Class 12 Question Paper 2022-23 Maths (Half Yearly)

HALF YEARLY EXAMINATION

Question Paper
2022-23

NCERT BASED SYLLABUS
FOR CBSE AND STATE BOARD
FOLLOWING NCERT

QUESTION PAPER
PT-
PT-2/HALF
2/HALF YEARLY EXAMINATION, 2022
2022-
22-23
APPLIED MATHEMATICS
Time – 3 hrs. Class – XII M.M. : 80
Date – 12.09.2022 (Monday)
Name of the student _________________________________________________ Section _____

General Instructions:
• This question paper contains two parts A and B.
• Part – A:
• It consists of three sections- I, II and III.
• Section I comprises of 7 MCQs type questions of 1 Mark each.
• Section II comprises of 6 MCQs type questions of 2 Marks each.
• Section III case study MCQs of 1 Mark each.
• Part – B:
• It consists of three sections- IV, V and VI.
• Section IV comprises of 8 questions of 2 marks each.
• Section V comprises of 4 questions of 4 marks each.
• Section VI comprises of 4 questions of 6 marks each.

Part-A
Section I : Comprises of 7 MCQs type questions of 1 Mark each
Q1 (18x 10) (mod 7) is 1
(a) 5 (b) 4 (c) 3 (d) 2
Q2 A boat running downstream covers a distance of 16 km in 2 hours while for 1
covering the same distance upstream it takes 4 hours. What is the speed of the
boat in still water?
(a) 4 km/h (b) 6 km/h (c) 8 km/h (d) 10 km/h
Q3 The solution set of 6 ≤ -3(2x-4) < 12, xє R is 1
(a) (0, 1 (b) [ 1, 0 ) (c) ( 0 , 1) (d) [0 , 1 ]
Q4 If y = Ae5x + Be-5x, then d2y/ dx² is equal to 1
(a) 25y (b) 5y (c)-25y (d) 15y

Q5 If y = x (x – 3 )²strictly decreases for the values of x given by 1
(a) 1<x<3 (b) x<0 (c) x>0 (d) 0<x< 3/2
Q6 The slope of the normal to the curve x = t² + 3t – 8 ; y = 2t² - 2t - 5, at t = 2 is 1
a) -7/6 (b) 6/7 (c) 1 (d) 5/6

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Q7 A die is rolled thrice. If the event of getting an even number is a success, then the 1
probability of getting at least two successes is
(a) 7/ 8 (b) 3/ 8 (c) 2/3 (d) 1/2
Section II : Comprises of 6 MCQs type questions of 2 Marks each.
Q8 Three types of wheat costing Rs. 18 per kg, Rs. 20 per kg and Rs. 25 per kg are 2
mixed together. If the mixed variety is sold at Rs. 22 per kg, then the ratio in which
these types of wheat should be mixed respectively is
(a) 1:2:3 (b) 2:2:3 (c) 2:3:1 (d) 1:1:2
Q9 Two pipes A and B together can fill a tank in 4 hours. Pipe B take 6 hours more 2
than A to fill the tank, if they opened separately. The time taken by A to fill the tank
alone is
(a) 2 hours (b) 4 hours (c) 6 hours (d) 8 hours.
Q10 A box contains 100 bulbs of which 10 are defective. The probability that out of a 2
sample of 5 bulbs drawn one by one with replacement none is defective is
5 5 5
1 9 9 1
(a)   (b) (c)   (d)  
2 10  10   10 
Q11 The total revenue received from the sale of x units of a product is given by R(x) = 2
3x² +6x +5. The marginal revenue, when x = 5 is
(a) 36 (b) 35 (c) 34 (d) 33
Q12 e
dx 2
Evaluate the integral  2
1 x (1 + log x )

(a) 0 (b) 1/2 (c) 2 (d) 2 -1
Q13 If A is a square matrix such that A2 = A, then (I - A)3 + A is equal to 2
(a) 0 (b) A (c) I - A (d) I
Section III: Case study MCQs of 1 Mark each
Q14 A farmer has a piece of land. He observed that he got 600 units of fruits per tree by 5
planting up to 25 trees and when 26 trees were grown, he received 15210 units of
fruits, for 27 trees he ended up with 15390 fruits, for 28 trees he got 15540 fruits
and this sequence of production of fruits continues in the same pattern as more
trees, in excess of 25, were grown.
Based on the above information answer the following questions:
i) If ‘x’ more trees, in excess of 25 are grown, then the number of fruits produced
per tree is

a) 600 – 15x b) 600 + 15x c) 600x – 15 d) 600x + 15
ii) The production of entire garden if ‘x’ more trees, in excess of 25, are planted
a) (25 + x) (600 + 15x) b) (25 – x) (600 – 15x)
c) (25 + x) (600 – 15x) d) (25 + x) (15x – 600)

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iii) The marginal production of the garden when ‘x’ more trees, in excess of 25,
are planted
a) 225 + 30x b) 225 – 30x c) 225x + 30 d) 225x –30
iv) The critical point of producing ‘x’ more units of trees is
a) 7 b) 8 c) 7.5 d) 8.5
v) The number of trees to be grown to get maximum production is
a) 30 or 31 trees b) 32 or 33 trees
c) 33 or 34 trees d) 34 or 35 trees
Part-B
Section IV: Comprises of 8 questions of 2 marks each.
Q15 Express the following matrix as the sum of a symmetric matrix and a skew 2
symmetric matrix.

A=

OR
Find the value of x from the following matrix equation:

. = 0.

Q16 Find the value of x and y from the following matrix equation: 2

. =

Q17 What positive values of x makes the following pair of determinants are equal 2

,

Q18 dy 2
Solve the differential equation xy = (x+2) (y+2); y = -1 when x = 1.
dx
Q19  1 2 2
Evaluate the integral  e x  2 −  dx. OR  log x dx.
x x3 

Q20 3
1 2
Evaluate the integral  dx
2 ( x + 1)( x + 2)

Q21 Find all the positive integers less than 30 forming the equivalence class of 5 for 2
modulo 7.

OR
Find the remainder when 862 × 783 × 671 x 549 is divided by 8 .
Q22 Priya can run 1 km in 3 min 10 sec and Preeti can cover the same distance in 3 min 2
20 sec. By what distance can Priya beat Preeti?

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Section V: Comprises of 4 questions of 4 marks each.
Q23 In a 10 km race, A, B and C each running at uniform speed get the gold, silver and 4
bronze medals respectively. If A beats B by 1 km and B beats C by 1 km, then find
by how many metres does A beat C?
OR
A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers
40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in
still water and that of the stream.
Q24 In 3 trials of a binomial distribution, the probability of 2 successes is 9 times the 4
probability of 3 successes. Find the probability of success in each trial.
OR
A class XII has 20 students whose marks (out of 30) are 14, 17, 25, 14, 21, 17, 17,
19, 18, 26, 18, 17, 17, 26, 19, 21, 21, 25, 14 and 19. If random variable X denotes
the marks of a selected student given that the probability of each student to be
selected is equally likely. Find mean, and variance of the probability distribution.
Q25 The demand and supply functions are Pd =25- x2 and Ps= 2x + 1 respectively. Find 4
the consumer's surplus and producer's surplus at equilibrium price.
dy 4
Q26 If x 1 + y + y 1 + x = 0, then find .
dx
Section VI: Comprises of 4 questions of 6 marks each
Q27 Using cramer’s rule, solve the following equations: 6
x – 2y + z = 3; 2x + y - z = 5 ; 3x – y + 2z =12
Q28 6
If matrix A = , then Prove that A2 – 4A – 5I=0, using it find A-1.

OR
Solve the following system of equation using matrix method
x + y + z = 10, 2x + y = 13, x + y - 4z = 0
Q29 In a bank, principal increases continuously at the rate of r% per year. Find the 6
value of r if Rs100 double itself in 10 years, using differential equation. (log, 2 =
0.6931)
Q30 Find the mean and variance of probability distribution of number of heads in three 6
tosses of a fair coin.
OR

Four bad eggs are mixed with 10 good ones. If three eggs are drawn one by one
with replacement, then find the probability distribution of the number of bad eggs
drawn.

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PT-
PT-2/HALF YEARLY EXAMINATION, 2022-
2022-23
MATHEMATICS STANDARD
Time – 3 hrs. Class – XII M.M. : 80
Date – 12.09.2022 (Monday)
Name of the student _________________________________________________ Section _____

General Instructions:
• This question paper contains six sections. Each section is
compulsory.
• Section A-7 Multiple Choice Questions (each of 1 mark)
• Section B-6 Multiple Choice Questions (each of 2 marks)
• Section C- 8 Subjective Type Questions(each of 2 marks)
• Section D- 4 Subjective Type Questions (each of 4 marks)
• Section E- 4 Subjective Type Questions (each of 6 marks)
• Section F- 1 Case Study based Question (5 marks)

Section-A
Multiple Choice Questions (1 mark each)
Q1. The principal value of tan-1(tan 3π/5) is
(a) 2π/5 (b) -2π/5 (c) 3π/5 (d) -3π/5
Q2. sin[π/3 – sin-1(-½)] is equal to:
(a) ½ (b) ⅓ (c) -1 (d) 1
k 8 
Q3. Value of k, for which A =   is a singular matrix is -
 4 2k 
(a) 4 (b) -4 (c) ± 4 (d) 0
 2 2 x + 3
Q4. Value of x if  is a symmetric matrix.
3 x − 2 2 

a) 3 b) 4 c) 5 d) 6
1 bc a (b + c)
Q.5 The value of 1 ca b(c + a ) will be equal to
1 ab c(a + b)

A) a3+b3+c3 B) 3abc C) a3+b3+c3 -3abc D) 0
2x −1 3 0
Q6. If = then x is
4 2 2 1

A) 3 B) 2/3 C) 3/2 D) -1/4

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Q7. If ∫ sec² (7 – 4x)dx = a tan (7 – 4x) + C, then value of a is
a) -4 b) -¼ c) 3 d) 7
Section-B
Multiple Choice Questions (2 marks each)
Q8 If sin–1x + sin–1y = π/2, then value of cos–1x + cos–1y is
(a) π/2 (b) π (c) 0 (d) 2π/3
Q9. The value of the expression sin [cot–1 (cos (tan–1 1))] is
(a) 0 (b) 1 (c) 1/√3 (d) √(2/3)
Q10. Let f : R → R be deGined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are
(a) φ, {4, – 4} (b) {3, – 3}, φ (c) {4, –4}, φ (d) {4, – 4}, {2, – 2}
Q11. If A(3 ,4) B(-7, 2) and C(x, y) are collinear then
A) x+5y+17 = 0 B) x+5y+13 = 0 C) x-5y+17 = 0 D) None of these
d2y
Q12. Given that x = at2 and y = 2at, then value of is
dx 2
1 1 1 − 2a
a) − b) − c) d)
2at 3 2at 2 t2 t
Q13. ∫(sin2x−cos2x) / sin2x cos2x dx is equal to
(a) tan x + cos x + c (b) tan x + cosec x + c (c) tan x + cot x + c (d) tan x+ sec x + c
SECTION - C
Subjective Type Questions (each of 2 marks)
Q14. Show that function f: R→R given by f(x)= 2x+3 is one-one onto.
Q15. Differentiate xx w.r.t x.
OR
Find dy/dx if y+sin y = cosx
d2y dy
Q16. If y = 3e2x + 2e3x, prove that 2
−5 + 6y = 0
dx dx
OR
If y = A sin x + B cos x, then prove that d2y/dx2 + y = 0
Q17. The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area
increasing when the length of an edge is 12 cm?
Q18. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the
ground, away from the wall, at the rate of 2 m/s. How fast is its height on the wall

decreasing when the foot of the ladder is 4 m away from the wall ?
OR
A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post
which is 6 metres high. Find the rate at which the length of his shadow increases.

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Q19. Solve :

Q20. Solve :

Q21. Solve :

OR
Integrate w.r.t x:

SECTION - D
Subjective Type Questions (4 marks each)

Q22. Express as a sum of a symmetric and a skew symmetric matrix.

Q23. If A= and B= then prove that (AB)T = BTAT .

Q24. Find k if f(x) is continuous

OR
Find values of a & b if f(x) is continuous at x= 2 and x=10

Q25. Find the intervals in which the function f given by f (x) = 4x3 – 6x2 – 72x + 30 is

(a) increasing (b) decreasing.
OR
Find local maximum and local minimum values of the function f given by
f (x) = 3x4 + 4x3 – 12x2 + 12

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SECTION - E
Subjective Type Questions (6 marks each)
Q26. Show that R = { (a,b) : |a-b| is divisible by 2 where a, b are integers} is an equivalence
relation.
Q27. Solve using inverse of matrices –
5x+3y+z= 16, 2x+y+3z= 19, x+2y+4z= 25
dy cos 2 ( a + y )
Q28. If cos y = x cos (a+y), with cos a ≠ ± 1 then prove that =
dx sin a
OR
Differentiate w.r.t x : cosx.cos2x.cos3x
Q29. Show that the maximum volume of a cone inscribed in a sphere of radius R cm is 8/27
times of the volume of the sphere.
OR
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere
2R
of radius R is . Also find the maximum volume of the cylinder.
3
SECTION - F
Case Study based Question (5 marks)
Q.30 Megha wants to prepare a handmade gift box for her friend's birthday at home. For
making lower part of the box, she takes a square piece of cardboard of side 20 cm.

Based on the above information, answer the following questions.
(i) If x cm be the length of each side of the square cardboard which is to be cut off from
corners of the square piece of side 20cm, then possible value of x will be given by the
interval :
(a) [0, 20] (b) (0, 10) (c) (0, 3) (d) none of these
(ii) Volume of the open box formed by folding up the cutting corners can be expressed as
(a) V = x(20 - 2x) (20 - 2x) (b) V = x/2 (20 + x) (20 - x)
(c) (b) V = x/3 (20 - 2x) (20 + 2x) (d) V = x(20 - 2x) (20 - x)
dV
(iii) The values of x for which = 0 , are
dx
(a) 3, 4 (b) 0, 10/3 (c) 0, 10 (d) 10, 10/3
(iv) Megha is interested in maximising the volume of the box, so, what should be the side of

the square to be cut off so that the volume of the box is maximum ?
(a) 12 cm (b) 8 cm (c) 10/3 cm (d) 2 cm
(v) The maximum value of the volume of the box is
17000 11000 8000 16000
(a) cm3 (b) cm3 (c) cm3 (d) cm3
27 27 27 27

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