(English Medium) ICSE Class 10 - CISCE Important Questions

Use ruler and compass only for answering this question.

Draw a circle of radius 4 cm. Mark the centre as O. Mark a point P outside the circle at a distance of 7 cm from the centre. Construct two tangents to the circle from the external point P.

Measure and write down the length of any one tangent.

Using ruler and compass construct a triangle ABC in which AB = 6 cm, ∠BAC = 120° and AC = 5 cm. Construct a circle passing through A, B and C. Measure and write down the radius of the circle.

A certain number of metallic cones, each of radius 2 cm and height 3 cm are melted and recast into a solid sphere of radius 6 cm. Find the number of cones.

A model of a ship is made to a scale 1: 300

1) The length of the model of the ship is 2 m. Calculate the lengths of the ship.

2) The area of the deck ship is 180,000 m2. Calculate the area of the deck of the model.

3) The volume of the model in 6.5 m3. Calculate the volume of the ship.

On a map drawn to a scale of 1: 50,000, a rectangular plot of land ABCD has the following dimensions. AB = 6 cm; BC = 8 cm and all angles are right angles. Find:

1) the actual length of the diagonal distance AC of the plot in km.

2) the actual area of the plot in sq. km.

Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed.

The surface area of a solid metallic sphere is 2464 cm². It is melted and recast into solid right circular cones of radius 3.5 cm and height 7 cm. Calculate:

1. the radius of the sphere.
2. the number of cones recast. (Take π = `22/7`)

A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of cones recast.

A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 2 cm and height 8 cm. Find the number of cones.

A solid cone of radius 5 cm and height 8 cm is melted and made into small spheres of radius 0.5 cm. Find the number of spheres formed.

A hemispherical and a conical hole is scooped out of a.solid wooden cylinder. Find the volume of the  remaining solid where the measurements are as follows:
The height of the solid cylinder is 7 cm, radius of each of hemisphere, cone and cylinder is 3 cm. Height of cone is 3 cm.

Give your answer correct to the nearest whole number.Taken`pi = 22/7`.

The model of a building is constructed with the scale factor 1 : 30. 
(i) If the height of the model is 80 cm, find the actual height of the building in meters.
(ii) If the actual volume of a tank at the top of the building is 27m³, find the volume of the tank on the top of the model. 

A solid sphere is cut into two identical hemispheres.

Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere.

Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere.

Which of the following is valid?

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`

The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships. Give your answer correct to the nearest whole number.

If `x/a=y/b = z/c` show that `x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)`.

Prove that `cosA/(1+sinA) + tan A =  secA`

Prove that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`

Prove that (1 + cot θ – cosec θ)(1+ tan θ + sec θ) = 2

Without using trigonometric tables evaluate:

`(sin 65^@)/(cos 25^@) + (cos 32^@)/(sin 58^@) - sin 28^2. sec 62^@ + cosec^2 30^@`