English Medium Class 10 - CBSE Question Bank Solutions for Mathematics

Show that a₁, a_2,..., a_n... form an AP where a_n is defined as below:

a_n = 9 − 5n

Also, find the sum of the first 15 terms.

If the sum of the first n terms of an AP is 4n − n², what is the first term (that is, S₁)? What is the sum of the first two terms? What is the second term? Similarly, find the 3^rd, the 10th, and the n^th terms.

Find the sum of first 40 positive integers divisible by 6.

Find the sum of first 15 multiples of 8.

Find the sum of the odd numbers between 0 and 50.

A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money does the contractor have to pay as a penalty  if he has delayed the work by 30 days.

A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.

In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant will be the same as the class, in which they are studying, e.g., a section of class I will plant 1 tree, a section of class II will plant 2 trees, and so on till class XII. There are three sections of each class. How many trees will be planted by the students?

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, .... as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take `pi = 22/7`)

[Hint: Length of successive semicircles is l₁, l₂, l₃, l₄, ... with centres at A, B, A, B, ...  respectively.]

200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed, and how many logs are in the top row?

In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.

A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 × 5 + 2 ×(5 + 3)]

Which term of the A.P. 121, 117, 113 … is its first negative term?

[Hint: Find n for a_n < 0]

The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.

A ladder has rungs 25 cm apart. (See figure). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and bottom rungs are 2 `1/2` m apart, what is the length of the wood required for the rungs?

[Hint: number of rungs = `250/25+ 1`]

A small terrace at a football field comprises 15 steps, each of which is 50 m long and built of solid concrete. Each step has a rise of `1/4` m and a tread of `1/2` m (See figure). Calculate the total volume of concrete required to build the terrace.

[Hint: Volume of concrete required to build the first step = `1/4 xx 1/2 xx 50  m^3`]

Find how many integers between 200 and 500 are divisible by 8.

Find the value of k for which the equation x² + k(2x + k − 1) + 2 = 0 has real and equal roots.

If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero

If the pth term of an A. P. is `1/q` and qth term is `1/p`, prove that the sum of first pq terms of the A. P. is `((pq+1)/2)`.

Find the value of p, for which one root of the quadratic equation px² – 14x + 8 = 0 is 6 times the other.