FIRST P U C ANNUAL EXAMINATION ( MODEL PAPER-1) MARCH 2024

SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes [ Total questions : 52 ] 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 15 multiple choice questions , 5 fill in the blank questions.

PART – A
I Answer ALL multiple choice questions 15 X 1 = 1 5
1. The interval form of { x : x∈ R , -12 < x < - 10 } is
(A) [ -12 ,- 10 ) (B) [ -12 ,- 10 ] (C) ( -12 ,- 10 ] (D) ( -12 ,- 10 )
2. If ( x + 1 , y – 2 ) = ( 3 , 1 ) then the value of x and y respectively is
(A) 1 , 2 (B) 2 , 1 (C) 2 , 3 (D) 3 , 2
3. The radian measure of 120 0 is
3π 2π 6π 7π
(A) (B) (C) (D)
4 3 7 6
19 π
4. The value of tan is
3
1 1
(A) (B) (C) √3 (D) 1
√2 √3
4. If A X B = { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) } then the set B is
(A) { 1 , 3 } (B) { 3 , 4 } (C) { 2 , 1} (D) { 2 , 4 }
5. The simplified form of (1 - i ) - ( -1 + i6) is
(A) 2 – 7 i (B) 2 + 7 i (C) 7– 2 i (D) 7+2i
6. The solution of 4x + 3 < 5x + 7 is
(A) ( -4 , ∞ ) (B) ( -2 , ∞ ) (C) ( -∞ , 4 ) (D) (- ∞ , ∞ )
7. A coin is tossed 3 times and the outcomes are recorded . Then the number of possible
outcomes are ..
(A) 3 (B) 6 (C) 8 (D) 0
8. Let Z = 3 + 2 i then the imaginary part of Z is
(A) 3 (B) 2 (C) – 2 (D) – 3
9. The third term of the sequence whose nth is given by an = (-1)n-1 5n+1 is
(A) 5 (B) 25 (C)125 (D) 625
10. The slope of the line making inclination of 60 0 with the positive direction of x -axis is
(A) 60 √3(B) (C) √ 2 (D) 1
11. The equation of the parabola with focus ( 6 , 0 ) and directrix is x = - 6 .
(A) y 2 = 6 x (B) y 2 = 24 x (C) x2 = 24 y (D) y 2 = - 24 x
12. The octant in which the point ( 2 , 4, -7) lie .
(A) 6 (B) 7 (C) 8 (D) 5
13. The distance between the parallel lines 3x – 4 y + 7 = 0 and 3x – 4 y + 5 = 0 is
3 3 2
(A) (B) (C) (D) 0
2 5 5
14. The mean of the data 6 , 7 , 10 , 12 , 13 , 4 , 8 , 12 is
(A) 4 (B) 3 (C) 2 (D) 9
2
15. If is the probability of an event , then the probability of an event ‘not A’ is ,
11
2 9 1
(A) (B) 13 (C) (D)
11 11 13
II . FILL IN THE BLANKS BY CHOOSING FROM THE GIVEN BOX : 5X1=5
1 4
[ , 64 , 2 ,4 , 9 , ]
√2 3

16. A function f is defined by f(x) = 2x – 5 then the value of f(7) is -------

17. The value of sin 765 0 is ------
1 1 x
18. If + = then the value of x is ___
6! 7! 8!

19. The slope of the line 4x – 3 y – 6 = 0 is

sin 4 x
20. lim = ____
x→0 sin 2 x
PART- B
III Answer any SIX questions 6 X 2 = 12
21. List all the elements of the following sets .
(i) A = { x : x is a letter in the word CATARACT }
(ii) B = { x : x∈N and x is a perfect cube }
22. Let A = {3 , 6 , 9 , 12 , 15 18 , 21} and B = { 4 , 8 , 12 , 16 , 20 } find A ∩ B and A U B .
23. A wheel makes 360 revolutions in one minute . Through how many radians does it turn in
one second ?
 a+ib
24. If x+iy = , prove that x 2 + y 2 = 1 .
a−ib
25. Find the multiplicative inverse of 4 – 3 i .
26. Solve 5x – 3 < 3x + 1 when (i) x is an integer (ii) x is a real number .
27. How many chords can be drawn through 21 points on a circle ?
28. Expand ( 1 – 2x )5 .
29. Find the equation of line through ( -2 , 3 ) with slope - 4 .
( x +1)5−1
30. Evaluate : lim
x→0 x
31. A and B are events such that P(A) = 0.42 , P(B) = 0.48 and P(A and B ) = 0.16 .
Determine (i) P( not B ) and (ii) P( A or B ) .

PART- C
IV Answer any SIX questions 6 X 3 = 18
32. Let U = { 1 , 2 , 3 , 4 , 5 , 6 } , A = { 2 , 3} and B = { 3 , 4 , 5 } .
Verify that ( A U B ) 1 = A1 ∩ B 1.
33. Let f( x ) = √ x and and g ( x ) = x be two functions defined over the set of
non-negative real numbers . Find ( f + g ) (x) , ( f – g ) (x) and ( f . g ) ( x ) .
3π π + 2 sec 2 π = 10 .
34. Show that 2 sin 2 + 2 cos 2
4 4 3
35. Prove that cos 3 x = 4 cos 2 x – 3 cos x .
( 3−2 i)(2+3 i)
36. Find the conjugate of
( 1+ 2i )( 2−i)
x x
37. Solve : x+ + < 11
2 3
38. Find the sum of first n terms and the sum of first 5 terms of the geometric series
2 4
1+ + + .....
3 9
39. Find the angles between lines √ 3 x + y = 1 and x + √ 3 y = 1 .
40. Find the equation of the ellipse whose vertices are ( ± 5 , 0 ) and foci are ( ± 4 , 0 ) .
41. Are the points A ( 3 , 6 , 9) , B ( 10 , 20, 30 ) and C ( 25 , -41 , 5 ) are the vertices of a
right angled triangle ?
42. Compute the derivative of cos x from first principle method .
 PART- D
Answer any FOUR questions 4 x 5 = 20
43. Define greatest integer function. Draw its graph . Write the domain and range .
cos 4 x+ cos 3 x+ cos 2 x
44. Prove that = cot 3x
sin 4 x+sin 3 x +sin 2 x
45. Find the number of arrangements of the letters of the word INDEPENDENCE .
In how many of these arrangement
a) do all the vowels always occur together ? b) do all the vowels never occur together.
46. Prove that for every positive integer n
(a + b ) n = n
C0 an +
n
C1 an-1 b+
n
C2 an -2 b2 + ---------+
n
Cn −1 a bn-1 + nCn b n
47. Derive the distance of a point ( x1 , y1 ) from the line A x + B y + C = 0 .
tan x
48. Prove that lim =1 where x is in radian measure .
x→0 x
49. Find the mean deviation about the mean for the following data .
Marks obtained 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Number of 2 3 8 14 8 3 2
students
50. A bag contains 9 discs of which 4 are red , 3 are blue and 2 are yellow . The discs are
similar in shape and size . A disc is drawn at random from the bag . Calculate the
probability that it will be (i) red (ii) yellow (iii) not blue (iv) either blue or red .

PART-E
Answer the following questions
51. Prove geometrically that cos ( x+ y ) = cos x cos y – sin x sin y 6
OR
2 2
x y
Define hyperbola and derive its equation in the standard form 2
− 2 =1 , a > b .
a b

52. Find the sum of the sequence 5 , 55 , 555 , 5555 , ........... to n terms . 4
OR
x +cos x
Find the derivative of With respect to x .
tan x
 FIRST P U C ANNUAL EXAMINATION (MODEL PAPER -2 ) FOR MARCH 2024
SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes [ Total questions : 52 ] 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 15 multiple choice questions , 5 fill in the blank questions.

PART – A
I Answer ALL multiple choice questions 15 X 1 = 1 5
1. The following is an example for infinite set.
(A) { x : x∈N and x2 = 4 } (B) { x : x∈N and x is a prime }
(C) { x : x∈N and 2x - 1 = 0 } (D) { x : x∈ N and (x – 1 ) ( x – 2 ) = 0}
2. If A X B = { ( p , q ) , ( p , r ) , ( m , q ) , ( m , r ) } then the set B is
(A) { p , q } (B) { m , r } (C) { q , r } (D) { m , q }
3. The radian measure of 210 0 is
3π 2π 6π 7π
(A) (B) (C) (D)
4 3 7 6
31 π
4. The value of sin is
3

(A)
1
(B)
1
(C) √3 (D) 1
2 √2 2
5. The simplified form of 3(7 + i 7 ) + i ( 7 + i7) is
(A) 14 – 28 i (B) 7 + 14 i (C) 49 - 7 i (D) 14 + 28 i
6. The solution of 4x + 3 < 6x + 7 is
(A) ( -2 , 4 ) (B) ( -2 , ∞ ) (C) ( -∞ , 4 ) (D) (- ∞ , ∞ )
7. If n
C9 =
n
C8 then the value of n
C17 is
(A) -1 (B) 1 (C) 17 (D) -17
8. The number of terms in the expansion of ( a+ b ) n is
(A) n (B) n–1 (C) n + 1 (D) 3
9. The fourth term whose n th term is given by a n = n (n + 2) is
(A) 24 (B) 10 (C) n + 2 (D) 15
10. The equation of a line with slope 2 and y intercept is -3 is
(A) y = 2 x –3 (B) y = 3 x – 2 (C) x = 2 y – 3 (D) y = 2 x
11. The equation of the parabola with focus ( 3 , 0 ) and directrix is x = - 3 is
(A) y 2 = 12 x (B) x2 = 12 y (C) x2 = 3 y (D) y 2 = 3 x
12. The octant in which the point ( -2 , 4, -7) lie .
(A) 6 (B) 7 (C) 8 (D) 5
13. lim x ( x +1) =
x→3

(A) 6 (B) 4 (C) 12 (D) 3

14. The mean of the data 4 , 7 , 8 , 9 , 10 , 12 , 13 , 9 .
(A) 8 (B) 9 (C) 7 (D) 10
15. Given P(A) = 3 and P(B) = 1 , if A and B are mutually exclusive events
5 5
then P(A or B ) is
2 4 3 1
(A) (B) (C) (D)
5 5 25 5
II . FILL IN THE BLANKS BY CHOOSING FROM THE GIVEN BOX : 5X1=5
3 3
[ , 4 , 28 , 0 , , 16 ]
2 5
16. The number of relations from A= { 1 , 2 } to B = { 3 , 4 } is ------

17. The value of cos ( - 1710 0 ) is ------

8!
18. The value of is -------
6 !2 !
19. The distance of the point ( 3 , - 5 ) from the line 3 x – 4 y - 26 = 0 is ___
15
x −1
20. lim 10
= ____
x→1 x −1
PART- B
III Answer any SIX questions 6 X 2 = 12
21. List all the elements of the following sets .
(i) A = { x : x is an odd natural number }
(ii) B = { x : x is a month of a year not having 31 days }
22. Let A = {1 , 2 , 3 , 4} and B = { 3 , 4 , 5 , 6 } find A ∩ B and A U B .
23. In a circle of diameter 40 cm , the length of a chord is 20 cm . Find the length of minor arc of the
chord .
24. Express 5+ √2 i in the form of a + i b .
1−√ 2 i
25. Find the multiplicative inverse of 2 – 3 i .
26. Solve 30 x < 200 when (i) x is a natural number , (ii) x is an integer.
n
P4 5
27. Find the value of n such that n −1 = ,n>4.
P4 3
3 4
28. Expand ( x2 + ) , x ≠0
x
29. Find the equation of the line parallel to the line 3 x – 4 y + 2 = 0 passing through
the point ( - 2 , 3 ) .
x 3−4 x 2 + 4 x
30. Evaluate : lim 2
x→2 x −4
1 1 1
31. If E and F are events such that P(E) = , P(F) = and P(E and F ) = , find
4 2 8
(i) P( E or F ) (ii) P( not E and not F ) .

PART- C
IV Answer any SIX questions 6 X 3 = 18
32. Let U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , A = { 2 , 4 , 6 , 8 } and B = { 2 , 3 , 5 , 7 } .
Verify that ( A U B ) 1 = A1 ∩ B 1.
33. Let f ( x ) = x 2 and and g ( x ) = 2 x + 1 be two real functions
then find ( f + g ) (x) , ( f – g ) (x) and ( f . g ) ( x ) .
34. Show that tan 3 x tan 2 x tan x = tan 3 x – tan 2 x – tan x
35. Prove that cos 4 x = 1 - 8 sin 2 x cos 2 x .
(3−2 i)(2+3 i)
36. Find the conjugate of
( 1+ 2i)( 2−i)
x x
37. Solve : > +1
3 2
38. A person has 2 parents , 4 grandparents , 8 great grandparents , and so on . Find the
number of his ancestors during the ten generations preceding his own .
39. Derive slope - intercept form of equation of a line .
40. Find the coordinates of the focus , axis, the equation of the directrix and latus rectum
of the parabola y 2 = 12 x .
41. Are the points A ( 0 , 7 , -10) , B ( 1 , 6 , -6 ) and C ( 4 , 9 , -6 ) are the vertices of an
isosceles triangle ?
42. Compute the derivative of log x from first principle method .
 PART- D
V Answer any FOUR questions 4 x 5 = 20

43. Define modulus function. Draw its graph . Write its domain and range .
sin 5 x−2 sin3 x +sin x
44. Prove that = tan x
cos 5 x −cos x
45. Find the number of arrangements of the letters of the word INDEPENDENCE .
In how many of these arrangement
a) do the words start with P? b) do the words begin with I and end in P ?
46. Prove that for every positive integer n
(a + b ) n = n
C0 an +
n
C1 an-1 b+
n
C2 an -2 b2 + ---------+
n
Cn −1 a bn-1 + nCn b n
47. Derive the distance of a point P( x1 , y1 ) from a line Ax + By + C = 0 .
sin x
48 . Prove that lim =1 where x is in radian measure .
x→0 x
49. Find the mean deviation about the median for the following data .

xi 5 7 9 10 12 15

fi 8 6 2 2 2 6

50. A committee of two persons is selected from two men and two women . What is the
probability that the committee will have (a) no man ? (b) one man (c) two men ?

PART-E
VI Answer the following questions

51. Prove geometrically that cos ( x+ y ) = cos x cos y – sin x sin y 6

OR
2 2
Define ellipse and derive its equation in the standard form x 2 + y2 =1 , a > b .
a b
52. Find the sum of the sequence 7 , 77 , 777 , 7777 , ........... to n terms . 4
OR
5
x −cos x
Find the derivative of With respect to x .
sin x
 FIRST P U C ANNUAL EXAMINATION (MODEL PAPER – 3 MARCH 2024
SUB: MATHEMATICS ( 35 )
TIME : 3 Hours 15 minutes [ Total questions : 52 ] 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 15 multiple choice questions , 5 fill in the blank questions.
PART – A
I Answer ALL multiple choice questions 15 X 1 = 1 5
1. The set – bilder form of the set { 2 , 3 } is
(A) { x : x∈ N and x is a prime } (B) { x : x∈N and (x + 3 ) ( x – 2 ) = 0}
(C) { x : x is a natural number and divisor of 6 }
(D) { x : x is a prime number and divisor of 6 }
2. The range of the function f(x) = x2 + 2 , x is a real number is
(A) ( 2 , ∞ ) (B) ( - ∞ , 2 ] (C) ( - ∞ , ∞] (D) [ 2 , ∞ )
3. The radian measure of 25 0 is
5π 5π 25 π π
(A) (B) (C) (D)
36 18 2 25
2π
4. The value of cos is
3

(A)
1
(B)
1
(C) √3 (D)
−1
2 √2 2 2
5. The simplified form of i 5 is
(A) – i (B) i (C) 1 (D) – 1
6. The solution of 3x – 7 > 5x – 1 is
(A) ( 3 , 4 ) (B) ( -3 , ∞ ) (C) ( -∞ , - 3) (D) (- ∞ , ∞ )
7. If n
P2 = 12 then the value of
n
C2 is
(A) 12 (B) 6 (C) 4 (D) 1
8. By using Binomial Theorem , The value of n
C0 +
n
C1 +
n
C2 + .......+
n
Cn is
(A) 0 (B) 2 (C) 2n (D) 3 n
n−3
9. The second term whose n th term is given by an= is
4
(A) 24 (B) (C) n + 2 (D) 15
 x y
10. The slope of the line + = 1 is
4 3
4 3 −3 −4
(A) (B) (C) (D)
3 4 4 3
11. The radius of the circle ( x + 5 )2 + ( y – 3 )2 = 36 is
(A) 3 (B) 6 (C) 36 (D) 9
12. The octant in which the point ( – 3 , – 1 , 6 ) lie .
(A) 2 (B) 3 (C) 6 (D) 7
13. The value of lim cos x
x→0

(A) 1 (B) π (C) -1 (D) 0

14. The mean deviation about the mean for the absolute values of the deviations
are given by 3,2,1,3,4,5,1,3.
(A) 2 (B) 3 (C) 3.75 (D) 2.75
15. The number of committee of two persons is selected from two men and two women is
(A) 2 (B) 3 (C) 6 (D) 7
II . FILL IN THE BLANKS BY CHOOSING FROM THE GIVEN BOX : 5X1=5
[ √ 3+1 , 4 , 1 , 715 , 14 , 2
]
2√ 2 5
16. The function ‘t’ which maps temparature in degree Celcius into temparature in
9C
degree Fahrenheit is defined by t(C) = +32 . then t(– 10 ) = __
5

17. The value of cos 15 0 is ------

13 !
18. The value of is -------
4 !9!
19. The distance between the parallel lines 3x – 4y + 7 = 0 and 3x – 4 y + 5 = 0 is ___
20. lim [x 3−x 2 +1] = ____
x→1

PART- B
III Answer any SIX questions 6 X 2 = 12
21. Let A = {1 , 2 , 3 , 4 , 5 , 6 } , B = {2 , 4 , 6 , 8 } . Find A – B and B – A .
22. Let A = { a , b } and B = { a , b , c } . Is A⊂B ? What is A U B .
23. The minute hand of a watch is 1.5 cm long . How far does its tip move in 40 minutes ?
(Use π = 3.14)
24. Express ( 1 – i ) 4 in the form of a + i b .
25. Find the multiplicative inverse of √5 + 3 i .
26. Solve 3x + 8 > 2 when (i) x is an integer (ii) x is a real number
2n n
27. Find the value of n such that C3 : C3 = 11 : 1 .

x 1 5
28. Expand ( + ) , x ≠0
3 x

29. Find the equation of the line through the points ( 1 , - 1 ) and ( 3 , 5 ) .
cos 2 x−1
30. Evaluate : lim
x→0 cos x−1

31. Check whether the following probabilities P(A) and P(B) are consistently defined

(i) P(A) = 0.5 , P(B) = 0.7 , P(A ∩ B ) = 0.6

(ii) P(A) = 0.5 , P(B) = 0.4 , P(A U B ) = 0.8

PART- C
32. Show that A U B = A ∩ B implies A = B .
33. Let A ={1 , 2 , 3 , 4 , 5 , 6 }. Define a relation R from A to A by R = {( x , y ) : y = x + 1 }
(i) Depict this relation using an arrow diagram .
(ii) Write down the domain and range of R .
34. Show that cos2 2x – cos 2 6x = sin 4x sin 6x .
35. Prove that sin 3x = 3 sin x – 4 sin 3 x .
36. Find the real numbers x and y if ( x – iy ) ( 3 + 5i ) is the conjugate of – 6 – 24 i .
3 (x−2) 5 (2−x)
37. Solve : ≤
5 3

38. In a G.P ., the 3rd term is 24 and 6th term is 192 . Find the 10 th term .
39. Derive two-point form of equation of a line .
39. Find the coordinates of the focus , axis, the equation of the directrix and latus rectum
of the parabola y2 = 8 x .
40. Find the equation of the set of points which are equidistant from the points ( 1 , 2 , 3 )
and ( 3 , 2 , -1 ) .
 1
41. Compute the derivative of by using the definition of derivative .
x
PART- D
Answer any FOUR questions 4 x 5 = 20
42. Define signum function. Draw its graph . Write its domain and range .
( sin7 x +sin 5 x)+(sin 9 x+sin 3 x)
44. Prove that = tan 6x
( cos 7 x+ cos 5 x )(cos 9 x +cos 3 x)

45. Find the number of different 8-letter arrangements that can be made from the letters of
the word DAUGHTER so that
a) All vowels occur together ? b) All vowels do not occur together ?
46. State and prove Binomial Theorem .
47. Derive the distance of a point P( x1 , y1 ) from a line ax + by + c = 0 .

48 . Prove that lim sin θ =1 where θ is in radian measure .

θ→0 θ
49. Find the mean deviation about the mean for the following data .
Marks 0-10 10-20 20-30 30-40 40-50 50-60
6 8 14 16 4 2
Number of Girls

50. In a class of 60 students , 30 opted for NCC , 32 opted for NSS and 24 opted for both NCC
and NSS . If one of these students is selected at random , find the probability that
(i) The student opted for NCC or NSS .
(ii) The student has opted neither NCC nor NSS .

PART-E
Answer the following questions
51. Prove geometrically that cos ( x+ y ) = cos x cos y – sin x sin y 6
OR
2 2
Define ellipse and derive its equation in the standard form x 2 + y2 =1 , a > b .
a b
52. Find the sum of the sequence 3 , 33 , 333 , 3333 , ........... to n terms . 4
OR
cos x
Find the derivative of With respect to x .
1+sin x