List of practice Questions

12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.

Five cards--the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. 
(i) What is the probability that the card is the queen? 
(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?

A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is 3.5 mm. What is the estimate on the thickness of hair ?

Which of the following is the most precise device for measuring length : 

a vernier calipers with 20 divisions on the sliding scale  
a screw gauge of pitch 1 mm and 100 divisions on the circular scale
an optical instrument that can measure length to within a wavelength of light ?

Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.

Explain this statement clearly : “To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary : 

1. atoms are very small objects 
2. a jet plane moves with great speed 
3. the mass of Jupiter is very large 
4. the air inside this room contains a large number of molecules 
5. a proton is much more massive than an electron 
6. the speed of sound is much smaller than the speed of light.

A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 \(\text {kg} \;\text m^2\; \text s^{-2}.\) Suppose we employ a system of units in which the unit of mass equals \(\alpha\) kg, the unit of length equals \(\beta\) m, the unit of time is \(\gamma\) s. Show that a calorie has a magnitude 4.2 \(\alpha^{-1}\; \beta^{-2}\; \gamma^{2}\) in terms of the new units.

In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line (see Fig. 5.6).

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?

A line passes through (x₁, y₁) and (h, k). If slope of the line is m, show that k-y₁=m(h-x₁)

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 (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?

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?

Check whether the following are quadratic equations : 
(i) \((x + 1)^2 = 2(x – 3)\)                      (ii) \(x^2 – 2x = (–2) (3 – x)\)
(iii) \((x – 2)(x + 1) = (x – 1)(x + 3)\)     (iv) \((x – 3)(2x +1) = x(x + 5)\)
(v) \((2x – 1)(x – 3) = (x + 5)(x – 1)\)       (vi) \(x^2 + 3x + 1 = (x – 2)^2\)
(vii) \((x + 2)^3 = 2x (x^2 – 1)\)               (viii) \(x^3 – 4x^2 – x + 1 = (x – 2)^3\)

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.

A contract on 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 the contractor has to pay as penalty, if he has delayed the work by 30 days.

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{sin(x+a)}{cos\,a}\)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{sin\,x+cos\,x}{sin\,x-cos\,x}\)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{cos\,x}{1+sin\,x}\)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): cosec x cot x

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): sin (x + a)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): (ax +b)ⁿ (cx + d)^m

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): (ax +b)ⁿ

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{px^2+qx+r}{ax+b}\)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers):\(\frac{ax+b}{px^2+qx+r}\)