Matrices & Determinants PYQs Set 11 (PDF)

Set 11 of 19 · Questions 301-330 of 565

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 11 (Q301-330)
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301. $Let A be a 2 \times 2 real matrix with entries from 0, 1 and (A) \neq = 0 . Consider the following two statements: (P) If A \neq = I_{2}, then (A) = - 1 (Q) If (A) = 1, then tr (A) = 2, where I_{2} denotes 2 \times 2 identity matrix and tr (A) denotes the sum of the diagonal entries A. Then:$ (1) $(P) is true and (Q) is false.$ (2) $Both (P) and (Q) are false.$ (3) $Both (P) and (Q) are true.$ (4) $(P) is false and (Q) is true.$ 302. $Let A, B and C be three 2 \times 2 matrices with real entries such that B = (I + A)^{- 1} and A + C = I . If B C = (1 - 1 - 5 2) and C B (x_{1} x_{2}) = (12 - 6), then x_{1} + x_{2} is$ (1) $2$ (2) $-2$ (3) $0$ (4) $4$ 303. $Let A = b^{2} + c^{2} b^{2} c^{2} a^{2} c^{2} + a^{2} c^{2} a^{2} b^{2} a^{2} + b^{2} . If a = sin π /6, b = cos π /4 and c = cot π /2 then A is$ (1) $Symmetric matrix$ (2) $Skew-symmetric matrix$ (3) $Singular matrix$ (4) $Non-singular matrix$ 304. $If A is a skew symmetric matrix, then A^{2021} is$ (1) $Row matrix$ (2) $Symmetric matrix$ (3) $Column matrix$ (4) $Skew symmetric matrix$ 305. $If the system of simultaneous linear equations x + y + z = a, x - y + bz = 2, 2 x + 3 y - z = 1 has infinitely many solutions, then b - 5 a =$ (1) $\frac{4}{5}$ (2) $3$ (3) $7$ (4) $-3$ 306. $If the system of equations x + y + z = 2 2 x + 4 y - z = 6 3 x + 2 y + λz = μ, has infinitely many solutions, then$ (1) $2 λ - μ = 5$ (2) $λ - 2 μ = - 5$ (3) $λ + 2 μ = 14$ (4) $2 λ + μ = 14$ 307. $Let A = (- cot θ cosec θ cosec θ - cot θ) . If A^{- 1} = A at θ = θ_{1} and A^{- 1} + A = O at θ = θ_{2}, then which one of the following is true?$ (1) $θ_{1} =π2,θ2=π$ (2) $θ_{1} =π2, such θ_{2} does not exist$ (3) $θ_{1} =π4,θ2=π2$ (4) $such θ_{1} does not exist, θ_{2} = π$ 308. $If A (adj A) = 5 I where I is the identity matrix of order 3 , then \| adj A \| is equal to$ (1) $5$ (2) $10$ (3) $125$ (4) $25$ 309. $Consider the system of equations ax + by + cz = 2 bx + cy + az = 2 cx + ay + bz = 2 where a, b, c are real numbers such that a + b + c = 0 . Then, the system$ (1) $has two solutions$ (2) $is inconsistent$ (3) $has unique solution$ (4) $has infinitely many solutions$ 310. $Let P = (p_{ij}) and Q = (q_{ij}) be two square matrices of order 3 such that q_{ij} = 2^{(i + j - 1)} p_{ij} and det (Q) = 2^{10} . Then the value of det (adj (adj P)) is:$ (1) $81$ (2) $16$ (3) $124$ (4) $32$ 311. $For some a, b, let f (x) = a + \frac{s i n x}{x} a a 1 1 + \frac{s i n x}{x} 1 b b b + \frac{s i n x}{x}, x \neq = 0, lim_{x \to 0} f (x) = λ + μ a + ν b . Then (λ + μ + ν)^{2} is equal to:$ (1) $16$ (2) $25$ (3) $9$ (4) $36$ 312. $Let the system of equations x + 2 y + 3 z = 5, 2 x + 3 y + z = 9, 4 x + 3 y + λz = μ have infinite number of solutions. Then λ + 2 μ is equal to :$ (1) $28$ (2) $17$ (3) $22$ (4) $15$ 313. $If A = (0010) then (a I + b A)^{n} is (where I is the identity matrix of order 2)$ (1) $a^{2} I + a^{n - 1} b \cdot A$ (2) $a^{n} I + n a^{n} bA$ (3) $a^{n} I + n a^{n - 1} bA$ (4) $a^{n} I + b^{n} A$ 314. $If A, B, and (adj (A^{- 1}) + adj (B^{- 1})) are non-singular matrices of same order, then the inverse of A (adj (A^{- 1}) + adj (B^{- 1}))^{- 1} B, is equal to$ (1) $A B^{- 1} + A^{- 1} B$ (2) $a d j (B^{- 1}) + a d j (A^{- 1})$ (3) $\frac{A B ^{- 1}}{∣ A ∣} + \frac{B A ^{- 1}}{∣ B ∣}$ (4) $\frac{1}{∣ A B ∣} (a d j (B) + a d j (A))$ 315. $G = {(x x x x) : x is a non-zero real number} is a group with respect to matrix multiplication. In this group, the inverse of (1/3 1/3 1/3 1/3) is$ (1) $(4/3 4/3 4/3 4/3)$ (2) $(3/4 3/4 3/4 3/4)$ (3) $(3333)$ (4) $(1111)$ 316. $The set of values of k for which the system of simultaneous equations x + y + kz = 1, 2 x + 2 y = 3 and x + 2 y + 2 kz = k has no real solution is$ (1) ${0}$ (2) $R - {0}$ (3) ${2}$ (4) ${- 1, 0, 1}$ 317. $Let A = a 52 3 - 1 3 53 - 4 and B = b 4 - 3 1 c 1 41 d . If the trace of A is -4 and A B = - 1 - 3 28 010 - 8 17253 then a + b + c + d =$ (1) $7$ (2) $-1$ (3) $3$ (4) $1$ 318. $The inverse of the matrix 20 - 1 510013 is$ (1) $3 - 15 5 - 1 6 - 2 1 - 5 2$ (2) $3 - 1 1 - 15 6 - 5 5 - 2 2$ (3) $3 - 1 1 - 5 - 6 - 5 5 - 2 2$ (4) $3 - 1 1 - 15 6 - 5 5 - 2 - 2$ 319. $If A is an 3 \times 3 non-singular matrix such that AA' = A'A and B = A-1A', then BB' equals :$ (1) $B^{- 1}$ (2) $(B^{- 1})^{'}$ (3) $I + B$ (4) $I$ 320. $Let A = (a_{ij}) be a matrix of order 3 \times 3, with a_{ij} = (2)^{i + j} . If the sum of all the elements in the third row of A^{2} is α + β 2, α, β \in Z, then α + β is equal to:$ (1) $280$ (2) $224$ (3) $210$ (4) $168$ 321. $If there are two values of ' a' which makes determinant Δ = 120 - 2 a 4 5 - 1 2 a = 86 Then the sum of these numbers is$ (1) $-4$ (2) $4$ (3) $9$ (4) $5$ 322. $If A = 1 - 2 3 - 3 12 23 - 1 then A^{2} Adj A =$ (1) $21 A$ (2) $- 42 A$ (3) $7 A - 1$ (4) $14(Adj A)$ 323. $If matrix A = (1423) such that A X = I, then X =$ (1) $\frac{1}{5} (12 3 - 1)$ (2) $\frac{1}{5} (44 2 - 1)$ (3) $\frac{1}{5} (- 3 4 2 - 1)$ (4) $\frac{1}{5} (- 1 - 1 24)$ 324. $Suppose n > 1 and A is an n \times n non-singular matrix such that ∣ Adj A ∣ = ∣ Adj (Adj A) ∣ . Then the matrix whose rank is n, is$ (1) $146257368$ (2) $222222222$ (3) $(120 - 13412 - 2325)$ (4) $120431 - 1 02$ 325. $If A = (0 sin α sin α 0) and det (A^{2} - \frac{1}{2} I) = 0 then a possible value of α is$ (1) $\frac{π}{2}$ (2) $\frac{π}{3}$ (3) $\frac{π}{4}$ (4) $\frac{π}{6}$ 326. $If f (x) = x^{3} - x x - a x - b a + x x^{2} - x x - c b + x c + x 0, then$ (1) $f (2) = 0$ (2) $f (0) = 0$ (3) $f (- 1) = 0$ (4) $f (1) = 0$ 327. $If the matrix (25 3 - 1) = A + B, where A is symmetric and B is skew symmetric, then B =$ (1) $(24 4 - 1)$ (2) $(02 - 2 0)$ (3) $(0 - 1 10)$ (4) $(01 - 1 0)$ 328. $If the system of equations x + 4 y - z = λ, 7 x + 9 y + μz = - 3, 5 x + y + 2 z = - 1 has infinitely many solutions, then (2 μ + 3 λ) is equal to :$ (1) $3$ (2) $-3$ (3) $-2$ (4) $2$ 329. $If the system of homogeneous equations tx + (t + 1) y + (t - 1) z = 0 (t + 1) x + ty + (t + 2) z = 0 (t - 1) x + (t + 2) y + tz = 0 in x, y, z has a non-trivial solution, then t is a root of the equation$ (1) $3 t^{2} - 4 t + 1 = 0$ (2) $2 t^{2} - 3 t + 1 = 0$ (3) $2 t^{2} + 3 t + 1 = 0$ (4) $3 t^{2} + 4 t + 1 = 0$ 330. $Let A = (a c b d) and B = (α β) \neq = (00) such that A B = B and a + d = 2021, then the value of a d - b c is equal to$