#### Alternate Sets

In Fig. 1, PQ is a tangent at a point C to a circle with centre O. if AB is a diameter and ∠CAB = 30^{°}, find ∠PCA.

Concept: Tangent to a Circle

For what value of k will k + 9, 2k – 1 and 2k + 7 are the consecutive terms of an A.P?

Concept: Arithmetic Progression

A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder

Concept: Pythagoras Theorem

A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen.

Concept: Probability Examples and Solutions

If -5 is a root of the quadratic equation 2x^{2} + px – 15 = 0 and the quadratic equation p(x^{2} + x)k = 0 has equal roots, find the value of k.

Concept: Nature of Roots

Let P and Q be the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4) such that P is nearer to A. Find the coordinates of P and Q.

Concept: Section Formula

In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA.

Concept: Number of Tangents from a Point on a Circle

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right angled isosceles triangle.

Concept: Pythagoras Theorem

The 4^{th} term of an A.P. is zero. Prove that the 25^{th} term of the A.P. is three times its 11^{th} term

Concept: Arithmetic Progression

In Fig.3, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP = 2r, show that ∠ OTS = ∠ OST = 30°.

Concept: Number of Tangents from a Point on a Circle

In Fig. 4, O is the centre of a circle such that diameter AB = 13 cm and AC = 12 cm. BC is joined. Find the area of the shaded region. (Take π = 3.14)

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

In Fig. 5, a tent is in the shape of a cylinder surmounted by a conical top of same diameter. If the height and diameter of cylindrical part are 2.1 m and 3 m respectively and the slant height of conical part is 2.8 m, find the cost of canvas needed to make the tent if the canvas is available at the rate of Rs. 500/sq. metre `( "Use "pi=22/7)`

Concept: Surface Areas and Volumes Examples and Solutions

If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.

Concept: Distance Formula

In Fig. 6, find the area of the shaded region, enclosed between two concentric circles of radii 7 cm and 14 cm where ∠AOC = 40°. (use `pi = 22/7`)

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

If the ratio of the sum of first n terms of two A.P’s is (7n +1): (4n + 27), find the ratio of their mth terms.

Concept: Sum of First n Terms of an AP

Solve for x

:`1/((x-1)(x-2))+1/((x-2)(x-3))=2/3` , x ≠ 1,2,3

Concept: Solutions of Quadratic Equations by Factorization

A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel. (use `pi=22/7`)

Concept: Surface Areas and Volumes Examples and Solutions

A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 3 `5/9` cm. Find the diameter of the cylindrical vessel.

Concept: Surface Areas and Volumes Examples and Solutions

A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of a hill as 30°. Find the distance of the hill from the ship and the height of the hill

Concept: Heights and Distances

Three different coins are tossed together. Find the probability of getting

(i) exactly two heads

(ii) at least two heads

(iii) at least two tails.

Concept: Probability Examples and Solutions

Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively offered to the state government to provide place and the canvas for 1500 tents to be fixed by the governments and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 cm and height 3.5 m, with conical upper part of same base radius but of height 2.1 m. If the canvas used to make the tents costs Rs. 120 per sq. m, find the amount shared by each school to set up the tents. What value is generated by the above problem? (use `pi =22/7`)

Concept: Surface Areas and Volumes Examples and Solutions

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

Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 60° to each other.

Concept: Construction of Tangents to a Circle

In Fig. 7, two equal circles, with centres O and O’, touch each other at X. OO’ produced meets the circle with centre O’ at A. AC is tangent to the circle with centre O, at the point C. O’D is perpendicular to AC. Find the value of `(DO')/(CO')`

Concept: Tangent to a Circle

Solve for x: `1/(x+1)+2/(x+2)=4/(x+4), `x ≠ -1, -2, -3

Concept: Nature of Roots

The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of tower is 45. Find the height of the tower PQ and the distance PX. (Use `sqrt3=1.73)`

Concept: Heights and Distances

The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to sum of the numbers of houses following X.

Concept: Sum of First n Terms of an AP

In Fig. 8, the vertices of ΔABC are A(4, 6), B(1, 5) and C(7, 2). A line-segment DE is drawn to intersect the sides AB and AC at D and E respectively such that `(AD)/(AB)=(AE)/(AC)=1/3 `Calculate th area of ADE and compare it with area of ΔABCe.

Concept: Area of a Triangle

A number x is selected at random from the numbers 1, 2, 3, and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16.

Concept: Probability Examples and Solutions

In Fig. 9, is shown a sector OAP of a circle with centre O, containing ∠θ. AB is perpendicular to the radius OQ and meets OP produced at B. Prove that the perimeter of shaded region is

`r[tantheta+sectheta+(pitheta)/180-1]`

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

A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32 km upstream than to return downstream to the same spot. Find the speed of the stream.

Concept: Pair of Linear Equations in Two Variables