Goa Board Class 12 Question Paper June 2018 Mathematics _New Pattern_

2018 VI 12 1430 Seat No. :

Time : 2½ Hours Mathematics (New Pattern)

Subject Code

H 7 5 4

Total No. of Questions : 30 (Printed Pages : 7) Maximum Marks : 80

Instructions : 1) All questions are compulsory.
2) The question paper consists of 30 questions divided into
five Sections A, B, C, D and E.
3) Section A contains 7 questions of 1 mark each which
are multiple choice type questions. Section B contains
7 questions of 2 marks each, Section C contains 7
questions of 3 marks each, Section D contains 7 questions
of 4 marks each and Section E contains 2 questions of 5
marks each.
4) There is no overall choice in the paper. However internal
choice is provided in 2 questions of 3 marks each,
2 questions of 4 marks each and 2 questions of 5 marks
each. In questions with choices, only one of the choices
is to be attempted.
5) Use of calculators is not permitted.

SECTION – A

Question numbers 1 to 7 carry 1 mark each. In each question, four options are
provided, out of which one is correct. Write the correct option.

dy
1. If xex = y + sinx then at x = 0 is
dx
• 0

• 1

• e

• –1

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

2. If A is a square matrix of order 3, such that |A| = –1 then |3A| is
• 3
• –3
• 27
• –27

3. The angle between the line r = a + λb and the plane r . n = d is

 b . n
• cos −1  
 b n 

−1
 a . n
• cos  
 a n

−1
 b . n
• sin  
 b n

−1
 a . n
• sin  
 b n

π
sec 2 x
4
4. The value of ∫ dx is
0
cosec 2 x
π
• 1 −
4
π
• 1+
4
π
• − 1
4
 π
• − 1 + 
 4

H-754 -2-

5. If A and B are two events such that P A ( B) = P (B A) then
• A⊂ B
• P(A) = P(B)
• B=A
• A ∩ B=φ

6. Equation of the normal to the curve y = sinx at the origin is
• x=0
• y=0
• x+y=0
• x–y=0

π
7. If sin–1x – cos-1x = − 2 then the value of x is
2
• sin 1 – cos 1
• cos 1
• cos1 – sin1
• –sin1

SECTION – B

Question numbers 8 to 14 carry 2 marks each.

 2 −2 
8. If A =  then find the value of (A + I2) (A – 4I2) where I2 – identity
 −3 1
matrix of order 2.

9. Find fοg(2) and gοf(–1), if f(x) = 2x + 1 and g(x) = x3 – 1.

10. If y = tan–1x, then prove that
d2 y
(1 + x2 ) dx 2
+ 2x
dy
dx
= 0.

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

11. The binary operation ∗ on N, then set of natural numbers is given by a ∗ b = lcm
(a, b). Find the identity element of (N, ∗) and which element of N is invertible.

12. If A and B are any two events of a sample space S and F is an element of S such
( ) ( )
that P(F) ≠ 0 then prove that P ( A ∪ B ) F  = P A + P B − P ( A ∩ B ) F  .
F F

13. For any three vectors a, b, c , find a b+c a + b + c  .

14. Find the perpendicular distance of the plane from the origin. Given that the
plane intercepts on the co-ordinate axes are 8, 4 and 4 respectively.

SECTION – C

Question numbers 15 to 21 carry 3 marks each.

 x2 − y2  dy x (1 − tan α )
15. If tan−1  2 2
= a , prove that = .
x + y  dx y (1 + tan α )

−1  3 sin 2α  1  π π
16. Evaluate tan   + tan−1  tan α  , − < α < .
 5 ± 3 cos 2α  4  2 2

( ) ( )
17. Find the value of 2a − 5b . 3a + b if a + b = 3 and a = 1, b = 1.

18. Form the differential equation of the following family of curves by eliminating
the arbitrary constants a, b and c :
y = aex + be2x + ce–3x.

19. Solve the differential equation (3xy + y2) dx – (x2 + xy) dy = 0.
OR
Solve the differential equation (tan–1 y – x) dy = (1 + y2) dx.

20. A letter is known to have come from ‘LONDON’ or ‘CLIFTON’. On the envelope
just two consecutive letters ‘ON’ are visible. What is the probability that the letter
has come from LONDON ?

H-754 -4-

21. By using the properties of determinants as far as possible, prove that

1 + a 2 − b2 2ab −2b
( )
3
2ab 1 − a 2 + b2 2a = 1 + a 2 + b2
2b −2a 1 − a 2 − b2

OR
Solve the following for x, using the properties of determinants.
x − 2 2x − 3 3 x − 4
x − 4 2x − 9 3 x − 16 = 0
x − 8 2x − 27 3 x − 64

SECTION – D

Question numbers 22 to 28 carry 4 marks each.

22. Using matrix method, solve the following system of equations :
x + 3y + 4z = 8, 2x + y + 2z = 5, 5x + y + z = 7

23. Using integration, prove that

a a

∫ f ( x ) dx = 2∫ f ( x ) dx , if f(x) is even function
−a 0
= 0, if f(x) is odd function.

24. Find the values of A and B, if the function
1 + sin x π
f (x) = −π ≤ x < −
A cos2 x 2
= A sin 2x + B −π ≤ x ≤0
2
x2
= 0<x≤π
e 4 x − 2e 2 x + 1

is continuous on [ − π, π ] .

H-754 -5- P.T.O.

x y
25. Using integration, find the area bounded by the line + = 1 and the ellipse
3 4
x2 y2
+ = 1.
9 16
dx
26. Evaluate ∫ .
sin x ( 5 − 4 cos x )

OR

dx
Evaluate ∫ .
sin x + cos4 x
4

x y−3 z−5
27. Find the equation of the plane which contains the line = = and
1 2 3
which is perpendicular to the plane 2x + 7y + 3z = 1.

OR

Find the equations of the planes through the intersection of the planes
2x + 6y + 12 = 0 and 3x – y + 4z = 0 which is at a unit distance from the
origin.

28. Solve the following linear programming problem graphically :

Maximise : Z = 6x + 3y

Subject to the constraints :

4x + y ≥ 80

x + 5y ≥ 115

3x + 2y ≤ 150

x ≥ 0, y ≥ 0

H-754 -6-

SECTION – E

Question numbers 29 to 30 carry 5 marks each.

29. Show that the height of the cylinder, open at the top of given surface area and
greater volume is equal to the radius of its base.
OR
Find the intervals in which the function f(x) = 5x3 – 15x2 – 120x + 3 is
i) strictly increasing and
ii) strictly decreasing.

−1  1 
30. Find ∫ tan  2
dx .
1− x + x 
OR
sin−1 x − cos −1 x
Find ∫ dx .
sin−1 x + cos −1 x

________________

H-754 -7- P.T.O.