CAT 2026 Slot 2 Quant Analysis: Expected Paper, Questions & Solutions

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. There are some direct paths between Q and R and between R and S. One can travel from P to S either via Q, via R, or via Q followed by R, in exactly 60 possible ways. One can also travel from Q to R either directly, via P, or via S, in exactly 28 possible ways. Then, the number of direct paths between Q and R is:

A function f maps the set of natural numbers to whole numbers such that f(xy) = f(x)f(y) + f(x) + f(y) for all natural numbers x, y, and f(p) = 1 for every prime number p. Then, the value of f(54000) is:

The sum of the infinite series (1/4)(1/4 − 1/6) + (1/4)^2[(1/4)^2 − (1/6)^2] + (1/4)^3[(1/4)^3 − (1/6)^3] + ⋯ is equal to:

If √(x + √5) − √(x − √5) = √2, then the value of x is:

If m and n are natural numbers such that n > 1, and m^n = 2^15 × 3^10, then the value of m − n is.

The roots α, β of the equation 2x² + λx − 1 = 0 satisfy 1/α² + 1/β² = 20. The value of (α³ + β³)² is:

All the values of x satisfying the inequality 1/(x + 4) ≤ 1/(3x − 2) are:

If a, b, c are positive real numbers such that a > 8 ≥ b ≥ c ≥ 2, and log₂(a + b) + log₂c = 6, then the greatest possible integer value of a is:

Consider the sequence t₁ = 1, t₂ = −1, and tₙ = [(n − 3)/(n − 1)]tₙ₋₂ for n ≥ 3. Then, the value of 1/t₂ + 1/t₄ + 1/t₆ + ⋯ + 1/t₂₀₂₄ + 1/t₂₀₂₆ is:

A bus starts at 8 am and follows a fixed route every day. One day, it travels at a constant speed of 60 km/hr and reaches its destination 3 hours later than its scheduled arrival time. The next day, it travels two-thirds of its route in one-third of its total scheduled travel time, and the remaining part of the route at 40 km/hr to reach exactly on time. The scheduled arrival time of the bus is:

Aman and Varun together can complete a task in 120 days, while Varun and Sameer together can complete the same task in 80 days. Aman starts working on the task and works for 20 days. Then Varun takes over and works for 120 days. Finally, Sameer takes over and completes the remaining task in 20 days. If instead Aman had worked on every day, Varun on every alternate day and Sameer on every third day, with all three starting on the first day, then on which numbered day would the task be completed?

The coordinates of the three vertices of a triangle are (3, 4), (15, 4) and (3, 13). Then, the radius of the incircle of the triangle is:

ABCD is a trapezium in which AB is parallel to CD. The sides AD and BC, when extended, intersect at point E. If AB = 3 cm, CD = 2 cm and the perimeter of ABCD is 10 cm, then the perimeter, in cm, of triangle AEB is:

Three circles of equal radius touch one another externally. Circle X encloses all three circles and is internally tangent to each. Circle Y lies in the central gap and is externally tangent to each of the three circles. Find the ratio of the radius of X to the radius of Y.

When 7^2025 is divided by 13, the remainder is.

A vessel contains a certain quantity of a solution of acid and water. When 3 litres of water is added to it, the new solution has 50% acid concentration. When 12 litres of acid is further added to this new solution, the final solution has 75% acid concentration. The ratio of water and acid in the original solution was:

A company has 40 employees whose names are listed in a certain order. In 2022, the average bonus of the first 30 employees was Rs. 36,000, of the last 30 employees was Rs. 56,000, and of the first 10 and last 10 employees together was Rs. 46,000. In the next year, the average bonus of the first 10 employees doubled, the average bonus of the last 10 employees became three times its previous value, and the bonuses of the remaining 20 employees remained unchanged. Then, the average bonus, in rupees, of all 40 employees together in 2023 was:

Riya incurs a 20% loss when she sells a product at Rs. 4800 to Meera, who in turn sells the product to Kavya. If Riya had sold the product to Meera at the purchase price of Kavya, she would have obtained a 15% profit. Then, the profit, in rupees, made by Meera 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, mangoes constitute 40% of his total stock. During the day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he finds that he has sold exactly 50% of the total fruits. The smallest possible total number of fruits in the stock at the beginning of the day is:

Arjun invests Rs. 25,000 for 4 years in a scheme offering 4% interest per annum, compounded half-yearly. Separately, Varun invests a certain amount in the same scheme for 3 years. At the end of 3 years, he reinvests the entire amount received for one more year at 8% simple interest. If the amounts received by both at the end of 4 years are equal, then the initial investment, in rupees rounded to two decimal places, made by Varun is:

The ratio of the number of learners enrolled in the forenoon batch and the afternoon batch of a coaching institute was 13 : 9. When 42 learners shifted from the forenoon batch to the afternoon batch, the ratio of the two batches became 19 : 14. Subsequently, a few fresh learners were admitted to the forenoon and afternoon batches in the ratio 3 : 8, after which the ratio of the number of learners in the forenoon batch to that in the afternoon batch became 5 : 4. What is the total number of fresh learners admitted?