mathematics model papers for class xi

MODEL PAPER CLASS-XI

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CLASS -XI Maths Sample Paper-1

Instructions

1. Questions 1 to 10 carry 1 mark each.

2. Questions 11 to 22 carry 4 marks each.


1. Write the following set in the set-builder form: {5, 25, 125, 625}.

2. Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A

and B, where x, y and z are distinct elements.

3. Let f be the subset of Z ×Z defined by f={(ab,a+b):a,b∈Z} Is f a function from Z to Z? Justify your

answer.

4. Express the given complex number in the form a + ib: (1 – i) – (–1 + i6).

5. If 2x+ i (x - y) = 5, where x and y are real numbers, find the values of x and y.

6. Solve the given inequality for real x: 4x + 3 < 5x + 7.

7. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be

repeated?

8. Find the sum of odd integer from 1 to 21.

9. Write the equations for the x and y-axes.

10. 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?

11. A wheel makes 500 revolutions per minute. How many radians it turns in one second?

12. Write the value of tan 75°.

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225

taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea

nor coffee?

14. Find the value of .

15. Solve the quadratic equation 25x2 – 30x + 11 = 0.



16. In how many of the distinct permutations of the letters in MISSISSIPPI do the four 1’s not come

together?

17. Let the sum of n, 2n, 3n terms of an A.P. be S1,S2and S3 respectively, show that : S3 = 3(S2 - S1).

18. Find coordinates of the foot of perpendicular from the point (3,-4) to line 4x – 15y + 17 = 0.

19. Find the equation of the circle which passes though the points (3,7), (5,5) and has its centre on

the line x - 4y = 1.

20. 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.

21. A group of two persons is to be selected from two boys and two girls. What

is the probability that the group will have

(a) two boys (b) one boy (c) only girls

22. Find the point in XY-plane which is equidistant from three points A(2,0,3), B(0,3,2) and C(0,0,1).

23. A market research group conducted a survey of 1000 consumers and reported that 720

consumers like product A and 450 consumers like product B. What is the leastnumber that must

have liked both products?

24. Solve : 2 cos2 x+3 sin x = 0.

25. Find centroid of a triangle, mid-points of whose sides are (1, 2, - 3), (2, 0, 1) and (- 1, 1, - 4).

26. If the ratio of the coefficients of 3rd and 4th terms in the expansion of is 1:2 then find

the value of n.

27. Find the value of n so that may be the geometric mean between a and b.

28. The mean and standard deviation of six observations are 8 and 4, respectively. If each

observation is multiplied by 3, find the new mean and new standard deviation of the resulting

observations.

29. P,Q,R shot to hit a target. If P hits it 3 times in 4 trials, Q hits it 2 times in 3 trials and R hits it 4

times in 5 trials, what is the probability that the target is hit by at least two persons?



CLASS -XI Maths Sample Paper -2

Instructions




1. If X = {a, b, c, d} and Y = {f, b, d, g}, find

(i) X – Y

(ii) Y – X

(iii) X ∩ Y

2. The figure, shows a relation between the sets A and B. Write the relation in roster form.

3. Let f, g: R → R defined by f(x) = 6x and g(x) = 3x. Find f - g.

4. Find the equation of the angle bisectors of the angle between the coordinate axes.

5. A bag contains 4 balls of different colours red, green, blue and yellow.A person draws two balls

from the bag, one after the another without replacement. Describe the sample space for the

experiment.

6. If cos x = cos y, then x = ?

7. How many different signals can be made using 3 flags at a time from 5 flags of different

colours?

8. Find the value of cos 35° + cos 145° + cos 330°.

9. Evaluate : sin 130° cos 110° + cos 130° sin 110°.

10. Find the sum of the series : 3 + 8 + 13 + ............... + 33.

11. Prove that : tan 30° + tan 15° + tan 30°.tan 15° = 1.

12. Solve cos 2θ – cos θ = 0.



13. If U = {1,2,3,4,……..,10} is the universal set for the sets A = {2,3,4,5} and B = {1,2,3,4,5,6}, then

verify that (A∪B)' = A'∩B'.

14. Find the domain and the range of the real function f defined by f (x) = |x – 1|.

15. Express the given complex number in the form a + ib: .

16. What is number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of

these

(i) four cards are of the same suits,

(ii) are face cards.

17. Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

18. Without using distance formula, show that points (–2, –1), (4, 0), (3, 3) and (–3, 2) are vertices of

a parallelogram.

19. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at

the base. How wide is it 2 m from the vertex of theparabola?

20. 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.

21. A and B are two students. Probabilities of solving a problem by A and B separately are 5/9

and 7/11 respectively. Find the probability that the problem will be solved.

22. Find lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

23. Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find

the range of f.

24. Show that

25. A owner of a land has 3000m of fencing with which he wishes to enclose a area of rectangular

piece of grazing land along a straight portion of a river. Fencing is not required along the river.

What could be the possible values of the breadth of the rectangular piece of grazing land if its

length is along the river.

26. The coefficients of 2nd, 3rd, 4th terms in the expansion of (1 + x)n are in A.P. Find n.

27. If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq-r, br-p and cp-q = 1.



28. Prove that 102n – 1 + 1 is divisible by 11 for all n ∈ N.

29. The mean and standard deviation of a group of 100 observations were found to be 20 and 3,

respectively. Later on it was found that three observations were incorrect, which were recorded as

21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.



XI Maths Sample Paper -3

Instructions




1. Write down all the subsets of the following set: {1, 2, 3}.

2. Let A = {1, 2, 3, 4} and B = {10, 12, 13, 14, 20}. Whether f: A → B defined by f(1) = 10, f(2) = 12, f(3) =

13 is a function?

3. If n(A) = 3 and n(B) = 3, then find n(A x B).

4. Find the multiplicative inverse of the complex number –i.

5. Write the first two terms in the following series. Tn = n3 - 1, if n is odd and Tn = n + 1 if n is even.

6. Tickets are numbered from 1 to 25. They are well shuffled and a ticket drawn at random . What is

the probability that the drawn ticket has a prime number?

7. Given and . Find P(A or B), if A and B are mutually exclusive events.

8. Find the slope of a line which passes through (1,2) and (-3,4)?

9. Write the value of tan 15°.

10. Evaluate : sin(40°+θ) cos(10°+θ) - cos(40°+θ) sin(10°+θ)

11. Prove that


13. Let A,B and C be the sets such that A∪B = A∪C and A∩B = A∩C. Show that B=C.

14. Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A

and B, where x, y and z are distinct elements.

15. If = 1, then find the least positive integral value of m.

16. How many numbers of 6 digits can be formed out of the digits of the numbers 567724? How

many of the numbers so formed is even?

17. Find the sum to n terms of the A.P., whose kth term is 5k + 1.



18. Find the equation of the line perpendicular to the line 2x – 3y + 7 = 0 and having x-intercept 4.

19. An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at the vertex of

the parabola. Find the length of the side of the triangle.

20. Find the equation of the hyperbola satisfying the give conditions: Foci , passing

through (2, 3).

21. The probability that a person visiting a doctor will have his blood testdone is 0.75 and the

probabilitythat he will be admitted is 0.30. The probabilitythat he will have his blood test done or be

admitted is 0.45. Find the probabilitythat a person visiting the doctor will have his blood test done

and be admitted?

22. Are the points A(3, 6, 9), B(10, 20, 30) and C(25, -41, 5) the vertices of a right angles triangle?

23. Let A = {1,2}, B = {1,2,3,4}, C = {5,6} and D = {5,6,7,8}

Verify that :

(a) A × (B∩C) = (A×B)∩(A×C)

(b) A × C is a subset of B × D.

24. Prove that: cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1.

25. How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that

the resulting mixture will contain more than 25% but less than 30% acid content?

26. If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive

integer.

27. Find the sum of all numbers between 200 and 400 which are divisible by 7.

28. Prove the following by using the principle of mathematical induction for all n ∈ N:

29. Find the mean deviation about the mean for the data

xi 10 30 50 70 90

fi 4 24 28 16 8



XI Maths Sample Paper -4

Instructions




1. Write the set A = {x : x ∈ N and x2 < 25} in roster form.

2. If A B = {(x, 1), (y, 2), (z, 1), (x, 2), (z, 2), (y, 1)}, find A and B.

3. If (x – 3, –6) = (2, y – 9), then find the values of x and y.

4. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one

second?

5. Express : i9 + i19 in the form a+bi.

6. How many 2-digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the digits can

be repeated?

7. Let A(6,4) and B(2,12) be two given points. Find the slope of a lineperpendicular to AB.

8. Show that the sequence n2 - 3 is not an A.P.

9. The odds in favour of an event are 4:7. What is the probability that this event will fail?

10. Find the value of cos 135°.

11. If in two circles, arcs of same length, subtend angles 120o and 150o at the centre, find the ratio

of their radii.

12. Solve sinθ – sin2θ = 0.



13. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis

only and not cricket? How many like tennis?

14. If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

15. Find the modulus and the argument of the complex number .

16. Find the sum to n terms of the series whose nth term is given by n (n + 1) (n + 4).

18. Find the coordinates of foot of perpendicular from the point (–1, 3) to line 3x – 4y – 16 = 0.

19. Find the equation of the ellipse that satisfies given conditions:Vertices (±6, 0), foci (±4, 0).

20. Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing

through (2, 3) and axis is along x-axis.

21. Find the probability that in a random arrangement of the word ‘society’ all the three vowels

come together.

22. Three vertices of a parallelogram ABCD are A(4, 0, 3), B (3, 4, – 2) and C (– 2, 0, 1). Find the

coordinates of the fourth vertex.

23. Find the domain and the range of the real function f defined by f (x) = |x – 1|.


25. The length of a rectangle is 5 more than the 2 times of its breadth. If the perimeter of the

rectangle is atleast 320 cm. Find the minimum value of its breadth.

26. Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate .

27. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of

terms occupying odd places, then find its common ratio.

28. Prove the following by using the principle of mathematical induction for all n ∈ N:

29. The mean and standard deviation of a group of 100 observations were found to be 20 and 3,

respectively. Later on it was found that three observations were incorrect, which were recorded

as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.



XI Maths Sample Paper – 5

Instructions




1. If A = {x : x is a prime number x N}, then find Ac.

2. If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Define a relation on a set A by R = {(x, y): 3x - y = 0, where x, y ∈

A}. Show this relationship using arrow diagram.

3. Examine the relation : R={(2,1),(3,1),(4,1)} and state whether it is a function or not?

4. Write the value of sin 75°.

5.

6. Express the given complex number in the form a + ib: (1 – i) – (–1 + i6).

7. How many permutations of the letters of the word ‘APPLE’ are possible?

8. Find the equation of a line which is parallel to Y-axis and passes through the point (2,3).

9. Write the 16th term of the sequence defined by an = n2 - n+1.

10. Find the sample space associated with the experiment of rolling a pair ofdice once. Find the

number of elements of this sample space.

11. Evaluate : sin(50°+θ)cos(10°-θ)+cos(50°+θ)sin(10°-θ).

12. Prove that cos 8θ = 1 – 8sin2 2θ cos2 2θ.

13. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked C. If 14

people liked products A and B, 12 people liked products C and A, 14 people liked products B and

C and 8 liked all the three products. Find how many liked product C only.

14. Prove by using the principle of mathematical induction for n∈N : (2n+7) < (n+3)2

15. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 +

B2.

16. If 8 parallel lines in a plane are intersected by a family of 10 parallel lines. Find the number of

parallelograms formed.



17. The ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio

of mth and nth term is (2m – 1): (2n – 1).

18. By using the concept of equation of a line, prove that the three points (3, 0), (–2, –2) and (8, 2)

are collinear.

19. Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the

eccentricity and the length of the latus rectum of the ellipse .

20. Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum

is of length 12.

21. Two cards are drawn from a pack of 52 cards. What is the probabilitythat either both are aces

or both are black?

22. Find the ratio in which the line joining the points (1, 2, 3) and (- 3, 4, - 5) is divided by the xy-

plane. Also, find the coordinates of the point of division.

23. Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find

the range of f.

24. Prove that tan 70° = 2 tan 50° + tan 20°.

25. IQ of a person is given by the formula .

Where MA is mental age and CA is chronological age. If 80 ≤ IQ ≤ 140 for a group of 12 years old

children, find the range of their mental age.

26. In the expansion of (1 + x)34, the coefficients of (2r+1)th and (r +2)th terms are equal, find r.

27. Let the sum of n, 2n, 3n terms of an A.P. be S1,S2and S3 respectively, show that : S3 = 3(S2 - S1).

28. Using the words “necessary and sufficient” rewrite the statement “The integer n is odd if and

only if n2 is odd”. Also check whether the statement is true.

29. Find the mean and variance for the data

xi 6 10 14 18 24 28 30

fi 2 4 7 12 8 4 3

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