Mathematics Tes...

Find the equation of the line passing through the intersection of the lines x + 2y - 3 = 0 and 4x – y + 7 = 0 and which is parallel to y – x + 10 = 0.

A line which is parallel to the x-axis and crosses the curve y =√x at an angle of 45^o is

The locus of the point of intersection of the lines x sin θ + (1 – cos θ) y = a sin θ and x sin  – (1 + cos θ ) y + a sin θ = 0 is

The equation of a circle touching the coordinate axes and the line x cos α + y sin α = 2 is x² + y² – 2gx + 2gy + g² = 0, where g is equal to

C₁ and C₂ are circles of unit radii with centres at (0, 0) and (1, 0), respectively. C₃ is a circle of unit radius, which passes through the centres of the circles C₁ and C₂ and has its centre above the x-axis. The equation of the common tangent to C₁ and C₃, which does not pass through C₂, is

A chord of the circle x² + y² – 4x – 6y = 0 passing through the origin subtends an angle tan^–1(7/4) at the point where the circle meets the positive y-axis. The equation of the chord is

If a circle passes through (a, b) and cuts the circle x² + y² = 4 orthogonally, then the locus of its centre is

The circles x² + y² – 10x + 16 = 0 and x² + y² = r² intersect each other at two distinct points if

Let f(x) = sin x + cos x. Then,

The sides of the rectangle of the greatest area, that can be inscribed in the ellipse x² + 2y² = 8, are given by

(1) f(x) has one point of inflexion.

(2) g(x) has one point of inflexion.

(3) f(x) has one point of local minima.

(4) g(x) has one point of local maxima.

Which one of the following is correct regarding the above statements?