UPSC (Prelims) 2026 Quantitative Aptitude Medium Questions

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

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.

The digit in the unit place of the number $6^{129}\times 7^{307}$ is

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.

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:

A person saves 10% of his salary every month. If his salary increases by 12% and the expenditure increases by 10%, then what will be the change in his saving per month?

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.

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?

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?

There are four types of weights, namely 1 kg, 2 kg, 5 kg and 10 kg. What is the maximum number of different ways one can measure 20 kg, if at least eight but not more than eleven weights of 1 kg are to be used while measuring?

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

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?

In an objective type question paper, 5 marks are awarded for a correct answer and 2 marks are deducted for a wrong answer. A student attempted all the questions and got a score of 69. Had he been awarded 4 marks for a correct answer and 1 mark deducted for a wrong answer, he would have scored 84. How many questions were there in the question paper?

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?

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 $x_4$?

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?

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

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

Statement II : $xz=261$

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?