(English Medium) ICSE Class 10 - CISCE Important Questions for Mathematics

Prove that `sqrt((1 - sin θ)/(1 + sin θ)) = sec θ - tan θ`.

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

How many terms of the A.P. : 24, 21, 18, ................ must be taken so that their sum is 78?

If the quadratic equation px² − 2√5px + 15 = 0 has two equal roots then find the value of p.

Solve for x: 1 + 4 + 7 + 10 + ... + x = 287.

Radha deposited ₹ 400 per month in a recurring deposit account for 18 months. The qualifying sum of money for the calculation of interest is ______.

Which of the following equations has 2 as a root?

The roots of equation (q – r)x² + (r – p)x + (p – q) = 0 are equal.

Prove that 2q = p + r; i.e., p, q, and r are in A.P.

If x, y and z are in continued proportion, Prove that:

`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`

A mixture of paint is prepared by mixing 2 parts of red pigments with 5 parts of the base. Using the given information in the following table, find the values of a, b and c to get the required mixture of paint.

While factorizing a given polynomial using the remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x³ + 7x² + 2x – 3.

1. Is the student’s solution correct in stating that (2x + 1) is a factor of the given polynomial?
2. Give a valid reason for your answer.

Also, factorize the given polynomial completely.

An Arithmetic Progression (A.P.) has 3 as its first term. The sum of the first 8 terms is twice the sum of the first 5 terms. Find the common difference of the A.P.

In the given figure, AC is the diameter of the circle with center O.

CD is parallel to BE.

∠AOB = 80° and ∠ACE = 20°

Calculate:

1. ∠BEC
2. ∠BCD
3. ∠CED

A manufacturing company prepares spherical ball bearings, each of radius 7 mm and mass 4 gm. These ball bearings are packed into boxes. Each box can have a maximum of 2156 cm³ of ball bearings. Find the:

1. maximum number of ball bearings that each box can have.
2. mass of each box of ball bearings in kg.
   (Use π = `22/7`)

Prove the identity (sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ.

Prove the following trigonometry identity:

(sin θ + cos θ)(cosec θ – sec θ) = cosec θ ⋅ sec θ – 2 tan θ

Jaya borrowed Rs. 50,000 for 2 years. The rates of interest for two successive years are 12% and 15% respectively. She repays 33,000 at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt.

Mr Lalit invested Rs. 5000 at a certain rate of interest, compounded annually for two years. At the end of the first year, it amounts to Rs. 5325. Calculate

1) The rate of interest

2) The amount at the end of the second year, to the nearest rupee.

In what time will Rs. 1500 yield Rs. 496.50 as compound interest at 10% per annum compounded annually?

The present population of the town is 2,00,000. The population is increased by 10% in the first year and 15% in the second year. Find the population of the town at the end of two years.