VEDANTHA PU COLLEGE

FIRST PUC PREPARATORY EXAMINATION JAN 2025

SUBJECT: MATHEMATICS (35)
Time: 3Hrs [Total questions: 47] Max Marks:80

Instructions:
1) Question paper has five parts namely A, B, C, D and E. Answer all the parts.
2) Part A has 15 multiple choice questions. 5 fill in the blanks questions.
3) Part A should be answered continuously at one or two pages of answer sheets and only
first answer is considered for the marks in I and II of part A.

PART – A
I. Answer all the multiple choice questions: 15x1=15
1. The {𝑥|𝑥 ∈ 𝑅, 6 < 𝑥 < 10} sets as intervals is
(a)(6,10) (b) [6,10] (c) {6,10} (d) (6,10]
2. If A = {1,2, 3,4,5} and B = {1,2, 3,4,5,6,7,8}
Statement I: The relation R = {(1,2),(2,3),(3,4), (4,5), (5,6), (5,7)} is a function.
Statement II: let f : A→B is a function, if every elements of A has one and only one elements of B
(a) Both statements are true
(b) Statement I is true and statement II is false
(c) Statement I is false and statement II is true
(d) Both statements are false.
3. The range of the real functions 𝑓(𝑥) = √𝑥 − 1
(a) [0, ∞) (b) (−∞, ∞) (c) (−∞, 0) (d) [ 0, ∞]
4. Which of the following is not correct?
−1 (b)𝑐𝑜𝑠𝜃 = 1 1 (d)𝑡𝑎𝑛𝜃 = 20
(a)sin 𝜃 = 5 (c)𝑠𝑒𝑐𝜃 =2
5. Match List I with List II
List I List II
a) 𝑠𝑖𝑛 (𝜋 − 𝜋 /4 ) i) 2
b) 𝑐𝑜𝑠𝑒𝑐 (𝜋 − 𝜋 /6 ) ii) −2
C)𝑠𝑒𝑐 (𝜋 − 𝜋 /3 ) iii) 1 /√2
Choose the correct answer from the options given below:
(a)a-iii, b-i, c-ii (b)a-iii, b-ii, c-i (c)a-ii, b-i, c-iii (d)a-i, b-iii, c-ii.
6. The modulus of the complex number − 1 + i√3.
(a)2 (b)√2 (c)4 (d)-2
7. The number of 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be
repeated is
(a)108 (b)216 (c)36 (d)60

8. Two numbers between 3 and 81 so that the resulting sequence is G.P.

(a)6 𝑎𝑛𝑑 36 (b)9 𝑎𝑛𝑑 18 (c)9 𝑎𝑛𝑑 27 (d)27 𝑎𝑛𝑑 54.

9. Equation of the passing through (0, 0) and slope m is

(a)y =mx +c (b)x=my +c (c)y=mx (d) x=my.
𝑥2 𝑦2
10. The latus rectum of the ellipse + = 1is
25 9
4 10 18 4
(a) 5 (b) (c) (d) 3
3 5
11. The eccentricity is never less than one for.
(a)circle (b)parabola (c)ellipse (d) hyperbola
 12. The derivative of f(x) = 𝑥 at x =100
(a)100 (b)99 (c) 1 (d) 0
√1+𝑋−1
13. lim =
𝑋→0 𝑋
1
(a)1 (b) 0 (c) 2 (d) √2

14. If A is any event associated with a sample space 𝑆 then

(a) 0 ≤ P(A) ≤ 1 (b) 0 < P(A) < 1
(c) ) P(A) ≥ 1 (d) P(A) ≤ 0)
15. Events E and F are such that P(not E or not F) = 0.25,then P(E and F) is
(a) 0 (b) 1
(c) 0.75 (d) 0.25

II. Fill in the blanks by choosing the appropriate answer from those given in the bracket 5x1=5

𝟏
( 𝟑, 𝟒, 𝟐, 𝟔, 𝟎, )
𝟐
16. The number of subsets of the set {𝑎} is ____________
17. The value of 1-sin 245° = ___________
18. The geometric mean of the numbers 4 and 9 is ____________
19. The derivative of the function f(x) = 𝑥3 − 27 at x = –1 is ___________.
1 𝑥
20. If 𝑃(𝐴) = 3 and 𝑃(𝑛𝑜𝑡 𝐴) = 6 , then x is ________.

PART-B
III. Answer any SIX Questions: 6x2=12
21. If A= {1, 2, 3, 4} B= {2, 3, 5} and C= {3, 5, 6}, find 𝐴 ∪ (𝐵 ∩ 𝐶 ).
22. Find the domain and range of the real function 𝑓 (𝑥 ) = √9 − 𝑥 2 .
23. Find the value of 𝑐𝑜𝑠(−17100 ).
2 tan 𝑥
24. Prove that sin 2𝑥 = 1+𝑡𝑎𝑛2 𝑥.
25. Find the value of x and y, if (𝑥 + 2𝑦) + 𝑖 (2𝑥 − 3𝑦)is the conjugate of 5 + 4𝑖.
26. If A. M. and G. M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the
quadratic equation.
27. Find the equation of the line passing through (-1, 1) and (2, -4).
28. Find the coordinates of focus, equation of directrix and length of the latus rectum of the parabola
−16𝑦 = 𝑥 2 .
29. A die is rolled. Let E be the event ‘die shows 4’and F be the event ‘die shows even number’. Are E and
F mutually exclusive?
PART-C
IV. Answer any SIX Questions: 6x3=18
30. Let U= {1, 2, 3, 4, 5, 6} A= {2, 3} and B= {3, 4, 5} show that (𝐴 ∪ 𝐵) = 𝐴 ∩ 𝐵 .
′ ′ ′

31. If 𝑓(𝑥 ) = 𝑥 2 𝑎𝑛𝑑 𝑔(𝑥 ) = 2𝑥 + 1 be two real functions find (i)(𝑓 + 𝑔)(𝑥) (ii) (𝑓 − 𝑔)(𝑥)(iii) (𝑓𝑔)(𝑥)
32. Prove that cot 4 𝑥(sin 5𝑥 + sin 3𝑥 ) = cot 𝑥 (sin 5𝑥 − sin 3𝑥 ).
1+𝑖 𝑚
33. If ( ) = 1, then find the least positive integral value of m.
1−𝑖
13
34. The sum of first three terms of a G. P. is 12 and their product is -1. find the common ratio and the
terms.
35. Find the equation of the line through the point (2, 2) and cutting off intercepts on the axes whose sum is
9.
 36. Reduce the equation of circle 𝑥 2 + 𝑦 2 − 8𝑥 + 10𝑦 − 12 = 0 into centre and radius form. Hence find
the centre and radius.
37. Find the derivative of cos x with respect to x from first principle.
1 1 1
38. A and B are events such that 𝑃(𝐴) = 4., 𝑃(𝐵) = 2and 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 8 find (i )P(A or B) (ii)P(not A
and not B).
PART –D
V. Answer any FOUR Questions: 4x5=20
39. Define modulus function. Draw the graph of modulus function, write the domain and range of the
function.
sin 5𝑥−2 sin 3𝑥+𝑠𝑖𝑛𝑥
40. Prove that = tan 𝑥.
cos 5𝑥−𝑐𝑜𝑠𝑥
𝜋 𝜋 3
41. Prove that 𝑐𝑜𝑠 2 𝑥 + 𝑐𝑜𝑠 2 (𝑥 + 3 ) + 𝑐𝑜𝑠 2 (𝑥 − 3 ) = 2.

42. A group consists of 4 girls and 7 boys, in how many ways can a team of 5 members be selected if the
team has (i) no girls (ii) at least one boy and one girl (iii)at least three girls
43. Derive a formula for the perpendicular distance of a point (𝑥1 , 𝑦1 ) from the line Ax + By + C = 0.
sin 𝑥 sin 𝑎𝑥
44. Prove that lim = 1 , (x being in radians), hence evaluate lim sin 𝑏𝑥 .
𝑥→0 𝑥 𝑥→0

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) diamond (ii) not an ace (iii) a black card
PART –E
VI. Answer the following Questions:
3 3𝜋 𝑥 𝑥 𝑥
46. If tanx = 4 , 𝜋 < 𝑥 < , find the value of sin 2 , 𝑐𝑜𝑠 2, and 𝑡𝑎𝑛 2.
2

OR 6
𝑥2 𝑦2
Define ellipse and derive the equation of the ellipse in standard form as + = 1. (𝑎 > 𝑏)
𝑎2 𝑏2
47. Find the sum of the sequence :7, 77, 777, 7777,……to n terms..
OR 4
𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥 𝑑𝑦 2
If 𝑓 (𝑥 ) = , show that =− .
𝑠𝑖𝑛𝑥−𝑐𝑜𝑠𝑥 𝑑𝑥 (𝑠𝑖𝑛𝑥−𝑐𝑜𝑠𝑥)2
ALL THE BEST