ISC (Arts) इयत्ता १२ - CISCE Important Questions for Mathematics

Verify Rolle’s theorem for the following function:

`f(x) = e^(-x) sinx " on"  [0, pi]`

if `x = tan(1/a log y)`, prove that `(1+x^2) (d^2y)/(dx^2) + (2x + a) (dy)/(dx) = 0`

Solve `cos^(-1)(sin cos^(-1)x) = pi/2`

Using L'Hospital's rule, evaluate : `lim_(x->0) (x - sinx)/(x^2 sinx)`

Verify Lagrange's Mean Value Theorem for the following function:

`f(x ) = 2 sin x +  sin 2x " on " [0, pi]`

`f(x)=(x^2-9)/(x - 3)` is not defined at x = 3. what value should be assigned to f(3) for continuity of f(x) at = 3?

 Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4. 

Verify the Lagrange’s mean value theorem for the function: 
`f(x)=x + 1/x ` in the interval [1, 3]

IF `y = e^(sin-1x)   and  z =e^(-cos-1x),` prove that `dy/dz = e^x//2`

Discuss the continuity of the function f at x = 0

If f(x) = `(2^(3x) - 1)/tanx`, for x ≠ 0

         = 1,   for x = 0

Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`

Using L ‘Hospital’s Rule, evaluate: 

`lim_("x"->pi/2) ("x"  "tan""x" - pi/4 . "sec"  "x")`

Solve : `"dy"/"dx" = 1 - "xy" + "y" - "x"`

Using L' Hospital's rule, evaluate:

`lim_("x"→0) (1/"x"^2 - cot"x"/"x")`

Verify Langrange’s mean value theorem for the function:

f(x) = x (1 – log x) and find the value of  c in the interval [1, 2].

If y = cos (sin x), show that: `("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y"  "cos"^2"x" = 0`

Solve the following differential equation: 
x² dy + (xy + y²) dx = 0, when x = 1 and y = 1

If the following function is continuous at x = 2 then the value of k will be ______.

f(x) = `{{:(2x + 1",", if x < 2),(                 k",", if x = 2),(3x - 1",", if x > 2):}`

The derivative of log x with respect to `1/x` is ______.

If y = 3 cos(log x) + 4 sin(log x), show that `x^2 (d^2y)/(dx^2) + x dy/dx + y = 0`