Q1. The number of distinct integer solutions (x, y) of the equation |x + y| + |x− y| = 2 is

Q2. For some constant real numbers p, k and a, consider the following system of linear equations in x and y:

p x − 4 y = 2

3 x + k y = a

A necessary condition for the system to have no solution for (x, y) is

Q3. If (a + b√3)² = 52 + 30√3, where a and b are natural numbers, then a + b equals

Q4. A circular plot of land is divided into two regions by a chord of length 10√3 metres such that the chord subtends an angle of 120° at the centre. Then the area (in m²) of the smaller region is

Q5. If 10⁶⁸ is divided by 13, the remainder is

Q6. The midpoints of sides AB, BC and AC in △ABC are M, N and P respectively. The medians from A, B and C intersect MP, MN and NP at X, Y and Z respectively. If area(△ABC) = 1440 cm², then area(△XYZ) is

Q7- After two successive increments, Gopal’s salary became 187.5 % of his initial salary. If the percentage increase in the second increment was twice that in the first increment, then the percentage increase in the first increment was

Q8 – Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

Q9- A certain amount of water was poured into a 300 litre 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 litres, that was initially poured into the container was

Q10- Aman invests ₹ 4000 at a rate 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 of the investment to exceed ₹ 20000 is

Q11- Gopi marks a price to make 20 % profit. Ravi gets 10 % discount on the marked price and thus saves ₹ 15. Then the profit (in ₹) made by Gopi on selling to Ravi is

Q12- The average of three distinct real numbers is 28. If the smallest is increased by 7 and the largest reduced by 10, the order remains unchanged, the new arithmetic mean becomes 2 more than the middle number, and the difference between largest and smallest becomes 64. Then the largest number in the original set is

Q13- The number of all positive integers up to 500 with non-repeating digits is

Q14- A train travelled a certain distance at uniform speed. Had the speed been 6 km/h more, it would have needed 4 hours less. Had the speed been 6 km/h less, it would have needed 6 hours more. The distance (in km) is

Q15- If 3ᵃ = 4, 4ᵇ = 5, 5ᶜ = 6, 6ᵈ = 7, 7ᵉ = 8 and 8ᶠ = 9, then the value of the product a b c d e f is

Q16- Consider the sequence t₁ = 1, t₂ = −1 and tn = ((n − 3)/(n − 1)) tn₋₂ for n ≥ 3. Then the value of the sum (1⁄t₂ + 1⁄t₄ + 1⁄t₆ + … + 1⁄t₂₀₂₂ + 1⁄t₂₀₂₄) is

Q17- The number of distinct real values of x satisfying max{x, 2} − min{x, 2} = |x + 2| − |x − 2| is

Q18- 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

Q19- For any non-zero real x, let f(x) + 2 f(1⁄x) = 3x. Then the sum of all possible values of x for which f(x) = 3 is

Q20- A regular octagon ABCDEFGH has side length 6 cm. Then the area (in cm²) of the square ACEG is

Q21- The sum of all distinct real values of x that satisfy 10ˣ + 4⁄10ˣ = 81⁄2 is

Q22- 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