English Medium इयत्ता १० - CBSE Important Questions for Mathematics

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]

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in given figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

 [Use `pi = 22/7`]

Prove that `sqrt3` is an irrational number.

If α and β are the zeroes of the polynomial x² – 1, then the value of (α + β) is ______.

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

Find the value of p, for which one root of the quadratic equation px² – 14x + 8 = 0 is 6 times the other.

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.

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

Assertion (A): The point (0, 4) lies on y-axis.

Reason (R): The x-coordinate of a point on y-axis is zero.

Assertion (A): The point (0, 4) lies on y-axis.

Reason (R): The x-coordinate of a point on y-axis is zero.

In the following figure, altitudes AD and CE of ΔABC intersect each other at the point P. Show that:

ΔABD ∼ ΔCBE

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that ΔABC ~ ΔPQR. 

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.

In the given figure, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller 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°

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

If a circle is touching the side BC of ΔABC at P and is touching AB and AC produced at Q and R respectively (see the figure). Prove that AQ = `1/2` (perimeter of ΔABC).

Prove that `tan θ/(1 - cot θ) + cot θ/(1 - tanθ)` = 1 + sec θ cosec θ

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.