Differentiation & Applications · JEE Main previous year questions

If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha }$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta }$ intersect, then the magnitude of the minimum value of $8\alpha \beta$ is ________.

Let a curve $y=f(x),x\in (0,\infty )$ pass through the points $P\left(1,\frac{3}{2}\right)$ and $Q\left(a,\frac{1}{2}\right)$. If the tangent at any point $R(b,f(b))$ to the given curve cuts the $y$-axis at the point $S(0,c)$ such that $bc=3$, then $(PQ{)}^2$ is equal to ____.

Let $f(x)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\sin 2x\end{array}\right|,x\in \left[\frac{\pi }{6},\frac{\pi }{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

1. A.${\alpha }^2-{\beta }^2=4\sqrt{3}$
2. B.${\beta }^2-2\sqrt{\alpha }=\frac{19}{4}$
3. C.${\beta }^2+2\sqrt{\alpha }=\frac{19}{4}$
4. D.${\alpha }^2+{\beta }^2=\frac{9}{2}$

If $f(x)=x^2+g^{\prime }(1)x+g^{\prime \prime }(2)$ and $g(x)=f(1)x^2+x{f}^{\prime }(x)+f^{\prime \prime }(x)$, then the value of $f(4)-g(4)$ is equal to ________.

Let $f(\theta )=3\left({\sin }^4\left(\frac{3\pi }{2}-\theta \right)+{\sin }^4(3\pi +\theta )\right)-2(1-{\sin }^{2}2\theta )$ and $S=\left\{\theta \in [0,\pi ]:f^{\prime }(\theta )=-\frac{\sqrt{3}}{2}\right\}$. If $4\beta =\sum _{\theta \in S}\theta$, then $f(\beta )$ is equal to

1. A.$\frac{9}{8}$
2. B.$\frac{3}{2}$
3. C.$\frac{5}{4}$
4. D.$\frac{11}{8}$

If the equation of the normal to the curve $y=\frac{x-a}{(x+b)(x-2)}$ at the point (1, $-3$) is $x-4y=13$, then the value of $a+b$ is equal to ____.

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S=\{x\in \mathbb{Z}:x(66-x)\ge \frac{5}{9}M\}$ and the event $\mathrm{A}=\{x\in S:x\text{ is a multiple of }3\}$. Then P(A) is equal to :

1. A.$\frac{1}{3}$
2. B.$\frac{1}{5}$
3. C.$\frac{7}{22}$
4. D.$\frac{15}{44}$

The distance of the point $(6,-2\sqrt{2})$ from the common tangent $y=\mathrm{mx}+\mathrm{c},\mathrm{m}>0$, of the curves $x=2y^2$ and $x=1+y^2$ is :

1. A.$\frac{1}{3}$
2. B.$5$
3. C.$\frac{14}{3}$
4. D.$5\sqrt{3}$

Let a tangent to the curve $9x^2+16y^2=144$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is ____

If $f(x)=x^3-x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)-f^{\prime \prime \prime }(3),x\in \mathbb{R}$, then

1. A.$2f(0)-f(1)+f(3)=f(2)$
2. B.$f(1)+f(2)+f(3)=f(0)$
3. C.$f(3)-f(2)=f(1)$
4. D.$3f(1)+f(2)=f(3)$

Let $f(x)=3^{{\left(x^2-2\right)}^3+4}$, $x\in \mathbf{R}$. Then which of the following statements are true? $\mathrm{P}:x=0$ is a point of local minima of $f$ $\mathrm{Q}:x=\sqrt{2}$ is a point of inflection of $f$ $\mathrm{R}:f^{\prime }$ is increasing for $x>\sqrt{2}$

1. A.Only P and Q
2. B.Only P and R
3. C.Only Q and R
4. D.All P, Q and R

If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\dots +\frac{1}{(180-a)(200-a)}=\frac{1}{256}$, then the maximum value of a is :

1. A.198
2. B.202
3. C.212
4. D.218

Let $\mathrm{p}$ and $\mathrm{p}+2$ be prime numbers and let $$\Delta =\left|\begin{array}{ccc}\mathrm{p}!& (\mathrm{p}+1)!& (\mathrm{p}+2)!\\ (\mathrm{p}+1)!& (\mathrm{p}+2)!& (\mathrm{p}+3)!\\ (\mathrm{p}+2)!& (\mathrm{p}+3)!& (\mathrm{p}+4)!\end{array}\right|$$ Then the sum of the maximum values of $\alpha$ and $\beta$, such that ${\mathrm{p}}^{\alpha }$ and $(\mathrm{p}+2{)}^{\beta }$ divide $\Delta$, is ____.

If the minimum value of $f(x)=\frac{5x^2}{2}+\frac{\alpha }{x^5},x>0$, is 14 , then the value of $\alpha$ is equal to :

1. A.32
2. B.64
3. C.128
4. D.256

Let $M$ and $N$ be the number of points on the curve $y^5-9xy+2x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M+N$ equals _________.

1. A.$(6,21)$
2. B.$(8,9)$
3. C.$(10,-4)$
4. D.$(12,-15)$

A common tangent $\mathrm{T}$ to the curves $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$ and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$ does not pass through the fourth quadrant. If $\mathrm{T}$ touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to ____.

If $\frac{\mathrm{d}y}{\mathrm{d}x}+2y\tan x=\sin x$, $0<x<\frac{\pi }{2}$ and $y\left(\frac{\pi }{3}\right)=0$, then the maximum value of $y(x)$ is :

1. A.$\frac{1}{8}$
2. B.$\frac{3}{4}$
3. C.$\frac{1}{4}$
4. D.$\frac{3}{8}$

Let ${\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3$ be three mutually exclusive events such that $\mathrm{P}({\mathrm{E}}_1)=\frac{2+3p}{6}$, $\mathrm{P}({\mathrm{E}}_2)=\frac{2-p}{8}$ and $\mathrm{P}({\mathrm{E}}_3)=\frac{1-p}{2}$. If the maximum and minimum values of p are ${\mathrm{p}}_1$ and ${\mathrm{p}}_2$, then $({\mathrm{p}}_1+{\mathrm{p}}_2)$ is equal to :

1. A.$\frac{2}{3}$
2. B.$\frac{5}{3}$
3. C.$\frac{5}{4}$
4. D.1

Let the function $f(x)=2x^2-{\log }_{\mathrm{e}}x$, $x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point P on it passes through the point $(8a,8a-1)$ but does not pass through the point $(-\frac{1}{a},0)$. If the equation of the normal at $P$ is : $\frac{x}{\alpha }+\frac{y}{\beta }=1$, then $\alpha +\beta$ is equal to ____ .

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x,y)$ on it is directly proportional to $\left(-\frac{y}{x}\right)$. If the curve passes through the points $(1,2)$ and $(8,1)$, then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to

1. A.$2{\log }_{e}2$
2. B.4
3. C.1
4. D.$4{\log }_{e}2$

If $x{y}^4$ attains maximum value at the point $(x,y)$ on the line passing through the points $(50+\alpha ,0)$ and $(0,50+\alpha )$, $\alpha >0$, then $(x,y)$ also lies on the line :

1. A.$y=4x$
2. B.$x=4y$
3. C.$y=4x+\alpha$
4. D.$x=4y-\alpha$

Let m and M respectively be the minimum and the maximum values of $f(x)={\sin }^{-1}2x+\sin 2x+{\cos }^{-1}2x+\cos 2x$, $x\in \left[0,\frac{\pi }{8}\right]$. Then $\mathrm{m}+\mathrm{M}$ is equal to :

1. A.$1+\sqrt{2}+\pi$
2. B.$(1+\sqrt{2})\pi$
3. C.$\pi +\sqrt{2}$
4. D.$1+\pi$

Consider a triangle ABC whose vertices are $\mathrm{A}(0,\alpha ,\alpha )$, $\mathrm{B}(\alpha ,0,\alpha )$ and $\mathrm{C}(\alpha ,\alpha ,0)$, $\alpha >0$. Let D be a point moving on the line $x+z-3=0=y$ and G be the centroid of $\Delta \mathrm{ABC}$. If the minimum length of GD is $\sqrt{\frac{57}{2}}$, then $\alpha$ is equal to ____ .

The lengths of the sides of a triangle are $10+x^2$, $10+x^2$ and $20-2x^2$. If for $x=k$, the area of the triangle is maximum, then $3k^2$ is equal to :

1. A.5
2. B.8
3. C.10
4. D.12