List of top Mathematics Questions asked in CBSE CLASS XII

Find the absolute maximum and minimum values of the function \(f\) given by \(f(x)=cos^2x+sinx,x∈[0,π]\)
Find the points at which the function \(f\) given by \(f(x)=(x-2)^4(x+1)^3\) has (i)local maxima (ii)local minima (iii)point of inflexion

Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is \((a^{\frac{2}{3}}+b^{\frac{2}{3}})^{\frac{3}{2}}\)

Find two numbers whose sum is \(24\) and whose product is as large as possible.

Find the maximum and minimum values of \(x+1 sin2x\) on \([0,2\pi]\).

A window is in the form of rectangle surmounted by a semicircular opening.The total perimeter of the window is \(10m\).Find the dimensions of the window to admit maximum light through the whole opening.
The sum of the perimeter of a circle and square is \(k\), where \(k\) is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

 

Find the equation of the normal at the point (am2, am3 ) for the curve ay2 = x3.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is \(2m\) and volume is \(8m^3\).If building of tank costs Rs 70per sq meters for the base and Rs 45per square metre for sides.What is the cost of least expensive tank?

 

If y=eacos-1x,-1≤x≤1,show that (1-x2)\(\frac{d^2y}{dx^2}\)-x\(\frac{dy}{dx}\)-a2y=0

Let I be any interval disjoint from (−1, 1). Prove that the function f given by \(f(x)=x+\frac{1}{x}\) is strictly increasing on I.

\(\int_0^\frac \pi2\)\(\frac {\pi}{20} cos\ 2x\ dx\)

\(y=\begin{vmatrix}  f(x)&g(x)  &h(x) \\   l&m  &n \\   a&b  &c  \end{vmatrix}\),prove that \(\frac{dy}{dx}=\begin{vmatrix}  f'(x)&g'(x)  &h'(x) \\   l&m  &n \\   a&b  &c  \end{vmatrix}\)

Using the fact that sin (A+B)=sinAcosB+cosAsinB and the differentiation, obtain the sum formula for cosines.

Using mathematical induction prove that \(\frac{d}{dx}\)(xn)=nxn-1 for all positive integers n

If f(x)=|x|3, show that f"(x)exists for all real x, and find it.

For the curve \(y = 4x^3 − 2x^5\) , find all the points at which the tangents passes through the origin.

If x=a(cost+tsint) and y=a(sint-tcost),find \(\frac{d^2y}{dx^2}\)

If cosy=xcos(a+y) with cosa≠±1,prove that \(\frac{dy}{dx}\)=\(\frac{cos^2(a+y)}{sin\,a}\)

Find the least value of a such that the function f given \(f(x)=x^2+ax+1\) is strictly increasing on \((1, 2)\).

If (x-a)2+(y-b)2=c2, for some c>0 prove that
[1+(\(\frac{dy}{dx}\))2]\(^{\frac{3}{2}}\)/\(\frac{d^2y}{dx^2}\) is a constant independent of a and b

Prove that the function f given by \(f(x) = x^2 − x + 1\) is neither strictly increasing nor strictly decreasing,on \((−1, 1)\).

\(x\sqrt{1+y}+y\sqrt{1+x}=0\), for -1<x<1,prove that \(\frac{dy}{dx}\)=\(-\)\(\frac{1}{(1+x)^2}\)

Prove that the logarithmic function is strictly increasing on \((0, ∞)\).