VELAMMAL NEW-GEN EDU NETWORK

12
MODEL PAPER [2024 - 2025]
041-MATHEMATICS

TIME: 3 Hrs DATE : MAX: 80 MARKS

General Instructions:
Read the following instructions very carefully and strictly follow them:
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into five Sections A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and
questions number 19 and 20 are Assertion-Reason based questions of 1 mark
each.
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type
questions, carrying 2 marks each.
(v)In Section C, Questions no. 26 to 31 are short answer (SA) type questions,
carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions
carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are case study based questions
carrying 4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and
2 questions in Section E.
(ix) Use of calculators is not allowed.

SECTION A
This section comprises multiple choice questions (MCQs) of 1 mark each.

1. The principal value of cos−1 (cos( − 680°)) is:

(a) 2π/9 (b) −2π/9 (c)34π/9 (d)π/9

2. How many 2x2 order matrices can be formed with entries 0 and 1 only.
(a) 8 (b) 16 (c) 4 (d) 32

2 �−3 �−2
3. . If A = 3 −2 −1 is a symmetric matrix then x is
4 −1 −5
(a) 3 (b) 6 (c) 8 (d) 0

4.If the sum of all the elements of a 3 × 3 scalar matrix is 9, then the product of
all its elements is:
(a) 0 (b) 9 (c) 27 (d) 729

5. If A is a square matrix such that �2 = A, then (I – A)3 + A is equal to

(a) I (b) 0 (c) I – A (d) I + A

6. If �2 − A + I = O, then the inverse of A is

(a) �−2 (b) I − A (c) O (d) A
7. The function f(x) = x|x| is
(a) continuous and differentiable at x = 0
(b) continuous but not differentiable at x = 0
(c) not continuous but differentiable at x = 0
(d) neither continuous nor differentiable at x = 0

8. The function f(x) = �3 + 3x is increasing in the interval

(a) (−∞, 0) (b) (0, ∞) (c) R (d) (0,1)

9.If sec2 (7 − 4x) dx = a tan(7 - 4x) + C , then value of a is

(a) 7 (b)-4 (c) 3 (d)-1/4
�
sin �
10 0
2
sin �+ cos �
dx
� � �
(a) 3
(b)� (c) 4
(d) 2

�
11. The area of the region bounded y = |sin �|, x axis and |x|= 2 is:
(a)1 sq. units (b) 0.5 sq. units (c) 2 sq. units (d) none of these

12. The sum of the degree and order of the differential equation
1
�� 2 �2 � 3
1 + ��
= c ��2
is

(a) 1 (b) 2 (c) 4 (d) undefined

��
13. The integrating factor of the differential equation�� - y = cos � is
(a) �� (b) �2� (c) �−� (d)�−2�

14. If � is a unit vector (�+ �).(�− �) = 24, then |�| is

(a) 16 (b) 4 (c) 2 (d) 5

15. If θ is the angle between vectors � and � and | �. � | = | � × � |, then the value
� �
(a) 0 (b)� (c) 4 (d) 2

16. The feasible region satisfied by the constraints

x + y ≤ 5, x ≤ 4, y ≤ 4, x ≥ 0, y ≥ 0, 5x + y ≥ 5, x+ 6y ≥ 6 is bounded by
(a)4 straight lines (b) 5 straight lines
(c)6 straight lines (d) 7 straight lines

17. Based on the given shaded region as the feasible region in the graph, at
which point(s) is the objective function Z = 3x + 9y maximum?
(a) Point B (b) Point C
(c) Every point on line segment CD (d) Point D

18. The probability of having 53 Mondays in a leap year chosen randomly

(a) 1/7 (b) 3/7 (c)4/7 (d) 2/7

Questions number 19 and 20 are Assertion and Reason based questions. Two
statements are given, one labeled Assertion (A) and the other labeled 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) : |sin �| is a continuous function.

Reason(R) : If f(x) and g(x) are both continuous functions, then f(x)ºg(x) is also
continuous.

20. A: The Principal value of cot−1 ( − √3 )+tan−1 ( 1) +sec−1 (2/√3) is equal to

5�/4.
R: Domain of cot−1 � and sin−1 � are respectively (0, �) and [−�/2 , �/2]

SECTION B
This section comprises very short answer (VSA) type questions of 2 marks each.
1 1
21. Find the principal value of tan (√3) − sec (−2).

22. Find the derivative of log (sec x + tan x)

(or)
��
Find �� if x = a cost, y = b sint.
23. Find the intervals in which the function f (x) = 2�2 -3x is strictly increasing and
strictly decreasing.

24. For the given vectors �=2� -� +2� and �=-� -� +� , find the unit vector in the
direction of �+�

(or)

For any three vectors � , � � are any three vectors,prove that � ×( �+ � )+ � ×( �

+ � )+ � ×( �+ � ) = 0.

25. Show that the points with position vectors � - 2 � + 3 � , -2 � + 3 � - � , 4 � - 7

� + 7 � are collinear.

SECTION C
This section comprises short answer (SA) type questions of 3 marks each.

26. A ladder is 5 m long leaning against the wall. The bottom of the ladder is
pulled along the ground away from the wall at the rate of 2cm/s. How fast is the
height on the wall decreasing when the foot of the ladder is 4 m away from the
wall.

27. Show that the function f(x) = sin x is

a) increasing in (0,π/2)
b) decreasing in (π/2,π)
c) neither increasing nor decreasing in (0,π)

28. Integrate
�
sin �−cos �
0
2
1+sin � cos �
dx
( or )
1
I = 1+�� dx

2 2 2 2
29. For any vector � ,prove that � × � + � ×� + � ×� =2�

( or )

(x−5) (2y+6) (z+3) (x−2) (y+1) (z−6)

Find the angle between the lines : 1
= −2
= 1
and : 3
= 4
= 5

30. Solve the following Linear Programming Problem graphically: Minimize:

z=x+2y , subject to the constraints: x + 2y ≥100, 2x -y≤0, 2x +y ≤200, x,y≥0 .
31. A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4
black balls. One bag is selected at random. From the selected bag, one ball is
drawn. Find the probability that the ball drawn is red
( or )

The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/8
respectively. If all three try to solve the problem simultaneously, find the
probability that exactly one of them can solve it.

SECTION D
This section comprises long answer (LA) type questions of 5 marks each.
1
32. Sketch the graph of y = |x+1| and hence evaluate −3 |x + 1| dx .

1 2 2
33. If A = 2 1 2 find �−1 and hence prove that �2 - 4A - 5I = 0 .
2 2 1
�2 � ��
34. If y = (tan−1 � )2 . Show that (�2 + 1)2 ��2 + 2x(�2 + 1)�� = 2.

( or )

1+�� − 1−��
, −1≤�<0
If � � = �
2x + 1
is continuous at x = 0
, 0≤�≤1
�−2

(4−x) (y) (1−z)

35. Find the foot of the perpendicular from (2,3,4) to the line 2
= 6
= 3

( or )

(x+1) (y+3) (z+5) (x−2) (y−4) (z−6)

Show that the lines 3 = 5
= 7
and 1
= 3
= 5
intersect and find
the point of intersection.

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

36. In order to set up a rainwater harvesting system, a tank to collect rainwater is

to be dug. The tank should have a square base and a capacity of 250�3 . The
cost of land is Rs. 5000 per �2 and cost of digging increases with depth and for
the whole tank, it is 40000ℎ2 ,where h is the depth of the tank in meters. x is the
side of the square base of the tank in meters.
Based on above information, answer the following questions
(i) Find the total cost C of digging the tank in terms of x
(ii) Find �� ��
(iii) (a) Find the value of x for which cost C is minimum.
OR
(b)Check whether the cost function c(x) is expressed in terms of x is
increasing or not, where x > 0

( or )

Read the following passage and answer the question given below. The relation
between the height of the plant (y in cm) with respect to its exposure to the
1
sunlight is governed by the following equation y = 4x− 2 �2 , where x is the
number of days exposed to the sunlight for
x ≤ 3.
(i) Find the rate of growth of the plant with respect to the number of days
exposed to the sunlight.
(ii) Does the rate of growth of the plant increase or decrease in the first three
days?
(iii) What will be the height of the plant after 2 days?

37. Kriti and Kirat are two friends studying in class XII in a school at Chandigarh.
While doing their mathematics project on Relations and Functions they have to
collect the name of five metro cities and four cities other than metro cities of India;
and present the name of cities in the form of sets. They have collected the name
of cities and write in the form of sets given as follows: A={ five metro cities of
India}= { Delhi, Mumbai, Bangalore, Calcutta, Pune} and B = {four non metro
cities of India} = { Patiala, Agra, Jaipur, Ahmedabad}
Answer the following questions using the above information
(i) How many functions exist from A to B.
(ii) Riya wants to know how many relation are possible from A to B.
(iii) Karan wants to know how many reflexive relation on set B.

(or)

The math teacher of class XII dictates a math problem as follows: Draw the
graph of the function, f of x is equal to modulus of x plus three minus one in the
closed interval -3 to +3
Three students Rakesh, Sravya and Navya have interpreted the same dictation
in three different ways and they have noted the function as f(x)= |x +3 -1|,
f(x)=|x|+3 -1 and f(x)=|x+3|-1 respectively.
Based on the above information answer the followings:
(i) Sravya ' s graph is in ' V shape ' with vertex
(a) (-3,1) (b) (3,-1) (c) (-2,0) (d) (0,2)
 −x − 4 , �� � ≤ − 3
(ii) The function f(x)= is another form of the function of…?
x + 2, �� � > − 3
(a) Rakesh (b) Sravya (c) Navya (d) None of them

(iii) Find the domain and range of the function interpreted by Sravya.

38. A coach is training three players .He observed that player A hits 4 times the
shot in 5 tries, player B hits 3 times the shot in 4 and player C is able to hit twice
the shot in 3 tries.
Based on above information, answer following questions
i) A,B,C all hit. It is a shot. What is the probability that the shot is hit by B only?
ii) If A,B,C all try , what is the probability that it was hit by none?
iii)Find the probability that the shot was hit by exactly 2 players.