List of top Mathematics Questions

In triangle ABC, right-angled at B, if tan A =\(\frac{ 1}{\sqrt{3}}\). find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to

If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = \(\frac{3}{7}\) AB and P lies on the line segment AB.

In \(Δ\)PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is

State whether the following are true or false. Justify your answer.
(i) The value of tan A is always less than 1.
(ii) sec A = \(\frac{12}{5}\) for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sin \(\theta\) = \(\frac{4}{3}\) for some angle \(\theta\).

For the following APs, write the first term and the common difference:
(i) 3, 1, − 1, − 3 …
(ii) − 5, − 1, 3, 7 …
(iii) \(\frac{1}{3}, \frac{5}{3},\frac{9}{3},\frac{13}{3}\) ,…
(iv) 0.6, 1.7, 2.8, 3.9 …

Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Write first four terms of the A.P. when the first term a and the common difference d are given as follows
(1) a = 10, d = 10
(2) a = − 2, d = 0
(3) a = 4, d = − 3
(4) a = − 1 d =\(\frac{1}{2}\)
(5) a = − 1.25, d = − 0.25

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use \(\pi = 3.14\) and \(\sqrt3 = 1.73\))

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: 
(i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :

Fill in the blanks :

1. A tangent to a circle intersects it in ____ point(s).
2. A line intersecting a circle in two points is called a ____ .
3. A circle can have ____ parallel tangents at the most.
4. The common point of a tangent to a circle and the circle is called ____ .

How many tangents can a circle have?

A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use π = 3.14)

In which of the following situations, does the list of numbers involved make as arithmetic progression and why?
(1) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.
(2) The amount of air present in a cylinder when a vacuum pump removes \(\frac{1}{4}\) of the air remaining in the cylinder at a time.
(3) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.
(4) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8% per annum.

Which term of the AP : 3, 15, 27, 39, ……… will be 132 more than its 54^th term?

The 17^th term of an AP exceeds its 10^th term by 7. Find the common difference.

If the 3^rd and the 9^th terms of an AP are 4 and –8 respectively, which term of this AP is zero?

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

An AP consists of 50 terms of which 3^rd term is 12 and the last term is 106. Find the 29^th term.