Board Paper of Class 10 2012 Maths (SET 3) - Solutions

If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.

A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.

The volume of a hemisphere is . Find its curved surface area.

Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in Fig. 3. If AP = 15 cm, then find the length of BP.

In Fig. 4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

OR

In Fig. 5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.

In Fig. 6, OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region.

Find the sum of all three digit natural numbers, which are multiples of 7.

Find the value(s) of k so that the quadratic equation 3x² − 2kx + 12 = 0 has equal roots.

A point P divides the line segment joining the points A (3, −5) and B (−4, 8) such that

. If P lies on the line x + y = 0, then find the value of K.

If the vertices of a triangle are (1, −3), (4, p) and (−9, 7) and its area is 15 sq. units, find the value(s) of p.

Prove that the parallelogram circumscribing a circle is a rhombus.

OR

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

From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid.

OR

A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.

In Fig. 7, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and

3.5 cm and centre O. If ∠POQ = 30°, then find the area of the shaded region.

Solve for x: 4x² − 4ax + (a² − b²) = 0

OR

Solve for x:

A kite is flying at a height of 45 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is

60°. Find the length of the string assuming that there is no slack in the string.

Draw a triangle ABC with side BC = 6 cm, ∠C = 30° and ∠A = 105°. Then construct another triangle whose sides are times the corresponding sides of ΔABC.

The 16^th term of an AP is 1 more than twice its 8^th term. If the 12^th term of the AP is 47, then find its n^th term.

A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face card (iii) the queen of diamonds.

A bucket is in the form of a frustum of a cone and it can hold 28.49 litres of water. If the radii of its circular ends are 28 cm and 21 cm, find the height of the bucket.

The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of depression from the top of the tower to the foot of the hill is 30°. If the tower is 50 m high, find the height of the hill.

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

OR

A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

A shopkeeper buys some books for Rs 80. If he had bought 4 more books for the same amount, each book would have cost Rs 1 less. Find the number of books he bought.

OR

The sum of two numbers is 9 and the sum of their reciprocals is. Find the numbers.

Sum of the first 20 terms of an AP is −240, and its first term is 7. Find its 24^th term.

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.