MATHEMATICS. 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81

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1 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81

2 We know that KA = A If A is n th Order 3AB =3 3 A. B = = 81 3

3 2. If A= then adj A = 1)9 2) 1/9 3) 81 4)

4 A = = = 9 If A is square matrix of order n, then adja = 1 adja = 3 1 = 2 = 9 2 =81 3

5 3. The sum of 2-3 and its multiplicative inverse is, ) 2) 3) 4)

7 = & 2 = 1 h =0 2 1 = =0 4

8 1 2 = 1 = = = = 1 =

9 5. b 2 c 2 bc b+c c 2 a 2 ca c+a = a 2 b 2 bc a+b 1) a+b+c 2) 0 3) ab + bc+ ca 4) 1/abc.(ab + bc+ca)

10 Multiply and divide 1 by a, 2 by b, 3 by c = =

11 If A = B = Then AB = ) 175x10 4 2) 175x10 6 3) 175x10 3 4) 0

12 = =2500 7= = = = h = = =

13 7. In a ABC, 1 Sin A Sin 2 A 1 SinB Sin 2 B = 0 1 SinC Sin 2 C 1) Right angled 2)Right angled isosceles 3) Isosceles 4) Equilateral

14 = = =0 = = = = 3

15 8. If α = 1 x yz and β = 1 x x 2 1 y zx 1 y y 2 Then 1 z xy 1 z z 2 1) α β 2) α=β 3) α=2β 4) α=-β

16 = = 2 1 h = 1 2 = 1 2 2

17 9. 1/a a 2 bc 1/b b 2 ca 1) 0 2) 1 3) -1 4) abc 1/c c 2 ab

18 =

19 10. Let 6i -3i 1 4 3i -1 = x + iy then (x, y) = 20 3 i 1) (0, 1) 2) (0, 0) 3) (1, 0) 4) (1, 1)

20 =0 h 3 20 x+iy = 0+i0 then (x,y) = (0,0) 2

21 If A = then (A-2I)(A-3I) = ) A 2) I 3) 0 4) 5I

23 12. Let W = - ½ + i 3/2 Then = w 2 w 2 = 1 w 2 w 4 1) 3w 2) 3w (w-1) 3) 3w 2 4) 3w (1-w)

24 = 1( 2 - ) 1( 2 + 1( = = =3 1 2

25 13. If a a 2 1+a 3 b b 2 1+b 3 = 0 and a,b,c are c c 2 1+c 3 distinct, then product abc = 1) 2 2) -1 3) 1 4) 0

26 = (1+abc)[(a-b)(b-c)(c-a)] = 0 = 1 h,, 2

27 14. If A = a a 0 then adja = 0 0 a 1) a 3 2) a 6 a 9 4) a 27

28 = 1 = 3 = 3 1 = 2 = 3 2 = 6 2

29 15. 0 c b 2 b 2 +c 2 ab ac c 0 a = ab c 2 +a 2 bc b a 0 ac bc a 2 +b 2 1) 4abc 2) 4a 2 b 2 c 2 3) a 2 b 2 c 2 4) 0

31 16. 4Sin 2 θ Cos 2θ -Cos2θ Cos 2 θ = 1) -1 2) 0 3) 1 4) Cos4θ

33 17. The roots of the equation 2+x x -4 = x 1) 0, 1 2) -2 3) 0, -1 4) -20

34 We know that + + = Then = =0 =0, = 1 3

35 = ) 2 2) -2 3) 20 4) -20

36 2 1

37 19.If a b then a + b + x + y = + = x 3 y ) 5 2) 20 3) -10 4) 0

38 = =2 +2=1 3 =1=> =1 = 1 a = 4 b = 6 Then a+b+x+y = 10 4

39 i+w 2 w 2 1-i -1 w 2-1 = w 1, w 3 =1 is -i -i+w-1-1 1) 1 2) -1 3) 6 4) None

40 = = =0 h 2 4

41 Cofactor of 200 in is ) 9 2) -9 3) 6 4) -6

42 => = =

43 A = and adj A = then (x, y, z ) = x y z ) (-1, 0-1) 2) (-1, 0, 1) 3) (0, 1, -1) 4) (-1, -1, 1)

44 = h., = 1,0,1 = 1 0 = 1 =+ 2 2 = 0 = 0 1 = 1 2

45 λ If the matrix is invertible then λ = ) -15 2) -16 3) -17 4) 17

46 = = 0 3 =51. = 17 3

47 24. If the matrix AB = 4 11 and A = Then B = 1) -6 2) -11 3) -7/2 4) 4

48 h = = = =4 = 6 1

49 25. If the three linear equations x+4ay+az=0, x+3by+bz=0 and x+2cy+cz=0 have a non-trivial solutions, then a,b,c are in 1) A.P. 2) G.P. 3) H.P 4) none

50 By Property = (3bc-2bc)-4a(c-b)+a(2c-3b) = 0 bc-4ca-4ab+2ac-3ab=0 There fore b = 2 + hence a,b,c are in HP 3

51 26. The Value of λ for which the following system of equations does not have a solution x + y + z = 6, 4x + λy + λz = 0 3x + 2y 4z = -8 1) 3 2) -3 3) 0 4) 4

52 By properties MATHEMATICS =0 1(-4-2 ) -1(-16-3 ) + 1(8-3 ) = 0-6 =-24. =4 4

53 27. If a 1, a 2. Form a G.P. a i > 0, i 1 loga m loga m+1 loga m+2 Then loga m+3 loga m+4 loga m+5 loga m+6 loga m+7 loga m+8 1) 2 2) 1 3) 0 4) -2

54 a,b,c are in GP. Then 2 = ac similarly, 2 +1 = +2 Log on both sides 2Log +1 = Log +4 = Log +7 = ½ = ½ (0) = 0 3

55 28. The Value of 1 log x y log x z log y x 1 log y z log z x log z y 1 1) 1 2) xyz 3) log xyz 4) 0

56 = = 1]

57 29. If 1 1+x 2+x 8 2+x 4+x is a singular matrix then x is 27 3+x 6+x 1) 2 2) -1 3) 1 4) 0

58 = = h = x[1(3-2)-1(24-14)+1(8-7)] = 0 x = 0 4

59 30. If = logx logy logz log2x log2y log2z log3x log3y log3z 1) 0 2) log (xyz) 3) log (6xyz) 4)6 log (xyz)

60 = log log => 1 2 log 2 3 log 2+ log log 3+ = 0 1

61 1. If A = 4-3 then eigen value of A -1 = 2-1 1) 1,2 2) 1, ½ 3) -1,-2 4) -1, -1/2

62 A 1 = Adj A A MATHEMATICS = Eigen Values = 1 2 A λi = λ λ =0 1 λ λ 2 λ2 3λ +2 =0 λ =1,2 Answer is (1)

63 32. If 1+a b 1 = 0 and abc 0, then a -1 +b -1 +c -1 = c 1) 0 2) 1 3) -1 4) abc

64 abc+bc+ca+ab=0 abc = 0 then abc 0 then =0 1 + b 1 + c 1 = 1 Answer is 3

65 33. A root of 0 x-a x-b x+a 0 x-c = 0, then x is x+b x+c 0 1) a 2) b 3) c 4) 0

66 put x=0 then determinent=0 0 0 = a[0 +bc]-b[ac-0] = 0 abc-abc = 0 h 4

67 34. If A is matrix of order 3 such that A. adj A = 10 I Then adj A = 1) 10 2) 100 3) ) none

68 n 1 =

69 35. If A= Cosπ/4 Sinπ/12 Then A -1 = Sinπ/4 Cosπ/12 1) 8 2) 4 3) 2 4) -2

70 A = Cos π 4.Cos π 12 Sinπ 4.Sin π 12 = Cos =Cosπ = A 1 = A 1 1 = A = =2 3

71 36. If x+3 x x+2 x x+1 x-1 = ax 3 +bx 2 +cx+d then the value of d = x+2 2x 3x+1 1) 1 2) 0 3)-1 4)2

72 3 0 2 put x=0 then d= = 3[1-0] 0 + 2[0-2] = 3 4 = -1 3

73 36. Which of the following is not invertible? 1) 1-1 2) 2-2 3) )

75 38. The sum of the products of the elements of any row (or col) of A with the corresponding co-factors of the same row ( or col ) is always equal to 1)0 2) A 3) A 4) None

76 If A = adja = By property A = a 11 A 11 + a 12 A 12 +a 13 A 13 and so on 3

77 39. The sum of the products of the elements of any row (or column) of A with the corresponding cofactors of any other row ( or column) is always equal to 1) 0 2) A 3) A 4) None

78 If A = adja = By property a 11 A 21 + a 12 A 22 +a 13 A 23 =0 and so on 3

79 40. x x 3 = 0 then x is 3 3 x 1) 3,4,7 2) 3, 4,-7 3) -3,4,7 4) 0

80 We know that = = = 0 X=4,3,-7 2

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