CAT Linear & Simple Equations questions (2023–2025) — actual Quant questions with solutions

If (x²+ 1/x²) = 25 and x > 0, then the value of (x⁷ + 1/x⁷) is:

The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins to be further shifted from box A to box B to make this ratio 1:1 is:

The monthly sales of a product from January to April were 120, 135, 150 and 165 units respectively. The cost per unit of the product was Rs 240 per unit and a fixed marked price was used for the product in all the four months. However, 20%, 10% and 5% discount on the marked price was offered in January, February, March respectively while no discount were given in April.If the total profit from January to April was Rs 138825, then the marked price (in Rs) is equal to, say:

The sum of all possible real values of x for which log x-3 (x² − 9) = log x-3 (x + 1) + 2 is :

In an arithmetic progression, if the sum of fourth, seventh and tenth term is 99, and the sum of the first fourteen terms is 497, then the sum of first five terms is :

Let p, q and r be three natural numbers such that their sum is 900, and r is a perfect square whose values lies between 150 and 500.If p is less than 0.3q and  not more than 0.7q, then the sum of the maximum and minimum possible values of p is: _____

If a, b, c and d are integers such that their sum is 46, then the minimum possible value of (a − b)² + (a − c)² + (a − d)² is ______

Breakup G Strategy Time Speed + APGP | MEDIUM | CAT 2025 Slot 1

Shruti travels a distance of 224 km in four parts for a total travel time of 3 hours. Her speeds in these four parts follow an arithmetic progression, and the corresponding time taken to cover these four parts follow another arithmetic progression. If she travels at a speed of 960 meters per minute for 30 minutes to cover the first part, then the distance, in meters, she travels in the fourth part is

Stocks A, B and C are priced at rupees 120, 90 and 150 per share, respectively. A trader holds a portfolio consisting of 10 shares of stock A, and 20 shares of stocks B and C put together. If the total value of her portfolio is rupees 3300, then the number of shares of stock B that she holds, is _______

If a −6b + 6c = 4 and 6a + 3b − 3c = 50, where a, b, and c are real numbers, the value of 2a + 3b − 3c is

If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is ____

System of Equations

For some constant real numbers p, k and a, consider the system of linear equations: px − 4y = 2 and 3x + ky = a. A necessary condition for the system to have no solution for (x,y) is:

Rajesh & Vimal Land

Rajesh owns 20 hectares, Vimal owns 30 hectares with wheat and mustard in 5:3 ratio. Total wheat:mustard is 11:9. Find Rajesh's ratio.

Aman Bank Investment

Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value after 5 years is 25:36, then the minimum number of years required for the value to exceed Rs 20000 is _____

A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is ______ TITA

MathJax Equation

Find the value of x? \[ \left( x + 6\sqrt{2} \right)^{\frac{1}{2}} - \left( x - 6\sqrt{2} \right)^{\frac{1}{2}} = 2\sqrt{2}, \]

Anil invests Rs 22000 for 6 years in a scheme with 4% interest per annum, compounded half-yearly. Separately, Sunil invests a certain amount in the same scheme for 5 years, and then reinvests the entire amount he receives 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 equal, then the initial investment, in rupees, made by Sunil is

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. What was the initial quantity, in kg, of grains?

System of equations: px − 4y = 2 and 3x + ky = a. Find condition for no solution.

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.

Let \( a \) and \( b \) be two sequences such that \( a_n = 13 + 6(n-1) \) and \( b_n = 15 + 7(n-1) \) for all natural numbers \( n \). Then, the largest three-digit integer that is common to both these sequences is:

A container has 40 liters of milk. Then, 4 liters are removed from the container and replaced with 4 liters of water. This process is repeated until the volume of milk becomes less than that of water. The smallest number of times this process is repeated to achieve this is:

If a person D joins a group of three people, the average weight of the group reduces by \( x \) kg. If another person E joins instead of D, the average increases by \( 2x \) kg. If E’s weight is 12 kg more than D’s, then the value of \( x \) is:

In a fruit market, there are initially 40% mangoes, 30% bananas, and 30% apples. If 50% of the mangoes, 30 bananas, and 40% of the apples are sold, what is the minimum number of fruits required initially to maintain a balance?

Let \( a_1, a_2, a_3, \ldots \) and \( b_1, b_2, b_3, \ldots \) be arithmetic progressions such that their common differences are prime numbers. If \( a_4 = b_4 \), \( a_7 = b_7 \), and \( b_1 = 0 \), then \( a_1 \) is equal to:

For some positive real number \( x \), if \( \log_{\sqrt{3}}(x) + \log_x(25) = \log_x(0.008) \), then the value of \( x \) is:

In a right-angled triangle, if the base is 12 cm and the height is 5 cm, then the length of the hypotenuse is:

Minu buys a pair of sunglasses at Rs. 1000 and sells to Kanu at 20% profit. Then, Kanu sells it back to Minu at 20% loss. Finally, Minu sells the same pair of sunglasses to Tanu. If the total profit made by Minu from all her transactions is Rs. 500, then the percentage of profit made by Minu when she sold the pair of sunglasses to Tanu is:

If \( \log_2(x + 12) = 4 \) and \( 3 \log_2 x = 1 \), then \( x + y \) equals:

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 \( n \)-th day exceeds one million, then the lowest possible value of \( n \) is:

In a lab experiment, the number of organisms increases exponentially. On day 1, the number of organisms is 2. What is the number on day 19?

The sum of all possible values of $x$ satisfying the equation $2^{4x^{2}}-2^{2x^{2}+x+16}+2^{2x+30}=0$, is

A sum of money is shared equally among n people, giving each Rs 352. If instead two people get Rs 506 each and the rest is shared equally among the other people, each of those receives at most Rs 330. What is the maximum possible value of n?

If $p^{2}+q^{2}-29=2pq-20=52-2pq$, then the difference between the maximum and minimum possible value of $p^{3}-q^{3}$ is

In a right-angled 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 ∆ABP, ∆ABQ, and ∆ABC are in arithmetic progression. If the area of ∆ABC is 1.5 times the area of ∆ABP, the length of PQ, in cm, is:

If \(\frac{\sqrt{5x + 9}}{\sqrt{5x - 9}} = 3(\sqrt{2} + 1)\), then find the value of \( \sqrt{10x + 9} \):

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 the 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:

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:

If \( \log_2 (x + 12) = 4 \) and \( 3 \log_2 x = 1 \), then \( x + y \) equals:

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:

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:

Gopi marks a price on a product in order to make 20% profit. Ravi gets 10% discount on this marked price, and thus saves Rs 15. Then, the profit, in rupees, made by Gopi by selling the product to Ravi, is:

Rajesh and Vimal own 20 hectares and 30 hectares of agricultural land, respectively, which are entirely covered by wheat and mustard crops. The cultivation area of wheat and mustard in the land owned by Vimal are in the ratio of 5 : 3. If the total cultivation area of wheat and mustard are in the ratio 11 : 9, then the ratio of cultivation area of wheat and mustard in the land owned by Rajesh is:

After two successive increments, Gopal's salary became 187.5% of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was:

The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64. What is the largest number in the original set of three numbers?

Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value of the investment after 5 years is \( 25 : 36 \), then the minimum number of years required for the value of the investment to exceed Rs 20000 is:

A certain amount of water was poured into a 300-liter container and the remaining portion of the container was filled with milk. Then an amount of this solution was taken out from the container which was twice the volume of water that was earlier poured into it, and water was poured to refill the container again. If the resulting solution contains 72% milk, then the amount of water, in liters, that was initially poured into the container was?