PRACTICE PAPER -2
Maximum Time:3hours MM: 80
GeneralInstructions:
ainternal
This Question paper contains -five sections A, B, C, Dand E. Each section is compulsory. However, there are
choices in some questions.
(ii) Section A has 18 MCQ's and 02Assertion-Reason based questions of 1 mark each.
(iüi) Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
() Section C has6 Short Answer (SA)-type questions of 3marks each.
(p)SectionD has 4Long Answer (LA)-type questions of 5marks each.
) Section Ehas 3source based/case based/passage based/integrated units of assessment (4 marks each) with
sub- parts.
SECTION-A
(A) discontinuous at only one point
Multiple Choice Questions [IMark Each]
(B) discontinuous at exactly two points
1, LetR be a relation in the set N given by () discontinuous at exactly three points
(D) none of the above 1
R= {(4, b) : a = b-2, b > 6}OEer
Then
6. If tan = k, then is equal to
(B) (6, 8) e R dx
(A) (8, 7)eR
(C) (3, 8) e R (D) (2, 4) e R 1
(A) (B)
2. Let f: R ’ R be defined as f(x) = x, Choose the
correct answer.
(C) (D) 1
(A)fis one-one onto.
(B) fis many-one onto.
(C) fis one-one but not onto.
7. The interval in which the function f(x) = 2x + 9xt
(D) fis neither one-one nor onto. 1 12x-1 is decreasing is
(A) (-1, o) (B) (-2, -1)
3. IfA = [a;] isa square matrix of order 2 such that (C) (- o, - 2) (D) (-1, 1) 1
[1, when i j then A is
0, when i =j' 8. Derivative of x with respect to x is
(A) (B)2x
2 3
(C) 3x (D) 2x
1
9. The position vector of three consecutive vertices of a
|6 0 -1
parallelogram ABCD are
4. The value of the determinant 21 4 is
|1 1 3 A(4i +2j -6k),B(5Î 3j+k) and C(12í +4j+5k).
(A) 10 (B) 8 The position vector of D is given by
(C) 7 (D) 7 1 (A) -3i-5,-10k (B) 21i +3,
5. The function f()= 4-x () 11i+9,-2k (D) -11f-9j+2k 1
4x
L0. The vector equation of the line -_yt0-Z 16. The angle between two vectors å and b
3 7 2 with
is magnitudes V3 and 4, respectively, andab= 2/3
(A) P=(si-4j+6i)+(3 +7j-zi) (A) (B)
2
(B) F-(+7j+ 2i) + (si-4j +6â) ( (D) 1
() i-(si-4j+6i) +(-i+7j-2â)
17. If Aand Bare two events such that P(B) = 0.6, PA/R
(D) Y=(-3-7j+2X) +a(-5i +4j -6X) 1 = 0.1, then P(A n B) =
11. ALPP isas follows: (A) 0.02 (B) 0.04
Minimise Z = 2x ty (C) 0.06 (D) 0.08 1
Subject to the constraints x >3, x<9, y20 18. IfA and B are two events such that B
x-y>0,x + ys14 A
The feasible region has and P(A) + P(B) =2 ,then P(B) is equal to
(A)5corner points including (0, 0) and (9, 5)
(B) 5corner points incuding (7, 7) and (3, 3)
(C) 5corner points including (14,0) and (9, 0) (4)
(D) 5 corner points including (3, 6) and (9, 5) 1
4 5
12. The value of (log 4+3sin x is ( (D) 9
1
+3cosy 4
ASSERTION-REASON BASED QUESTIONS
(A)2 (B) 1 In the following questions, a statement of assertion
(C) 0 (D) -2 1
(A) is followed by a statement of Reason (R). Choose
tanx-1
13. Anti-derivative of with respect to x is the correct answer out of the following choices.
tan +1
(A) Both A andR are true and R is the correct
explanation of A.
(A)
(B) Both A and R are true but R is not the correct
explanation of A.
(B) (C) A is true but R is false.
(D) A
is false but Ris true.
(9 19. Assertion (A): If a line makes angles a, B. y with
positive direction of the coordinate axes, then sin'a
1 + sin B+ sin y= 2.
Reason (R): The sum of squares of the direction
cosines of a line is 1.
cos 2x
14. The value of dx is ..
(sin x +cos x)² 20. Assertion (A): Feasible region is the set of points
(A) log |cos x + sin xl +C which satisfy all of the given constraints and objective
function too.
(B) log lcos x- sinx +C
(C) log lcos x+ sin x|'+C Reason (R): The optimal value of the objective
(D) log |cos x+ sin x|-+C 1
function is attained at the points on X-axis only. 1
15. Equation of a line passing through point (1, 2, 3) and SECTION-B
equally inclined to the coordinate axis, is This section comprises of very short answer type
Z questions (VSA)of 2 marks each.
(A) - (B)
1 2 3 12 3 21. If cosa+ cos- B+ cos-y =3n, then find the value
of a(B + )-B(y + a) + y(a + B).
OR
1
PracticePaper-2 333)
29. 1f A R- (3) and B=R-(1). Consider the function
Express tan in the
-sin I X-2
f:A’ Bdefined by fx) for all x e A.
simplest form. X-3
Then show that fis bijective. 3
22. The Sum of a matrix and its transpose OR
Find one such matrix for which this holds true. Show
Let N denote the set of all natural numbers andR
be the relation on N×Ndefined by (a, b) R(c, d) if
yourwork.
ad(b + c) = bo(a + d). Show that R is an equivalence
OR
relation.
End the value of a, b, c and d from the equation:
a-b 2a+c|-1 5 30. Acompany follows a model of bifurcating the tasks
into the categories shown below. 3
2-b 3c+d 0 13 IMPORTANT
NOT URGENT NOT URGENT
23.Ifa, b, are three non-zero unequal vectors such
that e.b = a.c, then find the angle between a urgent and not urgent but
important important
and b-c.
24. Events E and F are independent. Find P(F), if P(E)
3
-and P(E UF) =5. 2
urgent but not not urgent and not
important important
25. The twoequal sides of an isosceles triangle with fixed
base b are decreasing at the rate of 3 cm/s. How fast
is the area decreasing when the two equal sides are
equal to the base? 2
At the beginning of a financial year, itwas noticed
SECTION-C that:
40% of the total tasks were urgent and the rest were
This section comprises of short answer type questions
(SA)of 3marks each. not.
[1 2 31 half of the urgent tasks were important, and
26. IfA =3 -2 1 then show that A- 23A 401 = 0. 30% of the tasks that were not urgent, were not
|4 2 1 important
3 What is the probability that a randomly selected task
OR that is not important is urgent? Use Bayes' theorem
Find the value of k, so that the lines x = = kz and and show your steps.
z-2 =2y +1= -z +lare perpendicular to each 31. Show that the height of the right circular cylinder
othe.
of greatest volume which can be inscribed in a right
27. Solve the following differential equation: circular cone of height h and radius r is one-third of
the height of the cone, and the greatest volume of the
dx 4 3
cylinder is times the volume of the cone.
Given thaty= 1, when x= 1.
OR
Solve the differential equation:
SECTION-D
This section comprises of long answer-type
(1+e)dy +(1+y')e'dx =0 3 questions (LA) of 5 marks each.
olIIasin t-b cos t, y = a cos t + bsin t, then prove 32. A merchant plans to sell two types of personal
computers - a desktop model and a portable model
that 3
that will cost 25000 and 40000, respectively.
He estimates that the total monthly demand of () Let the target is hit by A,B, and C. Then find the
computers will not exceed 250 units. Determine the probability that A, Band, C all will hit. 1
number of units of each type of computers which (ii) What is the probability that 'none of them will hit
the merchant should stock to get maximum profit if the target'? 1
he does not want to invest more than 70 lakhs and (iii) What is the probability that B, C will hit andA
if his profit on the desktop model is 4500 and on will lose? 2
portable model is 5000. 5 OR
|1 2 31 What is the probability that any two of A, Band
33. If A = 2 3 -3 find A- and hence solve the Cwill hit?
|-3 2 4
37. Case-Study 2: Read the following passage and
system of equations answer the questions given below.
x+ 2y -3z = -4;2r + 3y + 2z = 14; 3x -3y - 4z The shape of atoy is given asf(x) =6(2x-x). To make
=-15 5 the toy beautiful2sticks which are perpendicular to
OR each other were placed at a point (2, 3), above the
Find the inverse of following matrices. toy.
|1 -1 1]
A=2 -1 0
1 0 0|
34. Find the equation of the line which intersect the lines
x+2 _y-3 z+1
and y-2 Z-3
and
1 2 4 2 3 4
passes through the point (1, 1, 1). 5
35. Find the area of the region in the first quadrant
enclosed by x-axis, the line x = V3y and the circle x (i) At what points the given function has maxima and
+y=4. 5 minima? 1
(iü) What is the maximum value of the given function? 1
SECTION-E (ii) Find the second order derivative of the function at
This section comprises of 3 case-study/passage-based X=5. 2
questions of with two sub-parts. First two case study OR
questions have three sub-parts (i), (iü), (ii) of marks 1, 1, Find the interval in which f(x) is increasing.
2respectively. The third case study question has two sub 38. Case-Study 3: Read the following passage and
parts of 2 marks each.
answer the questions given below.
36. Case-Study 1: Read the following passage and Today in the class of Mathematics, Mrs. Agrawal in
answer the questions given below.
explaining the inverse trigonometric function. She
A coach is training 3 players. He observes that the draws the graph of the sinx and and write down
player A can hit a target 4 times in 5 shots, player B the following about the principal value of branch
can hit 3 times in 4 shots and the player C can hit 2 function sin-1.
times in 3 shots.
Principal value of branch function sin: It
is a function with domain (- 1, 1] and range
I 3 and SO on
or
corresponding to each interval, we get a branch ot
the function sin x. The branch with range
is called the principal value branch. Thus, sin:
Practice Paper-2 335)
(i) Find the value 2
3
5r/2 (ii) Find the domain of sin Vx-1 and sin-[x]. 2
2r
3/2
X
2
-3r/2
-2r
-5r/2
y= sin-lx
