Let f : R → R be a continuous function defined
by f(x) = 1/e^x + 2e^-x
Statement - 1: f(c) = 1/3, for some c ∈ R.
Statement-2: 0 < f(x)≤ , for all x ∈ R.

• Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

• Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.

• Statement-1 is true, Statement-2 is false.

• Statement-1 is true, Statement-2 is false.

The equation of the tangent to the curve, that is parallel to the x-axis, is

• y = 0

• y = 1

• y = 3

• y = 3

The differential equation whose solution is Ax² + By² = 1, where A and B are arbitrary constants is of

• second order and second degree

• first order and second degree

• first order and first degree

• first order and first degree

The differential equation representing the family of curves  , where c > 0, is a parameter, is of order and degree as follows:

• order 1, degree 2

• order 1, degree 1

• order 1, degree 3

• order 1, degree 3

15.

Let f be differentiable for all x. If f(1) = - 2 and f′(x) ≥ 2 for x ∈ [1, 6] , then

• f(6) ≥ 8

• f(6) < 8

• f(6) < 5

• f(6) < 5

If f is a real-valued differentiable function satisfying |f(x) – f(y)| ≤ (x – y)² , x, y ∈ R and f(0) = 0, then f(1) equals

• -1

• 0

• 2

• 2

If then the solution of the equation is

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of 50 cm³ /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is

let  . If f(x) is continuous in  then f (π/4) is

• 1

• -1/2

• -1

• -1

A function y = f(x) has a second order derivative f″(x) = 6(x – 1). If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x – 5, then the function is

• (x-1)²

• (x-1)³

• (x+1)³

• (x+1)³