Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Find `bb(dy/dx)` for the given function:

(cos x)^y = (cos y)^x

Find `bb(dy/dx)` for the given function:

xy = `e^((x - y))`

Find the derivative of the function given by f(x) = (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) and hence find f′(1).

Differentiate (x² – 5x + 8) (x³ + 7x + 9) in three ways mentioned below:

1. By using the product rule.
2. By expanding the product to obtain a single polynomial.
3. By logarithmic differentiation.

Do they all give the same answer?

If u, v and w are functions of x, then show that `d/dx(u.v.w) = (du)/dx v.w + u. (dv)/dx.w + u.v. (dw)/dx` in two ways-first by repeated application of product rule, second by logarithmic differentiation.

Differentiate the function with respect to x:

x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

If cos y = x cos (a + y), with cos a ≠ ± 1, prove that `dy/dx = cos^2(a+y)/(sin a)`.

If x = a (cos t + t sin t) and y = a (sin t – t cos t), find `(d^2y)/dx^2`.

If y = `e^(acos^(-1)x)`, −1 ≤ x ≤ 1, show that `(1- x^2) (d^2y)/(dx^2) -x dy/dx - a^2y = 0`.

For given vectors,  `veca = 2hati - hatj + 2hatk` and `vecb = -hati  + hatj - hatk`, find the unit vector in the direction of the vector `veca +vecb`.

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

`x/a + y/b = 1`

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y² = a (b² – x²)

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = a e^3x + b e^{– 2x}

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = e^2x (a + bx)

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = e^x (a cos x + b sin x)

Find the particular solution of the differential equation (1 + e^2x) dy + (1 + y²) e^x dx = 0, given that y = 1 when x = 0.

Solve the differential equation  `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`

Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)