HSC Science कक्षा ११ - Tamil Nadu Board of Secondary Education Question Bank Solutions

Find the derivatives of the following:

If u = `tan^-1  (sqrt(1 + x^2) - 1)/x` and v = `tan^-1 x`, find `("d"u)/("d"v)`

Find the derivatives of the following:

Find the derivative with `tan^-1 ((sinx)/(1 + cos x))` with respect to `tan^-1 ((cosx)/(1 + sinx))`

Find the derivatives of the following:

If y = sin^–1x then find y”

Find the derivatives of the following:

If y = e^tan–1x, show that (1 + x²)y” + (2x – 1)y’ = 0

Find the derivatives of the following:

If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x²)y₂ – 3xy₁ – y = 0

Find the derivatives of the following:

If x = a(θ + sin θ), y = a(1 – cos θ) then prove that at θ = `pi/2`, yⁿ = `1/"a"`

Find the derivatives of the following:

If sin y = x sin(a + y), the prove that `("d"y)/("d"x) = (sin^2("a" + y))/sin"a"`, a ≠ nπ

Find the derivatives of the following:

If y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y₂ when x = 0

Choose the correct alternative:

`"d"/("d"x) (2/pi sin x^circ)` is

Choose the correct alternative:

If y = `1/4 u^4`, u = `2/3 x^3 + 5`, then `("d"y)/("d"x)` is 

Choose the correct alternative:

If y = `1/("a" - z)`, then `("d"z)/("d"y)` is

Choose the correct alternative:

If y = cos (sin x²), then `("d"y)/("d"x)` at x = `sqrt(pi/2)` is

Choose the correct alternative:
If f(x) = x tan^-1x then f'(1) is

Choose the correct alternative:

`"d"/("d"x) ("e"^(x + 5log x))` is

Choose the correct alternative:

If the derivative of (ax – 5)e^3x at x = 0 is – 13, then the value of a is

Choose the correct alternative:

x = `(1 - "t"^2)/(1 + "t"^2)`, y = `(2"t")/(1 + "t"^2)` then `("d"y)/("d"x)` is

Choose the correct alternative:

If x = a sin θ and y = b cos θ, then `("d"^2y)/("d"x^2)` is

Choose the correct alternative:

The differential coefficient of `log_10 x` with respect to `log_x 10` is

Choose the correct alternative:

If y = `(1 - x)^2/x^2`, then `("d"y)/("d"x)` is

Choose the correct alternative:

If f(x) = `{{:(x - 5,  "if"  x ≤ 1),(4x^2 - 9,  "if"  1 < x < 2),(3x + 4,  "if"  x ≥ 2):}` , then the right hand derivative of f(x) at x = 2 is