Goa Board Class 12 Question Paper 2022 Maths

2022 XI 14 0230 Seat No.

Time : 1½ Hours FIRST-TERM MATHEMATICS
Subject Code

H 4 7 5 4
Total No. of Questions : 40 (Printed Pages : 16) Maximum Marks : 40

INSTRUCTIONS : (i) The question paper consists of 40 questions.

(ii) All questions are compulsory.

(iii) All questions are of Multiple Choice Type and carry
one mark each.

(iv) For each question select only one correct option from

the alternatives given.

(v) Use of calculator is not allowed.

1. The matrix A [ aij ] of order 2 × 2 whose elements are given by aij 2i j

is ............................

1 0
(A) 0 1

1 0
(B) 3 2

1 0
(C) 2 3

1 0
(D)
3 2

H-4754 [FT] 1 P.T.O.

2. Matrices A and B will be inverses of each other if and only if .......... .

(A) AB = BA

(B) AB = BA = O

(C) AB = O and BA = I

(D) AB = BA = I

3 5
2
3. If A = 4 2 , A – 5A is ......................... .

(A) an identity matrix

(B) a row matrix

(C) a scalar matrix

(D) a zero matrix

4. For a skew symmetric matrix, all the diagonal elements are ................

(A) non-zero

(B) negative numbers

(C) positive numbers

(D) zero

5. If A is a square matrix such that A2 = I, then A 3 (A I)2 9A I2 ...... .

(A) – 6A + I

(B) – 6A

(C) 6A + I

(D) – 6A – I

H-4754 [FT] 2

6. A, B, C are 3 matrices such that the order of A is 4 × 3 and the order
of B is 4 × 5 and the order of C is 7 × 3. Then the order of (AT B)T CT
is ................... .

(A) 5 × 3

(B) 4 × 5

(C) 5 × 7

(D) 4 × 3

1 1 1
7. The value of 11 10 9 is ................... .
101 100 99

(A) 1

(B) –1

(C) 2

(D) 0

8. Given that A is a square matrix of order 3 and A 2 , then Adj A is
equal to .................... .

(A) 4

(B) –2

(C) –4

(D) 2

H-4754 [FT] 3 P.T.O.

4 3
9. The determinant which is equal to
5 1 is ................ .

3 1
(A) 2 1

6 5
(B) 5 1

6 5
(C) 5 1

3 4
(D) 1 5

a b
10. If A = , such that ad bc 0 , then A 1 = ........................... .
c d

1 a b
(A)
ad bc c d

1 d b
(B) ad bc c a

1 d b
(C)
ad bc c a

1 d b
(D) c a
ad bc

H-4754 [FT] 4

11. If R is a relation in the set a, b, c, d given by

R a, a , b, b , c, c , d, d , a, d , a, b , d, b , then ..................... .

(A) R is reflexive and symmetric but not transitive

(B) R is reflexive and transitive but not symmetric

(C) R is symmetric and transitive but not reflexive

(D) R is an equivalence relation

12. The function f : N N defined by f (x) = x3 + 12 is ................ .

(A) bijective

(B) injective but not surjective

(C) surjective but not injective

(D) neither injective nor surjective

13. * is a binary operation on R defined by a * b a , a, b R , then .......... .

(A) * is commutative but not associative

(B) * is both commutative and associative

(C) * is neither commutative nor associative

(D) * is associative but not commutative

14. f :R R is defined by f ( x) cos x and g : R R is defined by g( x) x2 .
Then ( gof ) ( x) ...................

(A) cos ( x 2 )
(B) cos2 x
(C) x 2 cos x
(D) x cos x

H-4754 [FT] 5 P.T.O.

4x 4
15. Let R 4 R be a function defined by f ( x) , x . The
3 3x 4 3
4
inverse of f is the map g : Range of f R given by :
3
3y
(A) g( y)
3 4y
4y
(B) g( y)
3 4y
3y
(C) g( y)
4 3y
4y
(D) g( y)
4 3y

sin 1 3 x
16. If f is a real function such that f ( x) , x 0 is continuous at
4x
x = 0, then f (0) = ........................... .
4
(A)
3
3
(B)
4
3
(C)
4
4
(D)
3
17. The value of ‘m’ for which the real function f where

5x 4 ,0 x 1
f ( x)
4 x 2 3mx , 1 x 2

is continuous at every point in its domain is ...................
(A) 7
(B) 0
(C) 1
(D) –1
H-4754 [FT] 6

18. To make the real function f continuous at x = 2, where

2x if x 2
f ( x) k if x 2
x2 if x 2

the value of k should be ........................ .

(A) 2

(B) –2

(C) 4

(D) –4

19. f :R R defined by

ax a x
f ( x) , x 0
x

= 3k , x = 0

is continuous at x = 0. Then k = ......................

2
(A) log a
3

2
(B) log a
3

3
(C) log a
2

3
(D) log a
2

H-4754 [FT] 7 P.T.O.

d2 y
20. If y x 2 log x, then ...........................
dx 2

(A) 2 log x

(B) 3 + 2 log x

(C) 2 + 2 log x

(D) 3 + log x

dy
21. If x ex y e y , then = .....................
dx

1 ex
(A)
1 ey

1 ey
(B)
1 ex

(C) 1 ex ey

1 ex
(D)
1 ey

4 dy
22. If y e log x , then at x = –1 is ..................... .
dx

(A) e

(B) –e

(C) 4

(D) –4

H-4754 [FT] 8

23. If x = a (1 – cos t), y = a (t + sin t) where ‘t’ is the parameter and ‘a’ is

dy
a constant, then ...........
dx t
2

(A) –1

(B) 1

(C) 2

(D) 2
dy
24. If y = (sin x)cos x, then ......................
dx
cos x
(A) sin x sin x cot x sin x log (sin x)
sin x
(B) cos x cos x cot x sin x log (sin x)
cos x
(C) sin x cos x cot x sin x log (sin x)
cos x
(D) sin x cos x cot x cos x log (sin x)

25. The derivative of y sec 2 ( x 3 ) with respect to x is ................. .

(A) 6 x 2 sec 2 ( x 3 ) tan ( x 3 )

(B) 6 x 2 sec ( x) tan ( x)

(C) 2 x sec ( x3 ) tan ( x 3 )

(D) 6 x 2 sec ( x3 ) tan ( x 3 )

26. If x [ 1, 1] , then sin 1 ( x) = ................ .

(A) sin 1 x

(B) sin 1 x

(C) sin 1 x

(D) cosec–1 x
H-4754 [FT] 9 P.T.O.

2 7
27. tan 1 tan 1 = ......................
11 24

(A) tan 1 (1)

1
(B) tan 1
2

3
(C) tan 1
4

2
(D) tan 1
3

28. If y cos 1 x , then ...........

(A) x [ 1, 1]; y [0, ]

(B) x R; y ,
2 2

(C) x [ 1, 1]; y ,
2 2

(D) x R [ 1, 1]; y 0,
2

5
29. The value of sec 2 tan 1 is ...................
11

25
(A)
121

96
(B)
121

146
(C)
121

121
(D)
146

H-4754 [FT] 10

30. The value of p for which the vectors a 3i 2j 9 k and b i pj 3k
are parallel vectors is .........................

2
(A)
3
3
(B)
2

(C) 2

(D) 3

31. If a and b are two unit vectors and is the angle between them, then

a b is a unit vector if = ................

(A)
4

(B)
3

(C)
2
2
(D)
3

32. If i , j and k are the three unit vectors, then the vector represented by

i j i j k j k i k = ..................

(A) i j k

(B) i j k

(C) i j k

(D) i j k

H-4754 [FT] 11 P.T.O.

33. The value of so that the vectors a 2i j k, b i 2j 3k,

c 3i j 5 k are complanar is ........................

(A) –1

(B) –2

(C) –3

(D) –4

34. Let r be the position vector of an arbitrary point p ( x, y, z) . The Cartesian
form of the equation of the line passing through two points ( x1 , y1 , z1 ) and
( x2 , y2 , z2 ) is ..................

x x1 y y1 z z1
(A) x2 x1 y2 y1 z2 z1

x x1 y y1 z z1
(B) x2 x1 y2 y1 z2 z1

x x1 y y1 z z1
(C) x2 x1 y2 y1 z2 z1

x x1 y y1 z z1
(D) x2 x1 y2 y1 z2 z1

x x1 y y1 z z1
35. The line is at right angles to the plane
a b c
Ax By Cz D 0 if .......................... .

(A) aA bB cC 0
(B) aA bB cC 1

(C) aA bB cC
a b c
(D)
A B C

H-4754 [FT] 12

36. The distance of the plane 2x + 3y – 6z + 2 = 0 from the origin is ...........

(A) 2

(B) 14

2
(C)
7

2
(D)
23

37. The equation of the plane passing through the intersection of the planes
x + 2y – 5z + 1 = 0 and 2x – y + 3z – 11 = 0 and also through the origin
is ....................... .

(A) 13x + 21y – 52z = 0

(B) 13x – 21y – 52z = 0

(C) 13x + 21y + 52z = 0

1
(D) 13x + 21y – 52z =
11

38. The direction cosines of the normal to the plane 2x + 3y – z = 5 are :

(A) 2, 3, –1

2 3 1
(B) , ,
14 14 14

(C) 2, 3, 1

2 3 1
(D) , ,
14 14 14

H-4754 [FT] 13 P.T.O.

39. The angle between the line r i 2 j k i j k and the plane

r 2i j k 6 is .....................

2 2
(A) sin 1
3

2
(B) sin 1
3

2
(C) cos 1
3

1
(D) sin 1
3

40. The equation of the plane through the point (–1, –1, 1) which is parallel to

the plane r i j k 0 is .................

(A) r i j k 1 0

(B) r i j k 1 0

(C) r i j k 3 0

(D) r i j k 3 0

H-4754 [FT] 14

Space For Rough Work

H-4754 [FT] 15 P.T.O.

Space For Rough Work

H-4754 [FT] 16