## Let a + b = 4, where a < 2, and let g(x) be a differentiable function. If dg/dx > 0 for all x ….

Q: Let a + b = 4, where a < 2, and let g(x) be a differentiable function. If dg/dx > 0 for all x , Prove that $\displaystyle \int_{0}^{a} g(x) dx + \int_{0}^{b} g(x) dx$ increases as (b – a) increases. Sol. Let b – a = t given a + b = … Read more

## A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area is to be removed ….

Q: A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area is to be removed from the sheet so that vessel has maximum volume. Sol: Lateral height of cone = Radius of circle = 1 Lateral area of cone = Area of circle with … Read more

## Find a polynomial f(x) of degree 5 which increases in the interval (-∞, 2] and [6, ∞) and decreases in the interval …

Q: Find a polynomial f(x) of degree 5 which increases in the interval (-∞, 2] and [6, ∞) and decreases in the interval [2, 6]. Given that f(0) = 3 and f ‘(4) = 0. Sol: The wavy cure of derivative will be like ⇒ f ‘(x)= k(x– 2) (x – 4)2(x – 6) and … Read more

## Find the shortest distance between the curves 9x^2 + 9y^2 – 30y + 16 = 0 and y^2 = x^3 ….

Q: Find the shortest distance between the curves 9×2 + 9y2 – 30y + 16 = 0 and y2 = x3 Sol: 9×2 + 9y2 – 30y + 16 = 0 can be rewritten as $\displaystyle x^2 + (y-\frac{5}{3})^2 = 1$ Any point on the curve y2 = x3 can be taken as (t2 , … Read more

## Let A(p^2 , – p), B(q^2, q), C(r^2, – r) be the vertices of a triangle ABC. A parallelogram AFDE is drawn with D, E and F on the line segments …..

Q: Let A(p2 , – p), B(q2, q), C(r2, – r) be the vertices of a triangle ABC. A parallelogram AFDE is drawn with D, E and F on the line segments BC, CA and AB respectively. Show that the maximum area of the parallelogram is $\displaystyle \frac{1}{4}(p+q)(q+r)(p-r)$ , given p > r. Sol: Let … Read more