Karnataka 2PUC Preparatory Exam Question Paper 2025 Maths

DISTRICT LEVEL II PUc PREPARATORY EXAM, JAN - 2025
Time: 3 Hours Sub: MATHEMATICS (35) Max. Marks: 80
General Instructions:
1. The question paper has Five parts, namely A,B, C, Dand E. Answer allthe parts.
2. PART-Ahas 15 Multiple choice questions (MCQ'S), 5 fill in the blanks of 1 mark each.
3. Use the graph sheet for the question on Linear programming in PART-E.
PART-A
15x 1 = 15
Answer ALL the guestions:
1. If Set A contains 5elements and the Set Bcontains 6 elernents, then the number of
one - one and onto mappings from A to B.
(A) 720 (B) 120 (C) 0 (D) None
2. The graph of the function y = sin" x isthe mirror image of the graph of the function
y=sin xalong the line
(A) x= 0 (B) y = 0 (C) x+y=0 (D) x- y=0
3. For matrices Aand B, AB =0, then
A) A =.0 or B =0 (B) It is not necessary that A=0 or B=0
C) A
= O and B=0 (D) All the above statements are wrong
|25 0 0
4. IfA isa square matrix of order3 and A. adj A=| 0 25 0 then |A=
0 25
(A)5 (B) 25 (C) 125 (D) None of these
5. Funtion which is continuous and differentiable at x=0
(A) (B) 1/x (C) logx (D) None

6. If y = log, 10then
d:
1
(A) (B) (C) 10 (D) None
10 e

7. Which of the following functions decreasing on
(A) sin x (B) CoSX (C) tanx (D) cos3x
8. The radius of circle is increasing at.the rate of 0.7 cm/ sec then the rate of increase of
its circumference is N Cm| sec
(A)2 (B) 1.4 (C) 0.7 (D) 4.9
9. sin x.cos x. dr
sin 2x cos 2x sin 4x
(A) sinx (B) (C) (D)
2 4 2

10. sin'x. cos*x. dx =
(A) -1 (B) 1 (C) 0 (D) 2

Page 1 P.T.O.

11.Direction cosines of negative y-axis
(A) -1,-l,0 (B) 0, 0, -1 (C) 0, -1, 0 (D) -1, 0, -1

12.Projection of vector a =2i+3j+4k on Z- axis
2 3 4
(A) (B) (C) (D) 4
V29 V29
13. the lines makes anale a. B. y with positive direction of the coordinate axes. hen
sin' a +sin B+sin" y =
(A) 2 (B) 1 (D) -1
(C)0
14.If P(4)=0.4, P(B) =0.5 and P(An B) =0.25 then P(A|B) is
(A) 3
(B) (c) (D) -
1
15.If A and B are two
independent events such that P(4)= P(B) = then

P(neither Anor B)=
(A) 3 7
3 (B) (c)
I. Fill in fhe blanks by choosing the appropriate answer
from those given in the bracket:
(-1, o, 2, 3, 4, 5) 5x1=5
16.If f(x) =(logx)´then f'(1) -
17. Minimumvalue of f(x)=|x+21 is
18. The number of arbitrary constants in the general solution of
diffrential equation of thrid
order is

19.If
(2a-36)×(3-26)= 2(xb) then the value of 2is.
20. Probability of solving a specific problem independently by A and B are½
and 1/3
respectively. If both try to solve the problem then the probability that the problem is
solved in k/3 then the value of k

PART -B
Answer any SIX guestions: 6x 2 12

21.Evaluate: cos 19
COS
6

(2 -2
22.Find the inverse of matrix A=|
4 3

23.If Vx +y =10 show that dx =0.

24.Prove that the function f)=e* do not have maxima or minima.
Page 2 P.T.O.

26. Find
x(logx)
solution of x +y.=0.
20. Verifythat the functiony =a'-r? rel-a. a)is
27. If 27 -37+4k and Ai +6j-8k are collinear then find .
and which is parallelto
28.Find Cartesian equation of the line through the point (5, 2, -4)
the vector 3i +2j -8k.
known
both children are males, it is
29.A couple hastwo children. Find the probability that
that at least one of the children is male.
PART-C 6x3= 18
Answer any $IX guestions:
relation R in R the set of real numbers defined as
30.Check whether the
R={(a,b) : asb'} is reflexive, symmetric and transitive.

31.Solve: tan Ean,*>0.
cosx -sin x 01
0 then show that F(x).F(y)= F(x+y).
32. If F(x)= sin x COSX

0 0 1
33.If x =y* find
d
=(-2)' (x+4) is
34. Find the intervals in which the function f given by f(x)
(a) decreasing (b) increasing.
sin x
35. Find sin(x+a).
and (0, 1, 2)
36. If the vertices A, B and C of a triangle are (1, 2, 3), (-1, 0, 0)
respectively, then find angle ABC.

37. Find the distance between the lines F=6i +2j+ 2k +a(i-2j+ 2k)
and

F=-4i-i+ (37 -2j-2E)
Black balls. One
38. Bag contains 4 Red and 4-Black balls, Bag l centains 2 Red and 6
probability
bag is selected at random and a ball is drawn as found to be red. What is the
that Baglis selected?
Page 3 P.T.O.

PART - D
Answer any FOUR of the following questions: 4x 5=20

X-2
39.Consider the function f:A’ Bdefined by f(x) = Where A=R-3} and
x-3

B=R-1 .Is f one-one and onto?
2 31
40. If A =|3 -2 1 then show that A - 23A-401 =0.
4 2

41.The sum of three numbers is 6. If we multiply the third number by 3 and add the second
number to it we get 11. By adding the first and third numbers, we get double the second
number. Represent if algebraically and find the numbers using matrix method.

42.If x=a(cost +t.sint) and y =a(sint- t. cost) find d'y
1 d
43.Find the integral of w.r.t. x and evaluate
x V7-6x-x
44. Find the area of the region bounded by the line y = 3x + 2 the x-axis and the ordinates
x=-1and x= 1 byintegration method.

45.Find equation of curve passing through (0, 2) given that the sum of the coordinates of
any point on the curve exceeds the magnitude of the slope of the tangent to the curve at
that point by 5.
PART -E
Answer the following guestions:
46. Solve the following linear programming problem graphically. Minimize and maximize
Z=x+2y subject to constraint x+2y100, 2x- ys0, 2x+ ys 200 and
x, y20. [6M]
OR

47.Prove that )de =[(*)ae +[f(x)dás hence evaluate
3 1
then verify that (AB) = B.A. [4M]

OR
49. Find the relation between 'a' and 'b' so that the function f defined by

ax +1 if x<3
f:)= | bx +3 if x>3 is continuous at x=3.

Page 4