Board Paper of Class 10 2010 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 whether on simplification gives a rational or an irrational number.
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 Question 2
If α, β are the zeroes of the polynomial 2y^{2} + 7y + 5, write the value of α + β+ αβ.
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 Question 3
If the sum of the first q terms of an A.P. is 2q + 3q^{2}, what is its common difference?
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 Question 4
In figure 1, CP and CQ are tangents from an external point C to a circle with O. AB are another tangent which touches the circle at R. If CP =11 cm and BR = 4 cm, find the length of BC.
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 Question 5
In Figure 2, DEBC in ΔABC such that BC = 8 cm, AB = 6 cm and DA = 1.5. Find DE.
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 Question 6
If and , find the value of .
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 Question 7
What is the distance between the points A(c, 0) and B(0, −c)?
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 Question 8
In ΔABC, rightangled at C, AC = 6 cm and AB = 12 cm. Find ∠A.
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 Question 9
The slant height of the frustum of a cone is 5 cm. If the difference the radii of its two circular ends is 4 cm, write the height of the frustum.
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 Question 10
A die is thrown once. What is the probability of getting a number greater than 4?
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 Question 11
For what value of k, is 3 a zero of the polynomial 2x^{2} + x + k?
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 Question 12
Find the value of m for which the pair of linear equations 2x + 3y − 7 = 0 and (m − 1) x + (m + 1) y = (3m − 1) has infinitely many solutions.
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 Question 13
Find the common differnece of an A.P. whose first term in 4, the lasta term is 49 and the sum of all its terms is 265.
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 Question 14
In figure 3, there are two concentric circles with centre O and of radii 5 cm and 3 cm. From an external point P, Tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.
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 Question 15
Without using trigonometric tables, evaluate the following:
OR
Find the value of sec60° geometrically.
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 Question 16
Prove that is an irrational number.
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 Question 17
Solve the following pair of liner equations for x and y:
x + y = 2ab
OR
The sum of the numerator and the denominator of a fraction is 4 more than twice the numerator. If 3 is added to each of the numerator and denominator, their ratio becomes 2:3 Find the fraction.
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 Question 18
In an A.P., the sum of its first ten terms is − 80 and the sum of its next ten terms is − 280. Find the A.P.
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 Question 19
In figure 4, ABC is an isosceles triangle in which AB = AC. E is a point on the side CB produced, Such that FE ⊥ AC. If AD ⊥ CB, prove that AB × EF = AD × EC.
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 Question 20
Prove the following:
(1 + cot A − cosec A) (1 + tan A + sec A) = 2
OR
Prove the following:
sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A
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 Question 21
Construct a triangle ABC in which AB = 8 cm, BC = 10 cm and AC = 6 cm. Then construct another triangle whose sides areof the corresponding sides of ABC.
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 Question 22
Point P divides the line segment joining the points A (−1, 3) and B (9, 8) such that If P lies on the line x − y + 2 = 0, find the value of k.
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 Question 23
If the points (p, q); (m, n) and (p − m, q − n) are collinear, show that pn = qm.
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 Question 24
The rainwater collected on the roof of a building, of dimensions 22 m × 20 m, is drained into a cylindrical vessel having base diameter 2 m and height 3.5 m. If the vessel is full up to the brim, find the height of rainwater on the roof
OR
In figure 5, AB and CD are two perpendicular diameters of a circle with centre O. If OA = 7 cm, find the area of the shaded region.
 Question 25
A bag contains cards which are numbered from 2 to 90. A card is drawn at random from the bag. Find the probability that it bears
(i) a two digit number,
(ii) a number which is a perfect square.
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 Question 26
A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.
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 Question 27
In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite the first side is a right angle.
Using the above, do the following:
In an isosceles triangle PQR, PQ = QR and PR^{2} = 2 PQ^{2}. Prove that ∠Q is a right angle.
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 Question 28
A man on the deck of a ship, 12 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Find the distance of the cliff from the ship and the height of the cliff. [Use = 1.732]
OR
The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of the cloud in the lake is 60°. Find the height of the cloud from the surface of the lake.
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 Question 29
The surface area of a solid metallic sphere is 616 cm^{2}. It is melted and recast into a cone of height 28 cm. Find the diameter of the base of the cone so formed.
OR
The difference between the outer and inner curved surface areas of a hollow right circular cylinder, 14 cm long, is 88 cm^{2}. If the volume of metal used in making the cylinder is 176 cm^{3}, find the outer and inner diameters of the cylinder.
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 Question 30
Draw ‘less than ogive’ and ‘more than ogive’ for the following distribution and hence find its median.
Class
20 − 30
30 − 40
40 − 50
50 − 60
60 − 70
70 − 80
80 − 90
Frequency
8
12
24
6
10
15
25

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