Board Paper of Class 10 2009 Maths (SET 1) - Solutions

Without drawing the graph, find out whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident:

The 17^th term of an A.P. exceeds its 10^th term by 7. Find the common difference.

Without using trigonometric tables, evaluate:

Show that the points (−2, 5); (3, −4) and (7, 10) are the vertices of a right angled isosceles triangle.

OR

The centre of a circle is (2α − 1, 7) and it passes through the point (−3, −1). If the diameter of the circle is 20 units, then find the values(s) of α.

If C is a point lying on the line segment AB joining A (1, 1) and B (2, −3) such that 3 AC = CB, then find the coordinates of C.

Show that the square of any positive odd integer is of the form 8m + 1, for some integer m.

OR

Prove that is not a rational number.

If the polynomial 6x⁴ + 8x³ − 5x² + ax + b is exactly divisible by the polynomial 2x² − 5, the find the values of a and b.

If 9^th term of an A.P. is zero, prove that its 29^th term is double of its 19^th term.

Draw a circle of radius 3 cm. From a point P, 6 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangents.

In figure 2, a triangle ABC is right angled at B. Side BC is trisected at points D and E. Prove that 8 AE² + 5 AD².

OR

In figure 3, a circle is inscribed in a triangle ABC having side BC = 8 cm, AC = 10 cm and AB = 12 cm. Find AD, BE and CF.

Prove that

Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

In figure 4, the shape of the top of a table in a restaurant is that of a sector of a circle with centre O and ∠BOD = 90°. If BO = OD = 60 cm, find

(i) the area of the top of the table.

(ii) the perimeter of the table top.

(Take π = 3.14)

OR

In figure 5, ABCD is a square of side 14 cm and APD and BPC are semicircles. Find the area of shaded region. (Take)

(i) an odd number,

(ii) a perfect square number,

(iii) a number divisible by 7.

A trader bought a number of articles for Rs 900. Five articles were found damaged. He sold each of the remaining articles at Rs. 2 more than what he paid for it. He got a profit of Rs. 80 on the whole transaction. Find the number of articles he bought.

OR

Two years ago the man’s age was three times the square of his son’s age. Three years hence his age will be four times his son’s age. Find their present ages.

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

Using the above theorem prove the following:

The area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.

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

A spherical copper shell, of external diameter 18 cm, is melted and recast into a solid cone of base radius 14 cm and height cm. Find the inner diameter of the shell.

OR

A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm³. The radii of the top and bottom circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of metal sheet used in making it.

Find the mode, median and mean for the following data: