Board Paper of Class 10 2009 Maths (SET 2) - Solutions

Find all the zeroes of the polynomial 2x³ + x² − 6x − 3, if two of its zeroes are and.

If , then evaluate

OR

Find the value of tan 60°, geometrically.

If the points A (4, 3) and B (x, 5) are on the circle with the centre O (2, 3), find the value of x.

Which term of the A.P. 3, 15, 27, 39… will be 120 more than its 21^st term?

In Figure 2, ΔABD is a right triangle, right-angled at A and AC ⊥ BD. Prove that AB² = BC . BD.

Prove that is an irrational number.

In Figure, 3, AD ⊥ BC and BD CD. Prove that 2CA² = 2AB² + BC².

OR

In Figure 4, M is mid-point of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2 BL.

The area of an equilateral triangle is cm². Taking each angular point as centre, circles are drawn with radius equal to half the length of the side of the triangle. Find the area of triangle not included in the circles. [Take= 1.73]

OR

Figure 5 shows a decorative block which is made of two solids − a cube and a hemisphere. The base of the block is a cube with edge 5 cm and the hemisphere, fixed on the top, has a diameter of 4.2 cm. Find the total surface area of the block. [Take π =]

Two dice are thrown simultaneously. What is the probability that

(i) 5 will not come up on either of them?

(ii) 5 will come up on at least one?

(iii) 5 will come up at both dice?

The sum of first six terms of an arithmetic progression is 42. The ratio of its 10^th term to its 30^th term is 1 : 3. Calculate the first and the thirteenth term of the A.P.

Evaluate:

Find the ratio in which the point (x, 2) divides the line segment joining the points (−3, −4) and (3, 5). Also find the value of x.

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, − 1), (2, 1) and (0, 3).

Solve for x and y:

ax − by = 2ab

OR

The sum of two numbers is 8. Determine the numbers if the sum of their reciprocals is

Draw a right triangle in which sides (other than hypotenuse) are of lengths 8 cm and 6 cm. Then construct another triangle whose sides are times the corresponding sides of the first triangle.

A juice seller serves his customers using a glass as shown in Figure 6. The inner diameter of the cylindrical glass is 5 cm, but the bottom of the glass has a hemispherical portion raised which reduces the capacity of the glass. If the height of the glass is 10 cm, find the apparent capacity of the glass and its actual capacity. (Use π = 3.14)

OR

A cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid cone of base diameter 7 cm and height 6 cm is completely immersed in water. Find the volume of

(i) water displaced out of the cylindrical vessel.

(ii) water left in the cylindrical vessel.

[Take π ]

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, prove that the angle opposite to the first side is a right angle.

Use the above theorem to find the measure of ∠PKR in the below figure.

From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30° and 60° respectively. Find

(i) the horizontal distance between the building and the lamp post

(ii) The height of the lamp post.

[Take ]

During the medical check-up of 35 students of a class their weights were recorded as follows:

Weight (in kg)	Number of students
38 − 40	3
40 − 42	2
42 − 44	4
44 − 46	5
46 − 48	14
48 − 50	4
50 − 52	3

Draw a less than type and a more than type ogive from the given data. Hence obtain the median weight from the graph.

Solve the following equation for x:

9x² − 9(a + b)x + (2a² + 5ab + 2b²) = 0

OR

If (−5) is a root of the quadratic equation 2x² + px − 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, then find the values of p and k.