Board Paper of Class 10 2008 Maths All India(SET 1)  Solutions
1. All questions are compulsory.
2. The question paper consists of 30 questions divided into four sections – A, B, C and D. Section A comprises of ten questions of 1 mark each, Section B comprises of five questions of 2marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6marks 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.
There is no overall choice. However, an internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
4. In question on construction, the drawing should be neat and as per the given measurements.
5. Use of calculators is not permitted.
 Question 1
Write a rational number between and.
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 Question 2
Write the number of zeroes of the polynomial y = f(x) whose graph is given in figure.
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 Question 3
Is x = −2 a solution of the equation x^{2} − 2x + 8 = 0?
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 Question 4
Write the next term of the A.P. .
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 Question 5
D, E, and F are the midpoints of the sides AB, BC, and CA respectively of ΔABC. Find.
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 Question 6
In figure, if ∠ATO = 40°, then find ∠AOB.
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 Question 7
If sin θ = cos θ , then find the value ofθ.
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 Question 8
Find the perimeter of the given figure, where AED is a semicircle and ABCD is a rectangle.
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 Question 9
A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball.
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 Question 10
Find the median class of the following data:
Marks obtained
0 − 10
10 − 20
20 − 30
30 − 40
40 − 50
50 − 60
Frequency
8
10
12
22
30
18
 Question 11
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
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 Question 12
In figure, OP is equal to diameter of the circle. Prove that ABP is an equilateral triangle.
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 Question 13
Without using trigonometric tables, evaluate the following:
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 Question 14
For what value of k are the points (1, 1), (3, k), and (−1, 4) collinear?
OR
Find the area of ΔABC with vertices A (−5, 7), B (−4, −5), and C (4, 5).
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 Question 15
Cards, marked with numbers 5 to 50, are placed in a box and shuffled thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken card is
a prime number less than 10
a number which is a perfect square
 Question 16
Prove that is an irrational number.
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 Question 17
Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
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 Question 18
The sum of the 4^{th} and 8^{th} terms of an A.P. is 24 and the sum of 6^{th} and 10^{th} terms is 44. Find the first three terms of the A.P.
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 Question 19
Solve for x and y:
(a − b) x + (a + b) y = a^{2}− 2ab − b^{2}
(a + b) (x + y) = a^{2} + b^{2}
OR
Solve for x and y:
37x + 43y = 123
43x + 37y = 117
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 Question 20
 Question 21
If the point P(x, y) is equidistant from the points A(3, 6) and B(−3, 4), then prove that 3x + y − 5 = 0.
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 Question 22
The point R divides the line segment AB, where A(−4, 0) and B(0, 6) are such that . Find the coordinates of R.
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 Question 23
In figure, ABC is a rightangled triangle, rightangled at A. Semicircles are drawn on AB, AC, and BC as diameters. Find the area of shaded region.
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 Question 24
Draw a ΔABC with side BC = 6 cm, AB = 5 cm, and ∠ABC = 60°. Construct a ΔAB'C' similar to ΔABC such that sides of ΔAB'C' are of the corresponding sides of ΔABC.
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 Question 25
D and E are points on the sides CA and CB respectively of ΔABC rightangled at C. Prove that AE^{2} + BD^{2} = AB^{2} + DE^{2}
OR
In the following figure, DB BC and AC BC. Prove that
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 Question 26
A motor boat whose speed is 18 kmph in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
OR
Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
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 Question 27
Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Using the above, do the following:
The diagonals of a trapezium ABCD, with ABCD, intersect each other at the point O. If AB = 2CD, then find the ratio of the area of ΔAOB to the area of ΔCOD.
OR
Prove that the lengths of the tangents drawn from external point to a circle are equal.
Using the above, do the following:
In the following figure, TP and TQ are the tangents drawn from T to the circle with centre O and R is any point on the circle. If AB is a tangent to the circle at R, then prove that TA + AR = TB + BR.
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 Question 28
A tent consists of frustum of a cone, surmounted by a cone. If the diameter of the upper and lower circular ends of the frustum are 14 m and 26 m respectively, the height of the frustum is 8 m and the slant height of the surmounted conical portion is 12 m, then find the area of the canvas required to make the tent. (Assume that the radii of the upper circular end of the frustum and the base of surmounted conical portion are equal.)
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 Question 29
The angle of elevation of a jet fighter from a point A on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the jet is flying at a speed of 720 km/hour, then find the constant height at which the jet is flying. [Use = 1.732]
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 Question 30
Find the mean, mode, and median of the following data:

Class
Frequency
0  10
5
10  20
10
20  30
18
30  40
30
40  50
20
50  60
12
60  70
5


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