Paper 1
Section A
41. State and prove reversal law of inverse
1 2
42. Find the adjoint of the matrix A = (
)and verify the result A (adj A) = (adj A)A = | A | . I
3 −5
⃗ ⃗b +c c +a⃗ ] =2[ a⃗ ⃗b c ]
43. Prove that [a⃗ +b
⃗
44.a) Find the angle between the vectors 𝑎 and 𝑏⃗ where 𝑎 = 𝑖 − 𝑗 and 𝑏⃗ = 𝑗 − 𝑘
⃗)𝑘
⃗
b) For any vector 𝑟 prove that 𝑟 =(𝑟. 𝑖) 𝑖 +(𝑟. 𝑗) 𝑗 +(𝑟. 𝑘
45. Find the real values of x and y for which the for ( 1 − i)x + (1 + i)y = 1 − 3i
46. Solve the equation x4 − 8x3 + 24x2 − 32x + 20 = 0 if 3 + i is a root.
47. Find the equation of the tangent and normal to the parabola y2=6x, parallel to the line 3x – 2y =5
48. At what θ do the curve y=ax and y=bx intersect (a≠b)
𝑙𝑖𝑚 x
49. Evaluate 𝑥→0
x
50. Use differentials to find an approximate value for the given number √36.1
51. The area of the region bounded by the curve xy = 1, x-axis, x = 1. Find the volume of the solid generated by
revolving the area mentioned about x-axis.
52. Solve (x2 +5x +7)dy + √9 + 8y − y 2 dx = 0
53.Prove that (C,+) is an infinite abelian group.
54. Find the probability distribution of the number of sixes in throwing three dice once.
55. In a hurdle race a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6 What
is the probability that he will knock down less than 2 hurdles.
Section c
2
2 1
56. If A = 3 [−2 1 2] prove that A-1 = A T
1 −2 2
1
57. Find the vector and cartesian equations of the plane through the point (2, - 1, - 3) and parallel to the lines
𝑥−2
𝑦−1
𝑧−3
𝑥−1
𝑦+1
𝑧−2
=
=
and
=
=
3
2
−4
2
3
2
⃗ ,2𝑖 − 2 𝑗 − 𝑘
⃗ ,7 𝑖 + 𝑘
⃗
58. Find the vector and cartesian equations of the plane passing throuqh 3𝑖 + 4 𝑗 + 2𝑘
𝑍+1
59. P represents the variable complex number z, find the locus of P if Re( 𝑍−𝑖 )=1
60. Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the
parabolas and hence draw their graphs.y2-8x+6y+9=0
61. The girder of a railway bridge is in the parabolic form with span 100 ft. and the highest point on the arch is
10 ft. above the bridge. Find the height of the bridge at 10 ft. to the left or right from the midpoint of the bridge.
62. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line x + 2y − 5 = 0
and passes through the points (6, 0) and (− 3, 0).
63. A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If
water is being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when
the water is 3m deep.
64. A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight river. He
needs no fence along the river.What are the dimensions of the field that has the largest area ?
65.Trace the curve y=x3
66.Find the area bounded by the curve y= x3 and the line y =x.
67. Find the perimeter of the circle with radius a.
68.Solve (D2 − 6D + 9) y = x + e2x
69. In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity
of the substance still untransformed at that instant. At the end of one hour, 60 grams remain and at the end of 4
hours 21 grams. How many grams of the substance was there initially?
70.a) Show that the set G of all rational numbers except − 1 forms an abelian group with respect to the
operation * given by a * b = a + b + ab for all a,
b) The probability density function of a random variable x is f(x)={𝑘𝑥
0
(ii) P(X > 10)
𝛼−1 −𝛽𝑥 𝛼
𝑒
𝑥,𝛼,𝛽>0
, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Find (i) k