List of practice Questions

Consider the sets $T_n = \{n, n+1, n+2, n+3, n+4\}$, where $n = 1, 2, 3, \dots, 96$. How many of these sets contain 6 or any integral multiple thereof (i.e., any one of the numbers 6, 12, 18, ...)?

In the figure below (not drawn to scale), rectangle ABCD is inscribed in the circle with center at O. The length of side AB is greater than side BC. The ratio of the area of the circle to the area of the rectangle ABCD is $\pi : \sqrt{3}$. The line segment DE intersects AB at E such that $\angle \text{DOC} = \angle \text{ADE}$. The ratio $AE : AD$ is:

If $\frac{1}{3} \log N + 3 \log M = 1 + \log 1000$, then:
Using only 2, 5, 10, 25, and 50 paisa coins, what will be the minimum number of coins required to pay exactly 78 paise, 69 paise and Rs. 1.01 to three different persons?
The length of the circumference of a circle equals the perimeter of a triangle of equal sides, and also the perimeter of a square. The areas covered by the circle, triangle, and square are $c$, $t$, and $s$, respectively. Then,
What is the remainder when $4^8$ is divided by 6?
If $x$ and $y$ are integers, then the equation $5x + 19y = 64$ has:
What is the sum of $n$ terms in the series $\log m + \log \left( \frac{m^2}{n} \right) + \log \left( \frac{m^3}{n^2} \right) + \log \left( \frac{m^4}{n^3} \right) + \dots$?
A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower. The top of the tower is observed from the car and, in the process, it takes 10 min for the angle of elevation to change from $45^\circ$ to $60^\circ$. After how much more time will this car reach the base of the tower?

In the figure (not drawn to scale) given below, if $AD = CD = BC$ and $\angle DBC = 96^\circ$, how much is the value of $\angle DBC$?

If both a and b belong to the set $\{1, 2, 3, 4\}$, then the number of equations of the form $ax^2 + bx + 1 = 0$ having real roots is:
If $\log_x x - \log_10 \sqrt{x} = 2 \log_10 10$, then the possible value of $x$ is given by:
What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7?
An intelligence agency forms a code of two distinct digits selected from 0, 1, 2, ..., 9 such that the first digit of the code is non-zero. The code, handwritten on a slip, can however potentially create confusion when read upside down — for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?
Consider two different cloth-cutting processes. In the first one, $n$ circular cloth pieces are cut from a square cloth piece of side $a$ in the following steps: the original square of side $a$ is divided into $n$ smaller squares, not necessarily of the same size, then a circle of maximum possible area is cut from each of the smaller squares. In the second process, only one circle of maximum possible area is cut from the square of side $a$ and the process ends there. The cloth pieces remaining after cutting the circles are scrapped in both the processes. The ratio of the total area of scrap cloth generated in the former to that in the latter is:
There are 12 towns grouped into four zones with three towns per zone. It is intended to connect the towns with telephone lines such that every two towns are connected with three direct lines if they belong to the same zone, and with only one direct line otherwise. How many direct telephone lines are required?

In the figure (not drawn to scale) given below, P is a point on AB such that $AP : PB = 4 : 3$. PQ is parallel to AC and QD is parallel to CP. In $\triangle ARC$, $\angle ARC = 90^\circ$, and in $\triangle PQS$, $\angle PQS = 90^\circ$. The length of QS is 6 cm. What is the ratio of $AP : PD$?

The Internet is a medium where users have nearly .......... choices and ............. constraints about where to go and what to do.
The best punctuation is that of which the reader is least conscious; for when punctuation, or lack of it, ................. itself, it is usually because it .................
The argument that the need for a looser fiscal policy to ............ demand outweighs the need to .............. budget deficits is persuasive.
The Athenians on the whole were peaceful and prosperous; they had ............ to sit at home and think about the universe and dispute with Socrates, or to travel abroad and ............... the world.
Their achievement in the field of literature is described as ...............; sometimes it is even called ...............
From the time she had put her hair up, every man she had met had grovelled before her and she had acquired a mental attitude toward the other sex which was a blend of ............ and ................
This simplified ............. to the decision-making process is a must read for anyone ............... important real estate, personal, or professional decisions.
Physicians may soon have ............ to help paralysed people move their limbs by bypassing the ............... nerves that once controlled their muscles.