Board Paper of Class 10 2010 Maths (SET 3)  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
If the sum of first p terms of an Α.P., is ap^{2} + bp, find its common difference.
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 Question 2
In fig. 1, S and T are points on the sides PQ and PR, respectively of ΔPQR, such that PT = 2 cm, TR = 4 cm and ST is parallel to QR. Find the ratio of the areas of ΔPST and ΔPQR.
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 Question 3
In the given figure, ΔAHK is similar to ΔABC. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find AC.
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 Question 4
If α, β are the zeroes of a polynomial, such that α + β = 6 and αβ = 4, then write the polynomial.
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 Question 5
Has the rational numbera terminating or a nonterminating decimal representation?
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 Question 6
If cosec θ = 2x and cot θ, find the value of
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 Question 7
A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting a red face card.
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 Question 8
The slant height of a frustum of a cone is 4 cm and the perimeters (circumferences) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.
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 Question 9
If A(1, 2), B(4, 3) and C(6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of the fourth vertex D.
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 Question 10
If P(2, p) is the midpoint of the line segment joining the points A(6, − 5) and B(− 2, 11), find the value of p.
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 Question 11
If and are two zeroes of the polynomial x^{3} + 3x^{2} − 5x − 15, find its third zero.
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 Question 12
If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.
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 Question 13
Without using trigonometric tables, find the value of the following expression:
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 Question 14
Find the value of k for which the following pair of linear equation have infinitely many solutions:
2x + 3y = 7; (k − 1) x + (k + 2)y = 3k
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 Question 15
In an A.P., first term is 2, the last term is 29 and sum of the terms is 155. Find the common difference of the A.P.
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 Question 16
Prove that is an irrational number.
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 Question 17
In figure 3, ABC is a right triangle, right angled at C and D is the midpoint of BC. Prove that AB^{2} = 4AD^{2} − 3AC^{2}.
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 Question 18
 Question 19
In an A.P., the sum of first ten terms is −150 and the sum of its next ten terms is −550. Find the A.P.
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 Question 20
The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes. Find the fraction.
OR
Solve the following pair of equations:
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$\frac{4}{x}+3y=8\phantom{\rule{0ex}{0ex}}\frac{6}{x}4y=5,x\ne 0$
 Question 21
Construct a triangle PQR in which QR = 6 cm, ∠Q = 60° and ∠R = 45°. Construct another triangle similar to ΔPQR such that its side are of the corresponding sides of ΔPQR.
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 Question 22
Cards bearing numbers 1, 3, 5, ..., 35 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card bearing
(i) a prime number less than 15.
(ii) a number divisible by 3 and 5.
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 Question 23
If the point P (m, 3) lies on the line segment joining the points and B (2, 8), find the value of m.
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 Question 24
Point P divides the line segment joining the points A (2, 1) and B (5, −8) such that. If P lies on the line 2x − y + k = 0, find the value of k.
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 Question 25
In figure 4, the boundary of shaded region consists of four semicircular arcs, two smallest being equal. If diameter of the largest is 14 cm and that of the smallest is 3.5 cm, calculate the area of the shaded region. [Use π =]
OR
Find the area of shaded region in figure 5, if AC = 24 cm, BC = 10 cm and O is the centre of the circle. [Use π = 3.14]
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 Question 26
A milk container is made of metal sheet in the shape of frustum of a cone whose volume is . The radii of its lower and upper circular ends are 8 cm and 20 cm respectively. Find the cost of metal sheet used in making the container at the rate of Rs. 1.40 per square centimetre.
Or
A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of base of the cone is 21 cm and its volume is of the volume of the hemisphere, calculate the height of the cone and the surface area of the toy. .
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 Question 27
Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using the above, prove the following:
Point D is the midpoint of the side BC of a right triangle ABC, right angled at C. Prove that 4AD^{2} = 4AC^{2} + BC^{2}
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 Question 28
Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers.
Or
The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.
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 Question 29
From the top of a 7 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of the foot of the tower is 30°. Find the height of the tower.
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 Question 30
Find the mean, mode and median of the following frequency distribution:
Class:
0 − 10
10 − 20
20 − 30
30 − 40
40 − 50
50 − 60
60 − 70
Frequency:
8
7
15
20
12
8
10

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