Goa Board Class 12 Question Paper June 2018 Mathematics _Old Pattern_

2018 VI 12 14301000 Seat No. :

Time : 2½ Hours mathematics (Old Pattern)

Subject Code

H 7 5 4
Total No. of Questions : 7 (Printed Pages : 5) Maximum Marks : 80

Instructions : i) The question paper contains 7 main questions.
ii) All questions are compulsory.
iii) Answer each question on a fresh page.
iv) Use of calculator is not allowed.
v) Log tables will be supplied on request.
vi) Graph should be drawn on answer paper only.
vii) For each main question, the sub-questions will carry the
following marks :
A = 1 mark; B = 2 marks; C = 3 marks; D = 4 marks and
E = 5 marks.
1. A) Select and write the correct alternative from those given below :
x
If [4 x ]   = [9] , then the value of x is
 −1
• 3
• –3
• –1
• 4
B) Using determinants, find the area of the triangle whose vertices are
(–2, 1), (–3, –5) and (2, 4).
–1 d2 y dy
C) If y = easin x, prove that (1 – x2) 2
−x − a2 y = 0 .
dx dx
D) Using integration, prove that
x 2 a2
∫ x 2 − a 2 dx = x − a2 − log x + x 2 − a 2 + c .
2 2

H-754 -1- P.T.O.

2. A) Select and write the correct alternative from those given below :
On R, the set of real numbers, a binary operation ∗ is defined by
ab
a∗b= ∀a, b ∈ IR.
3
If 2∗(x ∗ 3) = 4, then the value of x is
•   – 4
•    4
•   – 6
•    6

B) Prove that 2 tan −1 1 + tan −1 1 = tan −1 31 .
2 7 7
C) Using properties of determinants, prove that :

1 a bc
1 b ca = (a − b) (b − c ) (c − a) .
1 c ab

1 2 −3 
D) Find inverse of matrix A = 2 3 2  using the adjoint method. Hence
solve the equations : 3 −3 −4 
x + 2y – 3z = –4
2x + 3y + 2z = 2
3x – 3y – 4z = 11

dy
3. A) If x2 + 2y = 3, find at x = 2.
dx

B) Using derivatives, find the approximate value of 36.6 .

b b
C) Prove that ∫ f(x )dx = ∫ f(a + b − x )dx .
a a

H-754 -2-

D) Attempt any one of the following :

i) The function defined below is continuous on its domain. Find the values
of constants ‘a’ and ‘b’.

2 sin x − sin 2x
f( x ) = ;−π≤x <0
x3
π
= a sin x + b cos x ; 0 ≤ x ≤
2
3 cos x + cos 3x π
= ; <x≤π
(π − 2x )3 2

ii) Find the values of constants A and B if the function f(x) defined below is
π
continuous at x = .
2

1 − sin3 x π
f( x ) = 2
;x<
cos x 2
π
=A ;x=
2
B(1 − sin x ) π
= ;x>
(π − 2x )2 2

4. A) Find the direction cosines of the line passing through the points (1, 1, 1) and
(2, 3, 3).

B) Solve 2tan–1 (cosx) = tan–1 (2 cosecx).

C) If ∗ is a binary operation on the set N of natural numbers given by

a+b
a∗b= ∀ a, b ∈ IN, verify whether ∗ is (i) commutative
2
(ii) associative.

H-754 -3- P.T.O.

D) Solve the following linear programming problem graphically :
Minimize z = 6x + 3y subject to the constraints
4x + y ≥ 80
x + 5y ≥ 115
3x + 2y ≤ 150 ; x ≥ 0 ; y ≥ 0.

5. A) Select and write the correct alternative from those given below :

A B
If A and B are events such that P   = P   , then
B A
•   A ⊂ B but A ≠ B
•   A = B
•   A ∩ B = φ
•    P(A) = P(B)

dy
B) Find if y = (tanx)sinx.
dx
dy
C) Find the particular solution of the differential equation +2
dx
π
ytanx = sinx, given that y = 0 when x = .
3
D) A pair of dice is thrown thrice. Getting a sum of numbers on the upper most
faces of the two dice as 6 or 9 is considered as success. Find the probability
distribution of the number of successes.
 
6. A) The vectors a = ˆi − λˆj + 2kˆ and b = 8iˆ − 6ˆj − kˆ are at right angles. Find the
value of λ.
π
4
cos x
B) Evaluate
∫ (1 + sin x)3 dx .
0
 
C) Find the projection of 2 AC on BD if A = (1, 2, 3), B = (2, 1, 0), C = (3, 2, 1)
and D = (2, –1, 2).

D) A company manufactures cat food and wishes to pack the food in closed
cylindrical tins. What should be the dimensions of each tin, if each tin is to
have volume of 128 cu. cms and minimum surface area ?

H-754 -4-

E) Attempt any one of the following :
 ˆ + λ(iˆ − 3ˆj − k)
ˆ and
i) Show that the lines r = (2iˆ + 4ˆj + 3k)
 ˆ + µ(iˆ + ˆj + k)
ˆ intersect. Hence find the point of
r = (5iˆ + 3ˆj + 4k)
intersection.
ii) Show that the points 2iˆ + 5ˆj − 3kˆ and ˆi + 3ˆj + 3kˆ are equidistant from
 ˆ = 4 . write the cartesian and vector equation of
the plane r.(6iˆ + 3ˆj + 2k)
the line containing these two points. Find the angle between this line and
the given plane.

7. A) Select and write the correct alternative from those given below :
The probability that a number selected from 1, 2, 3, 4, ..., 20 is a prime
number if each of the 20 numbers is equally likely is
2
•
5
3
•
5
4
•
5
1
•
5
       
B) If a = ˆi + 4kˆ ; b = 2iˆ + 2kˆ ; c = 4iˆ + ˆj + 2kˆ and d = ˆi − ˆj , find (a + c).(b × d) .
C) At any point (x, y) of a curve, the slope of the tangent is twice the slope of
the line segment joining the point of contact to the point (– 4, – 3). Find the
equation of the curve given that it passes through (– 2, 1).

2x + 3
D) Find ∫ dx .
(x + 1) (x 2 + 4)
E) Attempt any one of the following :
i) Using integration, find the area of the triangle with vertices A(–1, 3), B(0, 6)
and C(3, 1).
ii) Find the area bounded by the curve y2 = 16x and chord BC where
B = (1, 4) and C = (9, 12) using integration.
_______________
H-754 -5-