This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2024 CAT exam Slot 1. The entire exam in pdf format can be downloaded from this link: 2024 CAT Quant Slot 1.
Question 1: If $x$ is a positive real number such that $4\log_{10} x + 4\log_{100} x + 8\log_{1000} x = 13$, then the greatest integer not exceeding $x$, is
Question 2: The selling price of a product is fixed to ensure $40\%$ profit. If the product had cost $40\%$ less and had been sold for 5 rupees less, then the resulting profit would have been $50\%$. The original selling price, in rupees, of the product is
- $\quad 15$
- $\quad 20$
- $\quad 10$
- $\quad 14$
Question 3: A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process of pouring out two-thirds the content and replacing with water is repeated three more times, then the final ratio of milk to water in the glass, is
- $\quad 1: 27$
- $\quad 1: 26$
- $\quad 1: 81$
- $\quad 1: 80$
Question 4: In the $XY-$plane, the area, in sq. units, of the region defined by the inequalities $y \geq x+4$ and $-4 \leq x^2+y^2+4(x-y) \leq 0$ is
- $\quad \pi$
- $\quad 3\pi$
- $\quad 2\pi$
- $\quad 4\pi$
Question 5: Suppose $x_1, x_2, x_3, \ldots, x_{100}$ are in arithmetic progression such that $x_5=-4$ and $2x_6+2x_9 = x_{11} + x_{13}$. Then, $x_{100}$ equals
- $\quad 206$
- $\quad -196$
- $\quad -194$
- $\quad 204$
Question 6: The surface area of a closed rectangular box, which is inscribed in a sphere, is $846$ sq cm, and the sum of the lengths of all its edges is $144$ cm. The volume, in cubic cm, of the sphere is
- $\quad 1125\pi \sqrt{2}$
- $\quad 1125\pi$
- $\quad 750\pi \sqrt{2}$
- $\quad 750\pi$
Question 7: Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
- $\quad 12$
- $\quad 18$
- $\quad 14$
- $\quad 16$
Question 8: The sum of all real values of $k$ for which $\left(\displaystyle \frac{1}{8}\right)^k \times \left(\displaystyle \frac{1}{32768}\right)^{\frac{1}{3}} = \displaystyle \frac{1}{8} \times \left(\displaystyle \frac{1}{32768}\right)^{\frac{1}{k}}$, is
- $\quad -\displaystyle \frac{4}{3}$
- $\quad \displaystyle \frac{2}{3}$
- $\quad \displaystyle \frac{4}{3}$
- $\quad -\displaystyle \frac{2}{3}$
Question 9: Let $x$, $y$, and $z$ be real numbers satisfying $4(x^2+y^2+z^2) =a$, $4(x-y-z)=3+a$. Then $a$ equal
- $\quad 1 \displaystyle \frac{1}{3}$
- $\quad 1$
- $\quad 4$
- $\quad 3$
Question 10: In September, the incomes of Kamal, Amal and Vimal are in the ratio of $8: 6: 5$. They rent a house together, and Kamal pays $15\%$, Amal pays $12\%$ and Vimal pays $18\%$ of their respective incomes to cover the total house rent in that month. In October, the house rent remains unchanged while their incomes increase by $10\%$, $12\%$, and $15\%$, respectively. In October, the percentage of their total income that will be paid as house rent, is nearest to
- $\quad 14.84$
- $\quad 13.26$
- $\quad 15.18$
- $\quad 12.75$
Question 11: $ABCD$ is a rectangle with sides $AB=56$ cm and $BC=45$ cm, and $E$ is the midpoint of side $CD$. Then, the length, in cm, of radius of incircle of $\triangle ADE$ is
Question 12: Consider two sets $A ={2,3,5,7,11,13}$ and $B={1,8,27}$. Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$, there is at least one element $a$ in $A$ such that $f(a)=b$. Then, the total number of such functions $f$ is
- $\quad 537$
- $\quad 540$
- $\quad 667$
- $\quad 665$
Question 13: A fruit seller has a total of 187 fruits consisting of apples, mangoes, and oranges. The number of apples and mangoes are in the ratio $5: 2$. After she sells 75 apples, 26 mangoes and half of the oranges, the ratio of number of unsold apples to number of unsold oranges becomes $3: 2$. The total number of unsold fruits is
Question 14: A shop wants to sell a certain quantity (in kg) of grains. It sells half the quantity and an additional 3 kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 3 kg of these grains to the second customer. Finally, when the shop sells half of the remaining quantity and an additional 3 kg of these grains to the third customer, there are no grains left. The initial quantity, in kg, of grains is
- $\quad 36$
- $\quad 50$
- $\quad 42$
- $\quad 18$
Question 15: There are four numbers such that average of first two numbers is 1 more than the first number, average of first three numbers is 2 more than the average of first two numbers, and average of first four numbers is 3 more than the average of first three numbers. Then, the difference between the largest and the smallest numbers, is
Question 16: For any natural number $n$, let $a_n$ be the largest integer not exceeding $\sqrt{n}$. Then the value of $a_1+a_2+\cdots + a_{50}$ is
Question 17: 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
Question 18: Renu would take 15 days working 4 hours per day to complete a certain task whereas Seema would take 8 days working 5 hours per day to complete the same task. They decide to work together to complete this task. Seema agrees to work for double the number of hours per day as Renu, while Renu agrees to work for double the number of days as Seema. If Renu works 2 hours per day, then the number of days Seema will work, is
Question 19: If the equations $x^2+mx+9=0$, $x^2+nx+17=0$ and $x^2+(m+n)x+35=0$ have a common negative root, then the value of $(2m+3n)$ is
Question 20: When $10^{100}$ is divided by 7, the remainder is
- $\quad 3$
- $\quad 4$
- $\quad 6$
- $\quad 1$
Question 21: If $(a+b\sqrt{n})$ is the positive square root of $(29-12\sqrt{5})$, where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is
- $\quad 4$
- $\quad 6$
- $\quad 18$
- $\quad 22$
Question 22: An amount of Rs $10000$ is deposited in bank A for a certain number of years at a simple interest of $5\%$ per annum. On maturity, the total amount received is deposited in bank B for another 5 years at a simple interest of $6\%$ per annum. If the interests received from bank A and bank B are in the ratio $10 : 13$, then the investment period, in years, in bank A is
- $\quad 6$
- $\quad 3$
- $\quad 4$
- $\quad 5$