#### Level – I

1(i) Let f be a real valued function satisfying f (x + y) = f (x) . f (y) , f (0) ≠ 0 , f ‘ (0) = 2 . Find the area bounded by y = f (x), its normal at (0, f (0)) and the line x = 2.

(ii) Let y = g(x) be the image of f(x) = x + sinx about the line x + y = 0. If the area bounded by y = g(x) , x-axis , x = 0 and x = 2π is A , then find the value of A/π^{2} .

2. Slope of the tangent to a curve y = f (x) at any arbitrary point on it is proportional to the square of the ordinate of that point. Slope of the curve at (1 , 1) is equal to 1. Calculate the area enclosed by y = f (-|x|) and y = 1/2 , x = -1 , x = 1 .

3. Let the sequence a_{1} , a_{2} , a_{3} , …. be in G.P. If the area bounded by the parabolas y^{2} = 4a_{n}x and y^{2} + 4a_{n}(x – a_{n}) = 0 be Δ_{n} , prove that the sequence Δ_{1}, Δ_{2}, Δ_{3} , …. is also in G.P.

4. A function y = f (x) is defined as f (x) = |x – r – 1| , r ≤ x ≤ r + 2 , r = 0 , 2 , 4 , 6 ,…. , 2n . Find the area bounded by y = f (x) and the x – axis.

5(i) A curve passing through (1 , 2) has its slope at any point (x , y) equal to 2/(y – 2) . Find the area of the region bounded by the curve and the line 2x – y – 4 = 0

(ii) Find the area of the figure bounded by the parabola y = – x^{2} – 2x + 3 , the line tangent to it at the point P(2, – 5) and the y – axis.

6. The graph of y = f(x) and y = sinx intersect at A(a , f(a)) , B (π , 0) and C (2π , 0). A_{i}( i = 1 , 2 , 3 ) is the area bounded by the curves y = f(x) and y = f(x) between x = 0 and x = a for i = 1, x = a and x = π for I = 2 and x = π and x = for i = 3 . If A_{1} = 1 – sina + (a – 1)cosa determine the function f(x). Also find A_{1} , A_{2} , A_{3}

7(i) For the circle x^{2} + y^{2}= r^{2} , find the value of r for which the area enclosed by the tangents drawn from the point P(6 , 8) to the circle and the chord of contact is maximum.

(ii) A closed right circular cylinder has volume 2156 cubic units. Then find the radius of its base so that its total surface area may be minimum .

8. Find the locus of the point of intersection of tangents at parabola y^{2} = 4ax such that the area of the triangle formed by the pair of tangents and the corresponding chord of contact is twice the area bounded by the parabola y^{2} = 4ax and the chord of contact.

9. Let c_{1} , c_{2} and c_{3} be the graphs of the function y = x^{2} , y = 2x, y = f (x), 0 ≤ x≤ 1 , f (0) = 0. For a point P on c_{1} , let the lines through P, parallel to the axes meet c_{2} and c_{3} at Q and R respectively (see figure). If for every position P (on c_{1}) the area of the shaded region OPQ and ORP are equal, determine the function f (x).