#### Alternate Sets

(i) All questions are compulsory.

(ii) The question paper consists of 34 questions divided into four sections -A, B, C and D.

(iii) Section A contains 8 questions of 1 mark each, which are multiple choice type questions, Section B contains 6 questions of 2 marks each, Section C contains 10

questions of 3 marks each and Section D contains 10 questions of 4 marks each.

(iv) Use of calculators is not permitted.

The probability that a number selected at random from the numbers 1, 2, 3, ..., 15 is a multiple of 4, is

`(A)4/15`

`(B)2/15`

`(C)1/5`

`(D)1/3`

Concept: Probability Examples and Solutions

The angle of depression of a car parked on the road from the top of a 150 m high tower is 30°. The distance of the car from the tower (in metres) is

`(A) 50sqrt3`

`(B) 150sqrt 3`

`(C) 150sqrt2`

`(D) 75`

Concept: Heights and Distances

Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of ∠ L APB is

(A) 30°

(B) 45°

(C) 60°

(D) 90°

Concept: Circles Examples and Solutions

If k, 2k- 1 and 2k + 1 are three consecutive terms of an A.P., the value of k is

(A) 2

(B) 3

(C) -3

(D) 5

Concept: Arithmetic Progression

A chord of a circle of radius 10 em subtends a right angle at its centre. The length of the chord (in em) is

`(A) 5sqrt 2`

`(B) 10 sqrt2`

`(C)5/sqrt2`

`(D) 10sqrt 3`

Concept: Circles Examples and Solutions

ABCD is a rectangle whose three vertices are B (4, 0), C(4, 3) and D(0,3). The length of one of its diagonals is

(A) 5

(B) 4

(C) 3

(D) 25

Concept: Pythagoras Theorem

In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is

(A) 4

(B) 3

(C) 2

(D) 1

Concept: Pythagoras Theorem

In a family of 3 children, the probability of having at least one boy is

`(A)7/8`

`(B)1/8`

`(C)5/8`

`(D)3/4`

Concept: Probability Examples and Solutions

In Figure 1, common tangents AB and CD to the two circles with centres 0_{1}and 0_{2 }intersect at E. Prove that AB = CD.

Concept: Circles Examples and Solutions

The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC.

Concept: Triangles Examples and Solutions

Two different dice are tossed together. Find the probability

(i) that the number on each die is even.

(ii) that the sum of numbers appearing on the two dice is 5.

Concept: Probability Examples and Solutions

If the total surface area of a solid hemisphere is 462 cm2 , find its volume.[Take π=22/7]

Concept: Surface Area of a Combination of Solids

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

Concept: Sum of First n Terms of an AP

Find the values of k for which the quadratic equation 9x^{2} - 3kx + k = 0 has equal roots.

Concept: Nature of Roots

The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds the angle of elevation becomes 300 If the aeroplane is flying at a constant height of 3000 3 m, find the speed of the aeroplane.

Concept: Heights and Distances

The largest possible sphere is carved out of a wooden solid cube of side 7 em. Find the volume of the wood left

Concept: Surface Area of a Combination of Solids

Water in a canal, 6 m wide and 1.5 m deep, is flowing at a speed of 4 km/h. How much area will it irrigate in 10 minutes, if 8 cm of standing water is needed for irrigation?

Concept: Surface Area of a Combination of Solids

In Figure 2, ABCD is a trapezium of area 24.5 sq. cm. In it, AD|| BC, ∠ DAB = 900, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a circle, find the area of the shaded region. [Take π=22/7]

Concept: Surface Area of a Combination of Solids

Find the ratio in which the line segment joining the points A(3,- 3) and B(- 2, 7) is divided by x-axis. Also find the coordinates of the point of division.

Concept: Division of a Line Segment

In Figure , two concentric circles with centre O, have radii 21cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region. [use π=22/7]

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle

Solve for x:

`16/x-1=15/(x+1);x!=0,-1`

Concept: Solutions of Quadratic Equations by Completing the Square

The sum of the 2^{nd} and the 7^{th} terms of an AP is 30. If its 15^{th} term is 1 less than twice its 8^{th} term, find the AP.

Concept: Arithmetic Progression

Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Concept: Construction of Tangents to a Circle

Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6), are equal and bisect each other.

Concept: Pythagoras Theorem

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Concept: Tangent to a Circle

150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel.

Concept: Surface Area of a Combination of Solids

A container open at the top, is in the form of a frustum of a cone of height 24 cm with radii of its lower and upper circular ends, as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container. at the rate of 21 per litre. [use π=22/7]

Concept: Frustum of a Cone

The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [use √3=1.73]

Concept: Heights and Distances

A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method

In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1to 12 classes in the school and each class has two sections, find how many trees were

planted by the students. Which value is shown in this question?

Concept: Probability Examples and Solutions

Solve for x: `(x-3)/(x-4)+(x-5)/(x-6)=10/3; x!=4,6`

Concept: Solutions of Quadratic Equations by Factorization

All the red face cards are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find the probability that the drawn card is

(i) of red colour

(ii) a queen

(iii) an ace

(iv) a face card

Concept: Probability Examples and Solutions

A(4, - 6), B(3,- 2) and C(5, 2) are the vertices of a 8 ABC and AD is its median. Prove that the median AD divides Δ ABC into two triangles of equal areas.

Concept: Area of a Triangle

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Concept: Number of Tangents from a Point on a Circle