#### Level – I

1. Find the equation of normal to the curve y = (1 + x)^{y} + sin^{-1}(sin^{2}x) at x = 0.

2. A point P is given on the circumference of circle of radius r. The chord QR is parallel to the tangent line at P. Find the maximum area of the triangle PQR.

3. Find the equations of the tangent drawn to the curve y^{2} – 2x^{3} – 4y + 8 = 0 from the point (1, 2).

4(i) Show that the normal to the curve 5x^{5} – 10x^{3} + x + 2y + 6 = 0 at P(0, -3) meets the curve again at two points. Find the equations of the tangents to the curve at these points.

(ii). Find the length of the normal at ‘ t ‘ on the curve x = a ( t + sint ) , y = a ( 1 – cost ).

5. Prove that

(i) sin^{2}θ < θsin(sinθ) for 0 < θ < π/2

(ii) cos(sinx) > sin(cosx), 0 < x < π/2

(iii) (a + b )^{p} ≤ a^{p} + b^{p} ; a , b > 0 , 0 < p < 1

6. Prove that if a_{o} , a_{1} , a_{2} , … , a_{n} are real numbers such that

then there exists at least one real number x between 0 and 1 such that

a_{o}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + …..+ a_{n} = 0

7. Show that

b – a ≠ kπ , k ∈ I, has no points of extrema.

8. Find the largest term in the sequence ,

for all x ≥ 1 then prove that P(x) > 0 for all x > 1

10(i) Find the interval in which the function f(x) = sin(lnx) – cos(lnx) is strictly increasing.

(ii) For what values of ‘ a ‘ the point of local minima of

f(x) = x^{3} – 3ax^{2} + 3(a^{2} – 1)x + 1 is less than 4 and point of local maxima is greater than – 2