2010-2011 March

The roots of the equation *x*^{2} − 3*x* − *m* (*m* + 3) = 0, where *m* is a constant, are

**A. ***m*, *m* + 3

**B. **− *m*, *m* + 3

**C. ***m*, − (*m* + 3)

**D. **−*m*, − (*m* + 3)

Chapter: [2.03] Quadratic Equations

If the common differences of an A.P. is 3, then *a*_{20} − *a*_{15} is

**A. **5

**B. **3

**C. **15

**D. **20

Chapter: [2.02] Arithmetic Progressions

In figure 1, O is the centre of a circle, PQ is a chord and PT is the tangent at P.

If ∠POQ = 70°, then ∠TPQ is equal to

**A. **55°

**B. **70°

**C. **45°

**D. **35°

Chapter: [3.01] Circles

In Figure 2, AB and AC are tangents to the circle with centre O such that

∠BAC = 40°. Then ∠BOC is equal to

**A. **40°

**B. **50°

**C. **140°

**D. **150°

Chapter: [3.01] Circles

The perimeter (in cm) of a square circumscribing a circle of radius *a* cm, is

**A. **8 *a*

**B. **4 *a*

**C. **2 *a*

**D. **16 *a*

Chapter: [3.01] Circles

The radius (in cm) of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is

**A. **4.2

**B. **2.1

**C. **8.4

**D. **1.05

Chapter: [4.01] Heights and Distances

A tower stands vertically on the ground. From a point on the ground which is 25 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 45°. Then the height (in meters) of the tower is

**A.** `25sqrt2`

**B.** `25sqrt3`

**C. **25

**D. **12.5

Chapter: [4.01] Heights and Distances

If `P(a/2,4)`is the mid-point of the line-segment joining the points A (−6, 5) and B(−2, 3), then the value of a is

**A. **−8

**B. **3

**C. **−4

**D. **4

Chapter: [6.01] Lines (In Two-dimensions)

If A and B are the points (−6, 7) and (−1, −5) respectively, then the distance

2AB is equal to

**A. **13

**B. **26

**C. **169

**D. **238

Chapter: [6.01] Lines (In Two-dimensions)

A card is drawn from a well-shuffled deck of 52 playing cards. The probability that the card will not be an ace is

**A.** `1/13`

**B.** `1/4`

**C.** `12/13`

**D.**`3/4`

Chapter: [5.01] Probability

Find the value of *m* so that the quadratic equation *mx* (*x* − 7) + 49 = 0 has two equal roots.

Chapter: [2.03] Quadratic Equations

How many two-digit number are divisible by 6?

Chapter: [2.02] Arithmetic Progressions

In Figure 3, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of the side AD.

Chapter: [3.01] Circles

Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that `(AP)/(AB)=3/5`.

Chapter: [3.03] Constructions

Find the perimeter of the shaded region in Figure 4, if ABCD is a square of side 14 cm and APB and CPD are semicircles. Use `pi22/7`

Chapter: [7.01] Areas Related to Circles

Two cubes each of volume 27 cm^{3} are joined end to end to form a solid. Find the surface area of the resulting cuboid.

Chapter: [7.02] Surface Areas and Volumes

A cone of height 20 cm and radius of base 5 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the diameter of the sphere.

Chapter: [7.02] Surface Areas and Volumes

Find the value of *y* for which the distance between the points A (3, −1) and B (11, *y*) is 10 units.

Chapter: [6.01] Lines (In Two-dimensions)

A ticket is drawn at random from a bag containing tickets numbered from 1 to 40. Find the probability that the selected ticket has a number which is a multiple of 5.

Chapter: [5.01] Probability

Find the roots of the following quadratic equation:

`x^2-3sqrt5x+10=0`

Chapter: [2.03] Quadratic Equations

Find the A.P. whose fourth term is 9 and the sum of its sixth term and thirteenth term is 40.

Chapter: [2.02] Arithmetic Progressions

In Figure 5, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of ΔPQR = 189 cm^{2}, then find the lengths of sides PQ and PR.

Chapter: [3.01] Circles

Draw a pair of tangents to a circle of radius 3 cm, which are inclined to each other at an angle of 60°.

Chapter: [3.03] Constructions

Draw a right triangle in which the sides (other than the hypotenuse) are of lengths 4 cm and 3 cm. Now construct another triangle whose sides are `3/5` times the corresponding sides of the given triangle.

Chapter: [3.03] Constructions

A chord of a circle of radius 14 cm subtends an angle of 120° at the centre. Find the area of the corresponding minor segment of the circle. `[User pi22/7 and sqrt3=1.73]`

Chapter: [3.01] Circles

An open metal bucket is in the shape of a frustum of a cone of height 21 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket at Rs. 30 per litre. `[User pi22/7]`

Chapter: [7.02] Surface Areas and Volumes

Point P(*x*, 4) lies on the line segment joining the points A(−5, 8) and B(4, −10). Find the ratio in which point P divides the line segment AB. Also find the value of *x*.

Chapter: [6.01] Lines (In Two-dimensions)

Find the area of the quadrilateral ABCD, whose vertices are A(−3, −1), B (−2, −4), C(4, − 1) and D (3, 4).

Chapter: [6.01] Lines (In Two-dimensions)

Find the area of triangle formed by joining the mid-points of the sides of the triangle whose vertices are A(2, 1), B(4, 3) and C(2, 5).

Chapter: [7.01] Areas Related to Circles

From the top of a vertical tower, the angles depression of two cars in the same straight line with the base of the tower, at an instant are found to be 45° and 60° . If the cars are 100 m apart and are on the same side of the tower, find the height of the tower.

Chapter: [4.01] Heights and Distances

Two dice are rolled once. Find the probability of getting such numbers on the two dice, whose product is 12.

Chapter: [5.01] Probability

A box contains 80 discs which are numbered from 1 to 80. If one disc is drawn at random from the box, find the probability that it bears a perfect square number.

Chapter: [5.01] Probability

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

Chapter: [3.01] Circles

The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?

Chapter: [2.02] Arithmetic Progressions

How many multiples of 4 lie between 10 and 250?

Chapter: [2.02] Arithmetic Progressions

A train travels 180 km at a uniform speed. If the speed had been 9 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Chapter: [2.03] Quadratic Equations

Find the roots of the equation .`1/(2x-3)+1/(x+5)=1,x≠3/2,5`

Chapter: [2.03] Quadratic Equations

In Figure 6, three circles each of radius 3.5 cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these three circles (shaded region). `[Use pi=22/7]`

Chapter: [3.01] Circles

Water is flowing at the rate of 15 km/hour through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?

Chapter: [7.02] Surface Areas and Volumes

The angle of elevation of the top of a vertical tower from a point on the ground is 60° . From another point 10 m vertically above the first, its angle of elevation is 30° .Find the height of the tower.

Chapter: [4.01] Heights and Distances

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