Matrices & Determinants PYQs Set 1 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Matrices & Determinants · Set 1 (Q1-30)
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1. $For the matrices A = (31 - 4 - 1) and B = (- 29 - 13 4918), if (A^{15} + B) (x y) = (00), then among the following which one is true?$ (1) $x = 5, y = 7$ (2) $x = 18, y = 11$ (3) $x = 11, y = 2$ (4) $x = 16, y = 3$ 2. $The inverse of the matrix 135032 00 - 1 is$ (1) $- \frac{1}{3} - 3 39 012 00 - 3$ (2) $- \frac{1}{3} - 3 3 - 9 0 - 1 - 2 003$ (3) $- \frac{1}{3} 33 - 9 0 - 1 - 2 003$ (4) $- \frac{1}{3} - 3 - 3 - 9 0 - 1 - 2 003$ 3. $If the system of equations x + 2 y + 3 z = 3 4 x + 3 y - 4 z = 4 8 x + 4 y - λz = 9 + μ has infinitely many solutions, then the ordered pair (λ, μ) is equal to$ (1) $(\frac{72}{5}, \frac{21}{5})$ (2) $(\frac{- 72}{5}, \frac{- 21}{5})$ (3) $(\frac{72}{5}, \frac{- 21}{5})$ (4) $(\frac{- 72}{5}, \frac{21}{5})$ 4. $Let α \in (0, \infty) and A = (12 α 101012) . If det (adj (2 A - A^{T}) \cdot adj (A - 2 A^{T})) = 2^{8}, then (det (A))^{2} is equal to:$ (1) $36$ (2) $16$ (3) $1$ (4) $49$ 5. $If P is a 3 \times 3 real matrix such that P^{T} = aP + (a - 1) I, Where a > 1, then$ (1) $P is a singular matrix$ (2) $∣ Adj P ∣ > 1$ (3) $∣ Adj P ∣ = \frac{1}{2}$ (4) $∣ Adj P ∣ = 1$ 6. $If A = (0030) and f (x) = x + x^{2} + x^{3} + \dots + x^{2023} then f (A) + I =$ (1) $(0000)$ (2) $(1030)$ (3) $(1031)$ (4) $(1131)$ 7. $The value of determinant a - b b - a c - a b + c c + a a + b a b c is$ (1) $a^{3} + b^{3} + c^{3}$ (2) $3 abc$ (3) $(a + b + c) (a - b) (c - b)$ (4) $none of the above$ 8. $If A = (12 - 1 - 102120) and B = (- 3 - 2422 - 1 - 203), then A^{2} =$ (1) $A - B$ (2) $B - A$ (3) $A + B$ (4) $B^{2}$ 9. $If A = (21 1 - 2 32 - 1 - 3), B = (21 1 - 1 0233) and 2 A + 3 B - 5 C = 0, then C =$ (1) $(21 1 7/5 6/5 2 7/5 3/5)$ (2) $(- 2 1 1 - 7/5 6/5 2 7/5 3/5)$ (3) $(- 2 1 1 7/5 6/5 2 7/5 3/5)$ (4) $(21 1 - 7/5 6/5 2 7/5 3/5)$ 10. $The value of lo g_{5} 729 lo g_{5} 27 lo g_{3} 5 lo g_{9} 25 \times lo g_{3} 5 lo g_{5} 9 lo g_{27} 5 lo g_{5} 9 is$ (1) $1$ (2) $6$ (3) $log_{5} 9$ (4) $(log_{3} 5) \times (log_{5} 81)$ 11. $The characteristic roots of the matrix 124035006 are$ (1) $4, 5, 6$ (2) $2, 4, 6$ (3) $1, 3, 6$ (4) $1, 2, 4$ 12. $The area of a triangle with vertices (- 3, 0), (3, 0) and (0, k) is 9 sq. units, the value of k is$ (1) $6$ (2) $9$ (3) $3$ (4) $-9$ 13. $The value of 1 lo g_{y} x lo g_{z} x lo g_{x} y 1 lo g_{z} y lo g_{x} z lo g_{y} z 1 =$ (1) $xyz$ (2) $0$ (3) $log xyz$ (4) $1$ 14. $If (α β γ) = (cos θ - sin θ 0 sin θ cos θ 0001) (x y z), then \frac{x ^{2} + y ^{2} + z ^{2}}{γ} =$ (1) $\frac{α ^{2} + β ^{2} + γ ^{2}}{z}$ (2) $0$ (3) $αβ + βγ + γα$ (4) $1 + α^{2} + β^{2} + γ^{2}$ 15. $Let m and M be respectively the minimum and maximum values of (cos^{2} x 1 + sin^{2} x sin 2 x 1 + cos^{2} x sin^{2} x sin 2 x cos^{2} x sin^{2} x 1 + sin 2 x) . Then the ordered pair (m, M) is equal to:$ (1) $(1, 3)$ (2) $(- 3, - 1)$ (3) $(- 4, - 1)$ (4) $(- 3, 3)$ 16. $The constant term in the expansion of 3 x + 1 5 x - 1 7 x - 2 2 x - 1 3 x + 2 3 x + 1 x + 2 x + 1 4 x - 1 is$ (1) $0$ (2) $-10$ (3) $2$ (4) $6$ 17. $If the system of equations 3 x + y + 4 z = 32 x + αy - z = - 3 x + 2 y + z = 4 has no solution, then the value of α is equal to :$ (1) $19$ (2) $4$ (3) $13$ (4) $23$ 18. $If A = 1 - 3 2 305 - 2 - 5 0 and A (adj A) = K I, then the value of K is (where I is unit matrix of order 3)$ (1) $-25$ (2) $-85$ (3) $85$ (4) $25$ 19. $Let A be a 3 \times 3 matrix such that adj A = (2 - 11 - 1021 - 2 - 1) and B = adj (adj A) . If (A) = λ and ((B^{- 1})^{T}) = μ, then the ordered pair, ((λ), μ) is equal to:$ (1) $(3, 81)$ (2) $(9, \frac{1}{9})$ (3) $(3, \frac{1}{81})$ (4) $(9, \frac{1}{81})$ 20. $Let A = (\frac{1}{10} \frac{- 3}{10} \frac{3}{10} \frac{1}{10}) and B = (10 - i 1), where i = - 1 . If M = A^{T} B A, then the inverse of the matrix A M^{2023} A^{T} is$ (1) $(10 - 2023 i 1)$ (2) $(1 - 2023 i 01)$ (3) $(1 2023 i 01)$ (4) $(10 2023 i 1)$ 21. $Let M be 2 \times 2 symmetric matrix with integer entries, then M is invertible if$ (1) $the first column of M is the transpose of second row of M$ (2) $the second row of M is the transpose of first column of M$ (3) $M is diagonal matrix with non-zero entries in the principal diagonal$ (4) $the product of entries in the principal diagonal of M is the product of entries in the other diagonal$ 22. $If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then x y z a + y b + y c + y x + a y + b z + c is equal to:$ (1) $y (b - a)$ (2) $y (a - b)$ (3) $y (a - c)$ (4) $0$ 23. $The values of m, n, for which the system of equations x + y + z = 4, 2 x + 5 y + 5 z = 17, x + 2 y + m z = n has infinitely many solutions, satisfy the equation:$ (1) $m^{2} + n^{2} - mn = 39$ (2) $m^{2} + n^{2} - m - n = 46$ (3) $m^{2} + n^{2} + m + n = 64$ (4) $m^{2} + n^{2} + mn = 68$ 24. $Let X and Y be two arbitrary, 3 \times 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 \times 3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?$ (1) $Y^{3} Z^{4} - Z^{4} Y^{3}$ (2) $X^{44} + Y^{44}$ (3) $X^{4} Z^{3} - Z^{3} X^{4}$ (4) $X^{23} + Y^{23}$ 25. $Let A be a 3 \times 3 matrix such that ∣ adj (adj (adj A)) ∣ = 1 2^{4} . Then ∣ A^{- 1} adj A ∣ is equal to$ (1) $23$ (2) $6$ (3) $12$ (4) $1$ 26. $Evaluate cos 1 5^{\circ} sin 7 5^{\circ} sin 1 5^{\circ} cos 7 5^{\circ}$ (1) $0$ (2) $3$ (3) $1$ (4) $2$ 27. $A, P, B are 3 \times 3 matrices. If ∣ - B ∣ = 5, ∣ B A^{T} ∣ = 15, ∣ P^{T} A P ∣ = - 27, then one of the values of ∣ P ∣ is$ (1) $3$ (2) $-5$ (3) $9$ (4) $6$ 28. $If A = (20 3 - 1) then the value of det (A^{4}) + det (A^{10} - (Adj (2 A))^{10}) is equal to$ 29. $Let A = (x 110), x \in R and A^{4} = (a_{ij}) . If a_{11} = 109, then a_{22} is equal to$ 30. $If the system of linear equations x + y + z = 6 x + 2 y + 3 z = 10 3 x + 2 y + λz = μ has more than two solutions, then μ - λ^{2} is equal to\dots..$