# 10th Maths Paper Solutions Set 1 : CBSE All India Previous Year 2010

General Instructions:
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.
Q1 :

Write whether on simplification gives a rational or an irrational number.

The given expression can be simplified as: Since 6 is a rational number, the expression on simplification gives a rational number.

Q2 :

If α, β are the zeroes of the polynomial 2y2 + 7y + 5, write the value of α + β+ αβ.

Since α and βare the zeroes of the polynomial 2y2 + 7y + 5: Thus, the value of α + β+ αβ is −1.

Q3 :

If the sum of the first q terms of an A.P. is 2q + 3q2, what is its common difference?

Let a and d respectively be the first term and common difference of the A.P.

It is given that Sq = 2q + 3q2

∴ S1 = 2 × 1 + 3 × 12 = 5

a1 + S1 = 5

S2 = 2 × 2 + 3 × 22 = 4 + 12 = 16

a1 + a2 = 16

a + (a + d) = 16

⇒ 2 × 5 + d = 16

d =16 − 10 = 6

Thus, the common difference of the A.P. is 6.

Q4 :

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. Q5 :

In Figure 2, DE||BC in ΔABC such that BC = 8 cm, AB = 6 cm and DA = 1.5. Find DE. Q6 :

If and , find the value of .

Q7 :

What is the distance between the points A(c, 0) and B(0, −c)?

Q8 :

In ΔABC, right-angled at C, AC = 6 cm and AB = 12 cm. Find ∠A.

Q9 :

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.

Q10 :

A die is thrown once. What is the probability of getting a number greater than 4?

Q11 :

For what value of k, is 3 a zero of the polynomial 2x2 + x + k?

Q12 :

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.

Q13 :

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 is265.

Q14 :

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. Q15 :

Without using trigonometric tables, evaluate the following: OR

Find the value of sec60° geometrically.

Q16 :

Prove that is an irrational number.

Q17 :

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.

Q18 :

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.

Q19 :

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. Q20 :

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

Q21 :

Construct a triangle ABC in which AB = 8 cm, BC = 10 cm and AC = 6 cm. Then construct another triangle whose sides are of the corresponding sides of ABC.

Q22 :

Point P divides the line segment joining the points A (−1, 3) and B (9, 8) such that If P lies on the line xy + 2 = 0, find the value of k.

Q23 :

If the points (p, q); (m, n) and (pm, qn) are collinear, show that pn = qm.

Q24 :

The rain-water 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 rain-water 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.  Q25 :

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.

Q26 :

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.

Q27 :

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 PR2 = 2 PQ2. Prove that ∠Q is a right angle.

Q28 :

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.

Q29 :

The surface area of a solid metallic sphere is 616 cm2. 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 cm2. If the volume of metal used in making the cylinder is 176 cm3, find the outer and inner diameters of the cylinder. 