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

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Concept: Tangent to a Circle

Prove that a parallelogram circumscribing a circle is a rhombus.

Concept: Number of Tangents from a Point on a Circle

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

Concept: Number of Tangents from a Point on a Circle

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

Concept: Arithmetic Progression

Solve the following quadratic equation for x: `4x^2 + 4bx – (a^2 – b^2) = 0`

Concept: Solutions of Quadratic Equations by Completing the Square

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

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

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm^{3}. The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (use *π* = 3.14)

Concept: Heights and Distances

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

In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region.

Concept: Areas of Sector and Segment of a Circle

What is the HCF of the smallest prime number and the smallest composite number?

Concept: Fundamental Theorem of Arithmetic

Given that `sqrt2` is irrational prove that `(5 + 3sqrt2)` is an irrational number

Concept: Revisiting Irrational Numbers

Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Concept: Fundamental Theorem of Arithmetic

Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream.

Concept: Pair of Linear Equations in Two Variables

In Fig. 1, ABCD is a rectangle. Find the value of *x* and *y*.

Concept: Equations Reducible to a Pair of Linear Equations in Two Variables

In an A.P., if S5 + S7 = 167 and S_{10=}235, then find the A.P., where Sn denotes the sum of its first n terms.

Concept: Sum of First n Terms of an AP

Find the 9^{th} term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185

Concept: Sum of First n Terms of an AP

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first *n* terms of the A.P.

Concept: Sum of First n Terms of an AP

The first three terms of an AP respectively are 3y – 1, 3y + 5 and 5y + 1. Then y equals:

(A) –3

(B) 4

(C) 5

(D) 2

Concept: Arithmetic Progression

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