# 10th Maths Paper Solutions Set 1 : CBSE All India Previous Year 2012

General instructions:
1. All questions are compulsory.
2. The question paper consists of 34 questions divided into four sections A, B, C and
D.
3. Section A contains 10 questions of 1 mark each, which are multiple choices type
questions, Section B contains 8 questions of 2 marks each, Section C contains 10
questions of 3 marks each, Section D contains 6 questions of 4 marks each.
4. There is no overall choice in the paper. However, internal choice is provided in one
question of 2 marks, 3 questions of 3 marks each and two questions of 4 marks each.
5. Use of calculators is not permitted.
Q1 :

The roots of the quadratic equation 2x2x − 6 = 0 are

A.

B.

C.

D.

The given quadratic equation is 2x2x − 6 = 0.

Roots of the equation can be found by factorizing it as follows:

The roots of given quadratic equation are 2 and.

Hence, the correct answer is B.

Q2 :

If the nth term of an A.P. is (2n + 1), then the sum of its first three terms is

A. 6n + 3

B. 15

C. 12

D. 21

Given: The nth term of A.P. i.e., an = 2n + 1

To find: Sum of first three terms

On putting n = 1, 2 and 3, we obtain:

a1 = 2 × 1 + 1 = 2 + 1 = 3

a2 = 2 × 2 + 1 = 4 + 1 = 5

a3 = 2 × 3 + 1 = 6 + 1 = 7

Sum of first three terms a1+ a2+ a3= 3 + 5 + 7 = 15

Hence, the correct answer is B.

Q3 :

From a point Q, 13 cm away from the centre of a circle, the length of tangent PQ to the circle is 12 cm. The radius of the circle (in cm) is

A. 25

B.

C. 5

D. 1

The given information can be represented diagrammatically as follows:

Let O be the centre of the circle.

Given: PQ = 12 cm and OQ = 13 cm.

To find: Radius of the circle

PQ is a tangent drawn from the external point Q to the circle.

OPQ = 90° (Radius is perpendicular to the tangent at the point of contact)

On applying Pythagoras theorem in ΔOPQ, we obtain:

OQ2 = OP2 + PQ2

OP2= OQ2− PQ2

OP2= (13 cm)2− (12 cm)2

OP2= 169cm2− 144 cm2

OP2 = 25 cm2

OP = 5 cm

Thus, the radius of circle is 5 cm.

Hence, the correct answer is C.

Q4 :

In Figure 1, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC

= 4 cm, then the length of AP (in cm) is

A. 7.5

B. 15

C. 10

D. 9

Q5 :

The circumference of a circle is 22 cm. The area of its quadrant (in cm2) is

A.

B.

C.

D.

Q6 :

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is

A. 1 : 2

B. 1 : 4

C. 1 : 6

D. 1 : 8

Q7 :

A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is

A. 45°

B. 30°

C. 60°

D. 90°

Q8 :

The Distance of the point (−3, 4) from the x-axis is

A. 3

B. −3

C. 4

D. 5

Q9 :

In Figure 2, P (5, −3) and Q (3, y) are the points of trisection of the line segment joining A (7, −2) and B (1, −5). Then y equals

A. 2

B. 4

C. −4

D.

Q10 :

Cards bearing numbers 2, 3, 4, ..., 11 are kept in a bag. A card is drawn at random from the bag. The probability of getting a card with a prime number is

A.

B.

C.

D.

Q11 :

Find the value of p for which the roots of the equation px (x − 2) + 6 = 0, are equal.

Q12 :

How many two-digit numbers are divisible by 3?

Q13 :

In Figure 3, a right triangle ABC, circumscribes a circle of radius r. If AB and BC are of lengths of 8 cm and 6 cm respectively, find the value of r.

Q14 :

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Q15 :

In Figure 4, ABCD is a square of side 4 cm. A quadrant of a circle of radius 1 cm is drawn at each vertex of the square and a circle of diameter 2 cm is also drawn. Find the area of the shaded region. (Use π = 3.14)

Q16 :

A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm.

Find the number of cones so formed.

Q17 :

Find the value of k, if the point P (2, 4) is equidistant from the points A(5, k) and B (k, 7).

Q18 :

A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability of getting

(i) a red king.

(ii) a queen or a jack.

Q19 :

Solve the following quadratic equation for x:

x2 − 4axb2 + 4a2 = 0

Q20 :

Find the sum of all multiples of 7 lying between 500 and 900.

Q21 :

Draw a triangle ABC with BC = 7 cm, B = 45° and C = 60°. Then construct another triangle, whose sides are times the corresponding sides of ΔABC.

Q22 :

In Figure 5, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR =

12 cm. Find the lengths of QM, RN and PL.

Q23 :

In Figure 6, O is the centre of the circle with AC = 24 cm, AB = 7 cm and BOD = 90°.

Find the area of the shaded region. [Use π = 3.14]

Q24 :

A hemispherical bowl of internal radius 9 cm is full of water. Its contents are emptied in a cylindrical vessel of internal radius 6 cm. Find the height of water in the cylindrical vessel.

Q25 :

The angles of depression of the top and bottom of a tower as seen from the top of a m high cliff are 45° and 60° respectively. Find the height of the tower.

Q26 :

Find the coordinates of a point P, which lies on the line segment joining the points A (−2, −2), and B (2, −4), such that .

Q27 :

If the points A (x, y), B (3, 6) and C (−3, 4) are collinear, show that x − 3y + 15 = 0.

Q28 :

All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn from it. Find the probability that the drawn card is

(i) a black face card.

(ii) a red card.

Q29 :

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by .Find the fraction.

OR

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.

Q30 :

Find the common difference of an A. P. whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.

Q31 :

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Q32 :

A hemispherical tank, full of water, is emptied by a pipe at the rate of litres per sec.

How much time will it take to empty half the tank if the diameter of the base of the tank is 3 m?

OR

A drinking glass is in the shape of the frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.

Q33 :

A military tent of height 8.25 m is in the form of a right circular cylinder of base diameter

30 m and height 5.5 m surmounted by a right circular cone of same base radius. Find the

length of the canvas used in making the tent, if the breadth of the canvas is 1.5 m.

Q34 :

The angles of elevation and depression of the top and bottom of a light-house from the top of a 60 m high building are 30° and 60° respectively. Find

(i) the difference between the heights of the light-house and the building.

(ii) the distance between the light-house and the building.