Board Paper of Class 10 2007 Maths (SET 1) - Solutions

Solve the following system of equations graphically.

2x + y = 8

x + 1 = 2y

Simplify the following rational expression in the lowest terms.

If the sum to the first n terms of an AP is given by S_n = n(n + 1), find the 20^th term of the AP.

In a cyclic quadrilateral ABCD, diagonal AC bisects ∠C. Prove that the tangent to the circle at A is parallel to the diagonal BD.

OR

In figure 3, O is any point in the interior of ΔABC. OD, OE and OF are drawn perpendiculars to the sides BC, CA and AB respectively. Prove that

AF² + BD² + CE² = OA² + OB² + OC² − OD² − OE² − OF²

Construct a ΔABC in which base BC = 6 cm, ∠B = 45° and ∠C = 60°. Draw a

circumcircle of ΔABC.

The diameter of a solid copper sphere is 18 cm. It is melted and drawn into a wire of uniform cross section. If the length of the wire is 108 m, find its diameter.

The expenditure (in rupees) of a family for a month is as follows:

Represent the above data by a pie chart.

From a pack of 52 cards, red face cards are removed. After that a card is drawn at random from the pack. Find the probability that the card drawn is

(i) a queen

(ii) a red card

(iii) a spade card

Prove that:

OR

If A, B and C are the interior angles of a triangle ABC, show that

The coordinates of the mid-points of the sides of a triangle are (4, 3), (6, 0) and (7, −2). Find the coordinates of the centroid of the triangle.

If the distance of P(x, y) from two points with coordinates (5, 1) and (−1, 5) is equal, prove that 3x = 2y.

A loan of Rs. 24600 is to be paid back in two equal semi-annual instalments. If the interest is charged at 10% per annum, compounded semi-annually, find the instalment.

Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Using the above, prove the following.

In figure 4, O is the centre of the circle. If ∠BAO = 30° and ∠BCO = 40°, find the value of ∠AOC.

State and prove Pythagoras theorem.

Use the above to prove the following.

ABC is an isosceles right triangle, right angled at C. Prove that AB² = 2AC².

The side of a square exceeds the side of another square by 4 cm and the sum of the areas of the two squares is 400 sq. cm. Find the dimensions of the squares.

OR

A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hour less than that of the fast train, find the speeds of the two trains.

A hollow copper sphere of external and internal diameters 8 cm and 4 cm respectively is melted into a solid cone of base diameter 8 cm. Find the height of the cone.

OR

If the radii of the circular ends of a bucket 45 cm high are 28 cm and 7 cm, find the capacity and surface area of the bucket.

An observer in a lighthouse observes two ships on the same side of the lighthouse, and in the same straight line with the base of the lighthouse. The angles of depression of the ships approaching it are 30° and 60°. If the height of the lighthouse is 150 m, find the distance between the ships.

Satish (aged 67 years) has monthly income of Rs 30000 (excluding HRA). He donates Rs 80000 to a charitable orphanage (50% exemption). He contributes Rs 30000 towards Public Provident Fund and purchases NSCs worth Rs 20000. He pays Rs 1500 as income tax per month for 11 months. Calculate the income tax to be paid by him in the 12^th month of the year.

Use the following table to calculate the income tax.