CAT Modern Math Previous Year Questions (PYQs)

If $\log_{64} x^2 + \log_8 \sqrt{y} + 3\log_{512}\left(\sqrt{y}\,z\right) = 4$ , where $x, y$ and $z$ are positive real numbers, then the minimum possible value of $(x+y+z)$ is

A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is

The number of distinct integers $n$ for which $\log_{1/4}(n^2 - 7n + 11) > 0$ , is

The sum of all four-digit numbers that can be formed with the distinct non-zero digits $a, b, c,$ and $d$ , with each digit appearing exactly once in every number, is $153310 + n$ , where $n$ is a single digit natural number. Then, the value of $(a + b + c + d + n)$ is

If $x$ is a positive real number such that $4 \log _{10} x+4 \log _{100} x+8 \log _{1000} x=13$ , t hen greatest integer not exceeding x , is

P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is

If a, b and c are positive real numbers such that $a > 10 \ge b \ge c$ and $\frac{\log_8(a+b)}{\log_2 c} + \frac{\log_{27}(a - b)}{\log_3 c} = \frac{2}{3}$ , then the greatest possible integer value of a is

In a group of 250 students, the percentage of girls was at least $44\%$ and at most $60\%$. The rest of the students were boys. Each student opted for either swimming or running or both. If $50\%$ of the boys and $80\%$ of the girls opted for swimming while $70\%$ of the boys and $60\%$ of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

The sum of all distinct real values of $x$ that satisfy the equation $10^x + \frac{4}{10^x} = \frac{81}{2}$ , is

The number of all positive integers up to 500 with non-repeating digits is

If $x$ and $y$ are positive real numbers such that $\log_{x}\left(x^{2} + 12\right)=4$ and $3 \log _{y} x=1$ , then $x+y$ equals

The number of all natural numbers up to 1000 with non-repeating digits is

In a class of 150 students, 75 students chose physics, 111 students chose mathematics and 40 students chose chemistry. All students chose at least one of the three subjects and at least one student chose all three subjects. The number of students who chose both physics and chemistry is equal to the number of students who chose both chemistry and mathematics, and this is half the number of students who chose both physics and mathematics. The maximum possible number of students who chose physics but not mathematics, is

The sum of all possible real values of $x$ for which $\log_{x-3}(x^2-9)=\log_{x-3}(x+1)+2$ , is

For some positive real number $x , if$\log_{ $$\sqrt{3}$}(x) + $\frac{\log_x (25)}{\log_x (0.008)}$ = $\frac{16}{3}$ , then the value of $\log_3(3x^2)$ is

For a real number $x$ , if $\frac{1}{2}, \frac{\log _{3}\left(2^{x} - 9\right)}{\log _{3} 4}$ , and $\frac{\log _{5}\left(2^{x}+\frac{17}{2}\right)}{\log _{5} 4}$ are in an arithmetic progression, then the common difference is

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

The number of integers greater than 2000 that can be formed with the digits $0, 1, 2, 3, 4, 5$ , using each digit at most once, is

The number of distinct integer values of n satisfying $\frac{4-\log_2 n}{3-\log_4 n} < 0$ , is

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in $1421$ , including itself, is

If $5-\log _{10}$ $\sqrt{1+x} + 4 \log _{10}$ $\sqrt{1-x} = \log _{10} \frac{1}{\sqrt{1- x^{2}}}$ , then $100 \mathrm{x}$ equals

The number of groups of three or more distinct numbers that can be chosen from 1,2, $3,4,5,6,7$ and 8so that the groups always include 3 and $5$ , while 7 and 8 are never included together is

The number of ways of distributing 15 identical balloons, 6 identical pencils and 3 identical erasers among 3 children, such that each child gets at least four balloons and one pencil, is

For all possible integers $n$ satisfying $2.25 \leq 2+ 2^{n+2}$ $\leq$ 202 , the number of integer values of $3+3^{n+1}$ is