Bangalore University 2010-2nd Year M.C.A Entrance -Year -Maths-Practice -4 - Question Paper
Total Questions: 75 Time allotted 90 minutes
1. If a = 2i + 2j + 3k,b = -i + 2j + k and c = 3i + j, then a + tb is perpendicular to c, if t is equal to
(a) 8 (b) 4
(c) 6 (d) 2
2. The vectors i + 2j + 31c,i + 4j + 71c,-3i-2j-51c are collinear if =
(a) 3 (b) 4
(c) 5 (d) 16
3. If b = 2i + j -k,c = i + 3k and a is a unit vector then the Maximum value of [a,b,c] is
(a) V59 (b) >/69
(c) 3 (d) None of these
4. The equation of the sphere passing through the point (1, 3, -2) and the circle y2 + z2 = 25 and x =
0 is
(a) x2 + y2 + z2 + 11x + 25 = 0
(b) x2 + y2 + z2 - 11x + 25 = 0
(c) x2 + y2 + z2 + 11x - 25 = 0
(d) x2 + y2 + z2 - 11x - 25 = 0
(12 4
5. If a plane meets the coordinate axes at A, B, C such that centroid of triangle ABC is I 3,3,3
then equation of plane is
(a) 4x + 2y + z = 4 (b) 4x + 2y + z = 3
(c) x + y + z = 6 (d) None of these
6. The foot of perpendicular from (0, 2, 3) to the line x + 3 = y = z + 4 is
(a) (-2, 3, 4) (b) (2, -1, 3)
(c) (2, 3, -1) (d) (3, 2, -1)
7. The AM of nC0,nCj,nC2,____nCnis
n n
(a)---(b)
n +1 n
2n-1
(c)--(d) None of these
n +1
8. The mean of the values 0, 1, 2,____n, with the corresponding weights nC0,nC1,____nCn
respectively is
2n 2n+1 (a)---(b)
9. In a base containing 100 bulbs, 10 bulbs are defective, the probability that out of a sample of 5 bulbs, none is defective is
(a) 10-6 (b) 2-5
(c) (0.9)5 (d) 0.9
10. Two dice are thrown, the probability that the sum of the points on the dice is 7, is (a) 5/36 (b) 6/36
(c) 7/36 (d) 8/36
11. If forces P, Q, R acting at a point can be represented by the sides of a A, taken in order, then,
(a) P + Q + R = 0 (b) P - Q + R = 0 (c) P + Q - R = 0 (d) P - Q - R = 0
12. The sum of the two forces is 18 and their resultant perpendicular to the lesser of the force is 12, then the lesser force is
(a) 5 (b) 3
(c) 7 (d) 15
13. P and Q are like parallel forces. If P is moved parallel to itself through a distance x, then the resultant of P and Q moves through a distance
(a) A(b)-
(c)--(d) None of these
14. A man is running with a velocity of 5 km/hr in a shower of rain which is descending with a velocity of 10 km/hr. If a be the angle which the apparent direction of the rain which the apparent direction of the rain makes with the vertical then value of tan a is
(a) 1/3 (b) 2/3
(c) / (d) None of these
15. A gun can fire with a velocity u in all directions from a given position on a horizontal plane. The shots will fall on the plane within a circle of radius
(a) f (b) g
2g g
2 2 (c) f (d) JL
16. An engine and train weights 420 tons and the engine exerts a force of 7 tons. If the resistance to motion be 14 lbs wt per ton, then the time, the train will take to acquire a velocity of 30m/hr from rest is
(a) 2.2 min (b) 2.6 min
(c) 2.8 min (d) 3 min
17. A light string passing over a light smooth Pulley Carries masses of 3kg and 5 kg at its ends. If the string breaks after the masses have moved 9m, how much further the 3 kg man will rise? (take g = 10m/sec)
(a) 1.75 m (b) 1.95 m
(i - i yo
18. 11i I is equal to
(a) 1 (b) -1/2
(c) (d) -1
19. The unit places digit in the number 13 25 +1 125 - 3 25 is:
(a) 0 (b) 1
(c) 2 (d) 3
20. Number lying between 999 and 10000 that can be formed from the digits 0,2, 3, 6, 7, 8 (repetition of digits not allowed) are
(a) 100 (b) 200
(c) 300 (d) 400
21. If 310 + x = 123456798, then
(a) 2 is factor of x (b) 3 is a factor of x
(c) 6 is a factor of x (d) x has no factor, a sit is prime number.
22. The least common multiple of 6ab2 (a+b)2 and 4a2 b(a2 - b2) is
(a) 24a3b3 (a+b)2 (a2 - b2)
(b) 12a2b2 (a+b)2 (a2 -b2)
(c) 4ab (a + b )(a - b)
(d) 2ab (a+b)
23. The value of the recurring 0.49 is -
(a) 49/990 (b) 490/99
(c) 49/99 (d) None of these
Directions (Q. 24-25) The following items consists of two statements, one labeled as the Assertion (A) and the other as Reason (R). You are to examine these two statements carefully and select the answer to these items using the codes given below :
Codes:
(a) Both A and R are individually true and R is the correct explanations of A.
(b) Both A and R are individually true, and R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
24. Assertion (A):
(cos 9 + isin )3 = cos39 + isin3
Reason (R) :
25. Assertion (A):
In the equation ax2 + 3x + 5 = 0 if one root is reciprocal of the other then a is equal to 5. Reason (R):
Product of the roots is 1.
26. The roots of the equation 2x+2.3x-1 = 9 are given by (a) log2 | -2 j, -2 (b) 3, -3
(c) -2,1 - -log3 (d) 1 - log2 3,2
log2 2
a 2
27. If a and p are the roots of the equation x2 - Px + q = 0, then the equation whose roots are and
P2
is
a
(a) qx2 +(P3 - 3Pq)x + q2 = 0
(b) qx2-(3 + 3Pq)x + q2 = 0
(c) qx2 -(3 - 3Pq)x + q2 = 0
(d) None of these
28. If one root of x2 + Px + q = 0 is 5 - 3i, then the real values of P and q are (a) P = -10,q = -34 (b) P = -10,q = 34
(c) P = 10, q = -34 (d) None of these
29. The positive integer r, such that 15C3r = 12Cr+3 is equal to (a) 3 (b) 4
(c) 5 (d) None of these
30. The sum of the digits in the unit place of all the numbers formed with the help of 3, 4, 5, 6 taken all at a time
(a) 432 (b) 108
(c) 36 (d) 18
31. 4 letter lock consists of three rings each marked with 10 different letters, the number of ways in which it is possible to make an unsuccessful attempt to open the lock, is
(a) 899 (b) 999
(c) 479 (d) None of these
32. The 13th term of I 9x--is
3yfxy
(a) 17682 (b) 18564
(c) 18564 x6 (d) None of these
33. If (1 + x-2x2) = 1 + ajx + a2x2 +___+ a12x12, then a2 + a4 + a6 +___+ a12 =
(a) 21 (b) 16
(c) 10 (d) None of these
34. If the coefficients of (2r + 1)th term and (r + 2)th term in the expansions of (1 + x)43 are equal, then r =
35.
is
The power of x occurring in the 7 term in the expansion of
(b) 5 (d) 3
(a) -5 (c) -3
36.
is
3!
. 1 + x + (1 + x) + (1 + x)
The coefficient of xn in the series
1! 2!
2e
n!
4e
n!
(b)
(a)
(d) None of these
The sum of the series log4 2 - log8 2 + log16 2___x is
(a) e2
(c) loge 2 - 1
. 1 2)
If A = | 2 1 I, then adj A is equal
(c) n
n!
37.
(b) loge 2 + 1 (d) 1 - loge 2
38.
-1 2 2 -1 2 1 1 1
1 -2 -2 1
1 -2 -2 1
(b)
(d)
(a)
(c)
cos a sin a - sin a cos a sin 2a cos 2a cos 2a - sin 2a cos 2a sin 2a - sin 2a cos 2a
then A2 is equal to (b)
If A = (a)
(c)
39.
cos 2a - sin a - sin a cos 2a 1 0
(d)
0 1
40.
If the system of equations Xx + 2y - 2z = 1,4x + 2Xy - z = 2,6x + 6y + Xz = 3 has a unique solution, then
(a) X * 1 (b) X * 2
(c) X * 3 (d) None of these
1 1 + ac 1+bc
A = 1 1 + ad 1+bd
1 1 + ae 1+be
(a) a + b + c (c) 1
41.
(b) 3 (d) 0
42. Let A {a,b, c}, B = {b, c, d}, C = {a,b, d, e} , then A n (B u C) (a) c (b) {a,b,c}
(c) {b,c,d} (d) [a,b,d,e]
43. If a = (2, 1, - 1), b = (1, -1, 0) and c = (5, -1, 1), then the unit vector parallel to a + b - c, but in the
opposite direction is
-i i (a) (2i - j + 2k) (b) 3 (2i - j + 2k)
(c) 3 (i + j - 2k) (d) None of these
44. Given two vectors i - j and i + 2j, the unit vector coplanar with the two vectors and perpendicular to first is
(c) (j + j) (d) None of these
45. The direction cosines of the line which is perpendicular to the line whose direction cosines are proportional to (1, -1, 2) and (2, 1, -1) are
1 -5 3 -1 5 3
(a) -=,=,= (b)
V35V35V35 V35V35V35
(c) (d) None of these
V35 v35 V35
x - y y - 7 z - 4
46. A plane which passes through the point (3, 2, 0) and the line 1 = 5 = ~ is
(a) x - y + z = 1 (b) x + y + z = 5
(c) x + 2y - z = 1 (d) 2x - y + z = 5
47. The probability of getting the sum as a prime number when two dice are through together is
17 (a) - (b)
2 12
(c) (d) None of these
12
48. If the mean of numbers 27, 31, 89, 107, 156 is 82, then the mean of 130, 126, 68, 50, 1 is (a) 80 (b) 82
(c) 157 (d) 75
49. Two forces act an angle of 1200. It the greater force is represented by 80 kg and the resultant is at right angles to the smaller then the smaller force is
(a) 10 kg (b) 15 kg
(c) 20 kg (d) 40 kg
50. A jet plane is rising vertically with a velocity of 10 m/s. It has reached a certain height when the pilot drops a coin, which makes 4 seconds to hit the ground. Assuming that there is no resistance to the motion of the coin, the height of the plane and the velocity of the coin on impact with the ground are
(a) 38.4m, 29.2m/sec (b) 38.4m, 28.7m/sec (c) 26.5m, 13.5m/sec (d) None of these
51. The term independent of x in the expansion of f x2 - 3-J is equal to
28
81
28
243
. n 28
(c)--
243
28
(d)--
81
52.
If the coefficients of x and x in | 2 + | are equal, then the value of n is:
(a) 56 (c) 64
(b) 63
(d) None of these
53. If A and B are coefficients of xn in the expansions of (1 + x )2n and (1 + x )2 n 1 respectively, then
(a) A = B (b) 2A = B
(c) A = 2B (d) None of these
The value of log13 + log is
13
54.
(b) zero
(d) None of these
(a) 13 (c) 1
If log (2 x - 3) + log (x +1) = 2log5, the value of x is (a) 3 (b) -7/
55.
-5/
(d)
(c)
'12
56. If log2 [log3 (log2 x) = 1, the x is equal
(a) 0 (c) 128
(b) 12 (d) 512
The value of log4128 is
(a) 2 (b) 3
57.
(d)
(c)
x x + y x + y + z
58. If 2x 3x + 2y 4x + 3y + 2z = 64,
3x 6 x + 3 y 10 x + 6 y + 3z
then the value of x is equal to (a) 6 (b) 4
(c) 3 (d) 2
log xy log xz
1
log yx 3 log zx log zy
log yz
5
59. For positive numbers x, y and z, the numerical value of the determinant:
is:
(a) 0 (c) 1
(b) log x, log y.log z (d) 8
For a matrix
(a) -2 (c) 2
60.
2 4
3 5
10
to be singular, X has the value:
(b) 4 (d) -4
The value of sin500 cos100 + cos500 cos8
is
61.
A
(b)
(a)
2
(c) sin400
(d) cos 400
If A + C = B, then tan A tan B tan C is equal to (a) tan B + tan A + tan C (b) tan B - tan A + tan C (c) tan B - tan A - tan C (d) tan B + tan A - tan C
62.
63.
The value of sin500 - sin700 + sin100 is (a) 1
(c) -1 (d) 0
If cos#-sin# = V2sin# ,then cos# + sin# is (a) V2cos# (b) V2sin#
(c) V2 (d) 1
In a triangle a = 13, b = 14, c = 15, then r is: (a) 4 (b) 8
(c) 2 (d) 6
64.
65.
66.
The eccentricity of the ellipse 5 x2 + 9 y2 = 45 is
(d)
(c)
The straight line y + x - k = 0 touches the curve y = x - x2 for k =
67.
68.
69.
(a) 0 (b) 1
(c) -1 (d) None of these
The ratio in which the yz - Plane divides the join of the points (-2,4,7) and (3, -5,8) is:
(a) 2 : 3 (b) 3 : 2
(c) -2 : 3 (d) 4 : -3
A (3, 2, 0) , B (5, 3, 2),C (-9, 6, -3) are the vertices of a triangle ABC. If the bisectors of ZABC meets BC at D, then the coordinates of D are
19 57 17 8 16 16
-19 57 17 8 16,16
(b)
(a)
19 57 -17
(c) 1 - 7!
(d) 119,.iZ
1 " 16 16
|
3r + 2 70. If f (x ) =-, then V 5 x - 3 (a) f-1 (x ) = f (x) (c) f -1 (x ) = - f (x) |
(b) (fof) x = - x (d) f -1 (x ) = f (x) |
71.
(b) 3
<d> 4
T x - 4x + 3 .
72. Lt - is:
x1 x + 2 x - 3
(a) 1 (c) - 3
x -tan2x .
73. u - is:
x1 x + 2 x - 3
1
(b) 2
(a) -
2
If y = loge (x - 2)2 forx 0, 2, then the value of y (3) is:
74.
2
(a)
3
(b)
3
1
(d) None of these
1 + sin x (a) 0
3
1. (a) 16. (b) 31. (b) 46. (a) 61. (a)
2. (a) 17. (d) 32. (b) 47. (c) 62. (c)
3. (a) 18. (a) 33. (c) 48. (d) 63. (d)
4. (c) 19. (b) 34. (a) 49. (d) 64. (a)
5. (a) 20. (c) 35. (c) 50. (a) 65. (a)
6. (d) 21. (b) 36. (c) 51. (b) 66. (d)
7. (b) 22. (b) 37. (d) 52. (b) 67. (b)
8. (d) 23. (c) 38. (d) 53. (c) 68. (a)
9. (c) 24. (a) 39. (c) 54. (b) 69. (a)
10. (b) 25. (d) 40. (b) 55. (b) 70. (a)
11. (a) 26. (c) 41. (d) 56. (d) 71. (c)
12. (a) 27. (c) 42. (b) 57. (d) 72. (b)
13. (c) 28. (b) 43. (b) 58. (b) 73. (c)
14. (c) 29. (a) 44. (a) 59. (d) 74. (d)
15. (c) 30. (b) 45. (b) 60. (b) 75. (c)
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