Board Paper of Class 10 2009 Maths (SET 3) - Solutions

If the polynomial is divided by another polynomial, the remainder comes out to be, find a and b.

Simplify:

Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that ∠APB = 2 ∠OAB.

Fig. 3

OR

Prove that the parallelogram circumscribing a circle is a rhombus.

If S_n, the sum of first n terms of an A.P. is given by S_n = , then find its nth term.

Prove that is an irrational number.

Solve the following pair of equations:

Find the value of sin 30° geometrically.

OR

Without using trigonometrical tables, evaluate:

Construct a ΔABC in which BC = 6.5 cm, AB = 4.5 cm and ∠ACB = 60°. Construct another triangle similar to ΔABC such that each side of new triangle is of the corresponding sides of ΔABC.

The king, queen and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of (i) heart (ii) queen (iii) clubs.

If P (x, y) is any point on the line joining the points A (a, 0) and B (0, b), then show that .

In Fig. 4, ΔABC is right angled at C and DE ⊥ AB. Prove that ΔABC ∼ ΔADE and hence find the lengths of AE and DE.

Fig. 4

OR

In Fig, 5, DEFG is a square and ∠BAC = 90°. Show that DE² = BD × EC.

Fig. 5

Find the point on y-axis which is equidistant from the points (5, −2) and (−3, 2)

OR

The line segment joining the points A (2, 1) and B (5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by
2x − y + k = 0, find the value of k.

The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th terms is 44. Find A.P.

In Fig. 6, PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Find the area of shaded region (take π = 3.14)

Fig. 6

From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm, is hollowed out. Find the volume of the remaining solid correct to two places of decimals. Also find the total surface area of the remaining solid. (take π = 3.1416)

OR

In Fig. 8, ABC is a right triangle right angled at A. Find the area of shaded region if AB = 6 cm, BC = 10 cm and O is the centre of the incircle of ΔABC.

(take π = 3.14)

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

In a trapezium ABCD, AC and BD are intersecting at O. AB || DC and AB = 2 CD. If area of ΔAOB = 84 cm², find the area of ΔCOD.

A pole of height 5 m is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 60° and the angle of depression of the point A from the top of the tower is 45°. Find the height of the tower. [Take]

The following table gives the daily income of 50 workers of a factory:

Daily income (in Rs.)	100−120	120−140	140−160	160−180	180−200
Number of workers	12	14	8	6	10

Find the Mean, Mode and Median of the above data.

The sum of the squares of two consecutive odd numbers is 394. Find the numbers.

OR

Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?