Class 10 Mathematics • Sample Paper Set 4

If $\tan A = \frac{4}{3}$, then the value of $\cos A$ is:

The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. The angle of elevation of the sun is:

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:

If $\triangle ABC \sim \triangle DEF$ such that $AB = 1.2$ cm and $DE = 1.4$ cm, the ratio of the areas of $\triangle ABC$ and $\triangle DEF$ is:

The value of $(\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ)$ is:

In the given figure, if $\angle AOB = 125^\circ$, then $\angle COD$ is equal to:

If $x = a \cos \theta$ and $y = b \sin \theta$, then $b^2x^2 + a^2y^2$ is equal to:

The distance between two parallel tangents to a circle of radius 5 cm is:

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is:

The value of $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}$ is:

A ladder 10 m long reaches a window 8 m above the ground. The distance of the foot of the ladder from the base of the wall is:

If $\sin A = \frac{1}{2}$, then the value of $\cot A$ is:

The area of a sector of a circle with radius 6 cm if angle of the sector is $60^\circ$ is:

In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ respectively such that $DE \parallel BC$. If $AD/DB = 3/5$ and $AC = 4.8$ cm, then $AE$ is:

The value of $\sin^2 60^\circ + 2\tan 45^\circ - \cos^2 30^\circ$ is:

If the angle between two tangents drawn from an external point P to a circle of radius $a$ and centre O is $60^\circ$, then $OP$ is equal to:

The distance of the point $P(2, 3)$ from the x-axis is:

If $\cos(\alpha + \beta) = 0$, then $\sin(\alpha - \beta)$ can be reduced to:

Prove that: $\frac{1 - \sin \theta}{1 + \sin \theta} = (\sec \theta - \tan \theta)^2$.

In the given figure, PA and PB are tangents to the circle with centre O such that $\angle APB = 50^\circ$. Write the measure of $\angle OAB$.

Find the value of $x$ if $\tan 3x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ$.

ABC is an isosceles triangle right angled at C. Prove that $AB^2 = 2AC^2$.

Find the area of a quadrant of a circle whose circumference is 22 cm.

Prove that the lengths of tangents drawn from an external point to a circle are equal.

If $\sin(A - B) = 1/2$, $\cos(A + B) = 1/2$, $0^\circ < A + B \le 90^\circ$, $A > B$, find A and B.

The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is $30^\circ$. Find the height of the tower.

Prove that: $(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$.

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 $\angle AOB = 90^\circ$.

Find the area of the shaded region in the figure, where ABCD is a square of side 14 cm and APD and BPC are semicircles.

State and prove the Basic Proportionality Theorem (Thales Theorem).

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

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 $30^\circ$, 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 $60^\circ$. Find the time taken by the car to reach the foot of the tower from this point.

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Case Study: Trigonometry (Kite Flying)
A boy is flying a kite. The string is 100m long and makes an angle of $60^\circ$ with the horizontal. Another boy is standing 20m away from the first boy in the direction of the kite.

Case Study: Geometry (Circles)
A Ferris wheel has a diameter of 50 meters. The center of the wheel is 30 meters above the ground. A passenger gets into a cabin at the bottom of the wheel.

Case Study: Coordinate Geometry
To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD.