Goa Board Class 12 Question Paper 2024 Maths

2024 III 15 0930 Seat No.

Time : 3 Hours MATHEMATICS

Subject Code

H 4 7 5 4

Total No. Of Questions : 36 (Printed Pages : 10) Maximum Marks : 80

INSTRUCTIONS : (i) The question paper consists of 36 questions.

(ii) All questions are compulsory.

(iii) Question numbers 1 to 8 are multiple choice type
questions of one mark each.

(iv) Question numbers 9 to 16 are very short answer type
questions of one mark each.

(v) Question numbers 17 to 22 are short answer type-I
questions of two marks each.

(vi) Question numbers 23 to 28 are short answer type-II
questions of three marks each.

(vii) Question numbers 29 to 34 are long answer type-I
questions of four marks each.

(viii) Question numbers 35 to 36 are long answer type-II
questions of five marks each.

(ix) There is no overall choice. However an internal choice
has been provided in two questions of 4 marks each and
2 questions of 5 marks each.

(x) Use of calculator is not permitted.

(xi) Log tables will be supplied on request.

(xii) Graph should be drawn on the answer paper only.

H-4754 1 P.T.O.

3 4
d2 y dy
1. The degree of the differential equation xy is ................ .
dx2 dx

• 1

• 2

• 3

• 4

2. If f : R R and g : R R are the two real functions defined by

f ( x) 3 x2 1 and g( x) 1 x , then ( gof ) ( 2) is .................. .

• 12

• 28

• –12

• –28

3. The value of iˆ ˆj 2 ˆj 3kˆ is ................... .

• 0

• 6

• 3

• 5

H-4754 2

1
4. If sin tan 1 x cot 1 1 , then the value of x is .................. .
3

• 3

• –3

•
2

1
•
3

5. If R is a relation in the set A 3 as R {( x, y) / x, y A and x2 y 4 is
a perfect square}, then R is ............... .

• reflexive, symmetric but not transitive

• reflexive but neither symmetric nor transitive

• an equivalence relation

• symmetric but neither reflexive nor transitive

3 4 3 1
6. If X , then the matrix X is ................ .
1 2 5 0

6 5
•
4 4

0 5
•
4 2

0 5
•
4 2

0 5
•
4 2

H-4754 3 P.T.O.

3

7. The value of | x 3| dx is ............... .
1

• 1

• –2

• 2

• 0

1

8. The value of x4 sin3 x dx is ................. .
1

• 1

• 0

• –1

• 2

9. Using determinants, show that the points (1, 3), (2, 2) and (0, 4) are

collinear.

10. Find the slope of tangent to the curve 2y = 3 – x3 at the point (1, 1).

11. Find the distance between the two planes :

x + y + 3z = 4 and 2x + 2y + 6z = 10.

H-4754 4

12. Find the area of parallelogram whose adjacent sides are given by the vectors

a iˆ jˆ and b 2iˆ 3 kˆ .

13. Find the principal value of cosec 1 ( 2) .

14. If

1
A 2
and B 2 0 11 3
4 3 1

then find the matrix (AB)', where (AB)' is the transpose of matrix (AB).

15. The random variable X has the following probability distribution :

X P(X)

0 K

1 2K

2 3K

3 4K

Find P(X < 2).

dy
16. If y ex y2 , then find .
dx

H-4754 5 P.T.O.

17. Find the angle between the following pairs of lines :

r 2iˆ 5 ˆj kˆ (3iˆ 2 ˆj 6kˆ ) and

r 7iˆ 6kˆ µ (2iˆ 2 ˆj kˆ ) .

18. If E and F are the events of a sample space S, such that P(F) 0 , then

prove that :

P(E'/F) 1 P(E/F) ,

where E' is the complement of event E.

19. Form the differential equation of family of curves represented by

A( y A)2 x3 , by eliminating arbitrary constant A.

20. Let * be a binary operation defined on set A = {1, 2, 3, 6, 12} as

a * b = H.C.F {a, b}. Prepare composition table for the binary operation *.

Also, compute 3* (6 * 12).

21. Prove that :

x y
tan 1 x tan 1 y tan 1 , xy 1.
1 xy

22. If A(1, –2, 3) and B(–1, –4, 3) are the given points and d 3iˆ 4 ˆj 5kˆ is

a given vector, then find the scalar projection of vector AB on d .

H-4754 6

23. Using properties of determinants, prove that :

a b c 2a 2a
2b b c a 2b ( a b c)3 .
2c 2c c a b

24. Find the general solution of the differential equation :

dy
x2 x2 xy 2 y2 .
dx

25. One third of the students in a class are boys and the rest are girls. It is

known that the probability of a girl getting first class is 0.32 and that a

boy getting first class is 0.22. If a student chosen at random gets first

class marks in the subject, what is the probability that the chosen student

is a boy ?

26. If

y ( x )sin 2 x (log 3 x) x 1 ,

dy
find .
dx

27. Find the equation of the plane passing through the line of intersection of

planes r . (2iˆ ˆj kˆ ) 3 and r . (5iˆ 3 ˆj 4 kˆ ) 9 and parallel to the line

r (iˆ 3 ˆj 5kˆ ) (2iˆ 4 ˆj 5kˆ ) .

28. Find :

x 5
dx .
x2 3x 7

H-4754 7 P.T.O.

29. Prove that :

2a a
f ( x) dx [ f ( x) f (2a x)] dx
0 0

Hence, show that :

2a a
f ( x) dx 2 f ( x) dx, if f (2a x) f ( x)
0 0
0 , if f (2a x) f ( x) .

30. Using integration, find the area of the smaller region bounded by the ellipse

x2 y2 x y
1 and the straight line 1.
16 9 4 3

Or

Using integration, find the area of the smaller region enclosed between

parabola y2 = 16x and the line x – y + 3 = 0.

31. Solve the following Linear Programming Problem graphically :

Minimize Z = 5x + 7y

Subject to constraints :

2x + y 8

x + 2y 10

2x + 3y 24

x 0, y 0.

H-4754 8

32. If x sin t , y sin( pt) prove that :

d2 y dy
(1 x 2 ) 2
x p2 y 0
dx dx

where t is a parameter.

Or

If y Ae kx cos( px 4) , prove that :

d2 y dy
2
2k ( p2 k2 ) y 0.
dx dx

33. If the function f(x) defined by :

1 sin x
f ( x) ; x
A cos2 x 2

3sin 2 x B ; x 0
2
e5 x e3 x
; 0 x
x

is continuous on [ , ] , then find the values of A and B.

1 2 5
34. Find the inverse of matrix A 1 1 1 .
2 3 1

Hence, solve the system of equations :

x + 2y + 5z = 10

x – y – z = –2

2x + 3y – z = –11.

H-4754 9 P.T.O.

35. The perimeter of an isosceles triangle is 200 cm. If its base is changing at

the rate 5 cm/sec, then find the rate at which the altitude is changing when

the base is 40 cm.

Or

Show that the semivertical angle of the cone of maximum volume and of

1 1
given slant height is cos .
3

36. Find :

2 cos2 x cos x
dx .
(sin x 2) (sin2 x 3)

Or

Find :

cos 1 x dx .

H-4754 10