Class 12 Pre Board Question Paper 2024 Maths

2024

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Pre-Board Exam 2024 Question Paper

PREBOARD EXAM 2023-24
CLASS – XII
MATHEMATICS (041)
Time Allowed : 3 Hours Maximum Marks : 80
General Instructions: This Question paper contains - five sections A, B, C, D and E. Each section
is compulsory. However, there are internal choices in some questions.
1. Section A has 18 MCQs and 02 Assertion-Reason based questions of 1 mark each.
2. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
3. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
4. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
5. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks
each) with sub parts.
SECTION A

(Multiple Choice Questions) Each question carries 1 mark

1. If A is any square matrix of order 3 × 3 such that |𝐴| = 3 , Then the value of |𝑎𝑑𝑗𝐴| is
1
(a) 3 (b) 3
(c) 9 (d) 27
2. If A = [ aij] is a symmetric matrix of order n, then
(a) aij = 1/aij for all i, j (b) aij ≠ 0 for all i,j
(c) aij = aji for all i,j (d) aij = 0 for all i, j

3. Write the element a23 of a 3 x 3 matrix A = (aij) whose elements aij are given by ai j= .

(a) 2 × 3 (b) (c) (d) None of these
4. If for any square matrix A, A(adjA) = [6 0 0 6 ] then value of |𝐴|
(a) 3 (b) 6 (c) 8 (d) 1
5. If a matrix A is both symmetric and skew symmetric then matrix A is
(a) scaler matrix (b) a diagonal matrix (c) a zero matrix (d) rectangular matrix
2 3 𝑑𝑦
6. If 𝑥 = 𝑡 and 𝑦 = 𝑡 , then 𝑑𝑥 is equal to
2𝑡 3𝑡
(a) 3 (b) 2𝑡 (c)3𝑡 (d) 2
7. If the function f(x) is continuous at x= 0, then the value of k is

(a) (b) (c) (d)

𝑎
3
8. The value of ∫ 𝑠𝑖𝑛 𝑥 𝑑𝑥 is
−𝑎

(a) a (b) a/3 (c) 1 (d) 0
2
9. Evaluate∫ 1+cos𝑐𝑜𝑠 2𝑥 𝑑𝑥

(a)tan 𝑡𝑎𝑛 𝑥 +c (b) 𝑙𝑜𝑔 tan 𝑡𝑎𝑛 𝑥 +c (c) tan(x/2) +c (d) log(1+cos2x)+c
𝑑𝑦
10. The integrating factor of differential equation cos 𝑐𝑜𝑠 𝑥 𝑑𝑥 + y sin x = 1 is
(a) cos x (b) tan x (c) sec x (d) sin x
11. If m is the order and n is the degree of given differential equation,
3

( ) 𝑑𝑦 5
2
𝑑𝑦
𝑑𝑥
2 + ( ) + 𝑥 = 0 then what is the value of m + n
𝑑𝑥
4

(a) 5 (b) 9 (c) 4 (d) 7
→ → → → → →
12. If |𝑎| = 10 ,|𝑏| = 2 and 𝑎. 𝑏 = 12 , then the value of |𝑎 × 𝑏| is
(a) 5 (b) 10 (c) 14 (d) 16

13. The value of λ for which the vectors are parallel is
( a) 2/3 (b). 3/2 (c) 5/2 (d) 2/5
^ ^ ^ ^ ^ ^
14. The scalar projection of the vector 2 𝑖 + 3𝑗 - 5𝑘 on the vector 5 𝑖 + 5𝑗 + 5𝑘 is
2
(a) 5√3
(b) 0 (c) 25 (d) None of these
15. The value of i. (𝑗 × 𝑘) + 𝑗. (𝑘 × 𝑖) + 𝑘. (𝑗 × 𝑖) is
(a) 1 (b) 3 (c) 0 (d) − 1
16. The corner points of the feasible region determined by the following system of linear
inequalities: 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0,0), (5,0), (3,4), (0,5). Let Z= px + qy, where
p,q > 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is
(a) p = q ( b) p = 2q (c) p = 3q (d) q = 3p
17. The Solution set of system 3x + 6y ≥ 80, 4x + 3y ≥ 100, x, y ≥ 0 is
(a) Lies in I quadrant (b) Lies in II quadrant
(c) Lies in III quadrant (d) Lies in IV quadrant
18. Three balls are drawn from a bag containing 2 red and 5 black balls, if the random variable X
represents the number of red balls drawn, then X can take values
(a) 0, 1, 2 (b) 0, 1, 2, 3 (c) 0 (d) 1, 2

ASSERTION-REASON BASED QUESTIONS

In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.

(a) Both 𝐴 and 𝑅 are true and 𝑅 is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) 𝐴 is true but 𝑅 is false.
(d) A is false but 𝑅 is true.
19. Assertion (A) : f: 𝑁→𝑁 given by 𝑓(𝑥) = 5𝑥 is injective but not surjective
Reason (R) : If co-domain ≠ range , then the function is not surjective.

20. Assertion (A) : The direction-cosines of the line joining the points (1, 0, 0) and (0, 1, 1) is

( , , )
−1
3
1
3
1
3
Reason (R) : direction ratios = direction cosines.
SECTION B

This section comprises of very short answer type-questions (VSA) of 2 marks each

21. Find the value of 𝑐𝑜𝑠 ⁡ 𝑐𝑜𝑠⁡ 6
−1
( ( )). 7π

22. Find the rate of change of the area of a circle with respect to its radius 'r' when r = 6 cm
OR
Show that the function f given by f(x) = x3 – 3x2 + 4x, x ∈ R is increasing on R
23. Find dy/dx if 2x + 3y = sin y.
^ ^ ^ ^ ^ ^
24. For what value of 'a' the vectors: 2 𝑖 - 3𝑗 + 4𝑘 and a𝑖 + 6𝑗 - 8𝑘 are collinear?
OR
→ → → → → ^ ^ ^ →
Find the value of λ if 𝑎 + 𝑏 and 𝑎 - 𝑏 are orthogonal given that 𝑎 = 𝑖 − 𝑗 + 7𝑘 and 𝑏 =5
^ ^ ^
𝑖 − 𝑗 + λ𝑘

25. Find the Vector equation of the line which passes through the point (-2,4,-5) and is parallel to the
𝑥+3 2−𝑦 2𝑧+3
line 2
= 5
= 6

SECTION C
(This section comprises of short answer type questions (SA) of 3 marks each)
2
𝑥 +𝑥+1
26. Find : ∫ 𝑑𝑥.
(2 )
( 𝑥+2) 𝑥 +1

27. If y =xcosx + (cos x)x, find .
OR

If y = , show that .
2 𝑑𝑦 2
28. Solve the differential equation:2𝑥 𝑑𝑥
− 2𝑥𝑦 + 𝑦 = 0.
OR
𝑑𝑦 2 π
Solve differential equation 𝑐𝑜𝑠 𝑥 𝑑𝑥 + 𝑦 = 𝑡𝑎𝑛𝑥 (0≤𝑥 < 2 )
π
2

sin𝑠𝑖𝑛 𝑥
29. Evaluate : ∫ 𝑑𝑥
sin𝑠𝑖𝑛 𝑥 + cos𝑐𝑜𝑠 𝑥
0

OR

Evaluate
30. Solve the following Linear Programming Problem graphically:

Maximize Z = 17.5x + 7y
subject to the constraints,
x + 3y ≤ 12
3x + y ≤ 12
x, y ≥ 0
31. A die is thrown twice and the sum of numbers appearing is observed to be 7. What is the conditional
probability that the number 2 has appeared at least once.

SECTION D
(This section comprises of long answer-type questions (LA) of 5 marks each)
32. Let 𝐴 = {𝑥 ϵ 𝑍; 0≤𝑥≤12 }. Check whether the relation 𝑅 = {(𝑎, 𝑏); 𝑎, 𝑏∈𝐴 |𝑎 − 𝑏| is divisible by
4} in the set A is reflexive , symmetric or transitive.
OR
Let L be the set of lines in XY plane and R be the relation in L defined as R={
(𝑙1,𝑙2): 𝑙1, 𝑖𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑙2}. Prove that the relation R is an equivalence relation. Find the set of
all the lines which are related to the line 𝑦 = 2𝑥 + 4.

33. Find the product of A= and hence solve the system of
linear equations :
x - y+ z = 4
x - 2y – 2z = 9
2x + y + 3z = 1
2
34. Using integration, find the area bounded by the curves 𝑦 = 𝑥 , 𝑎𝑛𝑑 𝑦 = |𝑥|.
OR
Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = –1
and x = 1.
35. Find the coordinates of the image of the point (1, 6, 3) with respect to the line
→ ^ ^ ^ ^ ^
( )
𝑟 = 𝑗 + 2𝑘 + λ (𝑖 + 2𝑗 + 3𝑘); where ' λ ' is a scalar. Also, find the distance of the image from the
y axis.

SECTION E

(This section comprises of 3 case-study/passage-based questions of 4 marks each)

36.Case study 1: Read the following passage and answer the questions given below:

You want to make two gardens in the shape of square and circle in front of your house. If you purchase
a wire of length 28m to fence these gardens and you have used x meters of wire to fence circular garden.

(i) What is the Radius of the circular garden and side of square garden?

(ii) If you want to minimize the combined area of both gardens without wasting the wire of length
28m. Then How much length of the wire will be needed to fence the circular garden. And how
much length of the wire will be needed to fence the squared garden

37. Case study 2: Read the following passage and answer the questions given below.

The Relation between the height of the plant (y in cm) with respect to exposure to sunlight is governed
1 2
by the following equation 𝑦 = 4𝑥 − 2
𝑥 where x is the number of days exposed to sunlight.

(i) What is the number of days it will take for the plant to grow to the maximum height?
(ii) What is the maximum height of the plant?
(iii) What will be the height of the plant after 2 days?
OR
7
If the height of the plant is 2 cm, then what is the number of days it has been exposed to the
sunlight ?

38. Case study 3: Read the text carefully and answer the questions:
A doctor is to visit a patient. From the past experience, it is known that the probabilities that he
will come by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1, and 0.4. The
probabilities that he will be late are 0.25, 0.3, 0.35, and 0.1 if he comes by cab, metro, bike and other
means of transport respectively.

i. When the doctor arrives late, what is the probability that he comes by metro?
ii. When the doctor arrives late, what is the probability that he comes by other means of
transport?

PRE-BOARD EXAM 2023-24
CLASS: XII
MATHEMATICS
TIME: 3 hrs. MARKING SCHEME Max Marks: 80
QUESTION ANSWER/SOLUTION MARKS
NUMBER
1 c 1
2 c 1
3 b 1
4 b 1
5 c 1
6 d 1
7 c 1
8 d 1
9 a 1
10 c 1
11 a 1
12 d 1
13. a 1
14 b 1
15 a 1
16 d 1
17 a 1
18 a 1
19 a 1
20 c 1
21 1
( ( )) = 𝑐𝑜𝑠 ⁡(𝑐𝑜𝑠⁡(2π −
−1
𝑐𝑜𝑠 ⁡ 𝑐𝑜𝑠⁡ 6
7π −1 5π
6 ))
= 𝑐𝑜𝑠 ⁡(𝑐𝑜𝑠⁡( )) =
−1 5π 5π
6 6 1

22 A = π r2 ½

dA/ dr = 2πr ½
at r = 6 dA / dr = 12π cm2/cm 1
OR
2
f ′(x) = 3x – 6x + 4 1

= 3(x – 1)2 + 1 > 0, in every interval of R. 1
Therefore, the function f is increasing on R.
23 2x + 3y = sin y.
2 + 3 dy/dx = cos y dy/dx 1
dy/dx = 2/(cos y - 3) 1
24 2 -3 4 1
− = − = −
a 6 −8
1
a= -4
OR
𝑎⃗ + 𝑏¯ = 6 𝚤 − 2𝚥 + (7 + 𝜆)𝑘^ and 𝑎⃗ − 𝑏¯⃗ = −4𝚤 + (7 − 𝜆)𝑘^ 1
𝑎⃗ + 𝑏¯⃗𝑎𝑛𝑑𝑎⃗ − 𝑏¯ will be orthogonal if, (𝑎⃗ + 𝑏¯⃗). (𝑎⃗ − 𝑏¯ ) = 0
i.e., if, −24 + (49 − 𝜆2) = 0 ⟹𝜆2 = 25 1
i.e., if, 𝜆 = ±5

25 Direction of the required line = Direction of given line
→ ^ ^ ^
𝑏 = 2𝑖 − 5𝑗 + 3𝑘
→ → → 1
𝑟 = 𝑎+ λ𝑏
Required equation of line
→ ^ ^ ^
𝑟 = ( -2𝑖+ 4𝑗 -5𝑘 ) + λ(2𝑖 − 5𝑗 + 3𝑘)
^ ^ ^ 1

26 2
𝑥 +𝑥+1 𝐴 𝐵𝑥+𝐶
= +
(
( 𝑥+2) 𝑥 +1
2
) ( 𝑥+2) (𝑥2+1) 1
Solving A = 3/5, B = 2/5, C = 1/5
2
𝑥 +𝑥+1 3 𝑑𝑥 1 2𝑥𝑑𝑥 1 𝑑𝑥 1
∫ 2 𝑑𝑥 = 5 ∫ ( 𝑥+2) + 5 ∫ 2 + 5∫ 2
( 𝑥+2) 𝑥 +1( ) 𝑥 +1 ( 𝑥 +1 ( ) ( )
3 1 2 1
= 5 log |x + 2 | + 5 log | 𝑥 + 1 | + 5 tan-1x + c ( ) 1

27 Putting U = x cos x and V = (cos x)sin x
1

Finding 1

Finding
1

Writing the value of
OR 1.5

1.5
Finding (1 + x2) y1 = 2 tan-1x
Again differentiating & obtaining the result
28 𝑑𝑦 2𝑥𝑦−𝑦
2
Getting 𝑑𝑥 =
2𝑥
2
1

This is a homogeneous diff. eq.
𝑑𝑦 𝑑𝑣
So let y = vx ⇒ 𝑑𝑥 = 𝑣 + 𝑥 𝑑𝑥
2 1
1
Putting in (i) and getting − 2 𝑑𝑣 = 𝑥 𝑑𝑥
𝑣

( 2
Integrating we get – − 𝑣 = log 𝑙𝑜𝑔 |𝑥| + 𝑐) 1
2𝑥
Putting 𝑦
= log 𝑙𝑜𝑔 |𝑥| + 𝑐 is the required solution

OR

𝑑𝑦
+ 𝑦𝑠𝑒𝑐 𝑥 =
2 𝑡𝑎𝑛𝑥 1
𝑑𝑥
𝑡𝑎𝑛𝑥
2
𝑐𝑜𝑠 𝑥 1
𝐼. 𝐹 = 𝑒

𝑦𝑒
𝑡𝑎𝑛𝑥 𝑡
= ∫ 𝑡. 𝑒 𝑑𝑡 + 𝑐 𝑝𝑢𝑡 𝑡 = 𝑡𝑎𝑛𝑥 1
−𝑡𝑎𝑛𝑥
𝑦 = (𝑡𝑎𝑛𝑥 − 1) + 𝑐𝑒
29

1

1

1

OR

1
Writing I =

Taking 16 common & putting 1-sin 2x=(sin x – cos x)2. 1
Substitution of sin x – cos x = t & limit change
1
Integrating & getting the result

30. Maximize Z = 17.5x + 7y … (1) subject to the constraints,
x + 3y ≤ 12 … (2) 3x + y ≤ 12 … (3) x, y ≥
0 … (4)
The feasible region determined by the system of constraints is as follows.

1

The corner points are A (4, 0), B (3, 3), and C (0, 4). The values of Z at these
corner points are as follows.
1.5

0.5

The maximum value of Z is 73.50 at (3, 3).
31 A={(1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)}
B={(1,2),(2,1),(2,2),(2,3),(3,2),(4,2),(2,4),(5,2),(2,5),(6,2),(2,6)} 1
A B = (5,2) (2,5) 1
2 1
P(A/B)= 11

32 Reflexive :|𝑎 − 𝑎| = 0, which is divisible by 4, ∀ 𝑎 ϵ 𝐴
∴(𝑎, 𝑎)∈𝑅, ∀ 𝑎 ϵ 𝐴∴ R is reflexive 1.5
Symmetric : Let (𝑎, 𝑏) ϵ 𝑅
⇒ |𝑎 − 𝑏| is divisible by 4
⇒ |𝑏 − 𝑎| is divisible by 4 (∵ |𝑎 − 𝑏| = |𝑏 − 𝑎|)
⟹(𝑏, 𝑎) ϵ 𝑅∴ R is symmetric
Transitive :Let (𝑎, 𝑏), (𝑏, 𝑐) ϵ 𝑅
⇒ |𝑎 − 𝑏|&|𝑏 − 𝑐| are divisible by 4 1.5
⇒𝑎 − 𝑏 = ±4𝑚, 𝑏 − 𝑐 = ±4𝑛, 𝑚, 𝑛 ϵ 𝑍
Adding we get, 𝑎 − 𝑐 = 4 (±𝑚±𝑛)
⇒ |𝑎 − 𝑐| is divisible by 4 ∴(𝑎, 𝑐)∈𝑅
2
∴ R is transitive
OR
1

1.5

1.5

1.5

1

33
1.5
Finding AB = 8 = 8I

1.5
Writing given system as BX= C so that X = =
Putting values and getting sol as x= 3, y=-2, z= -1 2
2
34 y = x and y = |x|

2
1
Required area =2{area of OACO – area of ODACO}
2
= 1/3 sq. units
OR

The line y = 3x + 2 meets x-axis at x = -2/3 and its graph lies below x-axis for x 2
(
∈ − 1, 3
−2
) −2
and above x-axis for x ∈ 3 , 1 ( )
The required area = Area of the region ACBA + Area of the region ADEA 1
| −23 | |1 |
| | | |
= | ∫ ( 3𝑥 + 2 )𝑑𝑥 | + | ∫ ( 3𝑥 + 2 )𝑑𝑥 |
| −1 | | −2 | 2
| | |3 |
= 1/6 + 25/6 = 13/3 sq units
35 Let P (1, 6, 3) be the given point, and let ' L' be the foot of the perpendicular from '
P ' to the given line AB (as shown in the figure below). The coordinates of a
general point on the given line are given by

x-0 y-1 z-2
= = = λ,
1
1 2 3
λ is a scalar, i.e., x = λ, y = 2 λ + 1 and z = 3 λ + 2
Let the coordinates of L be (λ, 2 λ + 1, 3 λ + 2) .

So, direction ratios of PL are λ- 1, 2 λ + 1 - 6 and 3 λ + 2 - 3, i.e. λ- 1, 2 λ- 5 and 3
λ- 1.

Direction ratios of the given line are 1, 2 and 3, which is perpendicular to 1
PL .
Therefore, (λ- 1)1 + (2 λ- 5)2 + (3 λ- 1)3 = 0 => 14 λ- 14 = 0 => λ= 1
So, coordinates of L are (1, 3, 5).
Let Q ( x1 , y1 , z1 ) be the image of P (1,6, 3) in the given line. Then, L is the mid-point 1
of 1
PQ.
( x1 + 1) ( y1 + 6) (z1 + 3) 1
Therefore, = 1, = 3 and =5
2 2 2
=> x1 = 1, y1 = 0 and z1 = 7
Hence, the image of P (1,6, 3) in the given line is (1,0, 7).

Now, the distance of the point (1,0,7) from the y - axis is √12 + 72 =√50 units.

36 (i) x =2πr
r = x/2π m. 1.5
(ii) The length of the wire will be needed to fence the squared garden =
112/(4+π) m. 2.5
37 (i) x=4 1
(ii) Maximum height=8cm 1
(iii) Height after 2 days=6cm
OR 2
x=1

38. A: he will come by cab B: he will come by metro
C: he will come by bike D: he will come by other means
E: HE arrives late
P(A) = 0.3, P(B) = 0.2, P(C) = 0.1, P(D) = 0.4
P(E/A) = 0.25, P(E/B) = 0.3 P(E/C) = 0.35, P(E/D) =0.1 1
0.2×0.3 6
i)P(B/E) = 0.3×0.25+0.2 ×0.3+0.1 ×0.35+0.4 ×0.1
= 21
= 2 /7
1.5
0.4×0.1 4
ii) P(D/E) = 0.3×0.25+0.2 ×0.3+0.1 ×0.35+0.4 ×0.1
= 21
1.5

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