UPSC (Prelims) 2026 Quantitative Aptitude Previous Year Questions

How many times does 5 appear in all two-digit positive integers?

If $x,y$ and $z$ are integers, each greater than 1, then is $x$ a prime number?

Statement I : $x{y}^2=116$

Statement II : $xz=261$

Suppose $x$, $y$ and $z$ are variables taking positive real numbers as their possible values. It is given that $y$ is directly proportional to $x^2$ and $x$ is inversely proportional to $z$. For $z=\frac{7}{25}$, the values of $x$ and $y$ are 5 and 50, respectively. If $y=98$, what is $z$ equal to?

If $10^m\times 1000\times n=75^{25}\times 25^{32}\times 32^{75}$, where $n$ is not divisible by 10, then the value of $m$ is:

The class average $x$ in a test increases by 4 when the score of a student is rectified, whose corrected score is 100 instead of 0. Later, the score of another student was found to have been recorded as 81 in place of 56. If there are no other corrections and the final corrected average is $y$, then $y-x$ is

A toy T jumps forward or backward. In each forward jump, it moves 5' forward whereas in each backward jump, it moves 2' backward. If in 31 jumps, T moves exactly 15' forward, then what is the difference of the number of forward and backward jumps?

X and Y are two runners who run for the same duration of time on the same circular track. They started running at the same time in the same direction with uniform speeds. When X completed 7 rounds, Y did exactly 5. After completing 5 rounds, Y changed his direction and started running in the opposite direction with speed which is double of his earlier speed. On the other hand, X continued to run with the same speed. They stopped running when X completed exactly 21 rounds. How many times did X and Y meet after they had started and before they finally stopped?

For two distinct real numbers $x$ and $y$, which of them is bigger?

Statement I : $x^2<y<1$

Statement II : $y<\sqrt{x}<1$

The weight of X, in kg, is denoted by X. The weights of A, B, C, D, P, Q, R and S are measured. Given

$A+B+C+D=17$

$A+C=6$

$P+Q+S+D=15$

$P+Q+R+B=17$

$P=R$ and $Q=S$

Which one of the following statements is correct?

The speed of a train $T$ is $100$ km per hour and the speed of a person $P$ is $4$ km per hour. $T$ crosses $P$ in $15$ seconds, if $P$ travels along the direction of motion of $T$. If $P$ travels along the opposite direction of $T$, then in how much time does $T$ cross $P$, in seconds, approximately?

For $\frac{1}{3}<x<y<2$, which of the following statements is/are always correct?

I. $x+\frac{1}{x}<y+\frac{1}{y}$

II. $\frac{\sqrt{1+y^2}}{y}<\frac{\sqrt{1+x^2}}{x}$

Select the answer using the code given below.

A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, one single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying $n$ cuts is denoted by $x_n$. The experiment is performed on a solid cube where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is not a possible value for $x_4$?

X, Y and Z jump forward 4', 6' and 5', respectively. At 8 AM, they all land on mark 199'. How many times will they all land on the same mark (need not be at the same moment) between mark 195' and 1000', if all of them cross mark 1000' by 9 AM?

Three variables $x$, $y$ and $z$ take values 2, 3, 4 or 5 such that their values are always distinct. If $M$ and $N$ denote the largest possible value and the smallest possible value, respectively, for the expression $(x\times y)+z$, then $M-N$ is

X is a collection of certain odd numbers whereas Y is a collection of certain even numbers. T consists of the numbers all of which are either from X or from Y. Is every number of T from Y?

Statement I : The sum of any two numbers belonging to T is even.

Statement II : If both $p$ and $q$ are picked from T, then $(p-1)q$ is even.

What is the minimum number of times one needs to measure to get 298 litres of water from a tank, if the measuring cylinders have capacities 1 litre, 6 litres, 25 litres and 100 litres?

The top of a table is rectangular and its dimensions are 6' x 10'. Two rectangular portions of the table top are painted in blue colour; both these portions have dimensions 2.5' x 8' and each of them has exactly two sides common with two edges of the table top. If the table is fixed to the ground and the remaining portion of the table top is painted in white, how many different patterns are possible when observed from above?

X receives three coins of different denominations: 1, 2, 5, 10 and 20. If the total amount received by X is $m$, does X receive a coin of denomination 5?

Statement I : $m$ is not a prime number.

Statement II : The sum of the digits of $m$ is greater than 5.

If $x$ and $y$ are integers, then is $x$ even?

Statement I : $x^2{y}^2$ is even.

Statement II : $1+x^2+y^2$ is odd.

How many words can one form by shuffling the letters of the word QUEUE, if Q is always followed by U? The words thus formed need not necessarily have any meaning.