Tiêu đề 27. Matrices and Determinants Easy.pdf ✅

1. The value of 4 13 9 12 8 12 2 12 7 11 6 11 11 5 10 4 10 +  = + m m m C C C C C C C C C is equal to zero, where m is (a) 6 (b) 4 (c) 5 (d) None of these 2. If , , ,......., , ...... a1 a2 a3 an are in G.P. then the value of the determinant 6 7 8 3 4 5 1 2 log log log log log log log log log + + + + + + + + n n n n n n n n n a a a a a a a a a is (a) –2 (b) 1 (c) 2 (d) 0 3. The value of x x 2 x x 2 x x 2 x x 2 x x 2 x x 2 (2 2 ) (3 3 ) (5 5 ) (2 2 ) (3 3 ) (5 5 ) 1 1 1 − − − − − − − − − + + + (a) 0 (b) x 30 (c) −x 30 (d) None of these 4. If x, y, z are integers in A.P. lying between 1 and 9 and x51, y41 and z31 are three digit numbers then the value of x y z x 51 y 41 z 31 5 4 3 is (a) x + y + z (b) x − y + z (c) 0 (d) None of these 5. If a  b  c , the value of x which satisfies the equation 0 0 0 0 = + + + − − − x b x c x a x c x a x b is (a) x = 0 (b) x = a (c) x = b (d) x = c 6. The number of distinct real roots of 0 cos cos sin cos sin cos sin cos cos = x x x x x x x x x in the interval 4 4   −  x  is (a) 0 (b) 2 (c) 1 (d) 3 7. If               3 4 3 1 2 4 3 1 3 2 4 3 2 − + + − − + − + p + q + r + s + t = , then value of t is (a) 16 (b) 18 (c) 17 (d) 19 8. If p 25 10 p 35 9 p 15 8 D 3 2 p = , then D1 + D2 + D3 + D4 + D5 = (a) 0 (b) 25 (c) 625 (d) None of these 9. The value of = N n Un 1 , if n N N n N N n Un 3 3 2 1 2 1 1 5 3 2 2 = + + is (a) 0 (b) 1 (c) –1 (d) None of these 10. If a a x a x b x b b 1 = and a x x b 2 = are the given determinants, then (a) 2 1 2  = 3( ) (b) 1 3 2 ( ) =  dx d (c) 2 1 2 ( ) = 2( ) dx d (d) 3 / 2 1 = 32 11. If y = sin mx , then the value of the determinant 6 7 8 3 4 5 1 2 y y y y y y y y y , where n n n dx d y y = is (a) 9 m (b) 2 m (c) 3 m (d) None of these 12. If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the value of a is (a) –1 (b) 1 (c) 0 (d) No real values 13. If the system of equations ax + y + z = 0, x + by + z = 0 and x + y + cz = 0 , where a, b, c  1 has a non-trivial solution, then the value of a b − c + − + − 1 1 1 1 1 1 is (a) –1 (b) 0 (c) 1 (d) None of these 14. If the system of equations x + 2y − 3z = 1, (k + 3)z = 3, (2k + 1)x + z = 0 is inconsistent, then the value of k is (a) –3 (b) 2 1 (c) 0 (d) 2 15. Theequations x + y + z = 6, x + 2y + 3z = 10, x + 2y + mz = n give infinite number of values of the triplet (x, y, z) if (a) m = 3, n  R (b) m = 3, n  10 (c) m = 3, n = 10 (d) None of these 16. The matrix           − 0 0 9 0 3 11 2 5 7 is known as (a) Symmetric matrix (b) Diagonal matrix (c) Upper triangular matrix (d) Skew symmetric matrix 17. In an upper triangular matrix n×n, minimum number of zeros is
(a) 2 n(n − 1) (b) 2 n(n + 1) (c) 2 2n(n −1) (d) None 18. If [ ] A = aij is a scalar matrix then trace of A is (a)  i j aij (b)  i ij a (c)  j aij (d)  i ii a 19. If  =      −     = 2 ,then A sin cos cos sin A (a)               sin cos cos sin (b)          −      sin cos cos sin (c)       −         sin cos cos sin (d)       −  −  −       sin cos cos sin 20. If       = b a a b A and           = 2 A then (a)  = a + b ,  = ab 2 2 (b) a b , 2ab 2 2  = +  = (c) 2 2 2 2  = a + b ,  = a − b (d) 2 2  = 2ab,  = a + b 21. If n N i i A        = , 0 0 , then n A 4 equals (a)       0 1 1 0 (b)       i i 0 0 (c)       0 0 i i (d)       0 0 0 0 22. If       −  =      − − = 1 1 , 2 1 1 1 b a A B and 2 2 2 (A + B) = A + B then value of a and b are (a) a = 4 , b = 1 (b) a = 1,b = 4 (c) a = 0, b = 4 (d) a = 2,b = 4 23. The order of                     z y x g f c h b f a h g [x y z] is (a) 3×1 (b) 1×1 (c) 1×3 (d) 3×3 24. Let           − = 0 0 1 sin cos 0 cos sin 0 ( )     F  . Then F().F(') is equal to (a) F(') (b) F( /') (c) F( +') (d) F( −') 25. For the matrix           = 2 1 0 1 2 1 1 1 0 A , which of the following is correct (a) 3 0 3 2 A + A − I = (b) 3 0 3 2 A − A − I = (c) 2 0 3 2 A + A − I = (d) 0 3 2 A − A + I = 26. If       = 3 4 4 2 A , then | adjA| is equal to (a) 16 (b) 10 (c) 6 (d) None of these 27. The inverse of matrix       = c d a b A is (a)       − − c a d b (b)       − − − c a d b ad bc 1 ` (c)       0 1 1 0 | | 1 A (d)       − − d c b a 28. Let           − − = 1 1 1 2 1 3 1 1 1 A and           − = − 1 2 3 5 0 4 2 2 10.B  . If B is the inverse of matrix A, then  is (a) 5 (b) –1 (c) 2 (d) –2 29. Matrix           − = 0 1 2 1 3 1 0 K K A is invertible for (a) K =1 (b) K = −1 (c) K = 0 (d) All real K 30. Let be a non-singular matrix, 1 ....... 0 2 + + + + = n p p p (0 denotes the null matrix), then −1 p = (a) n p (b) n − p (c) (1 ..... ) n − + p + + p (d) None 31. Let , 0 0 1 sin cos 0 cos sin 0 ( )           − =     f  where   R ,then 1 [ ( )]− f  is equal to (a) f(−) (b) ( ) −1 f  (c) f(2) (d) None 32. If I is a unit matrix of order 10, then the determinant of I is equal to (a) 10 (b) 1 (c) 1/10 (d) 9 33. If       =   2 2 A and | | 125 3 A = then  = (a)  3 (b)  2 (c)  5 (d) 0 34. If |A| denotes the value of the determinant of the square matrix A of order 3, then | −2A| = (a) −8 | A| (b) 8| A| (c) −2| A| (d) None of these 35. If x 3 6 2 x 7 4 5 x 3 6 x = x 7 2 = 5 x 3 = 0 6 x 3 7 2 x x 4 5 = 0 then x is equal to (a) 9 (b) -9 (c) 0 (d) None of these 36. If f() = _ iθ iθ 1 1 -1 1 e 1 1 -1 -e then
(a)    −    =   / 2 / 2 / 2 0 f( )d 2 f( )d (b) f() is purely real (c) f(/2) = 2 (d) None of these 37. If a, b, & c are sides of a ABC and 2 2 2 2 2 2 2 2 2 a b c (a + 1) (b + 1) (c + 1) = 0 (a - 1) (b - 1) (c - 1) , then (a) ABC is an equilateral triangle` (b) ABC is a right angled triangle (c) ABC is an Isosceles triangle (d) None of these 38. Let ax7 + bx6 + cx5 + dx4 + ex3 + fx2 + gx + h = 2 2 2 2 2 2 (x + 1) (x + 2) (x + x) (x + x) (x + 1) (x + 2) (x + 2) (x + x) x + 1 . Then (a) g = 3 and h = – 5 (b) g = –3 and h = – 5 (c) g = –3 and h = –9 (d) None of these 39. If y + z x x y z + x y = k(xyz) z z x + y , then k is equal to (a) 4 (b) -4 (c) Zero (d) None of these 40. 109 102 95 6 13 20 1 -6 -13 is equal to (a) Constant other than zero (b) Zero (c) 100 (d) -1997 41.  = p 2 - i i + 1 2 + i q 3 + i 1 - i 3 - i r is always (a) Real (b) Imaginary (c) Zero (d) None of these 42. If  = 2 2 2 1 a bc 1 a a 1 b ca = 1 b b 1 c ab 1 c c then (a)  = (a-b) (b-c) (c-a) (b) a, b, c are in G.P. (c) b, c, a are in G.P. (d) a, c, b are in G.P. 43. If A and B are any two square matrices of the same order, then (a) (AB) = AB (b) adj(AB) = adj(a) adj(b) (c) (AB) = BA (d) AB = O  A = O or B = O 44. The system AX = B of n equations in n unknowns has infinitely many solutions if (a) det A  0 (b) det A = 0, (adjA)B  O (c) det A = 0, (adjA)B = O (d) det A  0, (adjA)B = O 45. If A =           -2 3 -1 -1 2 -1 -6 9 -4 and B =           1 3 -1 2 2 -1 3 0 -1 , then (a) A B = B A (b) AB | BA (c) A B = 1 2 B A (d) None of these. 46. If A =       1 tanx -tanx 1 , then the value of |ATA−1 | is (a) cos4x (b) sec2 x (c) − cos4x (d) 1 47. If =  n n 1 n = an2 + bn, where a, b are constants and 1, 2, 3  {1, 2, 3, ......., 9} and 251, 372, 493 be three digit numbers, then 1 2 3 1 2 3 25 37 49 5 7 9       = (a) 1 + 2 + 3 (b) 1 – 2 + 3 (c) 7 (d) 0 48. The number of values of  [0, ] such that  = cos3 + sin3 and system of equation 3x – y + 4z = 3, x + 2y – 3z = – 2, 6x + 5y + 5z = – 3 does not have a unique solution is / are - (a) 0 (b) Infinite (c) 1 (d) None of these 29. If a1, a2, a3, ....., a9 are in H.P. and a4 = 5, a5 = 4, then the value of 7 8 9 4 5 6 1 2 3 a a a a a a a a a is- (a) 31/15 (b) 41/18 (c) 50/21 (d) 61/27 50. If  = 1 cos (1 ) 1 sin 1 2 2 x x x n x e x x  + = a + bx + cx2 then the value of b is (a) 0 (b) –1 (c) –2 (d) None of these 51. Coefficient of x in f(x)= x (1 x) 0 1 log(1 x) 2 x (1 sin x) cosx 2 2 3 + + + is - (a) 0 (b) 1 (c) – 2 (d) Cannot be determined 52. If     x x x x x x x x x x x x = f(x) – x f '(x), then f(x) is -
(a) (x – ) (x – ) (x – ) (x – ) (b) (x + ) (x + ) (x + ) (x + ) (c) 2(x – ) (x – ) (x – ) (x – ) (d) None of these 53. If p, q, r are in A.P., then the determinant a 2 p b 2 2q c r 2 p 2 q 2q a a 2p b 2 3q c p 2 n 2 n 1 2 n n 1 2 2n 1 2 n 2 2 + + + + − + + + + + + + + + + + = (a) 1 (b) 0 (c) a 2 b 2 c 2 – 2 n (d) (a2 + b2 + c2 ) – 2 n q 54. Determinant of second order is made with the element 0 and 1. The number of determinants with non negative values is (a) 3 (b) 10 (c) 11 (d) 13 55. The system of equations x + ky + 3z = 0, 3x + ky –2z = 0, 2x + 3y – 4z = 0 possess a non-trivial solution over the set of rationals, then 2k, is an integral element of the interval: (a) [10, 20] (b) (20, 30) (c) [30, 40] (d) (40, 50) 56. Number of distinct roots of  = cot x cot x tan x cot x tan x cot x tan x cot x cot x = 0 in the interval         − 4 , 4 is (a) 0 (b) 2 (c) 1 (d) 3 57. If A is a square matrix of order n such that its elements are polynomials in x and its r-rows become identical for x = , then - (a) (x – ) r is a factor of |A| (b) (x – ) r –1 is a factor of |A| |(c) (x – ) r + 1 is a factor of |A|(d) (x – ) r is a factor of A 58. The value of the determinant p – q q – r r – p x – y y – z z x a – b b – c c – a − is - (a) 0 (b) abc + pqr + xyz (c) (a – x)(y – z)(r – p) (d) None of these 59. If , ,  are different from 1 and are the roots of ax3 + bx2 + cx + d = 0 and ( – ) ( – ) ( – ) = 2 25 , then the determinant  = 2 2 2 1 1 1       −   −  −   equals: (a) 2a 25d (b) a 25d (c) a b c d 25d + + + − (d) None of these 60. If  be a repeated roots of the quadratic equation f(x) = 0 and A(x), B(x), C(x) are polynomials of degree 3, 4, 5 respectively then determinant A ( ) B ( ) C ( ) A( ) B( ) C( ) A(x) B(x) C(x)          is divisible by (where A() = =       x dx dA , etc) - (a)f(x) (x – ) 3 (b) (f(x))2 (c) f(x) (d) None of these 61. If 2 2 2 ac bc – c ab – b bc – a ab ac =  a 2b 2 c 2 , Then  = (a) 1 (b) 2 (c) 3 (d) 4 62. If a, b, c are in AP; Then value of 3 4 c 2 3 b 1 2 a is - (a) a + c (b) 2 (a + c) (c) b (d) None 63. The value of = a b a 2b a a 2b a a b a a b a 2b + + + + + + is equal to - (a) 9a2 (a + b) (b) 9b2 (a + b)(c)a2 (a + b)(d) b2 (a + b) 64. If a, b, c be positive and not all equal, then the value of the determinant c a b b c a a b c is - (a) +ive (b) –ive (c) Depends on a, b, c (d) None 65. The value of the determinant 12 13 14 11 12 13 10 11 12 is (a) 2( 10 11) (b) 2( 10 13) (c) 2( 10 11 12 ) (d) 2( 11 12 13) 66. Consider the system of linear equation in x, y and z (sin 3 )x – y + z = 0 (cos 2 ) x + 4y + 3z = 0 2x + 7y + 7z = 0 If the system has non-trivial solution, then   [0, ] are - (a) 0, , /6 (b) 0, , /6, 5/6 (c) 0, /6, 5/6 (d) None