#### Alternate Sets

What is the common difference of an A.P. in which a_{21} – a_{7} = 84?

Concept: Arithmetic Progression

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP

Concept: Tangent to a Circle

If a tower 30 m high, casts a shadow `10sqrt3` m long on the ground, then what is the angle of elevation of the sun?

Concept: Heights and Distances

The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

Concept: Probability Examples and Solutions

Find the value of *p*, for which one root of the quadratic equation px^{2} – 14x + 8 = 0 is 6 times the other.

Concept: Nature of Roots

Which term of the progression 20, 19`1/4`,18`1/2`,17`3/4`, ... is the first negative term?

Concept: Sum of First n Terms of an AP

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

Concept: Number of Tangents from a Point on a Circle

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

Concept: Tangent to a Circle

A line intersects the *y*-axis and *x*-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.

Concept: Concepts of Coordinate Geometry

If the distances of P(*x, y*) from A(5, 1) and B(–1, 5) are equal, then prove that 3*x* = 2*y*

Concept: Distance Formula

If ad ≠ bc, then prove that the equation (a^{2} + b^{2}) x^{2} + 2 (ac + bd) x + (c^{2} + d^{2}) = 0 has no real roots.

Concept: Nature of Roots

The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Concept: Sum of First n Terms of an AP

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

Concept: Heights and Distances

A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

Concept: Simple Problems on Single Events

In what ratio does the point `(24/11, y)` divide the line segment joining the points P(2, –2) and Q(3, 7)? Also find the value of y.

Concept: Section Formula

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle

In Figure , two concentric circles with centre O, have radii 21cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region. [use π=22/7]

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle

Water in a canal, 5·4 m wide and 1·8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation?

Concept: Surface Area of a Combination of Solids

The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. find the curved surface area of the frustum.

Concept: Frustum of a Cone

The dimensions of a solid iron cuboid are 4·4 m × 2·6 m × 1·0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe

Concept: Surface Areas and Volumes Examples and Solutions

Solve for x :

`1/(x + 1) + 3/(5x + 1) = 5/(x + 4), x != -1, -1/5, -4`

Concept: Solutions of Quadratic Equations by Factorization

Two taps running together can fill a tank in `3 1/13` hours. If one tap takes 3 hours more than the other to fill the tank, then how much time will each tap take to fill the tank?

Concept: Solutions of Quadratic Equations by Completing the Square

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 9^{th} terms.

Concept: Sum of First n Terms of an AP

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

Concept: Number of Tangents from a Point on a Circle

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

Concept: Number of Tangents from a Point on a Circle

Construct a triangle ABC with sides BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are `3/4` times the corresponding sides of ∆ABC.

Concept: Application of Pythagoras Theorem in Acute Angle and Obtuse Angle

An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are 45° and 60° respectively. Find the width of the river. [Use `sqrt3` = 1⋅732]

Concept: Heights and Distances

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

Concept: Concepts of Coordinate Geometry

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

1) even sum, and

2) even product.

Concept: Probability Examples and Solutions

In the given figure, ABCD is rectangle of dimensions 21 cm × 14 cm. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure.

Concept: Areas Related to Circles Examples and Solutions

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.

Concept: Surface Areas and Volumes Examples and Solutions