New Exam Prep
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New Exam Prep
English 30 leveled MCQs Free
Shared by a Veda learner · Generated with Veda AI
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Sample questions with model answers
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5 marks Long answer
If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by $r = \frac{(a + b - c)}{2}$.
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Derive the formula using area and semi-perimeter.
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5 marks Long answer
In the given figure, PA, QB and RC are perpendicular to AC. If PA = x units, QB = y units and RC = z units, prove that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$.
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Use properties of similar triangles to prove the statement.
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5 marks Long answer
A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, then find the area of the canvas required for making the tent, keeping a provision of $26 \text{ m}^2$ of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 250 per $\text{m}^2$.
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Calculate the surface area and total cost.
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5 marks Long answer
A chord of a circle of radius 14 cm subtends a $120^\circ$ at the centre. Find the area of the corresponding: (i) minor segment (ii) major sector.
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Use the sector and triangle area formulas to find both areas.
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5 marks Long answer
Water is flowing at the rate of 15 km/hr through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm? What should be the speed of water if the rise in water level is to be attained in 1 hour?
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Calculate volume flow rate and time for the desired rise.
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3 marks Medium answer
In the given figure, CD is perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that CF/CD = FG/DG.
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Use similar triangles to prove CF/CD = FG/DG.
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3 marks Medium answer
Through the mid-point Q of side CD of a parallelogram ABCD, the line AR is drawn which intersects BD at P and produced BC at R. Prove that (i) AQ = QR (ii) AP = 2PQ.
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Use properties of parallelograms and midpoints to prove the statements.
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3 marks Medium answer
If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that AQ = $\frac{1}{2}$ (perimeter of the $\triangle ABC$).
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Apply the tangent-segment theorem to prove AQ = $\frac{1}{2}$ (perimeter).
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3 marks Medium answer
A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. (Take $\pi = 3.14$)
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Volume = $\frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3$.
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3 marks Medium answer
If $\tan\theta = \frac{1}{\sqrt{5}}$ then find the value of $\frac{\text{cosec}^2 \theta - \sec^2 \theta}{\text{cosec}^2 \theta + \sec^2 \theta}$.
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Use identities to simplify the expression.
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3 marks Medium answer
If $m = \tan\theta + \sin\theta$ and $n = \tan\theta - \sin\theta$ then prove that $m^2 - n^2 = 4\sqrt{mn}$.
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Expand and simplify to prove the identity.
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3 marks Medium answer
A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$. After some time, the angle of elevation reduces to $30^\circ$. Find the distance travelled by the balloon during the interval.
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Use trigonometry to find the distance travelled.
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3 marks Medium answer
Manish is standing on level ground and observes a kite flying 200 m away from him at an angle of elevation of $30^\circ$. Mahesh, standing on the roof of a 50 m high building on the opposite side of the kite, observes the same kite at an angle of elevation of $45^\circ$. Find the distance between the kite and Mahesh.
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Use trigonometric ratios to find the distance.
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3 marks Medium answer
Ridhi drew a polygon with n sides. The smallest exterior angle is $8^\circ$ and each subsequent exterior angle is $4^\circ$ more than the previous exterior angle. Find the number of sides of the polygon that Ridhi had drawn.
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Use the formula for the sum of exterior angles to find n.
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3 marks Medium answer
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if it is given that area $\triangle ABC = 90 \text{ cm}^2$.
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Use the inradius formula to find the sides.
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1 marks Short answer
$5\sin^2 30^\circ + \cos^2 45^\circ - 4 an^2 30^\circ$ is equal to
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$\frac{5}{12}$
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1 marks Short answer
If the perimeter of a circle is equal to that of a square, then the ratio of their areas is
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$7:22$
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1 marks Short answer
In the given figure, AB is a chord of the circle and AOC is its diameter, such that $\angle ACB = 50^\circ$. If AT is the tangent to the circle at the point A, then $\angle BAT$ is equal to:
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$60^\circ$
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1 marks Short answer
The value of $\frac{1}{\tan\theta + \cot\theta}$ =
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$\cos\theta \sin\theta$
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1 marks Short answer
If the circumference of a circle and the perimeter of a square are equal, then:
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Area of the circle > Area of the square
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1 marks Short answer
Two circles touch each other externally at C. AB and CD are two common tangents to the circles. If D lies on AB such that CD = 6 cm, then the length of AB is:
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12 cm
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1 marks Short answer
In a circle of radius 35 cm, an arc subtends an angle of $60^\circ$ at the centre, then the length of the arc is:
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37 cm
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1 marks Short answer
The area of the circle that can be inscribed in a square of side 6 cm is:
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$18\pi \text{ cm}^2$
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1 marks Short answer
The area of a quadrant of a circle whose circumference is 22 cm is:
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$\frac{77}{8} \text{ cm}^2$
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1 marks Short answer
ABC is a right-angled triangle, right-angled at B, such that BC = 6 cm and AB = 8 cm. The exact radius of its in-circle is:
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2 cm
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1 marks Short answer
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of $80^\circ$, then $\angle POA$ is equal to:
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$50^\circ$
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1 marks Short answer
Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
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8 cm
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1 marks Short answer
A quadrilateral ABCD is drawn to circumscribe a circle with centre O. If $\angle AOB = 125^\circ$, then $\angle COD$ is equal to:
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$55^\circ$
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1 marks Short answer
Assertion (A): Rampal decided to donate canvas for 10 tents conical in shape with base diameter 14 m and height 24 m to a centre for handicapped person's welfare. The slant height of the conical tent is 25. Reason (R): According to assertion, the surface area of 10 tents is $5500 \text{ m}^2$.
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Both (A) and (R) are true but (R) is not the correct explanation of the (A).
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1 marks Short answer
Assertion (A): In two triangles, if the corresponding angles are equal, then the triangles are similar. Reason (R): If the areas of two similar triangles are equal, then the triangles are congruent.
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(A) is true but (R) is false.
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