Sample Paper – 2010
Class – X
Subject – Mathematics
TIME: 3 HOURS MAX MARKS: 80
GENERAL INSTRUCTIONS:
1. All questions are compulsory.
2. The question paper consists of thirty questions divided into four sections A, B, C & D. Section A comprises of ten questions of 01 marks each, Section B comprises of five questions of 02 marks each, Section C comprises of ten questions of 03 marks each and section D comprises of five questions of 06 marks each.
3. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question.
4. There is no overall choice. However internal choice has been provided in one question of 02 marks each, three questions of 03 marks each and two questions of 06 mark each. You have to attempt only one of the alternatives in all such questions.
5. In question on construction, drawings should be neat and exactly as per the given measurements.
6. Use of calculators is not permitted.
SECTION  A
1. If cos θ = and θ + φ = 90^{0}, find the value of sin φ.
2. Find the quadratic polynomial, the sum and product of whose zeros are and respectively.
3. Find the discriminant for the equation 9x^{2} – 12x + 4 = 0.
4. If one root of the equation 3x^{2} + 11x + k = 0 is the reciprocal of the other, find the value of k.
5. If tan 2A = cot ( A – 18^{0} ), where 2A is an acute angle, find the value of A.
6. A bag contains 8 red, 2 black and 5 white balls. One ball is drawn at random. What is the probability that the ball drawn is neither black nor red?
7. State Euclid’s Division Lemma.
8. The curved surface area of a cylinder is 1760 cm^{2} and its base radius is 14 cm, find the height of the cylinder? A
9. In Δ ABC, DE // BC, so that AD = 2.4 cm, DB = 3.2 cm, D E
and AC = 9.6 cm, then find EC?
B C
10. Both the ogives (less than and more than) for a data intersect at P(30, 23). Find the median for the data.
SECTION  B
11. Anand Patil started working in a firm in 1995 at an annual salary of Rs. 5000 and received an increment of Rs. 200 each year. In what year did his annual salary will reach Rs. 7000?
12. If 4 sin θ = 3 cos θ, find the value of .
OR
Prove the following identity:
( Sin A + Cosec A )^{2} + ( Cos A + Sec A )^{2} = 7 + tan^{2}A + Cot^{2}A A
13. In the given figure,


Prove that ∆ ABC is isosceles.
B C
14. Find the value of x for which the distance between the points P(2, –3) and Q(x, 5) is 10 units.
15. A jar contains 54 marbles each of which is blue, green or white. The probability of selecting a blue marble at random is , and the probability of selecting a green marble at random is . How many white marbles does the jar contain?
SECTION  C
16. Using Euclid’s division algorithm find the H C F of 84, 90 and 120.
17. Find the values of k for which the quadratic equation x^{2} – 2x(1 + 3k) + 7(3 + 2k) = 0
has real and equal roots.
OR
Solve for x: , x ≠ –1, –2, –4
18. Find the zeros of the polynomial f(x) = , and verify the relationship between the zeros and its coefficients.
19. Prove that: 2 sec^{2} θ – sec^{4} θ – 2 cosec^{2} θ + cosec^{4} θ = cot^{4} θ – tan^{4} θ
Page 2 of 4
20. Three numbers are in A.P. If the sum of these numbers is 27 and their product is
648, find the numbers.
OR
Sum of first 7 terms of an A.P. is 20 and the sum of next 7 terms is 17. Find the A.P.
21. Determine the ratio in which the point P(m, 6) divides the join of A(–4, 3) and B(2, 8). Also find the value of m.
OR
If (x, y) be on the line joining the two points (1, –3) and (–4, 2),
prove that x + y + 2 = 0.
22. Find the area of the triangle formed by joining the midpoints of the sides of the
triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of the area of the
triangle formed to the area of the given triangle.
23. Construct a ∆ ABC in which AB = 5.5 cm, BC = 4 cm and B = 75^{0}. Construct a triangle similar to ∆ ABC, each of whose sides are times the corresponding sides of ∆ ABC.
24. On a square handkerchief, nine circular designs
each of radius 7 cm are made. Find the area of the
remaining portion of the handkerchief.
25. In the figure ∆ ABC is a right triangle, right angled A
at B. AD and CE are the two medians drawn from

AD = cm, find the length of CE.
B D C
SECTION  D
26. Form a pair of linear equations in two variables using the following information and solve it graphically.
Five years ago, Sagar was twice old as Vijay. Ten years later Sagar’s age will be ten
years more than Vijay’s age. Find their present ages. What was the age of Sagar
when Vijay was born?
A
27. State and prove the converse of Pythagoras theorem.
Use the above theorem to prove the following:
In the figure, AD BC. If AD^{2} = BD × DC,
Prove that ABC is a right triangle. B D C
OR
Prove that the lengths of two tangents drawn from an external point to a circle are
equal.
Use the above theorem to prove the following: A
A circle is touching the side BC of ∆ ABC at P
and touching AB and AC produced at Q and R
respectively. B P C
Prove that: AQ = (Perimeter of ∆ ABC) Q R
27. As observed from the top of a 75m high lighthouse from the sealevel, the angles of depression of two ships are 30^{0} and 45^{0}. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships.
OR
The angle of elevation of a jet plane from a point A on the ground is 60^{0}.
After a flight of 15 seconds, the angle of elevation changes to 30^{0}. If the jet plane is flying at a constant height of 1500 m, find the speed of the jet plane in km/h.
28. The radii of the ends of a frustum of a cone 45cm high are 28cm and 7cm. Find its volume and total surface area.
OR
Water in a canal, 6m wide and 1.5m deep, is flowing with a speed of 10km/hr. How much area will it irrigate in 30 minutes, if 8cm standing water is needed? The median of the following data is 20.75. Find the missing frequencies x and y if the total frequency is 100.
Class Interval 
0 – 5 
5 – 10 
10 – 15 
15 – 20 
20 – 25 
25 – 30 
30 – 35 
35 – 40 
Frequency 
7 
10 
x 
13 
y 
10 
14 
9 
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