Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`

If x = e^cos2t and y = e^sin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`

If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at  t" = pi/4) = "b"/"a"`

If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`

Differentiate `x/sinx` w.r.t. sin x

Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan^–1x, when x ≠ 0

If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0

If x = t², y = t³, then `("d"^2"y")/("dx"^2)` is ______.

Derivative of x² w.r.t. x³ is ______.

Find the angle of intersection of the curves y² = x and x² = y.

Find the condition for the curves `x^2/"a"^2 - y^2/"b"^2` = 1; xy = c² to interest orthogonally.

Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.

Find the angle of intersection of the curves y² = 4ax and x² = 4by.

Show that the equation of normal at any point on the curve x = 3cos θ – cos³θ, y = 3sinθ – sin³θ is 4 (y cos³θ – x sin³θ) = 3 sin 4θ

The abscissa of the point on the curve 3y = 6x – 5x³, the normal at which passes through origin is ______.

The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ = 2 ______.

The tangent to the curve given by x = e^t . cost, y = e^t . sint at t = `pi/4` makes with x-axis an angle ______.

The equation of the normal to the curve y = sinx at (0, 0) is ______.

The point on the curve y² = x, where the tangent makes an angle of `pi/4` with x-axis is ______.

Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.