Commerce (English Medium) Class 12 - CBSE Important Questions for Mathematics

If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`

`"If y" = (sec^-1 "x")^2 , "x" > 0  "show that"  "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`

If `"x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.

Prove that : `2sin^-1 (3/5) -tan^-1 (17/31) = pi/4.`

The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",",  if x ≠ 0),(k",",  if x = 0):}` is continuous at x = 0 is ______.

If `ysqrt(1 - x^2) + xsqrt(1 - y^2)` = 1, then prove that `(dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))`

The function f(x) = x | x |, x ∈ R is differentiable ______.

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

If f(x) = | cos x |, then `f((3π)/4)` is ______.

The set of all points where the function f(x) = x + |x| is differentiable, is ______.

Read the following passage and answer the questions given below:

1. Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
2. Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?

The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x³ + 0.02x² + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b.

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.

Find the equation of tangents to the curve y= x³ + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.

If the function f(x)=2x³−9mx²+12m²x+1, where m>0 attains its maximum and minimum at p and q respectively such that p²=q, then find the value of m.

Prove that `y=(4sintheta)/(2+costheta)-theta `

Show that the equation of normal at any point t on the curve x = 3 cos t – cos³t and y = 3 sin t – sin³t is 4 (y cos³t – sin³t) = 3 sin 4t