RAM MATHS YOUTUBE & TUITION CENTrE, Coimbatore – 4.

CLASS : XII MATHEMATICS MAX.MARKS : 90

DATE : FULL TEST – 3 DURATION : 3.00 HRS

SECTION – A
CHOOSE THE BEST OPTION FROM THE GIVEN FOUR ALTERNATIVES : 20 X 1 = 20
1. If z represents a complex number then arg (z) + arg (z ) is
(1) /4 (2) /2 (3) 0 (4) /6
1 −2 6 0
2. If A[ ]=[ ], then A =
1 4 0 6
1 −2 1 2 4 −1 4 2
(1) [ ] (2) [ ] (3) [ ] (4) [ ]
1 4 −1 4 2 1 −1 1
 /4

3. The value of 
0
cos3 2x dx is

(1) 2/3 (2) 1/3 (3) 0 (4) 2/3

4. If z is a non zero complex number, such that 2iz2 = z̅ then |z| is

1
(1) (2) 1 (3) 2 (4) 3
2

5. A polynomials equation in x of degree n always has

(1) n distinct roots (2) n real roots (3) n complex roots (4) at most one root
2π
6. If sin−1 x + sin−1 y = ; then cos −1 x + cos −1 y is equal to
3
2π π π
(1) (2) (3) (4) 
3 3 6

7. The directrix of the parabola y2 = x + 4 is

(1) x = 15/4 (2) x = –15/4 (3) x = – 17/4 (4) x = 17/4
8. The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x – 4y = m at two distinct points if
(1) 15 < m < 65 (2) 35 < m < 85 (3) – 85 < m < – 35 (4) – 35 < m < 15

9. If u = f(y/x) then x u/x + y u/y is equal to

(1) 0 (2) 1 (3) 2u (4) u
10. If a vector ⃗ lies in the plane of ⃗ and  , then
(1) [ ⃗ , ⃗ ,  ] = 1 (2) [⃗ , ⃗ ,  ] = – 1 (3) [⃗ , ⃗ ,  ] = 0 (4) [⃗ , ⃗ ,  ] = 2
A 1
11. If f ( x)     x   is a p.d.f of a continuous random variable X, then the value of A
 16  x 2
is (1) 16 (2) 8 (3) 4 (4) 1
12. A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the
balloon left the ground. The rate of change of the balloon’s angle of elevation in radian per
second when the balloon is 30 metres above the ground.
3 4 1 1
(1) radians/sec (2) radians/sec (3) radians/sec (4) radians/sec
25 25 5 3

13. The approximate change in the volume V of a cube of side x metres caused by increasing the
side by 1% is
(1) 0.3xdx m3 (2) 0.03x m3 (3) 0.03x2 m3 (4) 0.03x3 m3
π
14. The value of ∫06 cos3 3x dx is
2 2 1 1
(1) (2) (3) (4)
3 9 9 3

15. The differential equation representing the family of curves y = A cos (x + B), where A and B
are parameters, is
𝑑2𝑦 𝑑2𝑦 𝑑2𝑦 𝑑2𝑥
(1) 2
–y=0 (2) 2
+y=0 (3) 2
=0 (4) =0
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑦 2

16. The slope of the normal to the curve y = 3x2 at the point whose x coordinate is 2 is
(1) 1/13 (2) 1/14 (3) –1/12 (4) 1/12
17. If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the
probability of success is
(1) 0.125 (2) 0.25 (3) 0.375 (4) 0.75
18. The proposition p  ( p  q) is
(1) a tautology (2) a contradiction
(3) logically equivalent to p  q (4) logically equivalent to p  q
19. If A = (2 0 1) then the rank of AAT is (1) 1 (2) 2 (3) 3 (4) 4
20. a and b are two unit vectors and  is the angle between them. Then ( a + b ) is a unit vector
if (1)  = /3 (2)  = /4 (3)  = /2 (4)  = 2/3

SECTION – B

ANSWER ANY 7 [ Q.No.30 is compulsory ] : 7 X 2 = 14

−1 3
21. Find the rank by minor method [ 4 −7]
3 −4
3+4𝑖
22. Write in the x + iy form, hence find its real and imaginary parts.
5−12𝑖
1
23. Find the period and amplitude of y = – sin( 𝑥)
3

24. If y = 2√2 x+ c is a tangent to the circle x2 + y2 = 16, find the value of c.

25. The volume of the parallelepiped whose coterminus edges are 7 𝑖̂ + 𝑗̂ − 3𝑘̂ ,
𝑖̂ + 2𝑗̂ − 𝑘̂ , −3𝑖̂ + 7𝑗̂ + 5𝑘̂ is 90 cubic units. Find the value of  .
 y2 −2z2
26. Determine whether U(x, y, z) = xy + sin ( ).is homogeneous or not. If it is so, find the
xy
degree.
1
27. Evaluate ∫0 𝑥 3 (1 − 𝑥)4 𝑑𝑥
28. Find the differential equation corresponding to the family of curves represented by the
equation y = Ae8x + Be–8x, where A and B are arbitrary constants.
1 0 1 0 0 1 0 1 1 1 0 1
29. Let A = (0 1 0 1), B = (1 0 1 0), C = (0 1 1 0) Find (A  B)  C.
1 0 0 1 1 0 0 1 1 1 1 1
30. If X is the random variable with distribution function F(x) given by
0 −<𝑥 <0
1 2
𝐹 (𝑥) = { 2 (𝑥 + 𝑥) 0 ≤ 𝑥 < 1 then find (i) the probability density function f(x)
1 1≤𝑥<
(ii) P(0.3  X  0.6).

SECTION – C

ANSWER ANY 7 [ Q.No.40 is compulsory ] 7 X 3 = 21

2 −4 2
31. If adjA = [−3 12 −7], find A.
−2 0 2
32. Find the values of the real numbers x and y, if the complex numbers (3 – i)x – (2 – i)y + 2i + 5
and 2x + (– 1 + 2i)y + 3 + 2i are equal.
33. Find the sum of the squares of roots of the equation 2x4 – 8x3 + 6x2 – 3 = 0
34. Is cos–1(– x) =  – cos–1(x) true?
35. Find the angle between the straight line𝑟 = (2𝑖̂ + 3𝑗̂ + 𝑘̂ ) + t(𝑖̂ − 𝑗̂ + 𝑘̂ ) and the plane
2x – y + z = 5.
36. Prove that the function f(x) = x2 + 2 is strictly increasing in the interval (2, 7) and strictly
decreasing in the interval (–2, 0).
37. A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find
approximations for the following: (i) change in the volume (ii) change in the surface area
38. In a pack of 52 playing card, two cards are drawn at random simultaneously. If the number of
black cards drawn is a random variable, find the values of the random variable and number of
points in its inverse images.
−7
39. Let  be defined on R by (a  b) = a + b + ab – 7. Is  binary on R? If so, find 3  ( ).
15

40. The slope of the tangent to the curve at any point is the reciprocal of four times the ordinate
at that point. The curve passes through (2, 5). Find the equation of the curve.
 SECTION – D
ANSWER ALL : 7 X 5 = 35
41. (a) A fish tank can be filled in 10 minutes using both pumps A & B simultaneously. However,
pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse,
then the tank will be filled in 30 minutes. How long would it take each pump to fill the tank by
itself? ( Use Cramer’s rule to solve the problem). [OR]
(b) A particle moves along a line according to the law s(t) = 2t3 – 9t2 + 12t – 4, where t  0.
(i) At what times the particle changes direction? (ii) Find the total distance travelled by the
particle in the first 4 seconds. (iii) Find the particle’s acceleration each time the velocity is
zero.

42. (a) Show that the equation z3 +2𝑧̅ = 0 has five solutions. [OR]
𝑦 𝑦 𝑦
(b) Solve (1 + 3𝑒 𝑥 ) dy + 3 𝑒 𝑥 (1 − ) dx = 0, given that y = 0 when x = 1.
𝑥

43. (a) Determine k and solve the equation 2x3 – 6x2 + 3x + k = 0 if one of its roots is twice the
sum of the other roots. [OR]
𝑥
(b) For f(x, y) = tan–1 , find the fx, fy and show that fxy = fyx.
y

44. (a) Find the number of solutions of the equation

tan (x – 1) + tan x + tan (x + 1) = tan (3x)
–1 –1 –1 –1 [OR]
(b) Determine the intervals of concavity of the curve f(x) = (x – 1)3.(x – 5) , x  R and, points
of inflection if any.

45. (a) Identify the type of conic and find centre, foci, vertices and directrices of
9x2 – 36x – y2 – 6y + 18 = 0 [OR]
(b) Find the parametric vector , non – parametric vector and Cartesian form of the equations
of the plane passing through the three non–collinear points (3, 6, –2), (–1, –2, 6) and
(6, 4, –2).

3π⁄ 1
8
46. (a) Evaluate ∫π⁄ dx [OR]
8 1+ √tan x

(b) Find the population of a city at any time t, given that the rate of increase of population is
proportional to the population at that instant and that in a period of 40 years the population
increased from 3,00,000 to 4,00,000.

47. (a) The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find
(i) the probability mass function (ii) P(X = 3) (iii) P(X  2). [OR]
(b) Prove that p  ( q  r)   p  ( q  r) using truth table.