#### If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k – 1, 5k) are collinear, then find the value of k.

#### Construct a triangle ABC with side BC = 7 cm, ∠B = 45^{o}, ∠A = 105^{o}. Then construct another triangle whose sides are \\frac{3}{4}) times the corresponding sides of the △ABC.

#### Two different dice are thrown together. Find the probability that the numbers obtained have

(i) even sum, and

(ii) even product

#### In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangents AB with point of contact C, is intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90^{o}.

#### In a rain–water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3.5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

#### Prove that the lengths of two tangents drawn from an external point to a circle are equal.

#### If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms.

#### Solve for x: \\frac{x - 1}{2x + 1} + \frac{2x + 1}{x - 1} = 2), where ≠ \\frac{-1}{2}), 1

#### A takes 6 days less than B to do a work. If both A and B working together can do it in 4 days, how many days will B take to finish it?

#### From the top of a tower, 100 m high, a man observe two cars on the opposite sides of the tower and in same straight line with its base, with its base, with angles of depression 30^{o} and 45^{o}. Find the distance between the cars. Take √3 = 1.732

#### In the given figure, O is centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90^{o}. Find the area of the shaded region.