1. If a, b and c are positive integers, then (a – b – c)^{3} – a^{3} + b^{3} + c^{3} is always divisible by

2. In the following figure, if ABC and BDC are two triangles, then x + y + z equals

3. If the graph of the linear equation 24x + 7y = 168 cuts the x-axis and y-axis at A and B respectively, then length of AB is equal to

4. If (x ^{2} – 1) is a factor of ax ^{198} + bx ^{195} + cx ^{192} + dx ^{187} + e, where a, b, c, d and e are non-zero integers, then

5. In ∆ABC, AD and AE are respectively bisector of ∠A and perpendicular on BC. If ∠EAD = 10° and ∠ABC : ∠ACB = 2 : 1, then ∠ACB equals

6.

7.

8. A linear equation 7x + 11y = 77 (where x and y are natural numbers) have

9. If a = 62, b = 66 and c = 68, then the value of (a ^{2} + b ^{2} + c ^{2} – ab – bc – ca) is

10. A shopkeeper mixed two varieties of sugar costing ₹45/kg and ₹54/kg so that the mixture costs ₹50/kg. If he mixes 12 kg of sugar costing ₹45/kg, then the quantity of sugar costing ₹54/kg that he mixes is (in kg)

11. In ΔABC, O is the point of intersection of altitudes from vertex A and B and I is the point of intersection of bisectors of ∠A, ∠B and ∠C. If ∠A : ∠B : ∠C = 3 : 4 : 2, then ∠IAO is

12. If (2 ^{a} – 4) ^{3} + (4 ^{a} – 2) ^{3} = (4 ^{a} + 2 ^{a} – 6) ^{3}, then the greatest value of a (where 2 ^{a} + 4^{a}≠ 6) is

13. The sum of the digits of product 2^{2021} 5^{2024} is

14.

15. If P(5, –2) and R(–5, 2) are the coordinates of the opposite vertices of a rectangle PQRS, then the coordinates of the vertex Q and S can be