Board Paper of Class 10 2010 Maths (SET 1) - Solutions

For what value of k, is 3 a zero of the polynomial 2x² + x + k?

Find the value of m for which the pair of linear equations 2x + 3y − 7 = 0 and (m − 1) x + (m + 1) y = (3m − 1) has infinitely many solutions.

Find the common differnece of an A.P. whose first term in 4, the lasta term is 49 and the sum of all its terms is 265.

In figure 3, there are two concentric circles with centre O and of radii 5 cm and 3 cm. From an external point P, Tangents PA and PB are drawn to these circles. If AP = 12 cm, find the length of BP.

Without using trigonometric tables, evaluate the following:

OR

Find the value of sec60° geometrically.

Prove that is an irrational number.

Solve the following pair of liner equations for x and y:

x + y = 2ab

OR

The sum of the numerator and the denominator of a fraction is 4 more than twice the numerator. If 3 is added to each of the numerator and denominator, their ratio becomes 2:3 Find the fraction.

In an A.P., the sum of its first ten terms is − 80 and the sum of its next ten terms is − 280. Find the A.P.

In figure 4, ABC is an isosceles triangle in which AB = AC. E is a point on the side CB produced, Such that FE ⊥ AC. If AD ⊥ CB, prove that AB × EF = AD × EC.

Prove the following:

(1 + cot A − cosec A) (1 + tan A + sec A) = 2

OR

Prove the following:

sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A

Construct a triangle ABC in which AB = 8 cm, BC = 10 cm and AC = 6 cm. Then construct another triangle whose sides areof the corresponding sides of ABC.

Point P divides the line segment joining the points A (−1, 3) and B (9, 8) such that If P lies on the line x − y + 2 = 0, find the value of k.

If the points (p, q); (m, n) and (p − m, q − n) are collinear, show that pn = qm.

The rain-water collected on the roof of a building, of dimensions 22 m × 20 m, is drained into a cylindrical vessel having base diameter 2 m and height 3.5 m. If the vessel is full up to the brim, find the height of rain-water on the roof

OR

In figure 5, AB and CD are two perpendicular diameters of a circle with centre O. If OA = 7 cm, find the area of the shaded region.

A bag contains cards which are numbered from 2 to 90. A card is drawn at random from the bag. Find the probability that it bears

(i) a two digit number,

(ii) a number which is a perfect square.

A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

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

Using the above, do the following:

In an isosceles triangle PQR, PQ = QR and PR² = 2 PQ². Prove that ∠Q is a right angle.

A man on the deck of a ship, 12 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Find the distance of the cliff from the ship and the height of the cliff. [Use = 1.732]

OR

The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of the cloud in the lake is 60°. Find the height of the cloud from the surface of the lake.

The surface area of a solid metallic sphere is 616 cm². It is melted and recast into a cone of height 28 cm. Find the diameter of the base of the cone so formed.

OR

The difference between the outer and inner curved surface areas of a hollow right circular cylinder, 14 cm long, is 88 cm². If the volume of metal used in making the cylinder is 176 cm³, find the outer and inner diameters of the cylinder.

Draw ‘less than ogive’ and ‘more than ogive’ for the following distribution and hence find its median.