GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION ( PRE UNIVERSITY )
Model Question Paper -2
I P.U.C.MATHEMATICS (35) :2024-25
Time: 3 hours Max. Marks: 80
Instructions:
1) The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2) PART A has 15MCQ’s ,5 Fill in the blanks of 1 mark each.
3) For questions having figure/graph, alternate questions are given at the end of
question paper in separate section for visually challenged students.
PART – A
I Answer all the multiple-choice questions: 15 × 1 = 15
1. The interval form of {𝑥: 𝑥 𝑅, − 5 < 𝑥 7} is
A) (-5, 7) B) [-5, 7] C) {-5, 7} D) (-5, 7]
2. A function f is defined by 𝑓(𝑥) = 2𝑥 − 5, then the value of f(0) is
A) − 3 B) −4 C) −5 D) 7
3. The range of f (x) = sin x is
A) [0, ) B] (-, ) C) [-1, 1] D) (-1, 1)
4. The additive identity of 5 + 3i is
5 − 3i
A) 0 + 1i B) 0 + 0i C) 5 - 3i D)
14
1
5. The standard form of (-5i) i is
8
−5 5 5
A) + i0 B) 0 + i C) 5 + 8i D) + i0
8 8 8
6
6. The solution of 3x + 8 > 2, when x is a real number is
A) (-2, ) B) (-, -2) C) (-, -2] D) (-, -1)
7. The equation of the line, which has slope 2 and y-intercept -5 is.
A) 2𝑥 − 𝑦 − 5 = 0 B) 2𝑥 + 𝑦 − 5 = 0 C) 2𝑥 − 𝑦 + 5 = 0 D) 2𝑥 + 𝑦 + 5 = 0
8. Match List I with List II
List I List II
a) 5𝐶0 i) 20
b) 5𝑃2 ii) 10
c) 𝐶2
5 iii) 1
Choose the correct answer from the options given below:
A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i
C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii
9. The equation of line in the figure is
A) 5𝑥 + 3𝑦 = 15
B) 3𝑥 + 5𝑦 = 15
C) 3𝑥 + 5𝑦 + 15 = 0
D) 5𝑥 + 3𝑦 + 15 = 0
10. The fifth term whose nth term is an = n(n + 2) is
A) 30 B) 35 C) 40 D) 45
5
11. Statement 1:The eccentricity of hyperbola 9𝑥 2 − 16𝑦 2 = 144 is
4
𝑥2 𝑦2 √𝑎 2 +𝑏2
Statement 2:The eccentricity of hyperbola − 𝑏2 = 1 is .
𝑎2 𝑎
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
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12. The axis in which the point (0, 5, 0) lies is
A) 𝑥 − 𝑎𝑥𝑖𝑠 B) 𝑦 − 𝑎𝑥𝑖𝑠 C) 𝑧 − 𝑎𝑥𝑖𝑠 D) 𝑥 + 𝑦 = 0
13. lim (x3 – x2 + 1) is
x → −1
A) -1 B) 0 C) 1 D) 2
14. The median of 3, 10, 6, 7, 11, 13, 15 is
A) 7 B) 9 C) 10 D) 11
15. The probability of getting exactly two heads on tossing a coin thrice is
2 2 3 1
A) B) C) D)
3 5 8 2
II. Fill in the blanks by choosing the appropriate answer from those given in the
bracket (1, -1, 64, 2, 4, 20) 5 ×1 = 5
16. Let A = {x, y, z} and B {1, 2}, then the number of relations from A to B is ______.
17. The slope of the line passing through the points (4, 0) and (6, 4) is _______.
5
18. The value of sin is ________.
2
5
19. The second term in the expansion of (√2 + 1) is _______.
20. The number of solutions of 24x < 100 when x is a natural number is _______.
PART – B
Answer any SIX questions: 6 × 2 = 12
21. If A B = (a,1) (a,2) (a,3) (b,1) (b,2) (b,3) , find the sets A and B and hence find
𝐵 × 𝐴.
25
22.
1
Express i + in a + i b form.
18
i
23. Find the multiplicative inverse of 2 – 3i.
24. Using binomial theorem evaluate (102)5 .
13
25. The sum of first three terms of a G.P. is and their product is – 1. Find the common
12
ratio and the terms.
26. Find the angle between the lines √3 x + y = 1 and x + √3 y = 1.
8
x15 − 1
27. Evaluate: lim 10
x →1
x −1
28. Find the derivative of f(x) w. r. t x from first principal given that 𝑓(𝑥) = 𝑠𝑖𝑛𝑥.
2 1
29. If P(A) = and P(B) = , find P(A or B) and P(A and B) if A and B are mutually exclusive.
3 2
PART – C
Answer any SIX questions: 6 ×3 = 18
30. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verity that
(A B)| = A| B|.
31. Draw the Venn diagram for (i) 𝐴 ∪ 𝐵 (ii) 𝐴 − 𝐵 (iii) (𝐴 ∩ 𝐵)|
−5
32. Find the values of other five trigonometric functions if cot x = , x lies in second
12
quadrant.
33. Prove that sin 3x = 3 sin x – 4 sin3x.
34. Solve the inequality and show the graph of the solution on the number line
3x − 4 x + 1
- 1.
2 4
35. Find the number of Permutations of the letters of the word PERMUTATIONS. Among
them how many have vowels are all together?
𝑥 2 4
36. Expand using binomial theorem (1 + 2 − 𝑥) , 𝑥 ≠ 0.
37. Reduce the equation of the circle 𝑥 2 + 𝑦 2 − 4𝑥 − 8𝑦 − 45 = 0 into Centre-radius form and
hence find its centre and radius.
38. If the origin is the centroid of the triangle PQR with vertices P (2a, 4, 6), Q(−4,3b, −10)
and R (8,14, 2c) then find the values of a, b, c.
PART – D
Answer any FOUR questions: 4 × 5 = 20
39. Define modulus function, draw the graph. Write the domain and the range.
cos 4 x + cos 3 x + cos 2 x
40. Prove that = cot 3x.
sin 4 x + sin 3 x + sin 2 x
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41. A group consists of 7 boys and 5 girls. Find the number of ways in which a team of 5
members can be selected so as to have at least one boy and one girl.
x y
42. Derive the equation of a line with x-intercept ‘a’ and y-intercept ‘b’ in the form of +
a b
= 1. Hence find the equation of a line that cuts off equal intercepts on the coordinate
axes and passes through the point (2, 3).
sin x
43. Prove geometrically that lim = 1, x being measured in radians.
x→0 x
44. Find the mean deviation about median for the following data
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No.of girls 6 8 14 16 4 2
45. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally
likely, calculate the probability that the card will be (i) a diamond (ii) not an ace (iii) a
black card (i.e., a club or, a spade) (v) not a black card.
PART - E
Answer the following questions:
46. Prove geometrically that cos (A + B) = cosAcosB – sin A sin B. Hence prove that
cos 2A = cos2A – sin2 A.
OR (6)
𝑥2 𝑦2
Define Hyperbola. Derive its equation in the form − =1
𝑎2 𝑏2
47. If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the
numbers. (4)
OR
sin x + cos x
Differentiate with respect to ‘ x’.
sin x − cos x
PART F
(For Visually Challenged Students only)
9. The Equations for x and y axes are
(A) 𝑥 = 1 , 𝑦 = 1 (B) 𝑦 = 1 (C) 𝑥 = 0 𝑎𝑛𝑑 𝑦 = 0 (D) 𝑥 = 1 𝑎𝑛𝑛 𝑦 = 0.
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