Model Question Papers

MODEL QUESTION PAPER (GIVEN BY BOARD)

Time : 3 hours 15 Minutes Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part A has 10 Multiple Choice Questions, 5 Fill in the blanks and 5 Very Short Answers questions of
1 mark each
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 subsection I and II of Part A.
4. Use the graph sheet for the question on linear inequality in PART D.

PART – A
I Answer all the Multiple Choice Questions 10 x 1 = 10
1. The number of proper subsets of the set {1, 2, 3} is
(A) 8 (B) 7 (C) 6 (D) 5
2. Let R be a relation on N defined by � + 2� = 8 The domain of R is
(A) 2,4,8 (B) 2,4,6,8 (C) 2,4,6 (D) 1,2,3,4
5
3. If tan x =− 12, x lies in second �������� then sin� is
12 12 5 5
(A) − 13 (B) 13 (C) 13 (D) − 13
4. The multiplicative inverse of a complex number 4 – 3i
4−3i 4+3i −4−3i −4+3i
(A) 25
(B) 25
(C) 25
(D) 25
5. If n C9 = n C8 then the value of n C17
(A) 8 (B) 7 (C) 17 (D) 1
6. The 4th term of the sequence defined by a1 = a2 = 1 and an = an−1 + an−2 , n > 2.
(A) 1 (B) 2 (C) 3 (D) 4
0
7. The slope of the line which makes angle 30 with positive direction of y axis measured
anticlockwise.
1
(A) 3 (B) − 3 (C) 1 (D) 3
1 1
+
8. lim x 2
is equal to
x→−2 x+2
1 1
(A) −1 (B) 4 (C) − 4 (D) −4
9. The negative of the proposition “Every natural number is an integer”
(A) Natural number is not an integer
(B) At least one natural number is an integer
(C) At least one natural number is not an integer
(D) One natural number is not an integer
2
10. If 5 is the probability of an event A then what is the probability of the event “not A‟?
5 3 1 4
(A) 2 (B) 5 (C) 5 (D) 5

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 Model Question Papers

II Fill in the blanks by choosing the appropriate answer from those given in the bracket.
�, �, �, �, � 5 x 1 = 10
11. _________number of elements has P(A), if A = { 1 }.
12. The positive value of m =_________ for which the coefficient of �2 in the expansion (1 + �)� �� 6.
13. The length of the latus rectum of the parabola �2 = −8� �� ____________
14. The point (–3,1, -2) lies in________th octant
15. The derivative of f(x) = x at x = 100 is_______

III Answer the Following Questions 5x1=5

31π
16. Find the values of sin 3
17. Solve 4x + 3 < 6x +7.
1 1 X
18. If + 7! = 8! then find x
6!
19. Find distance between parallel lines 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0.
20. Find the mean for first n natural numbers.

PART – B
Answer any NINE questions 9 x 2 = 18
21. Let U= {1, 2, 3, 4, 5, 6}, A= { 2, 3}, and B={ 3, 4, 5}. Find A∁ ∩ B∁
22. Let A = {a, b} and B = {a, b, c} . Is A  B ? What is A  B ?
x 2 5 2
23. If 3
+ 1, y−3 = ,
3 3
, Find the values of x and y.
24. A wheel makes 360 revolutions in one minute. Through how many radians it turns in one second.
π π π 1
25. Prove that sin2 6 + cos2 3 − tan2 4 = 2
26. Find the modulus and argument of the complex number − 3 − i
27. Solve: 7x + 3  5x + 9 . Show the graph of the solution on the number line.
28. Write the contrapositive and converse of the following statements.
“If x is a prime number, then x is odd”.
29. Reduce the equation 3x + y − 8 = 0 into normal form. Find the values of p and α.
30. Find the equation of the line perpendicular to the line x-2y+3=0 and passing through the point (1,-2).
31. If (1,1,1)is the centroid of the triangle with (3,−5,7)and (−1,7,−6) as the two vertices, Find the third
vertex.
x+1 5 −1
32. Evaluate lim  x
,
x→0
33. Co-efficient of variation and their standard deviations of certain distribution is 60 and 21
respectively. Find the arithmetic mean.
34. A coin is tossed twice, what is the probability that at least one tail occurs?

PART – C
Answer any NINE questions 9 x 3 = 27
35. In a class of 35 students, 24 likes to play cricket and 16 likes to play football, Also each student likes
to play at least one of the games. How many students like to play both cricket and football?
36. Let A = {1, 2, 3, .........14} . Define a relation R from A to A by R = {(x, y): 3x − y = 0 and x, yA}.
Write down its domain and range.
37. Find the general solution of sin 2� + cos � = 0

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 Model Question Papers

3−2i 2+3i
38. Find the conjugate of 1+2i 2−i
2 20
39. Solve the equation 3x − 4x + 3 =0
40. In how many ways can the letters of the word PERMUTATIONS be arranged if the vowels are all
together.
x 10
41. Find the middle term in the expansion of 3
+ 9y
42. Find the sum of n terms of the sequence 7, 77, 777, 7777,....to n terms.
43. In an A.P., if mth term is n and the nth term is m, where m  n , find the pth term.
44. Find the equation of the circle which passes through the points (2, – 2), and (3,4) and whose centre
lies on the line x + y = 2.
45. Find the derivative of tan x from first principles.
46. Show that the statement “If x is a real number such that x3 + 4x = 0 then x is 0” is true by direct
method.
47. A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the
probability that the sum of numbers that turn up is ( i ) 3 ( ii ) 12
48. 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 A) ii) P(not B) and iii) P( A or B) .
PART - D
Answer any FIVE questions 5 x 5 = 30
49. Define Modulus function. Draw the graph of Modulus function and write its domain and range.
sin 7x+sin 5x+sin 9x+sin 3x
50. Prove that cos 7x+cos 5x+cos 9x+cos 3x
= tan 6x
n n+1 2
51. Prove by the principle of mathematical induction that 13 + 23 + …. . + n3 = 2
52. Solve the following system of inequalities graphically: 3x + 2 y  150 , x + 4 y  80 , x  15 , x  0, y 
0
.
53. A committee of 7 is to be formed from 9 boys and 4 girls. In how many ways can this be done when
the committee consists of (i) exactly 3 girls? (ii) at least 3 girls?
54. State and prove Binomial Theorem for any positive integer n.
55. Derive a formula for the perpendicular distance of a point x1 , y1 from the line Ax + By + C = 0.
56. Derive an expression for the coordinates of a point that divides the line joining the points
x1 , y1 , z1 and x2 , y2 , z2 internally in the ratio m : n
sin x
57. Prove geometrically, lim = 1 where x is in radian
x→0 x
58. Find the mean deviation about the mean for the following data:
Height in cm 95-105 105-115 115-125 125-135 135-145 145-155
No. of boys 9 13 26 30 12 10
PART-E
Answer the following questions
π
59. Prove that geometrically that cos x + y = cos x . cos y − sin x . sin y and hence find cos 2
+x
OR
x2 y2
Define Ellipse and derive its equation in the form a2 + b2 = 1 (6)
x+cos x
60. Find the derivative of tan x
with respect to x
OR
Find the sum to n terms of the series 5 + 11 + 19 + 29 + ………… (4)
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 Model Question Papers

MODEL QUESTION PAPER 1

Time : 3 hours 15 Minutes Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part A has 10 Multiple Choice Questions, 5 Fill in the blanks and 5 Very Short Answers questions of
1 mark each
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 subsection I and II of Part A.
4. Use the graph sheet for the question on linear inequality in PART D.

PART – A
I Answer all the Multiple Choice Questions 10 x 1 = 10
1. For any sets � and �, � ∪ � − � =
(A) � − � (B) � ∪ � (C) � ∩ � (D) � − �
2. If A = {1,2,3}, B = {3,8} then (AB) x (AB) is
(A) {(1,3) (2,3) (3,3) (8,3)} (B) {(3,1) (2,3) (3,3) (8,3)}
(C) {(3,1) (2,3) (3,3) (3,8)} (D) {(1,3) (3,2) (3,3) (3,8)}
3. A circular wire of radius 7.5cm is cut and bent so as to lie along the circumference of a loop whose
radius is 120cm. Then the angle which is subtended at the centre of the loop is
� � � �
(A) 4 (B) 8 (C) 3 (D) 6
4. The conjugate of (1 + 2�) (2 − 3�) =
(A) −4 + � (B) −4 − � (C) 8 + � (D) 8 − �
5. How many 3 digit numbers can be formed and non the digits 1,2,3,4 &5 if the repetition is allowed.
(A) 60 (B) 125 (C) 120 (D) 65
6. If the nth term of a sequence is �� = 4� − 3; then �17 is
(A) 64 (B) 65 (C) 17 (D) −65
7. The slope of the line passing through the points 1, − 1 & 3, 5 is
(A) 2 (B) 3 (C) −3 (D) −2
2
8. lim  �(�), where � � = � − 1 � ≤ 1 is
�→1− − �2 − 1 � > 1
(A) 1 (B) 0 (C) −1 (D) −2
9. Which of the following is not a correct statement?
(A) 3 is a prime. (B) The sun is a star.
(C) Mathematics is interesting. (D) 2 is irrational
10. A die is rolled. The probability of an event “a number less than 7” occurs.
5 1 1
(A) 1 (B) 6 (C) 3 (D) 6

II Fill in the blanks by choosing the appropriate answer from those given in the bracket.
�, �, �, �, �� 5 x 1 = 10
11. If A has 4 elements then number of subsets of A is________
1 5
12. The coefficient of �−8 in the expansion of �4 − �3 is_______
13. The radius of the circle �2 + �2 + 8� + 10� − 8 = 0 is________
14. The point ( − 4, 2, − 5) lies is________th octant
15. If � � = 1 + � + �2 + �3 + …………. + �50 then �' 0 =________
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 Model Question Papers

III Answer the Following Questions 5x1=5

2�
16. Convert 3
radians into degree measure
17. Solve 3� − 2 < 2� + 1 , where � ∈ �
18. If � + 2 ! = 42 �! Then find �.
19. Reduce 6� + 3� − 5 = 0 into slope-intercept form.
20. Find the mean of 1,2,3,4,5

PART – B
Answer any NINE questions 9 x 2 = 18
21. If X & Y are the two sets such that � � = 17 , � � = 23 & � � ∪ � = 38. Find � � ∩ �
22. Define union of two sets and write its venn diagram.
23. If � � = �2 , � � = 2� + 1 be two real valued functions then find � + � (�) , � − � (�)
24. Find the value of ���( − 17100 ) .
2 tan �
25. Prove that sin 2� = 1+tan2 �
2+�
26. If � + �� = 2−� prove that �2 + �2 = 1
27. The marks obtained by a student of class XI in first & second terminal examinations are 62 & 48
respectively, find the minimum marks he should get in the annual examination to have an average
of at least 60 marks.
28. By using the concept of equation of the line prove that the three points (3,0), (-2,-2) and (8,2) are
collinear.
29. Find the equation of the line passing through (-3,5) and perpendicular to the line through the
points (2,5) and (-3,6).
30. The centroid of a triangle ABC is at the point 1, 1, 1 . If the coordinates of A & B are
3, − 5, 7 & −1, 7, − 6 respectively. Find the coordinates of the point C.
��+bx cos �
31. Evaluate lim � sin �
�→0
32. Write the contrapositive and converse of “If two lines are parallel then they do not intersect in the
same plane”.
33. If the coefficient of variation and standard deviation are 60, 21 respectively, what is the arithmetic
mean of the distribution?
34. A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls.
If it shows head, we throw a die. Find the sample space for this experiment.

PART – C
Answer any NINE questions 9 x 3 = 27
35. There are 200 individuals with a skin disorder. 120 has been exposed to the chemical A, 50 to
chemical B & 30 to both chemical A & B. Find the number of individuals exposed to (i) chemical A
but not to chemical B (ii) chemical A (or) chemical B.
36. If � = 1,2,3,4 , � = 5,6 . Define a relation R from A to B by
� = �, � ; � ∈ �, � ∈ � , � − � �� ��� . Write down its domain & range.
37. Find the general solution of 2 cos2 � + 3 sin � = 0
20
38. Solve 3�2 − 4� + 3
=0
1+� 3
39. Convert 2
into polar form

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 Model Question Papers

40. How many words with or without meaning can be made from the letters of the word MONDAY
assuming that no letter is repeated, if (i) 4 letters are used at a time. (ii) All letters are used at a
time (iii) All letters are used but first letter is a vowel.
� 10
41. Find the middle term in the expansion of 3
+ 9�
42. Insert 3 arithmetic means between 8 and 24
3 3 3069
43. How many terms of the G.P 3,2 , 4 ,. . . are needed to give the sum 512
.
44. Find the equation of parabola with vertex at the origin, axis along x-axis and passing through the
point (2,3) and also find its focus.
45. Find the derivative of � � = tan � with respect to � from the first principle
46. Verify by the method of contradiction that 7 is irrational.
47. A die is rolled. Let E be the event” die shows 4” and F be the vent “die shows even number”.
Are E and F mutually exclusive?
48. The student Anil and Ashima appeared in the examination, the probability that Anil will qualify the
examination is 0.05 and that Ashima will qualify the examination 0.10. The probability that the
both will qualify the examination is 0.02. Find the probability that only one of them qualify the
examination.

PART - D
Answer any FIVE questions 5 x 5 = 30
49. Define an identity function. Draw the graph of the identity function and write down its range and
domain.
cos 4�+cos 3�+cos 2�
50. Prove that sin 4�+sin 3�+sin 2�
= cot 3�
�(�+1)(2�+1)
51. Using the principle of mathematical induction, prove that 12 + 22 + …… + �2 = 6
52. Solve the following system of inequalities graphically:
3� + 2� ≤ 150, � + 4� ≤ 80, � ≤ 15, � ≥ 0, � ≥ 0
53. A committee of seven has to be formed from 9 boys and 4 girls. In how many ways this can be done
when the committee consists of (i) exactly 3 girls (ii) at least 3 girls and (iii) at most 3 girls
54. State and prove Binomial theorem for all natural numbers.
55. Derive the equation of a straight line passing through the point (�1 , �1 ) having the slope m. Hence
deduce the equation of a line which passes through (2,1) which makes an angle 450 with positive
direction of x-axis.
56. Derive the formula for the distance between two points �1 , �1 , �1 and �2 , �2 , �2 and hence
find the distance between the points � 1, − 3,4 and � −4,1,2 .
sin �
57. Prove geometrically, lim = 1 where � is in radian
�→0 �
58. Find the mean deviation about the mean for the following data

�� 5 10 15 20 25
�� 7 4 6 3 5

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 Model Question Papers

PART-E
Answer the following questions
59. Prove that geometrically that cos � + � = cos � . cos � − sin � . sin � and hence prove that
cos � − � = cos � . cos � + sin � . sin �
OR
Define Hyperbola and derive the equation of hyperbola in standard form. (6)
2 �2
60. Find the derivative of �+1 − 3�−1 with respect to �
OR
Find the sum to n terms of the series (1.2.5) + (2.3.6) + (3.4.7) + …………. (4)

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MODEL QUESTION PAPER 2

Time : 3 hours 15 Minutes Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part A has 10 Multiple Choice Questions, 5 Fill in the blanks and 5 Very Short Answers questions of
1 mark each
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 subsection I and II of Part A.
4. Use the graph sheet for the question on linear inequality in PART D.

PART – A
I Answer all the Multiple Choice Questions 10 x 1 = 10
1
1. If sets A and B are defined as � = �, � | � =
�
,0 ≠ � ∈� ,�= �, � | � =− �, � ∈ � , then
(A) � ∩ � = � (B) � ∩ � = � (C) � ∩ � = � (D) � ∪ � = �
2. R is a relation on N given by � = { �, � : 4� + 3� = 20} which of the following belongs to �
(A) (2,4) (B) (3,4) (C) (5,0) (D) ( − 4,12)
3
3. If cot � = 4, � lies in third quadrant then the value of sin � =
4 3 4 3
(A) 5 (B) 5 (C) − 5 (D) − 5
�+�� 2
4. If � + �� = �+��
, �ℎ�� �2 + �2 =

�2 +�2 �2 −�2 �2 +�2 �2 +�2

(A) �2+�2 (B) �2 −�2
(C) �2 +�2
(D) �2+�2
7!
5. The value of 4! is equal to
(A) 210 (B) 215 (C) 201 (D) 220
2 7
6. − 7 , �, − 2, are in G. P then � =
(A) 1 (B) 2 (C) 7 (D) 3
� �
7. The slope of the line 3
+2= 1 is
2 3 3 2
(A) 3 (B) 2 (C) − 2 (D) − 3
8. lim  � − 5
�→5
(A) 1 (B) 0 (C) −1 (D) Not exist

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 Model Question Papers

9. The negation of the statement “All triangles are not equilateral triangle” is
(A) All triangles are equilateral triangle
(B) Some triangles are not equilateral triangle
(C) some triangles are equilateral triangle
(D) All triangles are isosceles triangle
10. If a coin is tossed twice, then the probability that atleast one tail occurs is
1 1 3
(A) 4 (B) 2 (C) 4 (D) 1

II Fill in the blanks by choosing the appropriate answer from those given in the bracket.
�
�
, �, �, �, ���, 5 x 1 = 10
11. If number of subsets of a set � is 16 then the number of elements in A is_______
12. The value of 8C0 + 8C2 + 8C4+ 8C6+ 8C8 is__________
�2 �2
13. The eccentricity of the ellipse 25 + 9
= 1 is________
14. The perpendicular distance of the point �(6, 7, 8) from �� - plane is_______
15. The derivative of x2 – 2 at x = 1 is_______

III Answer the Following Questions 5x1=5

7�
16. Convert 6
into degrees .
17. If � is any real number then solve 3 � − 1 ≤ 2 � − 3
18. Find "�" if ��7 = ��6
19. Find the distance between 3� + 4� + 5 = 0 and 6� + 8� + 2 = 0
20. Find the median of 8,1,3,5,22,17,12,13.

PART – B
Answer any NINE questions 9 x 2 = 18
21. In a school, there are 20 teachers who teach maths or physics of these 12 teach maths & 4 teach
both physics & maths. How many teach physics.
22. If � = 3, 6, 9, 12, 15, 18, 21 & � = 4, 8, 12, 16, 20 then find � − � & � − �
23. Find the domain of the real function f defined by � � = � − 1
24. Find the angle in radian measure through which a pendulum swings if its length is 75 cm & the tip
describes an arc of length 15 cm.
sin 2�
25. Prove that 1−cos 2� = cot �
26. Find the value of x & y, if � + 2� + � 2� − 3� is the conjugate of 5 + 4�
27. A student obtained 70 and 75 marks in first two unit test. Find the number of minimum marks he
should get in the third test to have an average of at least 60 marks.
28. Find the equation of the line passing through −1,1 and 2, − 4
� �
29. If (ℎ , 0), �, � and 0, � lie on the line then show that ℎ + � = 1
30. Find the ratio in which the YZ-plane divides the line segment formed by joining the points
−2, 4, 7 & 3, − 5, 8
1−cos �
31. Evaluate lim �
�→0
32. Write the component statement of the following compound statement “zero is less than every
positive integer and every negative integer”.

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 Model Question Papers

33. The mean & variance of heights of XI students are 162.6cm and 127.69cm2 respectively. Find the
coefficient of variation.
1 1 1
34. If E and F are event such that P(E)= 4 , P(F)= 2 and P(E and F) = 8
, find P(E or F)

PART – C
Answer any NINE questions 9 x 3 = 27
35. In a survey of 400 students in a school, 100 were listed as taking apple juice, 150 as taking orange
juice & 75 were listed as taking both apple & orange juice. Find how many students were taking
neither apple juice nor orange juice.
36. Let �: � → � be a relation defined by � = �, � |�, � ∈ �, � − � ∈ � . Show that
i) ∀� ∈ �, �, � ∈ �. ii) �, � ∈ � ⟹ �, � ∈ �. iii) If �, � ∈ �, �, � ∈ � ⟹ �, � ∈ �
4 tan � 1−tan2 �
37. Prove that tan 4� = 1−6 tan2 �+tan4 �
38. Solve 2�2 + 3� − 1 = 0
39. Express 3 + � in the polar form.
40. Find the number of arrangement of the letters of the word PERMUTATIONS. In how many of these
arrangements (i) word start with P and end with S (ii) vowels are all together.
1 18
3
41. Find the term independent of x in the expansion of �+ 3 , �>0
2 �
42. In an �. � if ��ℎ term is '�' and the ��ℎ term is '�' , where � ≠ � then find the ��ℎ term.
43. Insert 3 numbers between 1 and 256 so that the resulting sequence is a G.P.
44. Find the equation of parabola with vertex at origin has its focus on the centre of
�2 + �2 − 10� + 9 = 0. Find its directrix and latus rectum.
�+1
45. Find the derivative of � � = �
with respect to � from the first principle
46. By the method of contrapositive, Check the validity of the statement : ‘If �, � ∈ � such that �� is
odd then both � and � are odd.
47. Consider the experiment of rolling a die. Let A be the event ‘getting a prime number’, B be the
event ‘getting an odd number’. Write the sets representing the events
(i) A or B (ii) A and B iii) A but not B
48. Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 52 cards, it
contains i) all kings ii) 3 kings iii) atleast 3 kings

PART - D
Answer any FIVE questions 5 x 5 = 30
49. Define greatest integer function. Draw the graph of it and write down its range and domain.
sin α+sin 2α+sin 4α+sin 5α
50. Prove that cos α+cos 2α+cos 4α+cos 5α = tan 3α
51. Prove by the principle of mathematical induction that
� �+1 �+2
1.2 + 2.3 + …………. . + � � + 1 = 3
52. Solve the following system of inequalities graphically : � + 2� ≤ 10, � + � ≥ 1, � − � ≤ 0, �, � ≥ 0
53. What is the number of ways choosing four cards from a pack of 52 playing cards. In how many of
these (i) Four cards of the same suit (ii) are face cards
(iii) two red and two black card (iv) cards are of the same colour
2� 1.2.3……….. 2�−1
54. Show that the middle term in the expansion of 1 + � is �!
2 � ��

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 Model Question Papers

55. Derive normal form of straight line � cos � + ���� � = � hence find the equation of line if � = 4
and � = 450
56. Derive an expression for the coordinates of a point that divides the line joining the points
�1 , �1 , �1 and �2 , �2 , �2 internally in the ratio � : �
��−�� �4 −81
57. Prove that lim = ���−1 and hence evaluate lim  2�2−5�−3
�→� �−� �→3
58. Find the mean deviation about the mean for the following data
Income per
0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800
day
Number of
4 8 9 10 7 5 4 3
persons

PART-E
Answer the following questions
59. Prove that geometrically that cos � + � = cos � . cos � − sin � . sin � and hence prove that
�
cos 2
+ � =− sin �
OR
Define Ellipse and derive the equation of ellipse in standard form. (6)
cos �
60. Find the derivative of 1+sin � with respect to �
OR
Find the sum to n terms of the series 1 + 1 + 22 + 12 + 22 + 32 + ………….
2 2
(4)

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MODEL QUESTION PAPER 3

Time : 3 hours 15 Minutes Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part A has 10 Multiple Choice Questions, 5 Fill in the blanks and 5 Very Short Answers questions of
1 mark each
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 subsection I and II of Part A.
4. Use the graph sheet for the question on linear inequality in PART D.

PART – A
I Answer all the Multiple Choice Questions 10 x 1 = 10
1. Which of the following set is empty set?
(A) {x/x is real and x2-1 = 0} (B) {x/x is real and x2 + 1 = 0}
(C) {x/x is real and x2-9 = 0} (D) {x/x is real and x2 = x + 2}
2. If � � = �� + �, where a and b are integers, � −1 = − 5 and � 3 = 3, then a and b are equal
to
(A) � = − 3, � = − 1 (B) � = 2, � = − 3
(C) � = 0 , � = 2 (D) � = 2, � = 3
3. The value of tan1 .tan 2 . tan 3 ……………. tan 890
0 0 0

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 Model Question Papers

(A) 0 (B) 1 (C) −1 (D) 2

4. The multiplicate inverse of 7 + 24� is
7+24� 7−24� 24�−7
(A) 625
(B) 625
(C) 625
(D) None of these
18 � = 18 �
5. If � �+2 then � �5 is equal to
(A) 23 (B) 46 (C) 50 (D) 56
6. The tenth term of G.P : 5, 25, 125, .......
(A) 510 (B) 105 (C) 511 (D) 59
7. The points �, 0 , 0, � & 1,1 are collinear if
1 1 1 1 1 1 1 1
(A) � − � = 1 (B) � + � = 1 (C) � − � = 0 (D)
�
+�=0
cos �
8. lim  �−� =
�→0
1
(A) 1 (B) 0 (C) � (D) �
9. The negation of the statement “For all real numbers � and �, � + � = � + �” is
(A) for some real numbers � and �, � − � = � − �
(B) for all real numbers � and �, � + � ≠ � + �
(C) for some real numbers � and �, � + � = � + �
(D) for some real numbers � and �, � + � ≠ � + �
10. There are four men and six women on the city council. If one council member is selected for a
committee at random, how likely is it that it is a woman?
2 1 3 1
(A) 3 (B) 4 (C) 5 (D) 6

II Fill in the blanks by choosing the appropriate answer from those given in the bracket.
�, ��, �, �, � 5 x 1 = 10
11. The number of improper subsets of a set having ‘n’ elements is_______
1 13 �
12. The 12th terms in the expansion of � + � is �9 then � =_________
�2 �2
13. The length of transverse axis of the hyperbola 16 − 9
=− 1 is________
14. The length of the foot of the perpendicular drawn from the point � 3,5,4 on y-axis is_____
1
15. If � � = � − � then �' −1 =__________

III Answer the Following Questions 5x1=5

3
16. If tan � = 4
and � lies in the third quadrant, find sin �
17. If � is any real number then solve 5� − 3 < 3� + 1
18. How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no
letter can be repeated?
19. Find the slope of the line joining the points 3, − 2 and −1,4
20. Find the mean of first � natural numbers.

PART – B
Answer any NINE questions 9 x 2 = 18
21. Define power set and write the power set of the set � = {�, �}.
22. If X & Y are two sets such that � ∪ � has 50 elements, X has 28 elements & Y has 32 elements. How
many elements does � ∩ � have?

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 Model Question Papers

23. If � = 1,2 & � = 3,4 .write � × �. How many subsets will � × � have?
24. The minute hand of a clock is 2.1 cm long. How far does its tip move in 20 minute?
25. Prove that cos 4� = 1 − 8 sin2 � cos2 �
1+� 4�
26. Find the least positive integer m such that 1−�
= 1.
27. Solve 7� + 1 ≤ 4� + 5 and represent the solution graphically on the number line.
1
28. Find the equation of the line passing through −4,3 with slope 2
29. Find the equation of the line passing through the point −3,4 and parallel to the line
3� − 4� + 10 = 0
30. Find the ratio in which the line segment joining the points 4, 8, 10 and 6, 10, − 8 is divided
by YZ-plane.
3�2 −�−10
31. Evaluate lim �2 −4
�→2
32. By giving a counter example show that the following statement is false, ‘If � is an odd integer then
� is a prime’.
33. Coefficient of variation of two distributions are and 60 and 70 , and their standard deviations are
21 and 16, respectively. What are their arithmetic means.
34. One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will
be “not an ace”.

PART – C
Answer any NINE questions 9 x 3 = 27
35. In a survey, it was found that 21 people liked product A, 26 liked product B & 29 liked product C.
If 14 liked products A & B, 12 liked products C & A, 14 liked product B & C then 8 liked all the three
products. Find how many liked product C only.
36. Let � = 1,2,3,4,5,6 . Define a relation R from A to A by � = �, � ; � = � + 1 and write down
the domain, co-domain & range.
37. Find the general solution of sin 6x + sin 2x − sin 4x = 0
1−�
38. Express 1+� into polar form.
39. Solve 5�2 + � + 5 = 0
40. If 5 4 �� = 6. 5 ��−1 then find �.
6
3�2 1
41. Find the term independent of x in the expansion of 2
− 3�
1
42. If the sum of � terms of an A.P is �� + 2 � � − 1 � , where � & � are constant, find the common
difference.
43. Find the sum to n terms of the sequence : 7,77,777,7777,…………
44. Find the co-ordinates of the foci and equation of latus rectum of the hyperbola 3�2 − �2 = 3
45. Find the derivative of � � = sin � with respect to � from the first principle
46. Check whether the statement ‘If �, � ∈ � such that � & � are odd then �� is odd’ is true or false by
direct method.
47. Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die. B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤5. Describe the events
i) A’ ii) not B iii) A or B

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 Model Question Papers

48. 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 - D
Answer any FIVE questions 5 x 5 = 30
49. Define modulus function. Draw the graph of the modulus function and write down its range and
domain.
sin 5�−2 sin 3� +sin �
50. Prove that cos 5�−cos �
= tan �
51. Prove by mathematical induction that
� �+1 �+2 �+3
1.2.3 + 2.3.4 + ………. + � � + 1 � + 2 = 4
52. Solve the following system of inequalities graphically: � + � ≥ 5, � − � ≤ 3
53. 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.
54. State and prove Binomial theorem for all natural numbers.
55. Derive a formula for the perpendicular distance of a point �1 , �1 from the line �� + �� + � = 0.
56. Derive the formula for the distance between two points �1 , �1 , �1 and �2 , �2 , �2 and hence
find the distance between the points 2, − 1, 3 and −2, 1, 3 .
sin �
57. Prove geometrically, lim = 1 where � is in radian
�→0 �
58. Find the mean deviation about the median for the following data
Marks 0-10 10-20 20-30 30-40 40-50 50-60

Number of girls 6 8 14 16 4 2

PART-E
Answer the following questions
59. Prove that geometrically that cos � + � = cos � . cos � − sin � . sin � and hence prove that
cos 2� = ���2 � − ���2 �
OR
Define Parabola and derive the equation of parabola in standard form. (6)
� + ��, � < 1
60. Suppose � � = 4, � = 1 & lim � � = � 1 , what are the possible values of a & b.
�→1
� − ��, � > 1
OR
1 1
Find the sum to n terms of the series 1.2
+ 2.3 + ………… (4)

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 Model Question Papers

MODEL QUESTION PAPER 4

Time : 3 hours 15 Minutes Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part A has 10 Multiple Choice Questions, 5 Fill in the blanks and 5 Very Short Answers questions of
1 mark each
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 subsection I and II of Part A.
4. Use the graph sheet for the question on linear inequality in PART D.

PART – A
I Answer all the Multiple Choice Questions 10 x 1 = 10
1. If � = �������� ����� ������� , � = ���� ������� then � � ∩ � =
(A) 0 (B) 1 (C) 2 (D) 4
2. If the set A as � elements then the number of relation on A is
2
(A) � (B) �2 (C) 2� (D) 2�
3. tan 750 + cot 750 =
(A) 4 (B) 2 3 (C) −4 (D) 3 − 1
1 26
4. The value of �24 + �
(A) 1 (B) I (C) −� (D) 0
n n
5. If C7 = C6 then the value of � is
(A) 13 (B) 6 (C) 7 (D) 12
6. The third term of a G. P. is 9. The product of its first five terms is :
(A) 310 (B) 35 (C) 312 (D) 39
7. If 2� + 3� + 4 = 0 and ℎ� + �� + 2 = 0 are identical lines then 3ℎ − 2� =
(A) 1 (B) 0 (C) 1 (D) 2
1 �� � > 0
8. limit of the function � � = at � = 0
−1 �� � < 0
(A) 1 (B) 0 (C) −1 (D) Not exist
9. The negation of the statement “Intersection of disjoint sets is not an empty set.” Is
(A) Intersection of disjoint sets is singleton set
(B) Intersection of disjoint sets is an empty set.
(C) Union of disjoint sets is not an empty set.
(D) Union of disjoint sets is an empty set.
1 1 1
10. If E and F are event such that � � = 4 , � � = 2 and � � ∩ � = 8
, then � � ∪ � =
3 7 1 5
(A) 8 (B) 8 (C) 4 (D) 8

II Fill in the blanks by choosing the appropriate answer from those given in the bracket.
��, �, �, �, � 5 x 1 = 10
11. If A = a, b, c then number of elements in P(A)_______
12. The coefficient of a2b3 in (a + b)5 is________
13. The radius of the circle �2 + �2 + 2� + 10� + 26 = 0 is_________
14. The point 1, − 3, 4 lies is________th octant
�100 �99 �2
15. If the function �(�) defined by � � = 100
+ 99
+…+ 2
+ � + 1, then �' 0 =_________

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 Model Question Papers

III Answer the Following Questions 5x1=5

7�
16. Find the value of sec 6
17. Solve the inequality (2� − 5) > (1 − 5�) if � is any real number.
1 1 �
18. If 6! + 7! = 8! then find �
19. Find the slope of a line 3� − 4� + 10 = 0
20. Find the mean of the data 4, 7, 8, 9, 10, 12, 13, 17.

PART – B
Answer any NINE questions 9 x 2 = 18
21. If � = 3, 5, 7, 9, 11 , � = 7, 9, 11, 13 & � = 11, 13, 15 , then find � ∩ � ∪ � .
22. If X & Y are two sets such that � ∪ � has 18 elements, X has 8 elements & Y has 15 elements, then
how many elements does � ∩ � have?
23. If � = 3,5,7,9,11 , � = 7,9,11,13 & � = 11,13,15 find � × � ∪ � .
24. Find the area of sector of a circle of radius 20cms and the centre angle 450
25. Prove that ���3� = 3���� − 4 sin3 �.
3+� 2 3−� 2
26. Express 5+�2 − 5−�2
in the form of � + ��
27. Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their
sum is less than 40.
28. Find the equation of the straight line intersecting � − ���� at a distance of 2units above the origin
and making an angle 300 with the positive direction of � − ����.
29. Find the equation of the line perpendicular to the line � − 2� + 3 = 0 and passing through the
point 1, − 2
30. Without using distance formula show that the points � −2, 3, 5 , � 1, 2, 3 & � 7, 0, − 1 are
collinear.
1 1
+
31. Evaluate lim � 2
�→−2 �+2
32. Write the contrapositive and converse of “If a parallelogram is a square, then it is a rhombus”.
33. Two series A and B with equal means have standard deviation 9 and 10 respectively. Which series
more consistent?
34. An experiment involves rolling a pair of dice and recording the number that comes up Describe the
following events. A : The sum is greater than 8 B: 2 occurs on either die

PART – C
Answer any NINE questions 9 x 3 = 27
35. In a group of 65 people, no like cricket, 10 like both cricket & tennis. How many like tennis?
How many like tennis only & not cricket?
36. Write the relation R defined as � = �, � + 5 ; � ∈ 0,1,2,3,4,5 in roster system. Write down
its range & domain.
37. Find the general solution of sec2 2� = 1 − tan 2�.
�
38. Solve �2 + 2
+1 =0
−16
39. Convert the complex number 1+� 3
in to polar form.
40. A letter is chosen at random from the ward “ASSASSINATION”, Find the probability that the letter is
an i) vowel ii) consonant.
41. Find the coefficient of �5 of (� + 3)8
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 Model Question Papers

42. Insert five numbers between 8 and 26 such that resulting sequence is an A.P.
43. The sum of first three terms of G.P is 16 and the sum of next three terms is 128 then determine �
and � and also find sum up to n terms.
44. Find the equation of circle which passes through (1,0) and (0, − 1) and whose centre lies on the
line � − � + 2 = 0.
45. Find the derivative of � � = �3 with respect to � from the first principle
46. Verify by the method of contradiction that 5 is irrational.
47. 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 iii) The student has opted NSS but not NCC.
48. A coin is toosed three times, consider the following events.
A: ‘No head appears’, b: ‘Exactly one head appears’ and C: ‘Atleast two heads appear’. Do they form
a set of mutually exclusive and exhaustive events?

PART - D
Answer any FIVE questions 5 x 5 = 30
49. Define signum function. Draw the graph of the signum function and write down its range and
domain.
sin 9�+sin 7�+sin 3�+sin 5�
50. Prove that cos 9�+cos 7�+cos 3�+cos 5�
= tan 6�
1 1 1 �
51. Prove that 1.2
+ 2.3 + ……. . + � �+1 = �+1 , ∀� ∈ � by the principle of mathematical induction.
52. Solve the following system of inequalities graphically: 3� + 4� ≤ 60 , � + 3� ≤ 30, � ≥ 0, � ≥ 0
53. 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 girl (ii) At least 1 boy and 1 girl (iii) At least three girls
54. Find � if 17th and 18th expansion 2 + � 50 are equal.
55. Derive the equation of the line with slope m and y-intercept c . Also find the equation of the line
1 3
for which tan � = 2 and y-intercept is − 2
56. Derive an expression for the coordinates of a point that divides the line joining the points
�1 , �1 , �1 and �2 , �2 , �2 internally in the ratio � : �
sin �
57. Prove geometrically, lim = 1 where � is in radian
�→0 �
58. 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 students 2 3 8 14 8 3 2

PART-E
Answer the following questions
59. Prove that geometrically that cos � + � = cos � . cos � − sin � . sin � and hence find cos 75°
OR
Define Hyperbola and derive the equation of hyperbola in standard form. (6)
�+cos �
60. Find the derivative of tan �
with respect to �
OR
Find the sum to n terms of the series (1 × 2) + (2 × 3) + (3 × 4) + (4 × 5) + ………… (4)
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 Model Question Papers

MODEL QUESTION PAPER 5

Time : 3 hours 15 Minutes Max. Marks: 100
Instructions:
1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part A has 10 Multiple Choice Questions, 5 Fill in the blanks and 5 Very Short Answers questions of
1 mark each
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 subsection I and II of Part A.
4. Use the graph sheet for the question on linear inequality in PART D.

PART – A
I Answer all the Multiple Choice Questions 10 x 1 = 10
1. If A and B are two sets, then � ∩ � ∪ � =
(A) A (B) B (C) � (D) �  �
�2 +2�+1
2. The domain of the function � � = �2 −�−6
(A) � − {3, − 2} (B) � − {3, 2} (C) � − {3, − 2} (D) � − (3, − 2)
1
3. If cos � =− 2 , � lies in third quadrant then find the value tan �
1 1
(A) 3
(B) 3 (C) − 3 (D) − 3
4. If � = −1and n is a +ve integer, then �� + ��+1 + ��+2 + ��+3 =
(A) 1 (B) i (C) �� (D) 0
5. If ��8 = � �2 then the value of ��2
(A) 90 (B) 45 (C) 10 (D) 1
6. Which term of the series 3 + 8 + 13 + 18 + ... is 498-
(C) 102th (D) 101th
th th
(A) 95 (B) 100
7. The slope of the line making inclination of 600 with positive direction of � − ����.
1 1
(A) 1 (B) 3
(C) 3 (D) 2
|�|
�≠0
8. Find lim  �(�), where � � = � is
�→0+ 0 �=0
(A) 1 (B) 0 (C) −1 (D) 2
9. Which of the following is not a statement-
(A) Please do me a favour (B) 2 is an even integer
(C) 2 + 1 = 3 (D) The number 17 is prime
10. The number of elements in sample space for the experiment “a coin is tossed repeatedly three
times”.
(A) 1 (B) 8 (C) 4 (D) 16
II Fill in the blanks by choosing the appropriate answer from those given in the bracket.
�, ��, ��, �, � 5 x 1 = 10
11. If A = {1,2,3,4} then the number of nonempty subsets of A is______
12. The number of terms in the expansion � + � 51 + � − � 51 is_______
13. If the length of the latus rectum of �2 = 4�� is 8, the value of � =________
14. The point (2, − 4, − 7) lies is________th octant
15. If � � = 2�2 + 3� − 5 then �' 1 = ____________

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 Model Question Papers

III Answer the Following Questions 5x1=5

0
16. Convert 520 in to radian measure.
17. Solve 4� + 3 < 5� + 7
18. Given 5 flags of different colours how many different signals can be made if each signal requires
the use of 2 flags , one below the other.
1
19. Find the equation of the line passing through −4,3 with slope 2
20. If variance of some ungrouped data is 64 then find its standard deviation

PART – B
Answer any NINE questions 9 x 2 = 18
21. Let � = 1, 2, 3, 4, 5, 6, 7, 8, 9 , � = 2, 4, 6, 8 & � = 3, 4, 5, 6 find (� − �)'
22. In a class of 35 students, 24 likes to play cricket, 5 likes to play both cricket and football.
Find how many students like to play football?
23. If f: Z → Z is a linear function defined by �(�) = 1,1 , 0, − 1 , 2,3 , then find f x .
24. Prove that tan 7� − tan 5� − tan 2� = tan 7� tan 5� tan 2�
3� 3�
25. Prove that cos 2
+ � cos 2� + � cot 2
− � + cot 2� + � =1
26. If � = 3 + 4� then find the value of ��
27. If � is any real number then solve −8 ≤ 5� − 3 < 7.
28. Find the distance between the parallel lines 3� − 4� + 7 = 0 and 3� − 4� + 5 = 0
29. Find the equation of the line parallel to the line 3� − 4� + 2 = 0 and passing through the point
( − 2,3).
30. Show that −1, 2, 1 1, − 2, 5 4, − 7, 8 & (2, − 3, 4) are the vertices of a parallelogram.
�3 −2�2
31. Evaluate lim
�→2 �2 −5�+6
32. Given P: 25 is a multiple of 5; q: 25 is a multiple of 8. Write the compound statement Connecting
these two statements with “and ”, “or”. Also check the validity of the statement.
33. Find the median for the data 36, 72,46,42,60,45,53,46,51,49.
3 1
34. Given P(A)= 5 and P(B) = 5. Find P(A or B), if A and B are mutually exclusive events.

PART – C
Answer any NINE questions 9 x 3 = 27
35. In a survey of 600 students in a school, 150 students were found to be taking tea and 250 taking
coffee, 100 were taking both tea & coffee. Find how many student were taking neither tea nor
coffee.
36. Let � = 1,2,3, ……. . , 14 . Define a relation R from A to A by � = �, � ; 3� − � = 0, �, � ∈ �
write its domain & range?
37. Prove that cos 6� = 32 cos6 � − 48 cos4 � + 18 cos2 � − 1
−1+�
38. Express 2
in the polar form
39. Solve 27�2 − 10� + 1 = 0
40. Find the number of Permutations of the word ‘MISSISSIPPI’ in which all 4S’s are together and 2P’s
are together.
41. Find the coefficient of �6 �3 in the expansion of � + 2� 6
1 1 1
42. In an A.P ��ℎ term is � and ��ℎ term is � then prove that the sum of first �� terms is 2 �� + 1

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 Model Question Papers

43. The sum of first three terms of a G.P. is 13/12 and their product if -1. Find the common ratio and
the terms.
44. Find the equation of hyperbola whose foci is 0, ± 13 and length of conjugate axis is 24.
45. Find the derivative of � � = cos � with respect to � from the first principle
46. Verify by the method of contradiction that 2 is irrational.
47. A die is thrown. Describe the following events:
i) A: a number less than 7. ii) B: a number greater than 7 iii) C : a multiple of 3
48. In class XI of a school 40% of the student study mathematics and 30% of students study Biology and
10% of students study both Mathematics and Biology. If a student is selected at random from the
class, find the probability that he will be studying Mathematics or Biology.

PART - D
Answer any FIVE questions 5 x 5 = 30
49. Define a polynomial function. If the function �: � → � is defined as � � = �2 , then draw the graph
of ‘ f ’ and find the domain and range.
� � 3
50. Prove that ���2 � + ���2 � + 3 + ���2 � − 3 = 2
1 1 1 �
51. Prove by mathematical induction that 2.5
+ 5.8 + ………. . + 3�−1 3�+2
= 6�+4
52. Solve the following system of inequalities graphically: 2� + � ≥ 4 , � + � ≤ 3, 2� − 3� ≤ 6
53. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls. If each
selection consists of 3 balls of each other.
54. State and prove Binomial theorem for all natural numbers.
55. Derive the formula for the angle between two straight lines with slopes �1 and �2 hence find the
angle between the lines − 3� + � − 1 = 0 & � + 3� + 1 = 0
56. Derive an expression for the coordinates of a point that divides the line joining the points
�1 , �1 , �1 and �2 , �2 , �2 internally in the ratio � : �
��−�� �15 −1
57. Prove that lim = ���−1 and hence evaluate lim  �10−1
�→� �−� �→1
58. Find the mean deviation about the median for the following data

�� 3 6 9 12 13 15 21 22

�� 3 4 5 2 4 5 4 3

PART-E
Answer the following questions
59. Prove that geometrically that cos � + � = cos � . cos � − sin � . sin � and hence prove that
cos 2� = cos2 � − sin2 �
OR
Define Ellipse and derive the equation of ellipse in standard form. (6)
�5 −cos �
60. Find the derivative of sin �
with respect to �
OR
Find the sum to n terms of the series 5 + 11 + 19 + 29 + ………… (4)

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 Model Question Papers

ANSWER KEY
QUESTION NUMBERS
Model
Multiple Choice Questions Fill in the blanks
papers
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Board B C C B D C A C C B 2 4 8 6 1
1 B A B D A B B B C A 16 5 7 6 1
2 C A C D A A D B C C 4 128 4/5 8 2
3 B B B B D A B C D C 1 78 6 5 2
4 B D A D A A B D B D 8 10 0 4 1
5 A C B D B B C A A B 15 26 2 8 7

∗∗∗∗∗∗ ALL THE BEST ∗∗∗∗∗∗

20