1. The relation R = {(x, y): x, y ∈ Z and x +y is even}

2. The number of non-empty equivalence relations on the set {1, 2, 3} is

3. Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then, the total number of one-one maps f : A →B, such that  f (1) + f (3) = 14, is

4. Let R = {(1,2), (2,3) (3,3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is

5.

6.

7. Let A be a square matrix such that AA^T = I. Then, 1/2 A [(A+A^T)²+(A-A^T)²] is equal to

8.

9.

10.

11.

12. If the function f(x) = sin 3x + α sin x - β cos 3x/x³ , x ∈ R, is continuous at x = 0, then f(0) is equal to

13. Let f : [-1,2] →R be given by

f(x) = 2x²+x+[x²]-[x], where [t] denotes the greatest integer less than or equal to t. The number of points, where f is not continuous, is

14. Let [x] denote the greatest integer function and let m and n, respectively be the numbers of the points,  where the function f(x) = [x] + |x - 2|, -2

15. Let f(x) = |2x² +5|x|-3|, x ∈ R. If m and in denote the number of points where f is not continuous and not differentiable respectively, then m + n is equal to

16. Let f(x) = ∫x³ √3 - x² dx. If 5 f (√2) = -4, then f(1) is equal to

17. For x ∈ (-Π/2, Π/2), if

y (x) = ∫ cosec x + sin x /cosec x sec x + tan x sin²x  dx and lim x →(Π/2)^- y (x) = 0, then y (Π/4) is equal to

18. If I (x) = ∫e ^sin² x (cos x sin 2x - sin x ) dx and I(0) =1, then I (Π/3) is equal to

19. The area (in sq. units) of the region described by  {(x, y): y² ≤ 2x and y ≥ 4x - 1} is

20. The area (in square units) of the region enclosed by the ellipse x² +3y² = 18 in the first quadrant below

the line y = x is

21. The area bounded by the curve y = |x ² - 9| and the line y = 3 is

22. The area of the region {(x, y) : |x − y |≤ y ≤ 4√x} is

23. The area of the region in the first quadrant inside the circle x²+y² = 8 and outside the parabola y²=2x is equal to

24. The area (in sq units) of the region bounded by the parabola y² =4 (x-2) and the line y = 2x -8, is

25. The area of the region {(x, y): x² ≤ y ≤8-x², y ≤7} is