Assam Board Class 12 Sample Paper 2026 Maths

ASSAM BOARD

SAMPLE
PAPER
Practice Papers
Model Question Papers

MATHEMATICS
Paper-1
SECTION A

Question numbers 1 to 20 carry 1 mark each.
Question numbers 1 to 10 are multiple choice type questions. Select the correct option.
1. If f and g are two functions from R to R defined as f(x) = | x | + x and
g(x) = | x |  x, then fog (x) for x < 0 is Unit I 1
(A) 4x (B) 2x
(C) 0 (D)  4x
2. The principal value of cot  1 ( 3 ) is Unit I 1

(A) 
6
(B) 
6
(C) 2
3
(D) 5
6
 2 0 0 
 
3. If A = 0  2 0  , then the value of | adj A| is Unit II 1
0 0  2 
(A) 64 (B) 16
(C) 0 (D) 8
4. The maximum value of slope of the curve y =  x + 3x2 + 12x  5 is Unit III
3
1
(A) 15 (B) 12
(C) 9 (D) 0
ex (1 + x)
5.  cos2 (xe x ) dx is equal to Unit III 1

(A) tan (xex) + c (B) cot ( xex) + c
(C) cot (ex) + c (D) ten [ex (1 + x)] + c
3
The degree of the differential equation x 2 d y2   x dy  y  is
2
6. Unit III 1
dx  dx 
(A) 1 (B) 2
(C) 3 (D) 6
7. The value of p for which is a unit vector is Unit IV 1
(A) 0 (B) 1
3
(C) 1 (D) 3
8. The coordinates of the foot of the perpendicular drawn from the point

(2, 8, 7) on the XZ - plane is Unit IV 1
(A) (2, 8, 7) (B) (2, 8, 7)
(C) (2, 0, 7) (D) (0, 8, 0)

1

9. The feasible region for an LPP is shown below : Unit V 1
Let z = 3x 4y be the objective function. Minimum of z occurs at

(A) (0, 0) (B) (0, 8)
(C) (5, 0) (D) (4, 10)
Fill in the blanks in question numbers 11 to 15.
10. If y = tan1 x + cot1 x, x ∈ R, then dy is equal to ______________. Unit III 1
dx
OR
If cos (xy) = k, where k is a constant and xy  n, n ∈ Z, then dy is dy equal to
dx
_________. Unit III 1
11. The value of  so that the function f defined by Unit III 1

 x, if x  
f(x) = 
cos x, if x > 
is continuous at x =  is _______________.
12. The equation of the tangent to the curve y = sec x at the point (0, 1) is ___________ .
Unit III 1
14. The area of the parallelogram whose diagonals are and is _______ square
units. Unit IV 1
OR
The value of  for which the vectors and are
orthogonal is __________ . Unit IV 1
15. A bag contains 3 black, 4 red and 2 green balls. If three balls are drawn simultaneously at
random, then the probability that the balls are of different colours is ___________ .
Unit VI 1
Question numbers 16 to 20 are very short answer type questions.
16. Construct a 2 × 2 matrix A = [aij] whose elements are given by aij = | (i)2 – j |. Unit II 1
17. Differentiate sin² ( x ) with respect to x. Unit III 1
18. Find the interval in which the function f given by f(x) = 7 – 4x – x² is strictly increasing.
Unit III 1

19. Evaluate: Unit III 1
2
 | x | dx
2
OR

2

Find :
dx
 9  4x 2

20. An unbiased coin is tossed 4 times. Find the probability of getting at least one head.
Unit VI 1
SECTION B
Question numbers 21 to 26 carry 2 marks each.
21. Solve for x: Unit I 2

sin 1 4x + sin 1 3x = 
2
OR
Express in the simplest form.

4  3
Express A = 
 1
22. as a sum of a symmetric and a skew symmetric matrix.
2
Unit II 2
If y² cos   = a², then find
1 dy
23. . Unit III 2
x dx

24. Show that for any two non-zero vectors and , Unit IV 2
iff and are perpendicular vectors.
OR
Show that the vectors and form
the sides of a right-angled triangle. Unit IV 2
25. Find the coordinates of the point where the line through (1, 1,  8) and (5, 2, 10)
crosses the zx-plane. Unit IV 2

26. If A and B are two events such that P(A) = 0.4, P(B) = 0.3 and P(A UB) = 0.6, then find
P(B̕̕∩A). Unit IV 2
SECTION C
Question numbers 27 to 32 carry 4 marks each.
27. Show that the function f : (, 0) → (1, 0) defined by f(x) = x ,
1 | x |
x ∈ (, 0) is one-one and onto. Unit I 4
OR
Show that the relation R in the set A = {1, 2, 3, 4, 5, 6} given by
R = {(a, b) : | a – b | is divisible by 2} is an equivalence relation. Unit I 4
dy

28. If y = x3 (cos x)x + sin1 x , find . Unit III 4
dx

29. Evaluate : Unit III 4

3

30. Find the general solution of the differential equation Unit III 4
x²y dx  (x + y ) dy = 0.
3 3

31. Solve the following LPP graphically: Unit V 4
Minimise z = 5x + 7y
subject to the constraints
2x + y ≥ 8
x + 2y ≥ 10
x, y ≥ 0
32. A bag contains two coins, one biased and the other unbiased. When tossed, the biased coin
has a 60% chance of showing heads. One of the coins is selected at random and on tossing
it shows tails. What is the probability it was an unbiased coin? Unit VI 4
OR
The probability distribution of a random variable X, where k is a constant is given below :
0.1 if x=0
 2
P(X = x)  
kx , if x=1
Unit VI 4
kx, if x = 2 or 3
0, otherwise

Determine
(a) the value of k
(b) P(x ≤ 2)
(c) Mean of the variable X.

SECTION D
Question numbers 33 to 36 carry 6 marks each.
33. Solve the following system of equations by matrix method: Unit II 6
x  y + 2z = 7
2x  y + 3z = 12
3x + 2y  z = 5
34. Find the points on the curve 9y2 = x3, where the normal to the curve makes equal
intercepts with both the axes. Also find the equation of the normals. Unit III 6
35. Find the area of the following region using integration: Unit V 6
{(x, y) : y ≤ | x | + 2, y ≥ x )
2

OR
Using integration, find the area of a triangle whose vertices are (1,0), (2,2) and (3,1). 6
36. Show that the lines
x  2 y  2 z  3 x  2 y  3 z  4
  and   intersect.
1 3 1 1 4 2
Also, find the coordinates of the point of intersection. Find the equation of the plane

containing the two lines. Unit VI 6

4

Paper-2
SECTION A
Question numbers 1 to 20 carry 1 mark each.
Question numbers 1 to 10 are multiple choice type questions. Select the correct option.
1. The relation R in the set (1, 2, 3) given by R = {(1, 2), (2, 1), (1, 1)} is Unit I 1
(A) symmetric and transitive, but not reflexive
(B) reflexive and symmetric, but not transitive
(C) symmetric, but neither reflexive nor transitive
(D) an equivalence relation
 3  
2. tan1 3 + tan1 λ = tan1   is valid for what values of λ? Unit I 1
 1  3 
(A)     1 , 1 
 3 3

(B) 1
3
(C) 1
3
(D) All real values of λ
3. If A is a non-singular square matrix of order 3 such that A2 = 3A, then value of | A | is 1
(A) 3 (B) 3 Unit II
(C) 9 (D) 27
4. The function f : R → R given by f(x) =  |x 1| is Unit I 1
(A) continuous as well as differentiable at x = 1
(B) not continuous but differentiable at x = 1
(C) continuous but not differentiable at x = 1
(D) neither continuous nor differentiable at x = 1
5. Let A = {1, 3, 5}. Then the number of equivalence relations in A containing (1, 3) is 1
(A) 1 (B) 2 Unit I
(C) 3 (D) 4
2 x
6. The interval in which the function f given by f(x) = x e is strictly increasing, is 1
(A) () (B) ( , 0) Unit III
(C) (2, ) (D) (0, 2)
7. If | | = 4 and 3   2, then |  | lies in Unit IV 1
(A) [0, 12] (B) [2, 3]
(C) [8, 12] (D) [12, 8]
8. The vectors and are coplanar if value of
is Unit IV 1
(A) 2 (B) 0
(C) 2 (D) Any real number

9. The area of a triangle formed by vertices O, A and B, where and
is Unit IV 1
(A) 3 5 sq. units (B) 5 5 sq. units
(C) 6 5 sq. units (D) 4 sq. units

5

10. The coordinates of the foot of the perpendicular drawn from the point
(2, 3, 4) on the y-axis is Unit IV 1
(A) (2, 3, 4) (B) (2,3,4)
(C) (0,3, 0) (D) (2, 0, 4)
Fill in the blanks in question numbers 11 to 15.
11. The range of the principal value branch of the function y = sec x is _____. Unit I 1
OR
The principal value of cos   1  is _________ . Unit I 1
 2
0 a 1
12. Given a skew-symmetric matrix A =  1 b 1 , the value of (a + b + c) is ______.
2
1

 1 c 0 
Unit II
13. If the radius of the circle is increasing at the rate of 0.5 cm/s, then the rate of increase of
its circumference is ________. Unit III 1
15. The corner points of the feasible region of an LPP are (0, 0), (0, 8), (2, 7),
(5, 4) and (6, 0). The maximum profit P = 3x + 2y occurs at the point __________. 1
Unit V
Question numbers 16 to 20 are very short answer type questions.

16. Differentiate sec2 (x2) with respect to x2. Unit III 1
OR
x dy
If y = f(x2) and f (x) = e , then find . Unit III 1
dx
kx 2 + 5 if x  1
17. Find the value of k, so that the function f(x) = 
 2 if x > 1
is continuous at x = 1. Unit III 1

18. Evaluate: Unit III 1

2

 x cos x dx
2


2

dy yx
19. Find the general solution of the differential equation e = 1 . Unit III 1
dx
x  1 y+4 z+4
20. Find the coordinates of the point where the line = = cuts the xy-
3 7 2
plane. Unit IV

SECTION B
Question numbers 21 to 26 carry 2 marks each.
 3 2
21. If A =   and I = , find scalar k so that A2 + I = kA. Unit II 2
1  1 

6

sec x 1  
22. If f(x) = , find f    . Unit III 2
sec x + 1 3
OR
Find f (x) if f(x) = (tan x) tan x
. Unit III 2
23. Find : Unit III 2
tan 3 x
 cos3 x dx
24. Find a vector equally inclined to the three axes and whose magnitude is 3 3 units.
Unit IV 2
OR
Find the angle between unit vectors and so that 3  is also a unit vector. 2
Unit IV
25. Find the points of intersection of the line and
the plane Unit IV 2
26. A purse contains 3 silver and 6 copper coins and a second purse contains 4 silver and 3
copper coins. If a coin is drawn at random from one of the two purses, find the probability
that it is a silver coin. Unit VI 2

SECTION C

Question numbers 27 to 32 carry 4 marks each.

27. Check whether the relation R in the set N of natural numbers given by
R = {(a, b) : a is divisor of b} Unit I 4
is reflexive, symmetric or transitive. Also determine whether R is an equivalence relation.
OR
1 2 1 4
Prove that tan1 + tan1 = sin1   . Unit I 4
4 9 2 5
 y
28. If tan1   = log x 2 + y2 , prove that . Unit III
x
4
OR
If y = e cos x, 1 < x < 1, then show that
a 1 Unit I II 4
2
d y dy
(1  x2) 2
x  a2y = 0
dx dx
29. Find: Unit III 4
3
x +1
 x 3  x dx

30. Solve the following differential equation : Unit III 4
 y
1  e  dy + e
y/x y/x
1  x  dx = 0 (x  0) .
 
7

31. Find the shortest distance between the lines Unit IV 4

32. A cottage industry manufactures pedestal lamps and wooden shades. Both the products
require machine time as well as craftsman time in the making. The number of hour(s)
required for producing 1 unit of each and the corresponding profit is given in the
following table:
Item Machine Time Craftsman time Profit (in ₹)
Pedestal lamp 1.5 hours 3 hours 30
Wooden shades 3 hours 1 hour 20

In a day, the factory has availability of not more than 42 hours of machine time and 24
hours of craftsman time.
Assuming that all items manufactured are sold, how should the manufacturer schedule his
daily production in order to maximise the profit? Formulate it as an LPP and solve it
graphically. Unit V 4

SECTION D
Question numbers 33 to 36 carry 6 marks each.
5  1 4 
33. If A =  2 3 5  , find A and use it to solve the following system of
 
5  2 6 
equations : Unit II 6
5x  y + 4z = 5
2x + 3y + 5z = 2
5x  2y + 6z = - 1
OR
2
x x 1 + x3
If x, y, z are different and y y 2 1 + y3 = 0, then using properties of determinants
z z 2 1 + z3
show that 1 + xyz = 0. Unit II 6
34. Amongst all open (from the top) right circular cylindrical boxes of volume 125𝝅 cm³, find
the dimensions of the box which has the least surface area. Unit III 6
35. Using integration, find the area lying above x-axis and included between the circle
x2 + y2 = 8x and inside the parabola y2 = 4x. Unit III 6
OR
Using the method of integration, find the area of the triangle ABC, coordinates of whose

vertices are A(2, 0), B(4, 5) and C(6, 3). Unit III 6
36. Find the probability distribution of the random variable X, which denotes the number of
doublets in four throws of a pair of dice. Hence, find the mean of the number of
doublets (X). Unit VI 6

8

Paper-3
MATHEMATICS
SECTION A
Question numbers 1 to 20 carry 1 mark each.
Question numbers 1 to 10 are multiple choice type questions. Select the correct option.
1. If A is a square matrix of order 3 and | A | = 5, then the value of | 2A| is Unit II 1
(A)  10 (B) 10
(C)  40 (D) 40
2. If A is a square matrix such that A = A, then (I  A)3 + A is equal to
2
Unit II 1
(A) I (B) 0
(C) I A (D) I+A
 3 
3. The principal value of tan1  tan  is Unit I 1
 5 
(A) 2 (B) 2
5 5
(C)
3 (D)
3
5 5

4. If the projection of on zero, then the value of  is Unit IV
1
(A) 0 (B) 1
2 3
(C) (D)
3 2
5. The vector equation of the line passing through the point (1, 5, 4) and perpendicular to
the plane z = 0 is Unit IV 1
(A)

(B)

(C)

(D)
6. The number of arbitrary constants in the particular solution of a differential equation of
second order is (are) Unit III 1
(A) 0 (B) 1
(C) 2 (D) 3

4

 sec x dx is equal to
2
7. Unit III 1


4

(A) 1 (B) 0
(C) 1 (D) 2
8. The length of the perpendicular drawn from the point (4, 7, 3) on the y- axis is 1
Unit III
9

(A) 3 units (B) 4 units
(C) 5 units (D) 7 units
9. If A and B are two independent events with P(A) = 1 and P(B) = 1 , then
3 4
P(B' | A) is equal to Unit VI 1
(A) 1 (B) 1
4 3
(C) 3 (D) 1
4
10. The corner points of the feasible region determined by the system of linear inequalities are
(0,0), (4,0), (2,4) and (0,5). If the maximum value of
z = ax + by, where a, b > 0 occurs at both (2,4) and (4,0), then Unit V 1
(A) a = 2b (B) 2a = b
(C) a=b (D) 3a = b
Fill in the blanks in question numbers 11 to 15.
11. A relation R in a set A is called _________, if (a1, a2) ∈ R, implies
(a2, a1) ∈ R for all a1, a2 ∈ A. Unit I 1
12. The greatest integer function defined by f(x) = [x], 0 < x < 2 is not differentiable at
x = _________. Unit III 1

13. If A is a matrix of order 3  2, then the order of the matrix A' is _______. Unit II 1
OR
A square matrix A is said to be skew-symmetric, if ________. Unit II 1
2
14. The equation of the normal to the curve y = 8x at the origin is _______. Unit III 1
OR
The radius of a circle is increasing at the uniform rate of 3 cm/sec. At the instant when the
radius of the circle is 2 cm, its area increases at the rate of _______ cm²/s. Unit III 1
15. The position vectors of two points A and B are and
, respectively. The position vector of a point P which divides the line segment joining A and B in
the ratio 2 : 1 is _______. Unit IV 1

Question numbers 16 to 20 are very short answer type questions.
2 0 0
16. If A =  1 2 3 , then find A (adj A). Unit II 1

3 3 5 
17. Find :

 x log x dx
4
Unit III 1
OR
Find :
2x
 x + 1 dx

3 2
Unit III 1

18. Evaluate :

10

3

 | 2x  1| dx
1
Unit III 1

19. Two cards are drawn at random and one-by-one without replacement from a well-shuffled
pack of 52 playing cards. Find the probability that one card is red and the other is black. 1
Unit VI
20. Find :
dx
 9  4x 2
Unit III 1

SECTION B
Question numbers 21 to 26 carry 2 marks each.
21.  
Prove that sin 2x 1  x 2 = 2 cosx,
1
2
≤ x ≤ 1. Unit I 2

OR
Consider a bijective function f : R+ → (7, ) given by f(x) = 16 x2 + 24 x + 7, where R+ is
the set of all positive real numbers. Find the inverse function of f. Unit I 2
d2 y
22. If x = at2, y = 2at, then find . Unit III 2
dx 2
23. Find the points on the curve y = x3  3x2  4x at which the tangent lines are parallel to the
line 4x + y 3 = 0. Unit III 2
24. Find a unit vector perpendicular to each of the vectors and where
and Unit III 2
OR
Find the volume of the parallelopiped whose adjacent edges are represented by 2 ,-
and 3 , where Unit IV 2

25. Find the value of k so that the lines x =  y = kz and x  2 = 2y + 1 =  z + 1 are
perpendicular to each other. Unit IV 2
26. The probability of finding a green signal on a busy crossing X is 30%. What is the probability
of finding a green signal on X on two consecutive days out of three? Unit VI 2
SECTION C
Question numbers 27 to 32 carry 4 marks each.
27. Let N be the set of natural numbers and R be the relation on N  N defined by (a,b) R (c,d)
iff ad = bc for all a, b, c, d  N. Show that R is an equivalence relation. Unit I 4
dy
x 2 cos x
 (cos x) x , then find

28. If y = e . Unit III 4
dx
29. Find: Unit III 4

 sec x dx
3

11

30. Find the general solution of the differential equation Unit III 4
y 3 y
y e dx = (y + 2x e ) dy.
OR
Find the particular solution of the differential equation Unit III 4
x dy = y x tan  y  , given that y =  at x = 1 .
dx x 4
31. A furniture trader deals in only two items  chairs and tables. He has ₹ 50,000 to invest and a
space to store at most 35 items. A chair costs him ₹ 1,000 and a table costs him ₹ 2,000. The
trader earns a profit of ₹ 150 and ₹ 250 on a chair and table, respectively. Formulate the above
problem as an LPP to maximise the profit and solve it graphically. Unit V 4

32. There are two bags, I and II. Bag I contains 3 red and 5 black balls and Bag II contains 4
red and 3 black balls. One ball is transferred randomly from Bag I to Bag II and then a ball
is drawn randomly from Bag II. If the ball so drawn is found to be black in colour, then
find the probability that the transferred ball is also black. Unit VI 4
OR
An urn contains 5 red, 2 white and 3 black balls. Three balls are drawn, one-by-one, at random
without replacement. Find the probability distribution of the number of white balls. Also, find
the mean and the variance of the number of white balls drawn. Unit VI 4

SECTION D
Question numbers 33 to 36 carry 6 marks each.
1 2 3
33. If A = 3 2  2  , then find A and use it to solve the following
1

 2 1 1 
system of the equations: Unit II 6
x + 2y  3z = 6
3x + 2y  2z = 3
2x y + z = 2
OR
Using properties of determinants, prove that Unit II 6
(b + c) 2 a2 bc
(c + a) 2
b 2
ca = (a  b) (b  c) (c  a) (a + b + c) (a2 + b2 + c2).
(a + b) 2 c2 ab
34. Using integration, find the area of the region bounded by the triangle whose vertices are
(2, 2), (4, 5) and (6, 2). Unit III 6
35. Show that the height of the right circular cylinder of greatest volume which can be inscribed in
a right circular cone of height h and radius r is one-third of the height of the cone, and the
greatest volume of the cylinder is 4 times the volume of the cone. Unit III 6
9

36. Find the distance of the point P(2, 4, 7) from the point of intersection Q of the line a
and the plane Also write the
vector equation of the line PQ. Unit IV 6
12

Paper-4
SECTION A
Question numbers 1 to 9 are multiple choice questions of 1 mark each. Select the correct option :

1. The value of sin1  cos 3  is Unit I 1
 5 


(a)  (b) 3 (c)  (d)  3
10 5 10 5
3 2
2. 
If A = [2 3 4], B =  2  , X = [1 2 3] and Y =  3  , than AB + XY equals Unit II 1

2 4
   
(a) [28] (b) [24] (c) 28 (d) 24
2 3 2
3. If x x x + 3 = 0 then the value of x is Unit II 1
4 9 1
(a) 3 (b) 0 (c) 1 (d) 1
 /8

0 tan (2x) is equal to
2
4. Unit III 1

(a) 4   (b) 4 +  (c) 4   (d) 4  
8 8 4 2
5. If then the angle between and is Unit IV 1
(a) 0° (b) 30° (c) 60° (d) 90°
6. The two lines x = ay + b, z = cy + d; and x = ay + b, z = cy + d are perpendicular to
each other, if Unit IV 1
(a) a + c  1 (b) a + c  1 (c) aa  + cc = 1 (d) aa  + cc =  1
a c a c
7. In an LPP, if the objective function z = ax + by has the same maximum value on two corner
points of the feasible region, then the number of points at which zmax occurs is Unit V 1
(a) 0 (b) 2 (c) finite (d) infinite
8. From the set {1, 2, 3, 4, 5}, two numbers a and b (a ≠ b) are chosen at random. The
probability that is a an integer is: Unit VI 1
b
(a) 1 (b) 1 (c) 1 (d) 3
3 4 2 5
9. A bag contains 3 white, 4 black and 2 red balls. If 2 balls are drawn at random (without
replacement), then the probability that both the balls are white is Unit VI 1

(a) 18 (b) 1 (c) 1 (d) 1
36 12 24
In Q. Nos. 10 to 14, fill in the blanks with correct word / sentence :
10. If f : R → R be given by f(x) = (3  x3)1/3, then fof (x) = ________. Unit I 1

13

x+y 7  2 7
11. If   = , then x ∙ y = ________. Unit II 1
 9 x  y 9 4
12. The number of points of discontinuity of f defined by f(x) = | x |  | x + 1| is _______. 1
Unit III
13. The slope of the tangent to the curve y = x3  x at the point (2,6) is _______. 1
Unit III
OR
The rate of change of the area of a circle with respect to its radius r, when
r = 3 cm, is ______. Unit III 1
14. If is a non-zero vector, then equals ________. 1
Unit IV
OR
The projection of the vector – on the vector + is _______. 1
Unit IV
Q. 15 to 20 are very short answer questions.
 2  1
Find adj A, if A = 
3 
15. Unit II 1
4
2x + 1  5 x 1
16. Find  dx Unit III 1
10x
2
17. Evaluate  | sin x | dx Unit III 1
0


a
dx
18. If  2
 , then find the value of a. Unit III 1
0
1 + 4x 8

19. Find 
dx Unit III 1
x+x
20. Show that the function y = ax + 2a2 is a solution of the differential equation 1
2
 dy   dy 
2     y = 0 . Unit III
 dx   dx 
SECTION B
Q. Nos. 21 to 26 carry 2 marks each.
21. Check if the relation R on the set A = {1, 2, 3, 4, 5, 6} defined as
R = {(x, y) : y is divisible by x } is (i) symmetric (ii) transitive Unit I 2
OR
Prove that :
9  9 sin 1  1   9 sin 1  2 2 
    Unit I 2
8 4 3 4  3 

at  =  , if x = cos – cos 2, y = sin  – sin 2. Unit III 2
dy
22. Find the value of

dx 3
23. Show that the function f defined by f(x) = (x – 1) ex + 1 is an increasing function for all
x > 0. Unit III 2
24. Find | | and | |, if | | = 2| |and ( + ). ( – ) = 12. Unit IV 2
14

25. Find the unit vector perpendicular to each of the vectors and
Unit IV 2
26. Find [P(B/A) + P(A/B)], if P(A) = 3 . P(B) = 2 and P(AUB) = 3 Unit VI 2
10 5 5
SECTION C
Q. Nos. 27 to 32 carry 4 marks each.
27. Prove that the relation R on Z, defined by R {(x, y) : (x  y) is divisible by 5} is an
equivalence relation. Unit I 4
 1 x  1
If y = sin 1  1  x 
dy
28.
  , then show that = Unit III 4
 2  dx 2 1 x2
OR
 
Verify the Rolle's Theorem for the function f(x) = ex cos x in   ,  Unit III 4
 2 2

x sin x
29. Evaluate :  2
dx. Unit III 4
0
1 + cos x
30. For the differential equation given below, find a particular solution satisfying the given
condition
dy
(x + 1) = 2e  y + 1 ; y = 0 when x = 0. Unit III 4
dx
31. A manufacturer has three machines I, II and III installed in his factory. Machine I and II
are capable of being operated for atmost 12 hours whereas machine III must be operated
for atleast 5 hours a day. He produces only two items M and N each requiring the use of
all the three machines.

The number of hours required for producing 1 unit of M and N on three machines are
given in the following table : Unit V
Number of hours required on machines
Items I II III
M 1 2 1
N 2 1 1.25

He makes a profit of ₹600 and ₹400 on one unit of items M and N respectively. How
many units of each item should he produce so as to maximize his profit assuming that he
can sell all the items that he produced? What will be the maximum profit? 4

32. A coin is biased so that the head is three times as likely to occur as tail. If the coin is
tossed twice, find the probability distribution of number of tails. Hence find the mean of
the number of tails. Unit VI 4
OR

Suppose that 5 men out of 100 and 25 women out of 1000 are good orators. Assuming that
there are equal number of men and women, find the probability of choosing a good orator.

15

SECTION D
Q. Nos. 33 to 35 carry 6 marks each.
33. Using properties of determinates prove that: Unit II 6
a  b b+c a
b  c c+a b = a3 + b3 + c3  3 abc.
c  a a+b c
OR
1 3 2
If A =  2 
0 1 , then show that A³ 4A  3A+ 11 I = O. Hence
2

1 2 3 

find AUnit II 6
34. Find the intervals on which the function f(x) = (x  1) (x  2) is (a) strictly increasing
3 2

(b) strictly decreasing. Unit III 6
OR
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as
large as possible, when revolved about one of its side. Also, find the maximum volume. 6
35. Find the area of the region lying in the first quadrant and enclosed by the
x  axis, the line y = x and the circle x2 + y2 = 32. Unit III 6

16

Paper-5
SECTION A
Question numbers 1 to 10 are multiple choice questions of 1 mark each. Select the correct option :
1. If A is a square matrix of order 3, such that A (adj A) = 10 I, then | adj A | is equal to
(a) 1 (b) 10 (c) 100 (d) 101 Unit II 1
2. If A is a 3×3 matrix such that |A| = 8, then |3A| equals. Unit II 1
(a) 8 (b) 24 (c) 72 (d) 216
2
3. If y = Ae5x + Be5x, then d y2 is equal to Unit III 1
dx
(a) 25y (b) 5y (c) 25y (d) 15y

x e
2 x3
4. dx equals Unit III 1
1 x3 1 x4 1 x3 1 2
(a) e +c (b) e +c (c) e + c (d) ex + c
3 3 2 2
5. If are unit vectors along three mutually perpendicular directions, then Unit IV 1

6. ABCD is a rhombus whose diagonals intersect at E. Then equals 1
Unit IV
x 1 y  3 4  z x 1 y  4 z  5
7. The lines  = and  = are mutually
1 1 k k 2 2
perpendicular if the value of k is Unit IV 1
2 2
(a)  (b) (c)  2 (d) 2
3 3
8. The graph of the inequality 2x + 3y > 6 is Unit V 1
(a) half plane that contains the origin.
(b) half plane that neither contains the origin nor the points of the line
2x + 3y = 6.
(c) whole ΧΟΥ - plane excluding the points on the line 2x + 3y = 6
(d) entire ΧΟΥ plane.
9. A card is picked at random from a pack of 52 playing cards. Given that the picked card is
a queen, the probability of this card to be a card of spade is Unit VI 1
4 1 1
(a) 1 (b) (c) (d)
3 13 4 2
10. A die is thrown once. Let A be the event that the number obtained is greater than 3. Let B
be the event that the number obtained is less than 5. Then P(AUB) is Unit VI 1

(a) 2 (b) 3 (c) 0 (d) 1
5 5
Fill in the blanks in Questions from 11 to 15.
11. A relation in a set A is called ______ relation, if each element of A is related to itself. 1

Unit I
1 0
If A + B =  , then A = _________.
1
12. and A 2B = Unit II 1
1

17

13. The least value of the function f(x) = ax +
b (a > 0, b > 0, x > 0) is _______. Unit III 1
x
14. The integrating factor of the differential equation x dy + 2y = x 2 is _______. Unit V 1
dx
OR
2
The degree of the differential equation 1 +  dy  = x is ________. Unit V 1
 dx 
15. The vector equation of a line which passes through the points (3, 4,  7) and (1, 1, 6) is
_________. Unit IV 1
OR
The line of shortest distance between two skew lines is ______ to both the lines. Unit IV 1
Q. Nos 16 to 20 are of very short answer type questions.
  17π  
16. Find the value of sin1 sin    . Unit I 1
  8 
 4
For A = 
3
17.  write A1. Unit II 1
1  1
18. If the function f defined as Unit III 1
x - 92
 ,x 3
f(x) =  x  3
k ,x 3

is continuous at x = 3, find the value of k.
19. If f(x) = x4  10, then find the approximate value of f(2.1). Unit III 1
OR
Find the slope of the tangent to the curve y = 2 sin2 (3x) at x =
π. Unit III 1
6
4
20. Find the value of
1 | x  5 | dx . Unit III 1

Section B
Q. Nos. 21 to 26 carry 2 marks each.
4x + 3 2 then show that (fof) (x) = x. for all x  2 Also, write inverse
21. If f(x) = ,x
6x  4 3 3
of f. Unit I 2
OR
Check if the relation R in the set R of real numbers defined as
R = {(a, b) : a < b) is (i) symmetric, (ii) transitive Unit I 2
x
22. Find  x 2 + 3x + 2 dx . Unit III 2

2
If x = a cos ; y = b sin , then find d y .

23. Unit III 2
dx 2
Find the differential of sin2 x w.r.t. ecosx. Unit III 2
1 1 
2
24. Evaluate    2  e2 dx . Unit III 2
1  x 2x 
18

1
25. Find the value of  x (1  x)n dx . Unit III 2
0

26. Given two independent events A and B such that P(A) = 0.3 and
P(B) = 0.6, find P(A̕∩ B̕ ) Unit VI 2
Section C
Q. Nos. 27 to 32 carry 4 marks each.
π
27. Solve for x: sin1 (1  x)  2 sin1 (x) = . Unit I 4
2
28. If y = (log x)x + xlogx, then find dy . Unit III 4
dx
29. Solve the differential equation: Unit III 4
x sin  y  dy + x y sin  y  = 0
 x  dx x
π
Given that x = 1 when y = .
2
30. If and represent two adjacent sides of a
parallelogram, find unit vectors parallel to the diagonals of the parallelogram. Unit IV 4
OR
Using vectors, find the area of the triangle ABC with vertices A (1, 2, 3), B(2, 1, 4) and
C (4, 5, 1). Unit IV 4
31. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of
type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs
of type B require 8 minutes each for cutting and 8 minutes each for assembling. Given that
total time for cutting is 3 hours 20 minutes and for assembling 4 hours. The profit for type
A souvenir is ₹100 each and for type B souvenir, profit is ₹120 each. How many souvenirs
of each type should the company manufacture in order to maximize the profit? Formulate
the problem as an LPP and solve it graphically. Unit V 4
32. Three rotten apples are mixed with seven fresh apples. Find the probability distribution of
the number of rotten apples, if three apples are drawn one by one with replacement. Find
the mean of the number of rotten apples. Unit VI 4
OR
In a shop X, 30 tins of ghee of type A and 40 tins of ghee of type B which look alike, are
kept for sale. While in shop Y, similar 50 tins of ghee of type A and 60 tins of ghee of
type B are there. One tin of ghee is purchased from one of the randomly selected shop and
is found to be of type B. Find the probability that it is purchased from shop Y. Unit VI 4

Section D
Q. 33 to 36, carry 6 marks each.
33. Find the vector and cartesian equations of the line which is perpendicular to the lines with
equations

Unit IV
x+2 y-3 z+1
= = and
1 2 4
and passes through the point (1, 1, 1). Also find the angle between the given lines. 6

19

34. Using integration find the area of the region bounded between the two circles x2 + y2 = 9
and (x 3)2 + y2 = 9. Unit III 6
2
35. Find the minimum value of (ax + by), where xy = c . Unit III 6
th th th
36. If a, b, c are p , q and r terms respectively of a G.P, then prove that Unit II 6
log a p 1
log b q 1  0
log c r 1
OR
 2 3 5 
 
If A =  3 2  4  , then find A1. Unit II
 1 1  2 
6
Using A1, solve the following system of equations :
2x  3y 5z = 11
3x + 2y 4z = 5
x + y  2z = 



























20

Paper-6
SECTION A
This section comprises multiple choice questions (MCQs) of 1 mark each.
1. A function f : R+ → R (where R+ is the set of all non-negative real numbers) defined by
f(x) = 4x + 3 is : Unit I 1
(A) one-one but not onto
(B) onto but not one-one
(C) both one-one and onto
(D) neither one-one nor onto
2. If a matrix has 36 elements, the number of possible orders it can have, is: Unit II 1
(A) 13 (B) 3
(C) 5 (D) 9
3. Which of the following statements is true for the function Unit III 1
 x 2 + 3, x  0
f(x) =  ?
 1 , x=0
(A) f(x) is continuous and differentiable  x  R
(B) f(x) is continuous  x  R
(C) f(x) is continuous and differentiable  x  R  {0}
(D) f(x) is discontinuous at infinitely many points
4. Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function
f(x) is strictly increasing in (a, b) if Unit III 1
(A) f '(x) < 0, ∀ x ∈ (a, b)
(B) f'(x) > 0, ∀ x ∈ (a, b)
(C) f '(x) = 0, ∀ x ∈ (a, b)
(D) f(x) > 0, ∀ x ∈ (a, b)
 x  y . 2  6 2   24 24 
5. If   =   , then the value of  +  is: Unit II 1
 5 xy   5 8   x y 
(A) 7 (B) 6
(C) 8 (D) 18
b
6. a f (x) dx is equal to: Unit III 1
b

b
(A) f (a  x) dx (B) a f (a + b  x) dx
a
b
(C) a f (x  (a + b)) dx (D
3
7. Let  be the angle between two unit vectors and such that sin  = . Then, . is
5
equal to : Unit IV 1
3 3
(A)  (B) 

5 4
4 4
(C)  (D) 
5 3

21

8. The integrating factor of the differential equation (1 – x2)
dy + xy = ax,
dx
1 < x < 1, is: Unit III 1
1 1
(A) (B)
x  1
2
x2  1
1 1
(C) (D)
1  x2 1  x2
9. If the direction cosines of a line are 3 k, 3 k, 3 k, then the value of k is: Unit IV 1
(A)  1 (B)  3
1
(C) 3 (D) 
3
10. A linear programming problem deals with the optimization of a/an: Unit V 1
(A) logarithmic function (B) linear function
(C) quadratic function (D) exponential function
11. If P(A | B) = P(A | B), then which of the following statements is true? Unit VI 1
(A) P(A) = P(A) (B) P(A) = 2 P(B)
(C) P(A ∩ B) = 1 P(B) (D) P(A ∩ B) = 2P(B)
2
x+1 x-1
12. is equal to: Unit II 1
x + x + 1 x2  x + 1
2

(A) 2x3 (B) 2
(C) 0 (D) 2x3  2
13. The derivative of sin (x2) w.r.t. x, at x = π is: Unit III 1
(A) 1 (B) 1
(C)  2 π (D) 2 π
3
  dy  
2
d2 y
14. The order and degree of the differential equation 1 +     respectively are: 1
  dx   dx 2
(A) 1, 2 (B) 2, 3 Unit III
(C) 2, 1 (D) 2, 6
15. The vector with terminal point A (2, – 3, 5) and initial point B (3, – 4, 7) is: Unit IV 1

16. The distance of point P(a, b, c) from y-axis is: Unit IV 1
2
(A) b (B) b
(C) a 2 + c2 (D) a2 + c2
17. The number of corner points of the feasible region determined by constraints x ≥ 0, y ≥ 0,

x + y ≥ 4 is: Unit V 1
(A) 0 (B) 1
(C) 2 (D) 3

22

18. If A and B are two non-zero square matrices of same order such that
(A + B)2 = A2 + B2, then: Unit II
(A) AB = 0 (B) AB = – BA
(C) BA = 0 (D) AB = BA
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are
given, one labelled Assertion(A) and the other labelled Reason(R). Select the correct answer from
the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion(A) and Reason(R) are true and Reason(R) is the correct explanation of the
Assertion(A).
(B) Both Assertion(A) and Reason(R) are true, but Reason(R) is not the correct explanation of
the Assertion(A).
(C) Assertion(A) is true, but Reason(R) is false.
(D) Assertion(A)is false, but Reason(R) is true.

19. Assertion(A) : For matrix A = , where θ  [0, 2π],

| Α |  [12, 4].
Reason(R) : cos θ  [– 1, 1],  θ  [0, 2π]. Unit II 1
20. Assertion(A) : A line in space cannot be drawn perpendicular to x, y and z axes
simultaneously.
Reason(R) : For any line making angles, α, β, γ with the positive directions of x, y and z
axes respectively, cos2 α + cos2 β + cos2 γ = 1. Unit IV 1
SECTION B
This section comprises very short answer (VSA) type questions of 2 marks each.
21. (a) Check whether the function f(x) = x² | x | is differentiable at x = 0 or not. Unit III 2
OR
dy 1 + y4
(b) If y = tan x , prove that x  . Unit III 2
dx 4y
22. Show that the function f(x) = 4x3  18x2 + 27x  7 has neither maxima nor minima. 2
Unit III
23. (a) Find :

 x 1 + 2x dx Unit III 2

OR
(b) Evaluate :
π2
sin x
0
4
x
dx Unit III 2

24. If and are two non-zero vectors such that ( + ) and (2 + ) , then

then prove that | |= 2 | |. Unit IV 2

23

25. In the given figure, ABCD is a parallelogram. If and
, then find and hence find the area of parallelogram ABCD. 2

Unit IV

SECTION C
This section comprises short answer (SA) type questions of 3 marks each.
26. (a) A relation R on set A = {1, 2, 3, 4, 5} is defined as R = {(x, y) : | x2 – y2 | < 8}. Check
whether the relation R is reflexive, symmetric and transitive. Unit I 3
OR
(b) A function f is defined from R → R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3.
Find function f(x). Hence, check whether function f(x) is one-one and onto or not. 3
dy 1  y2
27. (a) If 1  x 2 + 1  y 2 = a (x – y), prove that = . Unit III 3
dx 1 x2
OR
(b) If y = (tan x) , then find dy .
x
Unit III 3
dx
28. (a) Find :
x2
  x + 4  x + 9  dx
2 2
Unit III 3

OR
(b) Evaluate :

1 | x  1| + | x  2 | + | x  3 | dx
3
Unit III 3

29. Find the particular solution of the differential equation given by
x2 dy – xy = x2 cos2  y  , given that when x = 1, y = π . Unit III 3
dx  2x  2
 
30. Solve the following linear programming problem graphically: Unit V 3
Maximise z = 500x + 300y,
subject to constraints
x + 2y ≤ 12
2x + y ≤ 12
4x + 5y ≥ 20

x ≥ 0, y ≥ 0
31. E and F are two independent events such that P(Ē) = 0.6 and P(EF) = 0.6. Find P(F)
and . Unit VI 3

24

SECTION D
This section comprises long answer type questions (LA) of 5 marks each.
1 2 0
32. (a) If A =  2 1  1 , find A and use it to solve the following system of equations :
1

0 2 1
x  2y = 10, 2x  y  z = 8,  2y + z = 7 Unit II 5
OR
 1 a 2
(b) 
If A = 1 2 x  and A1 = Unit II 5

 3 1 1 
find the value of (a + x)  (b + y).
33. (a) Evaluate :
π
sin x + cos x

0
4
9 + 16 sin 2x
dx Unit III 5

OR
(b) Evaluate :
π


0
2
sin 2x tan 1 (sin x) dx Unit III 5

x2 y2
34. Using integration, find the area of the ellipse + = 1, included
16 4
between the lines x = 2 and x = 2. Unit III 5
x y1 z 2
35. The image of point P(x, y, z) with respect to line = = is
1 2 3
P (1, 0, 7). Find the coordinates of point P. Unit IV 5

SECTION E
This section comprises 3 case study based questions of 4 marks each.

Case Study - 1
36. The traffic police has installed Over Speed Violation Detection (OSVD) system at various
locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m
and even function in the dark.

A camera is installed on a pole at the height of 5 m. It detects a car travelling away from
the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the
angle of elevation of the speed camera from the car C is 
25

On the basis of the above information, answer the following questions: Unit III
(i) Express  in terms of height of the camera installed on the pole
and x. 1
dθ
(ii) Find . 1
dx
(iii) (a) Find the rate of change of angle of elevation with respect to time at an
instant when the car is 50 m away from the pole. 2
OR
(iii) (b) If the rate of change of angle of elevation with respect to time of another
car at a distance of 50 m from the base of the pole is 3 rad/s, then find the speed of the
101
car. 2
Case Study -2

37. According to recent research, air turbulence has increased in various regions around the
world due to climate change. Turbulence makes flights bumpy and often delays the flights.

Assume that, an airplane observes severe turbulence, moderate turbulence or light
turbulence with equal probabilities. Further, the chance of an airplane reaching late to the
destination are 55%, 37% and 17% due to severe, moderate and light turbulence
respectively.

On the basis of the above information, answer the following questions: Unit VI
(i) Find the probability that an airplane reached its destination late. 2
(ii) If the airplane reached its destination late, find the probability that it was due to
moderate turbulence. 2

Case Study – 3
38. If a function f: X → Y defined as f(x) = y is one-one and onto, then we can define a
unique function g: Y → X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y. Function g
is called the inverse of function f.
26

The domain of sine function is R and function sine: R → R is neither one-one nor onto.
The following graph shows the sine function.

Let sine function be defined from set A to [1, 1] such that inverse of sine function exists,
i.e., sin1 x is defined from [1, 1] to A. Unit I

On the basis of the above information, answer the following questions:
(i) If A is the interval other than principal value branch, give an example of one such
interval. 1
(ii)  1
If sin (x) is defined from [1, 1] to its principal value branch, find the value of
 1
sin1    sin1 (1). 1
 2
(iii) (a) Draw the graph of sin x from [1, 1] to its principal value branch. 2
OR
(iv) (b) Find the domain and range of f(x) = 2 sin1 (1 x). 2

27

Paper-7
SECTION A
This section comprises multiple choice questions (MCQs) of 1 mark each.
1. If the sum of all the elements of a 3 x 3 scalar matrix is 9, then the product of all its
elements is: Unit II 1
(A) 0 (B) 9
(C) 27 (D) 729
2. Let f: R+ → [5, ∞) be defined as f(x) = 9x2 + 6x  5, where R+ is the set of all non-
negative real numbers. Then, f is: Unit I 1
(A) one-one
(B) onto
(C) bijective
(D) neither one-one nor onto
a b c
3. If a  b c = kabc, then the value of k is: Unit II 1
a b c
(A) 0 (B) 1
(C) 2 (D) 4
 |x| + 3, if x  3

4. The number of points of discontinuity of f(x) =   2x, if -3 < x < 3 is : Unit III 1
6x + 2, if x  3

(A) 0 (B) 1
(C) 2 (D) infinite
5. The function f(x) = x 3x + 12x  18 is:
3 2
Unit III 1
(A) strictly decreasing on R
(B) strictly increasing on R
(C) neither strictly increasing nor strictly decreasing on R
(D) strictly decreasing on (, 0)

6. is equal to: Unit III 1

(A) π (B) Zero (0)

 2
2sin x π2
(C) 0 1  sin x cos x
dx (D)
4
dy
7. The differential equation = F(x, y) will not be a homogeneous differential equation, if
dx
F(x, y) is: Unit III 1
y
cos x – sin  y 

(A) (B)
x x

x 2 + y2  y
(C) (D) cos2  
xy x

28

8. For any two vectors and , which of the following statements is always true? 1

Unit IV
9. The coordinates of the foot of the perpendicular drawn from the point
(0, 1, 2) on the x-axis are given by: Unit IV 1
(A) (1, 0, 0) (B) (2, 0, 0)
(C) ( 5 , 0, 0) (D) (0, 0, 0)
10. The common region determined by all the constraints of a linear programming problem is
called: Unit V 1
(A) an unbounded region (B) an optimal region
(C) a bounded region (D) a feasible region
11. Let E be an event of a sample space S of an experiment, then P(S|E) = Unit VI 1
(A) P(S∩E) (B) P(E)
(C) 1 (D) 0
12. If A = [aij] be a 3  3 matrix, where aij = i  3j, then which of the following is false? 1
(A) a11 < 0 (B) a12 + a21 =  6 Unit II
(C) a13 > a31 (D) a31 = 0
13.  1 2
The derivative of tan (x ) w.r.t. x is: Unit III 1
x 2x
(A) (B)
1+ x2 1+ x4
2x 1
(C)  (D)
1 + x4 1+ x4
14. The degree of the differential equation (y)2 + (y)3 = x sin (y) is: Unit III 1
(A) 1 (B) 2
(C) 3 (D) not defined

15. The unit vector perpendicular to both vectors + and - is: Unit IV 1

x  1 2z + 1
16. Direction ratios of a vector parallel to line = y= are: Unit IV 1
2 6
(A) 2, 1, 6 (B) 2, 1, 6
(C) 2, 1, 3 (D) 2, 1, 3
cos x  sin x 0
17. If F(x) = sin x cos x 0  and [F(x)]2 = F(kx), then the value of k is: Unit II 1


 0 0 1 
(A) 1 (B) 2
(C) 0 (D) 2

29

18. If a line makes an angle of 30° with the positive direction of x-axis, 120° with the positive
direction of y-axis, then the angle which it makes with the positive direction of z-axis is: 1
(A) 90° (B) 120° Unit IV
(C) 60° (D) 0°
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are
given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer
from the codes (A), (B), (C) and (D) as given below.

(A) Both Assertion(A) and Reason(R) are true and Reason(R) is the correct explanation of the
Assertion(A).
(B) Both Assertion(A) and Reason(R) are true, but Reason(R) is not the correct explanation of
the Assertion(A).
(C) Assertion(A) is true, but Reason(R) is false.
(D) Assertion(A) is false, but Reason(R) is true.
19. Assertion(A) : For any symmetric matrix A, BAB is a skew-symmetric matrix.
Reason(R) : A square matrix P is skew-symmetric if P =  P. Unit II 1

20. Assertion(A) : For two non-zero vectors and , .
Reason(R) : For two non-zero vectors and , Unit IV 1

SECTION B
This section comprises very short answer (VSA) type questions of 2 marks each.
 1  1  1  1   π 
21. (a) Find the value of tan1    + cot   + tan sin   2   . Unit I 2
 3  3   
OR
(b) Find the domain of the function f(x) = sin1 (x² 4). Also, find its range. Unit I 2
π
22. (a) If f(x) = | tan 2x | , then find the value of f (x) at x = . Unit III 2
3
OR
dy
(b) If y = cosec (cotx), then prove that 1  x2  x = 0. Unit III 2
dx
23. If M and m denote the local maximum and local minimum values of the function
1
f(x) = x + (x ≠ 0) respectively, find the value of (M m). Unit III 2
x
24. Find :
e4x  1
 e4x + 1 dx Unit III 2

25. Show that f(x) = ex ex + x  tan1 x is strictly increasing in its domain. Unit III 2
SECTION C
This section comprises short answer (SA) type questions of 3 marks each.

dy y logx
26. (a) If x = ecos 3t and y = esin 3t, prove that =  . Unit III 3
dx x log y
OR

30

(b) Show that :
d x
| x |  = ,x  0 Unit III 3
dx |x|
27. (a) Evaluate :
2x
2

2 2  x dx Unit III 3

OR
(b) Find :
1
 x [(log x)2  3 log x  4] dx Unit III 3

28. (a) Find the particular solution of the differential equation given by
dy
2xy + y2  2x2 = 0; y = 2, when x = 1. Unit III 3
dx
(b) Find the general solution of the differential equation:
y dx = (x + 2y2) dy Unit III 3
29. The position vectors of vertices of ABC are and
. Find all the angles of  ABC. Unit IV 3
30. A pair of dice is thrown simultaneously. If X denotes the absolute difference of the
numbers appearing on top of the dice, then find the probability distribution of X. 3
Unit VI
31. Find :

 x . sin (x ) dx
2 1 32
Unit III 3

SECTION D
This section comprises long answer (LA) type questions of 5 marks each.
2x
32. (a) Show that a function f : R → R defined by f(x) = is neither one-one nor onto.
1  x2
Further, find set A so that the given function f: R → A becomes an onto function. 5
Unit I
OR
(c) A relation R is defined on N N (where N is the set of natural numbers) as :
(a, b) R (c, d)  a – c = b  d
Show that R is an equivalence relation. Unit I 5
33. Find the equation of the line which bisects the line segment joining
points A(2, 3, 4) and B(4, 5, 8) and is perpendicular to the lines
and x  15 = y  29 = z  5 . Unit IV 5
3 8 5
34. (a) Solve the following system of equations, using matrices:
2 3 10 4 6 5 6 9 20
 + ,  + = 1, +  =2 Unit II 5
x y z x y z x y z

where x, y, z ≠ 0
OR
 1 cot x    cos 2x  sin 2x 
(b) If A =  , show that AA1 =  . Unit II
 cos 2x 
 5
  cot x 1   sin 2x
31

35. If A1 denotes the area of region bounded by y2 = 4x, x = 1 and x-axis in the first quadrant
and A2 denotes the area of region bounded by y2 = 4x, x = 4, find A1 : A2 . 5

SECTION E
This section comprises 3 case study based questions of 4 marks each.
Case Study - 1
36. Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre
rolling friction and air resistance. While vehicles reach optimal fuel economy at different
speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h.

The relation between fuel consumption F (l/100 km) and speed V (km/h) under some
V2 V
constraints is given as F =  + 14 .
500 4
On the basis of the above information, answer the following questions: Unit III
(i) Find F, when V = 40 km/h. 1
(ii) Find dF . 1
dV
(iii) (a) Find the speed V for which fuel consumption F is minimum. 2
OR
(iii) (b) Find the quantity of fuel required to travel 600 km at the speed V at which
dF
=  0.01 . 2
dV

Case Study - 2
37. The month of September is celebrated as the Rashtriya Poshan Maah across the country.
Following a healthy and well-balanced diet is crucial in order to supply the body with the
proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes
improved level of energy.

32

A dietician wishes to minimize the cost of a diet involving two types of foods, food X (x
kg) and food Y (y kg) which are available at the rate of 16/kg and ₹ 20/kg respectively.
The feasible region satisfying the constraints is shown in Figure-2.
On the basis of the above information, answer the following questions: Unit V
(i) Identify and write all the constraints which determine the given feasible region in
Figure-2. 2
(ii) If the objective is to minimize cost Z = 16x + 20y, find the values of x and y at
which cost is minimum. Also, find minimum cost assuming that minimum cost is
possible for the given unbounded region. 2

Case Study - 3
38. Airplanes are by far the safest mode of transportation when the number of transported
passengers are measured against personal injuries and fatality totals.

Previous records state that the probability of an airplane crash is 0.00001%. Further, there
are 95% chances that there will be survivors after a plane crash. Assume that in case of no
crash, all travellers survive.

Let E1 be the event that there is a plane crash and E2 be the event that there is no crash. Let
A be the event that passengers survive after the journey.

On the basis of the above information, answer the following questions: Unit VI
(i) Find the probability that the airplane will not crash. 1
(ii) Find P(A | E1) + P(A | E2). 1
(ii) (a) Find P(A). 2
OR
(iii) (b) Find P(E2 | A). 2

---------------- x -----------------

33

AglaSem Earn while Learn Program. Send your papers and get paid.
Contact: support@

Study Materials
Notes

Model Papers Class 6 Notes

Sample Papers Class 7 Notes
Half Yearly Sample Papers Class 8 Notes

Class 9 Notes
Important Resources
Class 10 Notes
Periodic Table
Class 11 Notes
Writing Skills / Formats

Maps of India / World Class 12 Notes

Books and Solutions

NCERT Books
NCERT Book Solutions
HC Verma Chapter Wise Solutions
RD Sharma Solutions
CGBSE Solutions