The two circles x² + y² = ax and x² + y² = c²(c > 0) touch each other if

• 2|a| = c

• |a| = c

• a = 2c

• a = 2c

If two tangents drawn from a point P to the parabola y²= 4x are at right angles, then the locus of P is

• X = 1

• 2x +1 = 0

• x = -1

• x = -1

The radius of a circle, having minimum area, which touches the curve y = 4 – x² and the lines, y = |x| is

Let y be an implicit function of x defined by x^2x – 2x^xcoty – 1 = 0. Then y′ (1) equals 

• -1

• 1

• log 2

• log 2

Given P(x) = x⁴+ ax³ + cx + d such that x = 0 is the only real root of P′ (x) = 0. If P(–1) < P(1),then in the interval [–1, 1].

• P(–1) is the minimum and P(1) is the maximum of P

• P(–1) is not minimum but P(1) is the maximum of P

• P(–1) is the minimum but P(1) is not the maximum of P

• P(–1) is the minimum but P(1) is not the maximum of P

Suppose the cube x³– px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds?

• The cubic has minima at  and maxima at –

• The cubic has minima at – and maxima at

• The cubic has minima at both  and- 

• The cubic has minima at both  and- 

The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is 

• (x – 2)y′² = 25 – (y – 2)² 

• (y – 2)y′² = 25 – (y – 2)²

• (y – 2)²y′²= 25 – (y – 2)² 

• (y – 2)²y′²= 25 – (y – 2)² 

18.

The function f(x) = tan^-1 (sinx + cosx) is an increasing function in

• (π/4, π /2)

• (–π/2, π /4)

• (0, π /2)

• (0, π /2)

If 

• x /x+1

• x-1/x

• 1/x

• 1/x

The normal to the curve x = a(1 + cosθ), y = asinθ at ‘θ’ always passes through the fixed point

• (a, 0)

• (0, a)No

• (0,0)

• (0,0)