MBOSE Class 12 Sample Question Paper 2027 Mathematics

FOR MEGHALAYA CLASS 12 PREPARATION

MBOSE Class 12
Sample Question
Paper 2027
Mathematics
EXAM YEAR TYPE SUBJECT

Meghalaya Class 12 2027 Sample Paper Maths

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Subject: Mathematics a
Sample Question Paper

Class – XII
Full Marks: 70 Time: 3 Hours
General Instructions:

Read the following instructions very carefully and strictly follow them:
1. This Question paper contains 38 questions. All questions are compulsory.
2. This Question paper is divided into five Sections - A, B, C, D and E.
3. In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) with only one correct
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option and Questions no. 19 and 20 are Assertion-Reason based questions of 1 mark each.

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4. In Section B, Questions no. 21 to 25 are Very Short Answer (VSA)-type questions, carrying 2
marks each.
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5. In Section C, Questions no. 26 to 31 are Short Answer (SA)-type questions, carrying 3 marks each.
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6. In Section D, Questions no. 32 to 35 are Long Answer (LA)-type questions, carrying 5 marks each.
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7. In Section E, Questions no. 36 to 38 are Case study-based questions, carrying 4 marks each.
8. 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 one subpart each in 2 questions of Section
E.
Q No. Question Marks
Section-A Marks) 20x1=20
This section comprises multiple choice questions (MCQs) of 1 mark each:

1. The principal value of 𝑠𝑖𝑛−1 (−½) is: 1
a) 𝜋/6 b). π/3 c) –π/6
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2.
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Let A= {a, b, c}. Then the total number of distinct relation on A is 1

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a) 64 b) 32
a d) 512

3. The value of 𝑐𝑜𝑠 −1 (½)+2𝑠𝑖𝑛−1 (½) is: 1

2𝜋 2𝜋 𝜋 𝜋
a)
3
b) − 3 c) 3 d) − 3

4. 𝑘 8] 1
The value of k, for which A=[ in a singular matrix is:
4 2𝑘
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a) 4 b) −4 c) ±4 d) 0
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s em If 2𝑥 + 3𝑦 = 𝑠𝑖𝑛 𝑦, then
𝑑𝑥
is:
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𝑑𝑦

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2
cos 𝑦−3
2
b)
sin 𝑦−3
2
c)
3−cos 𝑦
2
d)
3−sin 𝑦
6. 4 𝑑𝑥 1
If 𝑥 = 4𝑡, 𝑦 = 𝑡 , then
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𝑑𝑦

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1 1 4 4
a) 𝑡 2 b) − 𝑡 2 c) 𝑡 2 d) − 𝑡 2
7. The rate of change of the area of a circle with respect to its radius r at r=6 cm is: 1

a) 10π b) 12 π c) 8 π d) 11 π
8. The value of 𝑥 for which 𝓍(𝑖̂ + 𝑗̂ + 𝑘̂) is a unit vector, is: 1

a) ±1/3 b) ±√3 c) ±1 d) ±1/√3
9. A line makes angle 90˚, 60˚ and 30˚ with the positive direction of x-axis, y-axis 1
and z-axis respectively. Then the direction cosines of the line are:
3
a) 0, ½, √
2
3
b) 1, ½, √2
√3
c) 0, ,½
2
√3
d) ½, ,0
2

10. If P(B)=0.5 and P(A∩B) =0.32, then P(A/B) is: 1

a) 0.64 b) 1.6 c) 0.46 d) 0.66

1
11. The anti-derivative (or integral) of the function (√𝑥 + ) equals: 1
√𝓍

1
a) 𝓍 1/3+2 𝓍 1/2+c
3
2 2/3 1 2
b) 3 𝓍 +2 𝓍 +c
2 3/2
c) 3 𝓍 1/2
+2 𝓍 +c
3 1
d) 2 𝓍 3/2+2 𝓍 1/2+c

12. 1 1
∫ 𝑥2(1− ) 𝑑𝑥 is equal to?
𝓍2

𝓍3
a) a) − 𝓍+c
3
𝓍2
b) 2 − 𝓍 + 𝑐
𝓍3
c) +𝓍+𝑐
3
𝓍2
d) +𝓍+𝑐
2

13. The maximum value of the function f(𝓍) = −(𝓍 − 1)2 + 10 is 1
a) 1 b) −1 c) 10 d) −10
14 If |𝑎⃗| = √3 , |𝑏⃗⃗ | =2 and angle between |𝑎
⃗| and |𝑏⃗⃗ | is 60º, then 1

2

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a) 2 b) ½ c) √3 d) 1/√3
15. 1 𝑑𝑥 1
The value of ∫0 is
1+𝑥 2

a) 0 b) 1 c) π/2 d) π/4

16. 𝑑2 𝑦 𝑑𝑦 2 𝑑𝑦 1
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The order and degree of differential equation 𝓍𝑦 𝑑𝓍2 + 𝓍 (𝑑𝑥 ) − 𝑦 𝑑𝑥 = 0 is
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a) 1,2 c) 1,1 d) 2,2

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17. In thea 1 ag
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𝜋

a
2

a) Strictly increasing
b) Strictly decreasing
c) Neither increasing nor decreasing
d) d None of the above
18. If A and B are events such that P(A/B) = P(B/A), then:

a). A⸦B but A≠B b). A=B 1
c). A∩B=Ф o mP(A) = P(B)
c
. questions two statements are given, one
d).

labeled Assertion (A) and the other labeled Reason e
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Questions number 19 and 20 are Assertion and Reason based

l as (R). Select the correct answer from the codes (A),
(B), (C) and (D) as given below.
a) Both (A) and (R) are true and (R) a g
is the correct explanation of the (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): The matrix 1
0 1 3
[−1 0 5] is skew –symmetric
−3 −5 𝑏
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if b = 0.
Reason (R): A square matrix A is said to be skew-symmetric if A′= −𝐴
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Assertion (A): In the set A= {1,2,3}, a relation is defined as R= {(1,1), (2,2)} is 1
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reflexive.

ag Reason (R): A relation R is reflexive in A if (a, a)∈ 𝑅 𝑓𝑜𝑟 𝑎𝑙𝑙 a∈ 𝐴.
Section-B
This section comprises 5 very Short Answers (VSA) type questions of 2 marks
each.
21. 1 −2] [1 2 3] 2
(A). Compute the product: [
2 3 2 3 1

OR
1 5],
(B). For the matrix A= [ verify that (A+A’) is a symmetric matrix.
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22 (A). Find the values of 𝓍, if |2 4| = |2𝑥 4| 2
5 1 6 𝑥

OR

(B). Using determinant, find the area of the triangle with the vertices (1, 0), (6, 0) 2
and (4, 3).

23. 𝑘𝑥 + 1, 𝑖𝑓 𝑥 ≤ 5 2
Find the value of k so that the function f(𝑥)= {
3𝑥 − 5, 𝑖𝑓 𝑥 > 5
Is continuous at 𝑥 =5.

24 𝑐𝑜𝑠2𝑥−𝑐𝑜𝑠2𝛼 2
(A). Evaluate: ∫ 𝑐𝑜𝑠𝑥−𝑐𝑜𝑠𝛼 d𝑥

OR
𝑥2
(B). Evaluate: ∫ d𝑥
1−𝑥 6 2

25 (A). If A and B are two events such that P(A) = 1, P(B) = ½ and P(A ∩ B) = 1, 2
4 8
find P(not A and not B).
OR

(B). Probability of solving a specific problem independently by A and B are 1/2
2
and 1/3 respectively. If both try to solve the problem independently, find the
probability that the problem is solved.

Section-C
This section comprises 6 Short Answer (SA) type questions of 3 marks each.

If y = (tan⁻¹ 𝑥)², show that (𝑥² + 1)² y₂ + 2𝑥(𝑥² + 1)y₁ = 2, where y₁
26. 3

𝑑𝑦 𝑑2𝑦
= 𝑑𝑥 and y₂ = 𝑑𝑥 2 .

27 (A). Evaluate: ∫ (𝑥² + 1) log𝑥 d𝑥 3
OR
𝑥−3 x
(B) Evaluate: ∫ 𝑒 d𝑥 3
(x−1)3

28 (A). Find a unit vector perpendicular to each of the vectors 𝑎⃗ + 𝑏⃗⃗ and 𝑎⃗ − 𝑏⃗⃗, 3
where 𝑎⃗ = 3𝑖̂ + 2ĵ + 2k̂ and 𝑏⃗⃗ = î + 2ĵ - 2k̂.
OR
(B). Find the position vector of a point R which divides the line joining the two
points P and Q whose position vectors are î + 2ĵ - k̂ and -î + ĵ + k̂ respectively, in 3
the ratio 2:1
(i) internally
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a
(ii) externally

29. Find the vector and cartesian equation of the line passing through the point 3
(1, 2, 3) and parallel to the vector 3î + 2ĵ - 2k̂.

30 (A). A stone is dropped into a quiet lake and waves move in circles at the 3
speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm,
how fast is the enclosed area increasing?
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OR
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(B). Find the intervals in which the function f given by

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f(𝑥) = 4𝑥³ - 6𝑥² - 72𝑥 + 30 is 3
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(i) strictly increasing

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(ii) strictly decreasing
31 a
(A). Prove that ∫0
𝑎 √𝑥
𝑑𝑥 =
𝑎 3
√𝑥+√𝑎−𝑥 2

OR

2𝑥
(B). Evaluate: ∫ 𝑑𝑥
(𝑥 2 + 1)(𝑥 2+ 3) 3

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Section-D

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This section comprises 4 Long Answer (LA) type questions of 5 marks
each.
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32. Minimise Z = −3𝑥 + 4y subject to 𝑥 + 2y ≤ 8, 3𝑥 + 2y ≤ 12, 𝑥 ≥ 0, y ≥ 0. 5

33 (A). Evaluate ∫𝜋/2 𝑙𝑜𝑔 𝑠𝑖𝑛𝑥 𝑑𝑥 5
0
OR

(B). Show that the semi-vertical angle of the cone of maximum volume and
5
of given slant height is tan⁻¹√2.
34 (A). Find the shortest distance between the lines whose vector equations
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OR

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(B) Solve the following:

(i) If 𝑎⃗, 𝑏⃗⃗, 𝑐⃗ are unit vectors such that 𝑎⃗+ 𝑏⃗⃗+ 𝑐⃗ = 0
⃗⃗, find the value 3
of 𝑎⃗. 𝑏⃗⃗ + 𝑏⃗⃗. 𝑐⃗ + 𝑐⃗. 𝑎⃗

(ii) Find the area of the parallelogram whose adjacent sides are
2
determined by the vectors 𝑎⃗ = î - ĵ + 3k̂ and 𝑏⃗⃗ = 2î - 7ĵ + k̂.
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35 (A). Show that (𝑥 𝑑𝑦 − 𝑦𝑑𝑥) = √𝑥 2 + 𝑦2 𝑑𝑥 is a homogeneous 5
differential equation and solve it.
OR
𝑑𝑦 1
(B). Solve the linear differential equation (1 + 𝑥2 ) + 2𝑥𝑦 = ,
𝑑𝑥 1+𝑥2 5
given that y = 0 when x = 1.

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

36. Case Study - 1 Raju, Rahim and Ria go to the market to purchase grocery
items. Raju buys 4 kg onion, 3 kg wheat and 2 kg rice. Rahim buys 2 kg
onion, 4 kg wheat and 6 kg rice. Ria buys 6 kg onion, 2 kg wheat and 3 kg
rice. Raju pays ₹60, Rahim pays ₹90 and Ria pays ₹ 70. Based upon the
above information, answer the following questions:
(i) Form the equation required to solve the problem of finding the 1
cost of each item and express it in the form of A𝑥 = B.
(ii) Find |A| and confirm if it is possible to find A⁻¹. 2
(iii) (a) Find A⁻¹ if possible. 2
OR
(b) Find A² − 5I, where I is an identity matrix.
37. Case Study - 2 Suppose f is a real function on a subset of real number and
c is any point on the domain of f. Then f is continuous at c if,

lim 𝑓(𝑥) = lim+ 𝑓(𝑥) = f(c)
𝑥→𝑐 − 𝑥→𝑐

The function f is given by:

𝑎𝑥 + 5, 𝑖𝑓 𝑥 < 5
f(𝓍) = { 10, 𝑖𝑓 𝑥 = 5
𝑏𝑥 + 20, 𝑖𝑓 𝑥 > 5

The given function is continuous at x = 5. Based upon the above
information, answer the following questions:
2
(i) Find the relation between a and b.
1
(ii) Find the value of a.
1
(iii) Find the value of f(3).
38. Case Study - 3 A bag contains 4 red and 4 black balls; another bag
contains 2 red and 6 black balls. One of two bags is selected at random and
a ball is drawn from the bag.
Based upon the above information:
(i) Find the probability that the ball drawn was a red ball. 2
(ii) What is the probability that the red ball is drawn from the first bag? 2

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