Goa Board Class 12 Question Paper Mar 2019 Mathematics

2019 III 13 1000 Seat No.

Time : 2½ Hours MATHEMATICS
Subject Code

H 7 5 4

Total No. Of Questions : 30 (Printed Pages : 8) Maximum Marks : 80

INSTRUCTIONS : (i) All questions are compulsory.

(ii) The question paper consists of 30 questions divided into

five sections A, B, C, D and E.

(iii) 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.

(iv) 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.

(v) Use of calculator is not allowed.

H-754 1 P.T.O.

Section A

Question Nos. 1 to 7 carry 1 mark each. In each question, four options are

provided out of which one is correct. Write the correct option.

1 0 1 0
1. If A = and B = , then AB is :
0 1 1 0

a zero matrix

an identity matrix

matrix A

matrix B

2. a b is a vector which is :

parallel to a

parallel to b

perpendicular to both a and b

perpendicular to only a

3
3. cot cos 1 is equal to :
2

1
3

1
2

3

2

H-754 2

dy
4. If y 2 log e3 x 5, then .............
dx

2

3

5

6

5. If f ( x) 2 x2 , then f o f (x) = .............

x

x2

2 – x2

x

6. y 5e x 2e x x is a solution of the differential equation :

d2 y dy
y
dx2 dx

d2 y
x y
dx2

d2 y
y x
dx2

d2 y dy
x
dx2 dx

H-754 3 P.T.O.

7. The direction cosines of two perpendicular lines are l1, m1, n 1 and

l2, m2, n2 respectively. Then the direction cosines of a line which is

perpendicular to both these lines are :

l1 + l2, m1 + m2, n1 + n2

l1 – l2, m1 – m2, n1 – n2

m1n2 – n1m2, n1l2 – l1n2, l1m2 – l2m1

m1n1 – m2n2, l1n1 – l2n2, l1m2 – l2m1

Section B

Question Nos. 8 to 14 carry 2 marks each.

b c b
8. Prove that f ( x) dx f ( x) dx f ( x) dx for a < c < b.
a a c

9. Find the value of ‘x’ if the vectors iˆ 2 ˆj k, xiˆ 5 ˆj 3kˆ and 5iˆ 9 ˆj 4 kˆ
are coplanar.

1
10. The probability that a student is not a swimmer is . What is the probability
3
that out of 5 students, at least one is a swimmer ?

11. Find matrices X and Y if :

2 0 1 3
2X + Y and X Y= .
3 5 0 1

12. The cartesian equations of a line are 4x – 2 = 3y – 1 = 2 – 2z. Find the

vector equation of the line and also find its direction ratios.

13. Form the differential equation representing the family of curves

y = A(x + B)2 by eliminating arbitrary constants A and B.

H-754 4

14. A particle moves along the curve y = 2x3 + 1. Find the points on the curve

at which the y-coordinate changes six times as fast as the x-coordinate.

Section C

Question Nos. 15 to 21 carry 3 marks each.

15. Using integration prove that :

1
dx log| x x2 a2 | c.
2 2
x a

16. Prove that :

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

1 2 3

17. Find ‘adj.A’ if A = 0 2 4 . Hence find A(adj. A).

0 0 5

18. A unit vector a makes angle with b and vector a 3b is perpendicular
3
to vector 2a b . Find the magnitude of b .

x 2
19. Show that the function f : R – {3} R – {1} defined by f ( x) is
x 3

invertible.

20. Solve the differential equation :

x 2 dy y( x y) dx 0.

H-754 5 P.T.O.

Or

Find a particular solution of the differential equation :

xe x dy ( x3 2 ye x )dx

given that y = 0 when x = 1.

21. Find the equation of a plane passing through the points (1, 0, 0) and
6
(0, 2, 0) which is at a distance of units from the origin.
7

Or

Find the distance of the point (1, –2, 3) from the plane x – y + z = –1
x y z
measured along a line parallel to the line .
2 3 6

Section D

Question Nos. 22 to 28 carry 4 marks each.

22. Find the values of ‘A’ and ‘B’ if the function defined below is continuous in
its domain :

Ax3 3 x2 Bx
f ( x) 6x
; 1 x 0
e 1

1
; x = 0
2

Ax 4 2
;0 x 1
log(3 6 x) log 3

H-754 6

23. Solve the following linear programming problem graphically :
Minimise :
Z = 50x + 60y
Subject to the constraints :
3x + 4y < 24
x + y > 5
x + 4y > 8
x, y > 0.
24. Evaluate :

2
(2 log (sin x) log(sin 2 x)) dx.
0

25. An electronic component consists of 3 parts A, B and C. The probabilities
that each part perform satisfactorily are 0.4, 0.3 and 0.2 respectively. The
component fails if two or more parts do not perform satisfactorily. Assuming
that the parts perform independently, find the probability that the component
does not perform satisfactorily.
Or
Four defective bolts are accidently mixed with six good ones. If two bolts are
drawn successively without replacement from this lot, find the probability
distribution of the number of defective bolts. Also find the mean of the number
of defective bolts.
26. Using integration, find the area of the region :
{(x, y) : y2 < 3x and x2 + y2 < 4}.
27. If the sum of the lengths of the hypotenuse and one of the sides of a right-
angled triangle is constant, then find the angle between that side and the
hypotenuse so that the area of triangle is maximum.
Or
Find the points on the curve 9y2 = x3 where normal to the curve makes
equal intercepts with the coordinate axes.

H-754 7 P.T.O.

28. If a, b, c are positive and unequal such that :

a a3 a4

b b3 b4 0

c c3 c4

Then using properties of determinants prove that ab + bc + ca = 0.

Section E

Question Nos. 29 to 30 carry 5 marks each.

d2 y t2 1 t2
29. Find if x = tan–1 and y sin 2 cot 1 where ‘t’
dx2 1 1 t4 1 t2

is the parameter.

Or

If ( x a)2 (y b)2 c 2 prove that

2 3/ 2
dy
1
dx
d2 y
dx 2

is a constant independent of ‘a’ and ‘b’.

e3 x 2e x
30. Find dx.
e3 x 2 e2 x e x 2

Or

x4 1 {log ( x4 1) 4 log x}
Find 7
dx.
x

H-754 8