50 · Numerical
Let $A = cos θ 0 sin θ 010 - sin θ 0 cos θ$ . If for some $θ \in (0, π), A^{2} = A^{T}$ , then the sum of the diagonal elements of the…2025 · 4 Apr · Shift 1 · Q48 · Numerical
Let the matrix $A = 110001010$ satisfy $A^{n} = A^{n - 2} + A^{2} - I$ for $n ⩾ 3$ . Then the sum of all the elements of $A^{50}$ is :2025 · 4 Apr · Shift 2 · Q31 · MCQ
Let $A$ be a $3 \times 3$ matrix such that $∣ adj (adj (adj A)) ∣ = 81$ . If $S = {n \in Z : (∣ adj (adj A) ∣)^{\frac{( n - 1 ) ^{2}}{2}} = ∣ A ∣^{(3 n^{2} - 5 n - 4)}}$ …2025 · 7 Apr · Shift 1 · Q33 · MCQ
Let the system of equations : $2 x + 3 y + 5 z = 9 7 x + 3 y - 2 z = 8 12 x + 3 y - (4 + λ) z = 16 - μ$ have infinitely many solutions. Then the radius of the circle centred at $(λ, μ)$ and…2025 · 7 Apr · Shift 1 · Q36 · MCQ
The number of singular matrices of order 2 , whose elements are from the set ${2, 3, 6, 9}$ , is .2025 · 7 Apr · Shift 1 · Q46 · Numerical
Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + λz = μ λ, μ ∈ ℝ, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is :2025 · 7 Apr · Shift 2 · Q44 · MCQ
Let α be a solution of $x^{2} + x + 1 = 0$ , and for some a and b in $R, [4 a b] 1 - 1 - 2 16 - 1 - 14 132 - 8 = [000]$ …2025 · 8 Apr · Shift 2 · Q31 · MCQ
Let $A = 246 2 + p 6 + 2 p 12 + 3 p 2 + p + q 8 + 3 p + 2 q 20 + 6 p + 3 q$ . If $det (adj (adj (3 A))) = 2^{m} \cdot 3^{n}$ , $m, n \in N$ , then $m + n$ is equal to2025 · 8 Apr · Shift 2 · Q38 · MCQ
Let $A$ be a square matrix of order 3 such that $det (A) = - 2$ and $det (3 adj (- 6 adj (3 A))) = 2^{m + n} \cdot 3^{mn}, m > n$ . Then $4 m + 2 n$ is equal to .2025 · 22 Jan · Shift 1 · Q49 · Numerical
If the system of linear equations : $x + y + 2 z = 6 2 x + 3 y + az = a + 1 - x - 3 y + b z = 2 b$ where $a, b \in R$ , has infinitely many solutions, then $7 a + 3 b$ …2025 · 22 Jan · Shift 2 · Q28 · MCQ
For a $3 \times 3$ matrix $M$ , let trace $(M)$ denote the sum of all the diagonal elements of $M$ . Let $A$ be a $3 \times 3$ matrix such that $∣ A ∣ = \frac{1}{2}$ and trace $(A) = 3$ . If $B = adj (adj (2 A))$ , then the…2025 · 22 Jan · Shift 2 · Q34 · MCQ
If the system of equations $(λ - 1) x + (λ - 4) y + λ z = 5 λ x + (λ - 1) y + (λ - 4) z = 7 (λ + 1) x + (λ + 2) y - (λ + 2) z = 9$ has infinitely many solutions,…2025 · 23 Jan · Shift 1 · Q35 · MCQ
If $A, B, and (adj (A^{- 1}) + adj (B^{- 1}))$ are non-singular matrices of same order, then the inverse of $A (adj (A^{- 1}) + adj (B^{- 1}))^{- 1} B$ …2025 · 23 Jan · Shift 1 · Q44 · MCQ
The system of equations $x + y + z = 6, x + 2 y + 5 z = 9, x + 5 y + λ z = μ,$ has no solution if2025 · 23 Jan · Shift 2 · Q35 · MCQ
Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A 010 = 001, A 413 = 010$ …2025 · 23 Jan · Shift 2 · Q38 · MCQ
If the system of equations $2 x - y + z = 4 5 x + λ y + 3 z = 12 100 x - 47 y + μ z = 212$ has infinitely many solutions, then $μ - 2 λ$ is equal to2025 · 24 Jan · Shift 1 · Q44 · MCQ
Let A be a $3 \times 3$ matrix such that $X^{T} AX = O$ for all nonzero $3 \times 1$ matrices $X = x y z$ . If $A 111 = 14 - 5, A 121 = 04 - 8$ …2025 · 24 Jan · Shift 1 · Q46 · Numerical
For some $a, b,$ let $f (x) = a + \frac{s i n x}{x} a a 1 1 + \frac{s i n x}{x} 1 b b b + \frac{s i n x}{x}, x e q 0, x \to 0 lim f (x) = λ + μ a + u b .$ …2025 · 24 Jan · Shift 2 · Q31 · MCQ
If the system of equations $x + 2 y - 3 z = 2 2 x + λ y + 5 z = 5 14 x + 3 y + μ z = 33$ has infinitely many solutions, then $λ + μ$ is equal to :2025 · 24 Jan · Shift 2 · Q33 · MCQ
Let M denote the set of all real matrices of order $3 \times 3$ and let $S = {- 3, - 2, - 1, 1, 2}$ . Let $S_{1} = {A = [a_{ij}] \in M : A = A^{T} and a_{ij} \in S, \forall i, j}, S_{2} = {A = [a_{ij}] \in M : A = - A^{T} and a_{ij} \in S, \forall i, j}, S_{3} = {A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{ij} \in S, \forall i, j} .$ …2025 · 28 Jan · Shift 1 · Q46 · Numerical
Let $A = [\frac{1}{2} 0 - 2 1]$ and $P = [cos θ sin θ - sin θ cos θ], θ > 0$ . If $B = PAP^{⊤}, C = P^{⊤} B^{10} P$ …2025 · 28 Jan · Shift 2 · Q42 · MCQ
Let $A = [a_{ij}] = [lo g_{5} 128 lo g_{5} 8 lo g_{4} 5 lo g_{4} 25]$ . If $A_{ij}$ is the cofactor of $a_{ij}$ , $C_{ij} = k = 1 \sum 2 a_{ik} A_{j k}, 1 \leq i, j \leq 2$ …2025 · 29 Jan · Shift 1 · Q28 · MCQ
Let M and m respectively be the maximum and the minimum values of $f (x) = 1 + sin^{2} x sin^{2} x sin^{2} x cos^{2} x 1 + cos^{2} x cos^{2} x 4 sin 4 x 4 sin 4 x 1 + 4 sin 4 x, x \in R$ …2025 · 29 Jan · Shift 1 · Q44 · MCQ
Let $S = {m \in Z : A^{m^{2}} + A^{m} = 3 I - A^{- 6}}$ , where $A = [21 - 1 0]$ . Then $n (S)$ is equal to .2025 · 29 Jan · Shift 1 · Q47 · Numerical
Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in {0, 1}$ for all $i$ and $j$ . Let the random variable $X$ denote the possible values of the determinant of the matrix $A$ . Then, the variance of $X$ is:2025 · 29 Jan · Shift 2 · Q27 · MCQ
Let $α, β (α e q β)$ be the values of $m$ , for which the equations $x + y + z = 1$ , $x + 2 y + 4 z = m$ and $x + 4 y + 10 z = m^{2}$ have infinitely many solutions. Then the value of $n = 1 \sum 10 (n^{α} + n^{β})$ is equal to :2025 · 29 Jan · Shift 2 · Q31 · MCQ
Let $A = [a_{ij}]$ be a matrix of order $3 \times 3$ , with $a_{ij} = (2)^{i + j}$ . If the sum of all the elements in the third row of $A^{2}$ is $α + β 2, α, β \in Z$ , then…2025 · 29 Jan · Shift 2 · Q39 · MCQ
If $A = [2 - 1 12], B = [1101], C = ABA^{T}$ and $X = A^{T} C^{2} A$ …2024 · 1 Feb · Shift 1 · Q33 · MCQ
If the system of equations $2 x + 3 y - z = 5 x + α y + 3 z = - 4 3 x - y + β z = 7$ has infinitely many solutions, then $13 α β$ is equal to :2024 · 1 Feb · Shift 1 · Q42 · MCQ
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 :2024 · 1 Feb · Shift 2 · Q35 · MCQ
Let $A = I_{2} - 2 M M^{T}$ , where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^{T} M = I_{1}$ holds. If $λ$ is a real number such that the relation $A X = λ X$ holds for some non-zero real matrix $X$ of order $2 \times 1$ …2024 · 1 Feb · Shift 2 · Q57 · Numerical
Let $α \in (0, \infty)$ and $A = 110201 α 12$ . If $det (adj (2 A - A^{T}) \cdot adj (A - 2 A^{T})) = 2^{8}$ …2024 · 4 Apr · Shift 1 · Q33 · MCQ
If the system of equations $x + (2 sin α) y + (2 cos α) z = 0 x + (cos α) y + (sin α) z = 0 x + (sin α) y - (cos α) z = 0$ has a non-trivial solution,…2024 · 4 Apr · Shift 1 · Q48 · MCQ
Let $A$ be a square matrix of order 2 such that $∣ A ∣ = 2$ and the sum of its diagonal elements is $-$ 3 . If the points $(x, y)$ satisfying $A^{2} + x A + y I = O$ lie on a hyperbola, whose transverse axis is…2024 · 4 Apr · Shift 1 · Q53 · Numerical
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A 111 = 3111$ . Then the maximum value of $det (A)$ …2024 · 4 Apr · Shift 1 · Q55 · Numerical
Let $A = [1021]$ and $B = I + adj (A) + (adj A)^{2} + \dots + (adj A)^{10}$ . Then, the sum of all the elements of the matrix $B$ is:2024 · 4 Apr · Shift 2 · Q49 · MCQ
Let $A$ be a $2 \times 2$ symmetric matrix such that $A [11] = [37]$ and the determinant of $A$ be 1 . If $A^{- 1} = α A + β I$ , where $I$ is an…2024 · 4 Apr · Shift 2 · Q53 · Numerical
Let A and B be two square matrices of order 3 such that $∣A∣ = 3$ and $∣B∣ = 2$ . Then $∣ A^{T} A (adj (2 A))^{- 1} (adj (4 B)) (adj (AB))^{- 1} AA^{T} ∣$ …2024 · 5 Apr · Shift 1 · Q36 · MCQ
If the system of equations $11 x + y + λ z = - 5 2 x + 3 y + 5 z = 3 8 x - 19 y - 39 z = μ$ has infinitely many solutions, then $λ^{4} - μ$ is equal to :2024 · 5 Apr · Shift 1 · Q40 · MCQ
The values of $m, n$ , for which the system of equations $x + y + z = 4, 2 x + 5 y + 5 z = 17, x + 2 y + m z = n$ has infinitely many solutions, satisfy the equation :2024 · 5 Apr · Shift 2 · Q39 · MCQ
Let $α β e q 0$ and $A = β α - β α α α 3 β 2 α$ . If $B = 3 α - α - 2 α - 9 75 3 α - 2 α - 2 β$ …2024 · 5 Apr · Shift 2 · Q45 · MCQ
For $α, β \in R$ and a natural number $n$ , let $A_{r} = r 2 r 3 r - 2 123 \frac{n ^{2}}{2} + α n^{2} - β \frac{n ( 3 n - 1 )}{2}$ . Then $2 A_{10} - A_{8}$ …2024 · 6 Apr · Shift 1 · Q42 · MCQ
Let $α β γ = 45; α, β, γ \in R$ . If $x (α, 1, 2) + y (1, β, 2) + z (2, 3, γ) = (0, 0, 0)$ for some $x, y, z \in R, x y z e q 0$ , then $6 α + 4 β + γ$ is equal to …2024 · 6 Apr · Shift 1 · Q54 · Numerical
If $A$ is a square matrix of order 3 such that $det (A) = 3$ and $det (adj (- 4 adj (- 3 adj (3 adj ((2 A)^{- 1}))))) = 2^{m} 3^{n}$ …2024 · 6 Apr · Shift 2 · Q35 · MCQ
If the system of equations $2 x + 7 y + λ z = 3 3 x + 2 y + 5 z = 4 x + μ y + 32 z = - 1$ has infinitely many solutions, then $(λ - μ)$ is equal to :2024 · 6 Apr · Shift 2 · Q57 · Numerical
Let $A = 210 a 35 01 b$ . 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 to2024 · 8 Apr · Shift 1 · Q31 · MCQ
Let $A = [21 - 1 1]$ . If the sum of the diagonal elements of $A^{13}$ is $3^{n}$ , then $n$ is equal to .2024 · 8 Apr · Shift 1 · Q58 · Numerical
If $α e q a, β e q b, γ e q c$ and $α a a b β b c c γ = 0$ …2024 · 8 Apr · Shift 2 · Q37 · MCQ
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 :2024 · 8 Apr · Shift 2 · Q39 · MCQ
Let $λ, μ \in R$ . If the system of equations $3 x + 5 y + λ z = 3 7 x + 11 y - 9 z = 2 97 x + 155 y - 189 z = μ$ has infinitely many solutions, then $μ + 2 λ$ is equal to…2024 · 9 Apr · Shift 1 · Q47 · MCQ
Let $A$ be a non-singular matrix of order 3. If $det (3 adj (2 adj ((det A) A))) = 3^{- 13} \cdot 2^{- 10}$ and $det (3 adj (2 A)) = 2^{m} \cdot 3^{n}$ …2024 · 9 Apr · Shift 1 · Q59 · Numerical
Let $B = [1135]$ 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 to2024 · 9 Apr · Shift 2 · Q44 · MCQ
Consider the matrices : $A = [23 - 5 m], B = [20 m]$ and $X = [x y]$ . Let the set of all $m$ , for which…2024 · 9 Apr · Shift 2 · Q53 · Numerical
Consider the matrix $f (x) = cos x sin x 0 - sin x cos x 0 001$ . Given below are two statements : Statement I : $f (- x)$ is the inverse of the matrix $f (x)$ …2024 · 27 Jan · Shift 1 · Q41 · MCQ
Let $A = 211010101, B = [B_{1}, B_{2}, B_{3}]$ , where $B_{1}, B_{2}, B_{3}$ are column matrics, and $AB_{1} = 100, AB_{2} = 230, AB_{3} = 321$ …2024 · 27 Jan · Shift 1 · Q57 · Numerical
The values of $α$ , for which $11 2 α + 3 \frac{3}{2} \frac{1}{3} 3 α + 1 α + \frac{3}{2} α + \frac{1}{3} 0 = 0$ , lie in the interval2024 · 27 Jan · Shift 2 · Q48 · MCQ
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $∣ A - x I ∣ = 0$ be $- 1$ and 3, then the sum of the diagonal elements of the matrix $A^{2}$ is2024 · 27 Jan · Shift 2 · Q57 · Numerical
$Let A = 100 0 α β 0 β α and ∣2 A ∣^{3} = 2^{21} where α, β \in Z, Then a value of α is$ …2024 · 29 Jan · Shift 1 · Q38 · MCQ
Let $A$ be a square matrix such that $AA^{T} = I$ . Then $\frac{1}{2} A [(A + A^{T})^{2} + (A - A^{T})^{2}]$ is equal to2024 · 29 Jan · Shift 1 · Q44 · MCQ
Let $A = 2631232112$ and $P = 157201025$ . The sum of…2024 · 29 Jan · Shift 2 · Q50 · MCQ
Let for any three distinct consecutive terms $a, b, c$ of an A.P, the lines $a x + b y + c = 0$ be concurrent at the point $P$ and $Q (α, β)$ be a point such that the system of equations $x + y + z = 6, 2 x + 5 y + α z = β and$ …2024 · 29 Jan · Shift 2 · Q56 · Numerical
Consider the system of linear equations $x + y + z = 4 μ, x + 2 y + 2 λ z = 10 μ, x + 3 y + 4 λ^{2} z = μ^{2} + 15$ where $λ, μ \in R$ . Which one of the following statements is NOT correct ?2024 · 30 Jan · Shift 1 · Q39 · MCQ
Let $R = x 00 0 y 0 00 z$ be a non-zero $3 \times 3$ matrix, where $x sin θ = y sin (θ + \frac{2 π}{3}) = z sin (θ + \frac{4 π}{3}) e q 0, θ \in (0, 2 π)$ …2024 · 30 Jan · Shift 2 · Q37 · MCQ
Consider the system of linear equations $x + y + z = 5, x + 2 y + λ^{2} z = 9, x + 3 y + λ z = μ$ , where $λ, μ \in R$ . Then, which of the following statement is NOT correct?2024 · 30 Jan · Shift 2 · Q46 · MCQ
If the system of linear equations $x - 2 y + z = - 4 2 x + α y + 3 z = 5 3 x - y + β z = 3$ has infinitely many solutions, then $12 α + 13 β$ is equal to2024 · 31 Jan · Shift 1 · Q43 · MCQ
Let $A$ be a $3 \times 3$ real matrix such that $A 101 = 2101, A - 1 01 = 4 - 1 01, A 010 = 2010 .$ …2024 · 31 Jan · Shift 2 · Q33 · MCQ
Let A be a $3 \times 3$ matrix and $det (A) = 2$ . If $n = det (2024 - times adj (adj (\dots .. (adj A))))$ , then the remainder when $n$ is divided…2024 · 31 Jan · Shift 2 · Q59 · Numerical
Let $S$ denote the set of all real values of $λ$ such that the system of equations $λ x + y + z = 1 x + λ y + z = 1 x + y + λ z = 1$ is inconsistent, then $\sum_{λ \in S} (∣ λ ∣^{2} + ∣ λ ∣)$ is equal to2023 · 1 Feb · Shift 1 · Q26 · MCQ
For the system of linear equations $α x + y + z = 1, x + α y + z = 1, x + y + α z = β$ , which one of the following statements is NOT correct?2023 · 1 Feb · Shift 2 · Q28 · MCQ
If $A = \frac{1}{2} [1 - 3 31]$ , then :2023 · 1 Feb · Shift 2 · Q34 · MCQ
If the system of equations $x + y + a z = b 2 x + 5 y + 2 z = 6 x + 2 y + 3 z = 3$ has infinitely many solutions, then $2 a + 3 b$ is equal to :2023 · 6 Apr · Shift 1 · Q24 · MCQ
Let $A = [a_{ij}]_{2 \times 2}$ , where $a_{ij} e q 0$ for all $i, j$ and $A^{2} = I$ . Let a be the sum of all diagonal elements of $A$ and $b = ∣ A ∣$ …2023 · 6 Apr · Shift 1 · Q28 · MCQ
Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in N$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P$ , then $α + β + γ - δ$ is equal to :2023 · 6 Apr · Shift 2 · Q28 · MCQ
For the system of equations $x + y + z = 6 x + 2 y + α z = 10 x + 3 y + 5 z = β$ , which one of the following is NOT true?2023 · 6 Apr · Shift 2 · Q35 · MCQ
Let $A = 210 12 - 1 0 - 1 2$ . If $∣ adj (adj (adj 2 A)) ∣ = (16)^{n}$ , then $n$ is equal to :2023 · 8 Apr · Shift 1 · Q24 · MCQ
Let $P = [\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}], A = [1011]$ and $Q = P A P^{T}$ . If $P^{T} Q^{2007} P = [a c b d]$ …2023 · 8 Apr · Shift 1 · Q36 · MCQ
If $A = [1 λ 510], A^{- 1} = α A + β I$ and $α + β = - 2$ , then $4 α^{2} + β^{2} + λ^{2}$ is equal to :2023 · 8 Apr · Shift 2 · Q31 · MCQ
Let S be the set of all values of $θ \in [- π, π]$ for which the system of linear equations $x + y + 3 z = 0 - x + (tan θ) y + 7 z = 0 x + y + (tan θ) z = 0$ has non-trivial solution. Then $\frac{120}{π} \sum_{θ \in s} θ$ …2023 · 8 Apr · Shift 2 · Q34 · MCQ
For the system of linear equations $2 x - y + 3 z = 53 x + 2 y - z = 74 x + 5 y + α z = β$ , which of the following is NOT correct?2023 · 10 Apr · Shift 1 · Q34 · MCQ
Let $S$ be the set of values of $λ$ , for which the system of equations $6 λ x - 3 y + 3 z = 4 λ^{2}$ , $2 x + 6 λ y + 4 z = 1$ , $3 x + 2 y + 3 λ z = λ$ has no solution. Then $12 \sum_{i \in S} ∣ λ ∣$ is equal…2023 · 10 Apr · Shift 2 · Q36 · Numerical
Let $A$ be a $2 \times 2$ matrix with real entries such that $A^{'} = α A + I$ , where $α \in R - {- 1, 1}$ . If $det (A^{2} - A) = 4$ , then the sum of all possible values…2023 · 11 Apr · Shift 1 · Q33 · MCQ
Let $A = 0 a 1 10 c 230$ , where $a, c \in R$ . If $A^{3} = A$ and the positive value of $a$ belongs to the interval $(n - 1, n]$ , where $n \in N$ …2023 · 11 Apr · Shift 1 · Q41 · Numerical
If the system of linear equations $7 x + 11 y + α z = 13 5 x + 4 y + 7 z = β 175 x + 194 y + 57 z = 361$ has infinitely many solutions, then $α + β + 2$ is equal to :2023 · 11 Apr · Shift 2 · Q25 · MCQ
$x + 1 x x x x + λ x x x x + λ^{2} = \frac{9}{8} (103 x + 81)$ , then $λ, \frac{λ}{3}$ are the roots of the equation :2023 · 11 Apr · Shift 2 · Q30 · MCQ
Let $A = [10 \frac{1}{51} 1]$ . If $B = [1 - 1 2 - 1] A [- 1 1 - 2 1]$ ,…2023 · 12 Apr · Shift 1 · Q24 · MCQ
Let $D_{k} = 1 n n 2 k n^{2} + n + 2 n^{2} + n 2 k - 1 n^{2} n^{2} + n + 2$ . If $\sum_{k = 1}^{n} D_{k} = 96$ , then $n$ is equal to …2023 · 12 Apr · Shift 1 · Q35 · Numerical
For the system of linear equations $2 x + 4 y + 2 a z = b x + 2 y + 3 z = 42 x - 5 y + 2 z = 8$ which of the following is NOT correct?2023 · 13 Apr · Shift 1 · Q26 · MCQ
Let $B = 11 α 32 α α 34, α > 2$ be the adjoint of a matrix $A$ and $∣ A ∣ = 2$ . Then $[α - 2 α α] B α - 2 α α$ …2023 · 13 Apr · Shift 1 · Q32 · MCQ
The number of symmetric matrices of order 3, with all the entries from the set ${0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$ is :2023 · 13 Apr · Shift 1 · Q37 · MCQ
If the system of equations $2 x + y - z = 52 x - 5 y + λ z = μx + 2 y - 5 z = 7$ has infinitely many solutions, then $(λ + μ)^{2} + (λ - μ)^{2}$ is equal to2023 · 13 Apr · Shift 2 · Q30 · MCQ
If A and B are two non-zero n $\times$ n matrices such that $A^{2} + B = A^{2} B$ , then :2023 · 24 Jan · Shift 1 · Q33 · MCQ
If the system of equations $x + 2 y + 3 z = 34 x + 3 y - 4 z = 48 x + 4 y - λ z = 9 + μ$ has infinitely many solutions, then the ordered pair ( $λ, μ$ ) is equal to :2023 · 24 Jan · Shift 2 · Q27 · MCQ
Let S $_{1}$ and S $_{2}$ be respectively the sets of all $a \in R - {0}$ for which the system of linear equations $a x + 2 a y - 3 a z = 1 (2 a + 1) x + (2 a + 3) y + (a + 1) z = 2 (3 a + 5) x + (a + 5) y + (a + 2) z = 3$ has unique solution and…2023 · 25 Jan · Shift 1 · Q36 · MCQ
Let $A_{1}, A_{2}, A_{3}$ be the three A.P. with the same common difference d and having their first terms as $A, A + 1, A + 2$ , respectively. Let a, b, c be the $7^{th}, 9^{th}, 1 7^{th}$ terms of $A_{1}, A_{2}, A_{3}$ ,…2023 · 25 Jan · Shift 1 · Q44 · Numerical
Let A, B, C be 3 $\times$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements (S1) A $^{13}$ B $^{26} -$ B $^{26}$ A $^{13}$ is symmetric (S2) A $^{26}$ C $^{13} -$ C $^{13}$ A $^{26}$ is symmetric…2023 · 25 Jan · Shift 2 · Q27 · MCQ
Let $A = [\frac{1}{10} \frac{- 3}{10} \frac{3}{10} \frac{1}{10}]$ and $B = [10 - i 1]$ …2023 · 25 Jan · Shift 2 · Q36 · MCQ
Let $α$ and $β$ be real numbers. Consider a 3 $\times$ 3 matrix A such that $A^{2} = 3 A + α I$ . If $A^{4} = 21 A + β I$ , then2023 · 29 Jan · Shift 1 · Q29 · MCQ
Consider the following system of equations $α x + 2 y + z = 12 α x + 3 y + z = 13 x + α y + 2 z = β$ for some $α, β \in R$ . Then which of the following is NOT correct.2023 · 29 Jan · Shift 1 · Q38 · MCQ
Let A be a symmetric matrix such that $∣A∣ = 2$ and $[23 1 \frac{3}{2}] A = [1 α 2 β]$ . If the…2023 · 29 Jan · Shift 2 · Q44 · Numerical
Let the system of linear equations $x + y + k z = 22 x + 3 y - z = 13 x + 4 y + 2 z = k$ have infinitely many solutions. Then the system $(k + 1) x + (2 k - 1) y = 7 (2 k + 1) x + (k + 5) y = 10$ has :2023 · 30 Jan · Shift 1 · Q25 · MCQ
For $α, β \in R$ , suppose the system of linear equations $x - y + z = 5 2 x + 2 y + α z = 8 3 x - y + 4 z = β$ has infinitely many solutions. Then $α$ and $β$ are…2023 · 30 Jan · Shift 2 · Q24 · MCQ
For the system of linear equations $x + y + z = 6 α x + β y + 7 z = 3 x + 2 y + 3 z = 14$ which of the following is NOT true ?2023 · 31 Jan · Shift 1 · Q29 · MCQ
Let $A = 1000412 0 - 1 - 3$ . Then the sum of the diagonal elements of the matrix $(A + I)^{11}$ is equal to :2023 · 31 Jan · Shift 1 · Q36 · MCQ
The number of values of $α$ for which the system of equations : x + y + z = $α α$ x + 2 $α$ y + 3z = $-$ 1 x + 3 $α$ y + 5z = 4 is inconsistent, is2022 · 24 Jun · Shift 1 · Q26 · MCQ
Let the system of linear equations x + y + $α$ z = 2 3x + y + z = 4 x + 2z = 1 have a unique solution (x $^{*}$ , y $^{*}$ , z $^{*}$ ). If ( $α$ , x $^{*}$ ), (y $^{*}$ , $α$ ) and (x $^{*}$ , $-$ y $^{*}$ ) are collinear points, then the…2022 · 24 Jun · Shift 2 · Q24 · MCQ
Let $S = {(- 1 0 a b); a, b \in {1, 2, 3, ....100}}$ and let $T_{n} = {A \in S : A^{n (n + 1)} = I}$ . Then the number of elements in $n = 1 ⋂ 100 T_{n}$ …2022 · 24 Jun · Shift 2 · Q38 · Numerical
The number of $θ \in (0, 4 π)$ for which the system of linear equations $3 (sin 3 θ) x - y + z = 2 3 (cos 2 θ) x + 4 y + 3 z = 3 6 x + 7 y + 7 z = 9$ has no solution, is :2022 · 25 Jul · Shift 1 · Q24 · MCQ
Let $A = 211 - 1 0 - 1 - 1 - 1 0$ and $B = A - I$ . If $ω = \frac{3 i - 1}{2}$ , then the number of elements in the $set {n \in {1, 2, \dots, 100} : A^{n} + (ω B)^{n} = A + B}$ …2022 · 25 Jul · Shift 1 · Q36 · Numerical
The number of real values of $λ$ , such that the system of linear equations 2x $-$ 3y + 5z = 9 x + 3y $-$ z = $-$ 18 3x $-$ y + ( $λ$ 2 $-$ | $λ$ |)z = 16 has no solutions, is2022 · 25 Jul · Shift 2 · Q24 · MCQ
Let $A = 100 a 10 a b 1, a, b \in R$ . If for some $n \in N, A^{n} = 10048102160961$ …2022 · 25 Jul · Shift 2 · Q39 · Numerical
Let A be a 3 $\times$ 3 real matrix such that $A 110 = 110; A 101 = - 1 01$ …2022 · 25 Jun · Shift 1 · Q26 · MCQ
Let $A = [02 - 2 0]$ . If M and N are two matrices given by $M = k = 1 \sum 10 A^{2 k}$ and $N = k = 1 \sum 10 A^{2 k - 1}$ then MN2 is :2022 · 25 Jun · Shift 1 · Q31 · MCQ
The system of equations $- k x + 3 y - 14 z = 25 - 15 x + 4 y - k z = 3 - 4 x + y + 3 z = 4$ is consistent for all k in the set2022 · 25 Jun · Shift 2 · Q26 · MCQ
Let $A = (21 - 2 - 1)$ and $B = (- 1 - 1 22)$ . Then the number of elements in the set {(n, m) :…2022 · 25 Jun · Shift 2 · Q37 · Numerical
If the system of linear equations. $8 x + y + 4 z = - 2 x + y + z = 0 λ x - 3 y = μ$ has infinitely many solutions, then the distance of the point $(λ, μ, - \frac{1}{2})$ from the plane $8 x + y + 4 z + 2 = 0$ is…2022 · 26 Jul · Shift 1 · Q27 · MCQ
$Let A = 111 and B = 9^{2} 1 2^{2} - 1 5^{2} - 1 0^{2} 1 3^{2} 1 6^{2} 1 1^{2} - 1 4^{2} 1 7^{2}, then the value of A^{'} B A is:$ …2022 · 26 Jul · Shift 2 · Q22 · MCQ
The number of matrices $A = (a c b d)$ , where $a, b, c, d \in {- 1, 0, 1, 2, 3, \dots\dots, 10}$ , such that $A = A^{- 1}$ , is .2022 · 26 Jul · Shift 2 · Q39 · Numerical
The ordered pair (a, b), for which the system of linear equations 3x $-$ 2y + z = b 5x $-$ 8y + 9z = 3 2x + y + az = $-$ 1 has no solution, is :2022 · 26 Jun · Shift 1 · Q23 · MCQ
If the system of equations $α$ x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = $β$ has infinitely many solutions, then the ordered pair ( $α$ , $β$ ) is equal to :2022 · 26 Jun · Shift 2 · Q26 · MCQ
Let $X = 000100010, Y = α I + β X + γ X^{2}$ and $$Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma…2022 · 26 Jun · Shift 2 · Q42 · Numerical
Let $A = (1 - 2 2 - 5)$ . Let $α, β \in R$ be such that $α A^{2} + β A = 2 I$ . Then $α + β$ is equal to2022 · 27 Jul · Shift 1 · Q27 · MCQ
Let $S$ be the set containing all $3 \times 3$ matrices with entries from ${- 1, 0, 1}$ . The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A^{T} A$ is 6 is .2022 · 27 Jul · Shift 1 · Q44 · Numerical
Let the system of linear equations $x + 2 y + z = 2$ , $α x + 3 y - z = α$ , $- α x + y + 2 z = - α$ be inconsistent. Then $α$ is equal to :2022 · 27 Jun · Shift 1 · Q21 · MCQ
Let $A = [12 - 1 α]$ and $B = [β 1 10], α, β \in R$ . Let $α_{1}$ be the value of $α$ which…2022 · 28 Jul · Shift 1 · Q42 · Numerical
Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?2022 · 28 Jul · Shift 2 · Q23 · MCQ
If the system of linear equations $2 x + 3 y - z = - 2 x + y + z = 4 x - y + ∣ λ ∣ z = 4 λ - 4$ where, $λ \in$ R, has no solution, then2022 · 28 Jun · Shift 1 · Q25 · MCQ
If the system of linear equations $2 x - 3 y = γ + 5$ , $α x + 5 y = β + 1$ , where $α$ , $β$ , $γ \in$ R has infinitely many solutions then the value of | 9 $α$ + 3 $β$ + 5 $γ$ | is equal to …2022 · 28 Jun · Shift 2 · Q46 · Numerical
Let $A = (1 + i - i 10)$ where $i = - 1$ . Then, the number of elements in the set { n } $\in$ {1, 2, ......, 100} : An = A } is .2022 · 28 Jun · Shift 2 · Q47 · Numerical
Let A and B be two $3 \times 3$ non-zero real matrices such that AB is a zero matrix. Then2022 · 29 Jul · Shift 1 · Q27 · MCQ
Let p and p + 2 be prime numbers and let $Δ = p! (p + 1)! (p + 2)! (p + 1)! (p + 2)! (p + 3)! (p + 2)! (p + 3)! (p + 4)!$ …2022 · 29 Jul · Shift 1 · Q45 · Numerical
Which of the following matrices can NOT be obtained from the matrix $[- 1 1 2 - 1]$ by a single elementary row operation ?2022 · 29 Jul · Shift 2 · Q24 · MCQ
If the system of equations $x + y + z = 6 2 x + 5 y + α z = β x + 2 y + 3 z = 14$ has infinitely many solutions, then $α + β$ is equal to2022 · 29 Jul · Shift 2 · Q25 · MCQ
Let $X = 111$ and $A = - 1 00 210 36 - 1$ . For $k \in N$ , if $X^{'} A^{k} X = 33$ , then $k$ …2022 · 29 Jul · Shift 2 · Q38 · Numerical
If the system of linear equations 2x + y $-$ z = 7 x $-$ 3y + 2z = 1 x + 4y + $δ$ z = k, where $δ$ , k $\in$ R has infinitely many solutions, then $δ$ + k is equal to:2022 · 29 Jun · Shift 1 · Q25 · MCQ
Let $A = [a_{ij}]$ be a square matrix of order 3 such that $a_{ij} = 2^{j - i}$ , for all i, j = 1, 2, 3. Then, the matrix A2 + A3 + ...... + A10 is equal to :2022 · 29 Jun · Shift 1 · Q27 · MCQ
Let $M = [0 α - α 0]$ , where $α$ is a non-zero real number an $N = k = 1 \sum 49 M^{2 k}$ . If $(I - M^{2}) N = - 2 I$ , then the positive…2022 · 29 Jun · Shift 2 · Q42 · Numerical
Let $A = [1 α - 2 2 α - 1]$ and $B = 2 - 1 4 α 2 - 5, α \in C$ …2022 · 30 Jun · Shift 1 · Q26 · MCQ
Consider the system of linear equations $-$ x + y + 2z = 0 3x $-$ ay + 5z = 1 2x $-$ 2y $-$ az = 7 Let S1 be the set of all a $\in$ R for which the system is inconsistent and S2 be the set of all a $\in$ R for which the system has…2021 · 1 Sep · Shift 2 · Q26 · MCQ
Let $A = [i - i - i i], i = - 1$ . Then, the system of linear equations $A^{8} [x y] = [864]$ …2021 · 16 Mar · Shift 1 · Q28 · MCQ
The total number of 3 $\times$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to .2021 · 16 Mar · Shift 1 · Q43 · Numerical
Let $A = [a_{1} a_{2}]$ and $B = [b_{1} b_{2}]$ be two 2 $\times$ 1 matrices with real entries such that A = XB, where $X = \frac{1}{3} [11 - 1 k]$ …2021 · 16 Mar · Shift 2 · Q45 · Numerical
The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to :2021 · 17 Mar · Shift 1 · Q32 · MCQ
If x, y, z are in arithmetic progression with common difference d, x $e$ 3d, and the determinant of the matrix $345 4252 k x y z$ …2021 · 17 Mar · Shift 2 · Q32 · MCQ
Let $A = [a c b d]$ and $B = [α β] e [00]$ such that AB = B and…2021 · 17 Mar · Shift 2 · Q34 · Numerical
If 1, log10(4x $-$ 2) and log10 $(4^{x} + \frac{18}{5})$ are in arithmetic progression for a real number x, then the value of the determinant $2 (x - \frac{1}{2}) 1 x x - 1 01 x^{2} x 0$ …2021 · 17 Mar · Shift 2 · Q39 · Numerical
The solutions of the equation $1 + sin^{2} x cos^{2} x 4 sin 2 x sin^{2} x 1 + cos^{2} x 4 sin 2 x sin^{2} x cos^{2} x 1 + 4 sin 2 x = 0, (0 < x < π)$ …2021 · 18 Mar · Shift 1 · Q26 · MCQ
Let $α$ , $β$ , $γ$ be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c $\in$ R and a, b $e$ 0). If the system of equations (in u, v, w) given by $α$ u + $β$ v + $γ$ w = 0, $β$ u + $γ$ v + $α$ …2021 · 18 Mar · Shift 1 · Q27 · MCQ
Let $A + 2 B = 16 - 5 2 - 3 3 031$ and $2 A - B = 220 - 1 - 1 1 562$ …2021 · 18 Mar · Shift 1 · Q37 · MCQ
Let the system of linear equations 4x + $λ$ y + 2z = 0 2x $-$ y + z = 0 $μ$ x + 2y + 3z = 0, $λ$ , $μ \in$ R. has a non-trivial solution. Then which of the following is true?2021 · 18 Mar · Shift 2 · Q24 · MCQ
Let I be an identity matrix of order 2 $\times$ 2 and P = $[25 - 1 - 3]$ . Then the value of n $\in$ N for which Pn = 5I $-$ 8P is equal to .2021 · 18 Mar · Shift 2 · Q41 · Numerical
Let $A = [2 a 30]$ , a $\in$ R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible…2021 · 20 Jul · Shift 1 · Q26 · MCQ
Let $A = 100 - 1 10 0 - 1 1$ and B = 7A20 $-$ 20A7 + 2I, where I is an identity matrix of order 3 $\times$ 3. If B = [bij], then b13is…2021 · 20 Jul · Shift 1 · Q38 · Numerical
Let a, b, c, d in arithmetic progression with common difference $λ$ . If $x + a - c x - 1 x - b + d x + b x + c x + d x + a x + b x + c = 2$ …2021 · 20 Jul · Shift 1 · Q42 · Numerical
The value of k $\in$ R, for which the following system of linear equations 3x $-$ y + 4z = 3, x + 2y $-$ 3z = $-$ 2 6x + 5y + kz = $-$ 3, has infinitely many solutions, is :2021 · 20 Jul · Shift 2 · Q37 · MCQ
The values of $λ$ and $μ$ such that the system of equations $x + y + z = 6$ , $3 x + 5 y + 5 z = 26$ , $x + 2 y + λ z = μ$ has no solution, are :2021 · 22 Jul · Shift 2 · Q29 · MCQ
Let A = [aij] be a real matrix of order 3 $\times$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :2021 · 22 Jul · Shift 2 · Q31 · MCQ
Let $A = 010100001$ . Then the number of 3 $\times$ 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is …2021 · 22 Jul · Shift 2 · Q40 · Numerical
The system of linear equations 3x - 2y - kz = 10 2x - 4y - 2z = 6 x+2y - z = 5m is inconsistent if :2021 · 24 Feb · Shift 1 · Q35 · MCQ
Let P = $323 - 1 0 - 5 - 2 α 0$ , where $α \in$ R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k…2021 · 24 Feb · Shift 1 · Q36 · Numerical
Let M be any 3 $\times$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is .2021 · 24 Feb · Shift 1 · Q43 · Numerical
Let A and B be 3 $\times$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 $-$ B2A2) X = O, where X is a 3 $\times$ 1 column matrix of unknown variables and O is a 3 $\times$ …2021 · 24 Feb · Shift 2 · Q23 · MCQ
For the system of linear equations: $x - 2 y = 1, x - y + k z = - 2, k y + 4 z = 6, k \in R$ , consider the following statements : (A) The system has unique solution if $k e 2, k e - 2$ . (B) The system has unique solution if k = $-$ 2 (C) The…2021 · 24 Feb · Shift 2 · Q26 · MCQ
If $A = [0 tan (\frac{θ}{2}) - tan (\frac{θ}{2}) 0]$ and $(I_{2} + A) (I_{2} - A)^{- 1} = [a b - b a]$ …2021 · 25 Feb · Shift 1 · Q40 · Numerical
If the system of equations kx + y + 2z = 1 3x $-$ y $-$ 2z = 2 $-$ 2x $-$ 2y $-$ 4z = 3 has infinitely many solutions, then k is equal to .2021 · 25 Feb · Shift 1 · Q42 · Numerical
Let A be a 3 $\times$ 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 $\to$ 2R2 + 5R3 on 2A, then det(B) is equal to :2021 · 25 Feb · Shift 2 · Q32 · MCQ
If for the matrix, $A = [1 α - α β]$ , $A A^{T} = I_{2}$ , then the value of $α^{4} + β^{4}$ is :2021 · 25 Feb · Shift 2 · Q37 · MCQ
The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x $-$ y + 4z = 82021 · 25 Feb · Shift 2 · Q39 · MCQ
The values of a and b, for which the system of equations 2x + 3y + 6z = 8 x + 2y + az = 5 3x + 5y + 9z = b has no solution, are :2021 · 25 Jul · Shift 1 · Q34 · MCQ
The number of distinct real roots of $sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0$ in the interval $- \frac{π}{4} \leq x \leq \frac{π}{4}$ …2021 · 25 Jul · Shift 2 · Q33 · MCQ
If $P = [1 \frac{1}{2} 01]$ , then P50 is :2021 · 25 Jul · Shift 2 · Q37 · MCQ
Let $θ \in (0, \frac{π}{2})$ . If the system of linear equations $(1 + cos^{2} θ) x + sin^{2} θ y + 4 sin 3 θ z = 0 cos^{2} θ x + (1 + sin^{2} θ) y + 4 sin 3 θ z = 0 cos^{2} θ x + sin^{2} θ y + (1 + 4 sin 3 θ) z = 0$ …2021 · 26 Aug · Shift 1 · Q26 · MCQ
If $A = (\frac{1}{5} \frac{- 2}{5} \frac{2}{5} \frac{1}{5})$ , $B = (1 i 01)$ …2021 · 26 Aug · Shift 1 · Q32 · MCQ
Let $A = 101010010$ . Then A2025 $-$ A2020 is equal to :2021 · 26 Aug · Shift 2 · Q24 · MCQ
Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is :2021 · 26 Feb · Shift 1 · Q25 · MCQ
The value of $(a + 1) (a + 2) (a + 2) (a + 3) (a + 3) (a + 4) a + 2 a + 3 a + 4 111$ is :2021 · 26 Feb · Shift 1 · Q32 · MCQ
Consider the following system of equations : x + 2y $-$ 3z = a 2x + 6y $-$ 11z = b x $-$ 2y + 7z = c, where a, b and c are real constants. Then the system of equations :2021 · 26 Feb · Shift 2 · Q28 · MCQ
If the matrix $A = 103020 00 - 1$ satisfies the equation $A^{20} + α A^{19} + β A = 100040001$ …2021 · 26 Feb · Shift 2 · Q41 · Numerical
If the matrix $A = (0 K 2 - 1)$ satisfies $A (A^{3} + 3 I) = 2 I$ , then the value of K is :2021 · 27 Aug · Shift 1 · Q25 · MCQ
If the system of linear equations 2x + y $-$ z = 3 x $-$ y $-$ z = $α$ 3x + 3y + $β$ z = 3 has infinitely many solution, then $α$ + $β - α β$ is equal to .2021 · 27 Aug · Shift 1 · Q38 · Numerical
Let $A = [x + 1] [x] [x] [x + 2] [x + 3] [x + 2] [x + 3] [x + 3] [x + 4]$ , where [t] denotes the greatest integer less than or…2021 · 27 Aug · Shift 2 · Q24 · MCQ
Let [ $λ$ ] be the greatest integer less than or equal to $λ$ . The set of all values of $λ$ for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [ $λ$ ])z = [ $λ$ ] has a…2021 · 27 Aug · Shift 2 · Q28 · MCQ
Let $A = [1 - 1 24]$ . If A $-$ 1 = $α$ I + $β$ A, $α$ , $β \in$ R, I is a 2 $\times$ 2 identity matrix then 4( $α - β$ ) is equal to :2021 · 27 Jul · Shift 1 · Q31 · MCQ
For real numbers $α$ and $β$ , consider the following system of linear equations : x + y $-$ z = 2, x + 2y + $α$ z = 1, 2x $-$ y + z = $β$ . If the system has infinite solutions, then $α$ + $β$ is equal to …2021 · 27 Jul · Shift 1 · Q39 · Numerical
Let $f (x) = sin^{2} x 2 + sin^{2} x sin^{2} x - 2 + cos^{2} x cos^{2} x cos^{2} x cos 2 x cos 2 x 1 + cos 2 x, x \in [0, π]$ …2021 · 27 Jul · Shift 1 · Q43 · Numerical
If $A = 100110111$ and M = A + A2 + A3 + ....... + A20, then the sum of all the elements of the matrix M is equal to …2021 · 27 Jul · Shift 2 · Q44 · Numerical
If the following system of linear equations 2x + y + z = 5 x $-$ y + z = 3 x + y + az = b has no solution, then :2021 · 31 Aug · Shift 1 · Q31 · MCQ
If $a_{r} = cos \frac{2 r π}{9} + i sin \frac{2 r π}{9}$ , r = 1, 2, 3, ....., i = $- 1$ , then the determinant $a_{1} a_{4} a_{7} a_{2} a_{5} a_{8} a_{3} a_{6} a_{9}$ …2021 · 31 Aug · Shift 1 · Q36 · MCQ
If $α$ + $β$ + $γ$ = 2 $π$ , then the system of equations x + (cos $γ$ )y + (cos $β$ )z = 0 (cos $γ$ )x + y + (cos $α$ )z = 0 (cos $β$ )x + (cos $α$ )y + z = 0 has :2021 · 31 Aug · Shift 2 · Q22 · MCQ
The number of elements in the set ${A = (a 0 b d) : a, b, d \in {- 1, 0, 1} an d (I - A)^{3} = I - A^{3}}$ , where I is 2 $\times$ 2 identity matrix, is :2021 · 31 Aug · Shift 2 · Q40 · Numerical
Let S be the set of all $λ \in$ R for which the system of linear equations 2x – y + 2z = 2 x – 2y + $λ$ z = –4 x + $λ$ y + z = 4 has no solution. Then the set S :2020 · 2 Sep · Shift 1 · Q20 · MCQ
Let A be a 2 $\times$ 2 real matrix with entries from {0, 1} and |A| $e$ 0. Consider the following two statements : (P) If A $e$ I2 , then |A| = –1 (Q) If |A| = 1, then tr(A) = 2, where I2 denotes 2 $\times$ 2 identity matrix and tr(A)…2020 · 2 Sep · Shift 1 · Q25 · MCQ
Let a, b, c $\in$ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = $a b c b c a c a b$ satisfies ATA = I, then a value of abc can…2020 · 2 Sep · Shift 2 · Q23 · MCQ
Let A = {X = (x, y, z)T: PX = 0 and x2 + y2 + z2 = 1} where $P = 1 - 2 1 239 1 - 4 - 1$ , then the set A :2020 · 2 Sep · Shift 2 · Q36 · MCQ