English Medium कक्षा ९ - CBSE Question Bank Solutions

Find the following product:

(x² – 1)(x⁴ + x² + 1)

Without actually calculating the cubes, find the value of:

`(1/2)^3 + (1/3)^3 - (5/6)^3`

Without actually calculating the cubes, find the value of:

(0.2)³ – (0.3)³ + (0.1)³

Find the value of x³ + y³ – 12xy + 64, when x + y = – 4

Find the value of x³ – 8y³ – 36xy – 216, when x = 2y + 6

Give possible expressions for the length and breadth of the rectangle whose area is given by 4a² + 4a – 3.

Simplify (2x – 5y)³ – (2x + 5y)³.

Multiply x² + 4y² + z² + 2xy + xz – 2yz by (–z + x – 2y).

If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ – 3abc = – 25.

Prove that (a + b + c)³ – a³ – b³ – c³ = 3(a + b)(b + c)(c + a).

Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in major segment.

In the following figure, ∠ACB = 40º. Find ∠OAB.

In the following figure, AB and CD are two chords of a circle intersecting each other at point E. Prove that ∠AEC = `1/2` (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).

A circle has radius `sqrt(2)` cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in major segment is 45°.

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:

1. OB = OC
2. AO bisects ∠A

In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that

1. ΔABE ≅ ΔACF
2. AB = AC, i.e., ABC is an isosceles triangle.