List of top Mathematics Questions on Application of derivatives asked in TS EAMCET

Statement-I: The equation of the tangent to the curve \[ y=3x^2-5 \] drawn through the point \((1,2)\) is \[ y=6x-4. \] Statement-II: If \(L,M,N\) are respectively the lengths of tangent, normal and subnormal drawn to a curve at a point \((a,b)\), then \[ \frac{(L)(N)}{M}=b^2. \] Choose the correct option.

A function is defined as \[ f(x)= \begin{cases} 3x-1, & 0\le x\le 2,\\ \sqrt{25(x-1)}, & 2\le x<\infty. \end{cases} \] For \(f(x)\) in the interval \(\left[\frac13,3\right]\), choose the correct statement.

In the interval \([-5,5]\), if \[ f(x)=(x+3)^2(x-2)^3 \] is increasing on \[ S=\{x\mid -5\le x<\alpha \text{ and } \beta<x\le5\}, \] then \(f(\alpha)-f(\beta)=\) ?

If the angle between the curves \[ y^2=4x \] and \[ y=ax^2-5 \] at the point \((1,2)\) is \(\alpha\), then \[ (a-2)|\tan\alpha| = ? \]

The function \[ f(x)=x|x-1|+|x+2| \] is

If a tangent drawn at the point \(P(h,k)\), where \(h,k\in \mathbb{Z}\), on the curve \[ y=2x^3+3x^2-4x-1 \] passes through the point \(Q(2,8)\), then \(PQ=\)

If the base of an isosceles triangle is \(3\sqrt{2}\) feet and the two equal sides are increasing at the rate of \(1\) ft/s, then the rate of increase of its area (in sq.ft/sec) when the angle between the equal sides is a right angle is

If \[ f(x)=x^3-19x+30 \] is a real valued function with domain \([-4,1]\), then the value of \(c\) according to Lagrange's Mean Value Theorem for \(f(x)\) is

The function $f(x)=2x^3-9ax^2+12a^2x+1$ where $a>0$ attains its local maximum and local minimum at p and q respectively. If $p^2=q$ then a =

The height of a cone with semi vertical angle $\pi/3$ is increasing at the rate of 2 units/min. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

For the curve $(\frac{x}{a})^n + (\frac{y}{b})^n = 2$, ($n \in N$ & $n>1$) the line $\frac{x}{a}+\frac{y}{b}=2$ is

For a real number 'a', if a real valued function $f(x) = 4x^3 + ax^2 + 3x - 2$ is monotonic in its domain, then the range of 'a' is

There is a possible error of 0.03 cm in a scale of length 1 foot with which the height of a closed right circular cylinder and the diameter of a sphere are measured as 3.5 feet each. If the radii of both cylinder and sphere are same, then the approximate error in the sum of the surface areas of both cylinder and sphere is (in square feet)

A man of 5 feet height is walking away from a light fixed at a height of 15 feet at the rate of K miles/hour. If the rate of increase of his shadow is $\frac{11}{5}$ feet/sec, then K = (Take 1 mile = 5280 feet)

If the tangent and the normal drawn to the curve $xy^2 + x^2y = 12$ at the point (1,3) meet the X-axis in T and N respectively, then TN =

Consider all functions given in List-I in the interval [1,3]. The List-2 has the values of 'c' obtained by applying Lagrange's mean value theorem on the functions of List-1. Match the functions and values of 'c'.

A real valued function
$f:[4, \infty) \to \mathbb{R}$ is defined as $f(x) = (x^2+x+1)^{(x^2-3x-4)}$, then f is

The approximate value of $\sqrt{6560}$ is

If a balloon flying at an altitude of 30 m from an observer at a particular instant is moving horizontally at the rate of 1 m/s away from him, then the rate at which the balloon is moving away directly from the observer at the 40th second is (in m/s)

If the normal drawn at the point P on the curve $y^2 = x^2-x+1$ makes equal intercepts on the coordinate axes, then the equation of the tangent drawn to the curve at P is