Board Paper of Class 10 2008 Maths (SET 1)  Solutions
(i) All questions are compulsory.
(ii) The question paper consists of 30 questions divided into four sections A, B, C and D. Section A comprises of 10 questions of one mark each, Section B comprises of 5 questions of two marks each, Section C comprises of 10 questions of three marks each, and Section D comprises of 5 questions of six marks each.
(iii) All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
(iv) Use of calculators is not permitted.
 Question 1
Complete the missing entries in the following factor tree:
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 Question 2
If (x + a) is a factor of 2x^{2} + 2ax + 5x + 10, then find a.
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 Question 3
Show that x = −3 is a solution of x^{2} + 6x + 9 = 0.
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 Question 4
The first term of an A.P. is p and its common difference is q. Find its 10^{th} term.
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 Question 5
If, then find the value of (sin A + cos A) sec A.
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 Question 6
The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.
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 Question 7
 Question 8
The surface area of a sphere is 616 cm^{2}. Find its radius.
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 Question 9
A die is thrown once. Find the probability of getting a number less than 3.
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 Question 10
Find the class marks of classes 10 − 25 and 35 − 55.
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 Question 11
Find all the zeroes of the polynomial x^{4} + x^{3} − 34x^{2 }− 4x + 120, if two of its zeroes are 2 and −2.
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 Question 12
A pair of dice is thrown once. Find the probability of getting the same number on each die.
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 Question 13
If sec 4A = cosec (A − 20°), where 4A is an acute angle, then find the value of A.
OR
In a ΔABC, rightangled at C, if then find the value of sin A cos B + cos A sin B.
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 Question 14
Find the value of k, if the points (k, 3), (6, −2), and (−3, 4) are collinear.
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 Question 15
E is a point on the side AD produced of a ^{gm }ABCD and BE intersects CD at F. Show that ΔABE ∼ ΔCFB.
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 Question 16
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 17
Represent the following pair of equations graphically and write the coordinates of points where the lines intersect yaxis:
x + 3y = 6
2x − 3y = 12
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 Question 18
For what value of n are the n^{th} terms of two A.P.’s 63, 65, 67 … and 3, 10, 17 … equal?
OR
If m times the m^{th} term of an A.P. is equal to n times its n^{th} term, then find the (m + n)^{th} term of the A.P.
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 Question 19
In an A.P., the first term is 8, n^{th} term is 33, and sum to first n terms is 123. Find n and d, the common difference.
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 Question 20
Prove that:
(1+ cot A + tan A) (sin A − cos A) = sin A tan A − cot A cos A
OR
Without using trigonometric tables, evaluate the following:
 Question 21
If P divides the join of A(−2, −2) and B(2, −4) such that , then find the coordinates of P.
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 Question 22
The midpoints of the sides of a triangle are (3, 4), (4, 6), and (5, 7). Find the coordinates of the vertices of the triangle.
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 Question 23
Draw a right triangle in which the sides containing the right angle are 5 cm and 4 cm. Construct a similar triangle whose sides are times the sides of the above triangle.
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 Question 24
Prove that a parallelogram circumscribing a circle is a rhombus.
OR
In figure, AD ⊥ BC. Prove that AB^{2} + CD^{2} = BD^{2} + AC^{2}.
 Question 25
In the figure, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.
 Question 26
A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake, the peacock pounces on it. If their speeds are equal, then at what distance from the hole is the snake caught?
OR
The difference of two numbers is 4. If the difference of their reciprocals is, then find the two numbers.
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 Question 27
The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of m, then find the speed, in km/hour, of the plane.
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 Question 28
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Using the above, prove the following:
In figure, AB  DE and BC  EF. Prove that AC  DF.
OR
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Using the above, prove the following:
ABC is an isosceles triangle in which AB = AC, circumscribed about a circle, as shown in figure. Prove that the base is bisected by the point of contact.
 Question 29
If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, then find the capacity and total surface area of the bucket.
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 Question 30
Find mean, median, and mode of the following data:

Class
Frequency
0 − 20
6
20 − 40
8
40 − 60
10
60 − 80
12
80 − 100
6
100 − 120
5
120 − 140
3


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