CAT 2023 Slot 1 QA Previous Year Questions with Detailed Solutions

If $x$ and $y$ are positive real numbers such that $\log_x(x^2 + 12) = 4$ and $3\log_y x = 1$, then $x + y$ equals

If $x$ and $y$ are real numbers such that $x^2 + (x - 2y - 1)^2 = -4y(x + y)$, then the value of $x - 2y$ is

If $\sqrt{5x+9} + \sqrt{5x-9} = 3(2 + \sqrt{2})$, then $\sqrt{10x+9}$ is equal to

Let $n$ be the least positive integer such that 168 is a factor of $1134^n$. If $m$ is the least positive integer such that $1134^n$ is a factor of $168^m$, then $m + n$ equals

The number of integer solutions of the equation $2|x|(x^2 + 1) = 5x^2$ is _______

Let $\alpha$ and $\beta$ be the two distinct roots of the equation $2x^2 - 6x + k = 0$, such that $(\alpha + \beta)$ and $\alpha\beta$ are the distinct roots of the equation $x^2 + px + p = 0$. Then, the value of $8(k - p)$ is _______

The equation $x^3 + (2r+1)x^2 + (4r-1)x + 2 = 0$ has $-2$ as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of $r$ is _______

Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of $x$ km till then, where $x$ is a whole number and is palindromic (i.e., $x$ remains unchanged when its digits are reversed). At the end of the trip, the car had travelled a total of 26862 km till then, this number again being palindromic. If Brishti never drove at more than 110 km/h, then the greatest possible average speed at which she drove during the trip, in km/h, was

The minor angle between the hour hand and minute hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after 8:48 am when this angle increases by 50% is

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is