Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

The slope of the tangent to the curve x = 3t² + 1, y = t³ −1 at x = 1 is ___________ .

The curves y = ae^x and y = be^−x cut orthogonally, if ___________ .

The equation of the normal to the curve x = a cos³ θ, y = a sin³ θ at the point θ = π/4 is __________ .

If the curves y = 2 e^x and y = ae^−x intersect orthogonally, then a = _____________ .

The point on the curve y = 6x − x² at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .

The angle of intersection of the parabolas y² = 4 ax and x² = 4ay at the origin is ____________ .

The angle of intersection of the curves y = 2 sin² x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .

Any tangent to the curve y = 2x⁷ + 3x + 5 __________________ .

The point on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes is

(a) \[\left( 4, \frac{8}{3} \right)\]

(b) \[\left( - 4, \frac{8}{3} \right)\]

(c) \[\left( 4, - \frac{8}{3} \right)\]

(d) none of these

The slope of the tangent to the curve x = t² + 3t − 8, y = 2t² − 2t − 5 at the point (2, −1) is _____________ .

The line y = mx + 1 is a tangent to the curve y² = 4x, if the value of m is ________________ .

The normal at the point (1, 1) on the curve 2y + x² = 3 is _____________ .

The normal to the curve x² = 4y passing through (1, 2) is _____________ .

Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .

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

Find the area of a parallelogram whose adjacent sides are represented by the vectors\[2 \hat{i} - 3 \hat{k} \text { and } 4 \hat{j} + 2 \hat{k} .\]

Evaluate : \[\int\frac{x \cos^{- 1} x}{\sqrt{1 - x^2}}dx\] .

Form the differential equation representing the family of curves y = mx, where m is an arbitrary constant.

Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.