Roll No: g4
Name: <wka ul
DEVAMATHA CMIPUBLIC SCHOOL
FIRST MODEL EXAMINATION- 2023 -24
MATHEMATICS (041)
Class - XII Marks: 80
Time: 3 hrs.
General Instructions :
Read the following instructions very carefully and strictly follow them:
(i) The question paper contains 38 questions. All questions are compulsory.
(ii) The question paper is divided into five sections A, B. C,D and E.
(ii) In Section A, Question Nos. Ito 18 are Multiple C'hoice Questions (MCQs) and question number 19
and 20 are Assertion Reason Based Questions of l mark each.
(iv) neach.
Section B, Questions Nos. 21 to 25 are Very Short Answer (VS4) type guestions, carrying 2marks
() In Section C, Questions Nos. 26 to31 are ShortAnswer (SA) type questions, carrving 3 marks euch.
(i) In Section D, Questions Nos. 32 to 35 are long answer (LA) type questions carrying 5 marks each.
(vi) In Section E. Questions Nos. 36 to 38 are case study based questions carrying 4marks each.
(vii) There is no overall choice. However, an internal choice has been provided in 2questions in Section B,
3 questions in Section C, 2questions in Section Dand 2 yuestions in Section E.
(ix) Use of calculators is not allowed.
SECTION A (1x20=20)
0 a 11
1. If A b 1 is askew symmetric matrix.Then (a+b+c) is
C
a) I b c) 4 d) 2
3 0
2. IfA (adjA) =]0 3 0 then the value of |A| + \adjA| is
lo 0 3
b) 9 c) 3 d) 27
3. A and B are 2 matrices AB = A and BA =B then B is
b) A c) I d) 0
dy
2 +2 = 2** then da
c)2*y
a) 2y-x b) -2x
5. Aline makes the same angles Owith x andz axes and ß with yaxis and if sin'ß - 3sin'e
then cos 0 is
7
a) , b)
1
6. The sum of the order and deg. of the differential equation sinx (dx + dy) = cosx (dx - dy)
A) 1 B) 2 C)0 D)Not defined
7. The feasible region for the contraints s0,,> 0/; =0
G
D
H =0
X
A - B C
a) area DHF b) area AHC c) line segment EG d) line segment GI
8. The number of feasible solutions of the LPP maximise z = 15x + 30y subject to
3x + y< 12, x +2y< 10x >0y>0 is
a) 1 b) 2 c) 3 d) infinite
1 d
9. If[f(x)] =1+x2
dx
then
dx
[f(x'))
3x 3x2 6x5 -6x5
a)
1+x3
b) 1+x6
c) 1+x6 d)
1+x6
J0. If and b are 2 unit vectors and is the angle between them - b| is
b) 2cos 2 c) sin 2 d) cos 2
n. k is unit vector rto band then the second unit vector Lr to b and is
a) x h b) x c) b x
(sin5x
12. f (x) 3x is continuous at x =0.
X= 0)
c)5 d) 3
13. The integrating fator of (ylog y) ds =(logy - x)dy is
1
a) ogy b) log (logy) c) I tlogy d) logy
2
14. IfAandB are invertible matrices of the same order. I(AB)-|=8 and A| then |B| is
4
a) 6 6 d)
-3
15. and xyz = 7. x +y -6z = || then A(adjA) is
1
a) -5/ b)SI c) 41 d) 41
J6. If f(2a-x)da =m, o f(x) dx = nthen c24
f(x) dx =
a) 2m 2n b) 2m+2n C) m- 1 d) 2m - 2n
2
17.
Three vectors , b, such that projection of on - projection of on b, lal =2. |b|= I,
lcl =3, .h= 1l - 2b - c =
a) V30 b) V10 c) V12 d) V13
IfP (B)=0.2 P(A) =0.6, P()-0.5. Find P()
5
a) 10
b) d)
8
Question number 19 and 20 are Assertion (A) and Reason (R) based questions. Select the
correct answer from the codes (a)(b) (c) (d) as given below
a) Both Assertion (A), and Reason (R) are True and Reason (R ) is the
of the Assertion (A)
correct explanation
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 and Reason (R) is False.
d) Assertion (A) is False and Reason(R) is True.
19. Assertion (A):The number of onto functions from a set P containing 5 elements to a set O
containing2 elements in 30.
Reason (R): Number of onto functions froma set containing m elements to a set ontaining
n elements in n".
20. Assertion (A): Equation of aline passing through the points (1.2,3) and (3, -3. 3) is
X-3 y+1 Z-3
2 3
Reason (R ): Equation of a line passing thrOugh (e,y,z), (N2,y2,23) is
Z2 1
3
SECTION - B (2x5=10)
21. a) A function f: A ’B defined as f(x) =2x is both one-one and onto :if A={.2.3,4}.
Then find set B.
OR
by Evaluate sin' sin cos'cos +lan-'()
22, Find the sub intervals in which f(x) = log (2+x)
.X>-2 is increasing or
decreasing?
1
23. Ify =xx find dy
d
at x = 1.
OR
1fx =a sin2t, y = a (cos2t + log tant) prove that cot2t
dx
24. Find fe sin 2x dx.
25. The bottom of a rectangular swimming tank is 50cm x 20cm. Water is
at arate of 500cm'/mFind the rate at which the level of the pumped into the tank
water in the tank is rising?
SECTION -C (3r6=18)
2x+3
Evaluate f Vx2+4x+1 d.
sinx+cosx
27) Evaluate ? Vsin2x
dx
6
OR
S sin2x tan (sinr)dx
There are 2 coins one of them is a biased coin such that P (head): P(tail) is 1:3 and
other
coin is a fair coin. A coin is selected at random and tossed once. If the coin
showed head.
then find the probability that it is a biased coin.
24. Solve (1+y+ x²y) dx +(x+x')dy =0
OR
Xex - y tx dy 0
30,/ Ify =x* +(sinx) find ydx
Solvegraphically
minimise z -5x + 10y
X - 2y > 0, x +2y < 120, x+y> 60. x. y20.
OR
4
Minimise z 20x + 10y
A 2y s40
N y 30
4x +3y 60
SECTION -D (5x4-20)
32.
Using integration find area of the regionin the 1 guadrant enclosed by the x-axis the line by
yNandx´ +y 32.
A-{1.2.3,4,5,6,7.8.9} and R be the relation in A x A defined by (a.b) (c.d) if a td=btc
for (a,b), (c.d) E AXA. Prove that R is an cauivalence relation. Also obtain the equivalent
class [(2.5)).
OR
A=R- 3} and B-R-( consider f:A Bdefined by f(x) =x-3 Show that fis one
one and onto.
-1 0 2
IfA 3 4 2 -4 Find AB and use it to solve
1 2J 2 -1 5
X- y=3
2x + 3yt 4z= 17
y+ 2z =7
OR
1 2 3 -1 1
IfA=-15 3 -15 6 -s| find (ABj
2
-2 5 -2
Prove that _y-2 Z-3
35. 2 3 4
x-2 y+2 Z-3
are skew lines.
3 -4 4
OR
X-11 y+2 z+8
the line Alsofind
Find the foot of the Lr draw from (2, I, 5) on 10 4 -11
distance between (2.-1,5) and its image
5
(4x3=12)
SECTION - E
Case Study - 1
36.
P
.9..9.
the
Apoint P is given on the circumference of acircle of radius r. chordOR is parallel to
tangent at P, QOR =20
LYWrite the formula for finding the arca of APOR.
2) Express QR and PSin terms of rand
3) Determine the maximum possible arca of the triangle PQR.
OR
Find OOS so that area of APOR is maximum.
Case Study - 2
37. bz +2j
-3k
=7i
b
....
= 2i- j -2k
Ifb, parallel to k and b, l k
)Write b, in terms of
2) b, in terms of a and b
3)Find b, and b, so that b =b, tb2
38.
white balls black balls
white it is not replaced into the bag,
A ballis drawn at random from this bag. If it is colour. The process IS repeated.
otherwise it is replaced along with another ball of the same
n How many ways the first two balls drawn can occur?
im What are their probabilities?
iii) Find the probability that third ball is black.
OR
of first two balls are white?
fthethird ball is black what is the probability
