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

As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.

Prove that `(tan^2 theta)/(sec theta - 1)^2 = (1 + cos theta)/(1 - cos theta)`

Prove that (cosec A – sin A)(sec A – cos A) sec² A = tan A.

Without using trigonometric tables evaluate

`(sin 35^@ cos 55^@ + cos 35^@ sin 55^@)/(cosec^2 10^@ - tan^2 80^@)`

Prove that: 
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1

Simplify 

sin A `[[sinA   -cosA],["cos A"  " sinA"]] + cos A[[ cos A" sin A " ],[-sin A" cos A"]]`

Prove that:

`(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(2 sin^2 A - 1)`

If x = h + a cos θ, y = k + b sin θ.

Prove that `((x - h)/a)^2 + ((y - k)/b)^2 = 1`.

tan θ × `sqrt(1 - sin^2 θ)` is equal to:

(1 + sin A)(1 – sin A) is equal to ______.

Prove the following identity:

(sin²θ – 1)(tan²θ + 1) + 1 = 0

Statement 1: sin²θ + cos²θ = 1

Statement 2: cosec²θ + cot²θ = 1

Which of the following is valid?

Factorize: sin³θ + cos³θ

Hence, prove the following identity:

`(sin^3θ + cos^3θ)/(sin θ + cos θ) + sin θ cos θ = 1`

The daily wages of 80 workers in a project are given below.

Use a graph paper to draw an ogive for the above distribution. (Use a scale of 2 cm = Rs. 50 on x-axis and 2 cm = 10 workers on y-axis). Use your ogive to estimate:

1. the median wage of the workers.
2. the lower quartile wage of workers.
3. the numbers of workers who earn more than Rs. 625 daily.

The histogram below represents the scores obtained by 25 students in a mathematics mental test. Use the data to:

1. Frame a frequency distribution table.
2. To calculate mean.
3. To determine the Modal class.

The weight of 50 workers is given below:

Draw an ogive of the given distribution using a graph sheet. Take 2 cm = 10 kg on one axis and 2 cm = 5 workers along the other axis. Use a graph to estimate the following:

1) The upper and lower quartiles.

2) If weighing 95 kg and above is considered overweight, find the number of workers who are overweight.

The marks obtained by 100 students in a Mathematics test are given below:

Draw an ogive for the given distribution on a graph sheet.

Use a scale of 2 cm = 10 units on both axes.

Use the ogive to estimate the:

1) Median.

2) Lower quartile.

3) A number of students who obtained more than 85% marks in the test.

4) A number of students who did not pass in the test if the pass percentage was 35.

A Mathematics aptitude test of 50 students was recorded as follows:

Draw a histogram from the above data using a graph paper and locate the mode.

Use graph paper for this question. Estimate the mode of the given distribution by plotting a histogram. [Take 2 cm = 10 marks along one axis and 2 cm = 5 students along the other axis]

The given graph with a histogram represents the number of plants of different heights grown in a school campus. Study the graph carefully and answer the following questions:

1. Make a frequency table with respect to the class boundaries and their corresponding frequencies.
2. State the modal class.
3. Identify and note down the mode of the distribution.
4. Find the number of plants whose height range is between 80 cm to 90 cm.