This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2023 CAT exam Slot 1. The entire exam in pdf format can be downloaded from this link: 2023 CAT Quant Slot 1: Quantitative Ability.
Question 1: 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
- $\quad 1$
- $\quad 2$
- $\quad -1$
- $\quad 0$
Question 2: Let $n$ be the least positive integer such that $168$ is a factor of $1134^n$. If $m$ is the least positive integers such that $1134^n$ is a factor of $168^m$, then $m+n$ equals
- $\quad 24$
- $\quad 12$
- $\quad 9$
- $\quad 15$
Question 3: If $\sqrt{5x+9} + \sqrt{5x-9} = 3(2+\sqrt{2})$ then $\sqrt{10x+9}$ is equal to
- $\quad 3\sqrt{31}$
- $\quad 2\sqrt{7}$
- $\quad 3\sqrt{7}$
- $\quad 4\sqrt{5}$
Question 4: 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
- $\quad 10$
- $\quad 68$
- $\quad 20$
- $\quad 11$
Question 5: The number of integer solutions of equation $2|x|(x^2+1)=5x^2$ is
Question 6: 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
Question 7: 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
Question 8: 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
- $\quad 21$
- $\quad 19$
- $\quad 20$
- $\quad 22$
Question 9: The salaries of three friends Sita, Gita and Mita are initially in the ratio $5 : 6 : 7$, respectively. In the first year, they get salary hikes of $20\%$, $25\%$ and $20\%$, respectively. In the second year, Sita and Mita get salary hikes of $40\%$ and $25\%$, respectively, and the salary of Gita becomes equal to the mean salary of the three friends. The salary hike of Gita in the second year is
- $\quad 26$
- $\quad 30$
- $\quad 28$
- $\quad 25$
Question 10: The minor angle between the hours hand and minutes 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
- $\quad \displaystyle \frac{24}{11}$
- $\quad \displaystyle \frac{36}{11}$
- $\quad 4$
- $\quad 2$
Question 11: 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
- $\quad 90$
- $\quad 80$
- $\quad 100$
- $\quad 110$
Question 12: Gita sells two objects $A$ and $B$ at the same price such that she makes a profit of $20\%$ on object $A$ and a loss of $10\%$ on object $B$. If she increases the selling price such that objects $A$ and $B$ are still sold at an equal price and a profit of $10\%$ is made on object $B$, then the profit made on object $A$ will be nearest to
- $\quad 42\%$
- $\quad 30\%$
- $\quad 45\%$
- $\quad 47\%$
Question 13: A mixture $P$ is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount with cocoa powder. The same amount is again removed from mixture $P$ and replaced with same amount of cocoa powder to form a new mixture $Q$. If the ratio of coffee and cocoa in the mixture $Q$ is $16 : 9$, then the ratio of cocoa in mixture $P$ to that in mixture $Q$ is
- $\quad 4: 9$
- $\quad 1: 3$
- $\quad 5: 9$
- $\quad 1: 2$
Question 14: Anil invests Rs. $22000$ for $6$ years in a certain scheme with $4\%$ interest per annum, compounded half-yearly. Sunil invests in the same scheme for $5$ years, and then reinvests the entire amount received at the end of $5$ years for one year at $10\%$ simple interest. If the amounts received by both at the end of $6$ years are same, then the initial investment made by Sunil, in rupees, is
Question 15: The amount of job that Amal, Sunil and Kamal can individually do in a day, are in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for $4$ days and $9$ days, respectively, Kamal needs to work for $16$ days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone, is
Question 16: Arvind travels from town $A$ to town $B$, and Surbhi from town $B$ to town $A$, both starting at the same time along the same route. After meeting each other, Arvind takes $6$ hours to reach town $B$ while Surbhi takes $24$ hours to reach town $A$. If Arvind travelled at a speed of $54$ km/h, then the distance, in km, between town $A$ and town $B$ is
Question 17: A quadrilateral $ABCD$ is inscribed in a circle such that $AB : CD = 2 : 1$ and $BC : AD = 5 : 4$. If $AC$ and $BD$ intersect at the point $E$, then $AE : CE$ equals
- $\quad 2: 1$
- $\quad 1: 2$
- $\quad 8: 5$
- $\quad 5: 8$
Question 18: Let $C$ be the circle $x^2+y^2+4x-6y-3=0$ and $L$ be the locus of the point of intersection of a pair of tangents to $C$ with the angle between the two tangents equal to $60^{\circ}$. Then, the point at which $L$ touches the line $x=6$ is
- $\quad (6, 6)$
- $\quad (6, 4)$
- $\quad (6, 8)$
- $\quad (6, 3)$
Question 19: In a right-angled triangle $\triangle ABC$, the altitude $AB$ is $5$ cm, and the base $BC$ is $12$ cm. $P$ and $Q$ are two points on $BC$ such that the areas of $\triangle ABP$, $\triangle ABQ$ and $\triangle ABC$ are in arithmetic progression. If the area of $\triangle ABC$ is $1.5$ times the area of $\triangle ABP$, the length of $PQ$, in cm, is
Question 20: For some positive and distinct real numbers $x$, $y$ and $z$, if $\displaystyle \frac{1}{\sqrt{y} + \sqrt{z}}$ is the arithmetic mean of $\displaystyle \frac{1}{\sqrt{x} + \sqrt{z}}$ and $\displaystyle \frac{1}{\sqrt{x} + \sqrt{y}}$, the relationship which will always hold true, is
- $\quad x, y,$ and $z$ are in arithmetic progression
- $\quad \sqrt{x}, \sqrt{y},$ and $\sqrt{z}$ are in arithmetic progression
- $\quad y, x,$ and $z$ are in arithmetic progression
- $\quad \sqrt{x}, \sqrt{z},$ and $\sqrt{y}$ are in arithmetic progression
Question 21: The number of all natural numbers up to $1000$ with non-repeating digits is
- $\quad 738$
- $\quad 648$
- $\quad 504$
- $\quad 585$
Question 22: A lab experiment measures the number of organisms at $8$ am every day. Starting with $2$ organisms on the first day, the number of organisms on any day is equal to $3$ more than twice the number on the previous day. If the number of organisms on the $n^{\textrm{th}}$ day exceeds one million, then the lowest possible value of $n$ is