Differentiation & Applications · JEE Main previous year questions

Let $f:\mathbb{R}\to \mathbb{R}$ be a twice differentiable function such that $f^{\prime \prime }(x)>0$ for all $x\in \mathbb{R}$ and $f^{\prime }(a-1)=0$, where $a$ is a real number. Let $g(x)=f({\tan }^{2}x-2\tan x+a),0<x<\frac{\pi }{2}$. Consider the following two statements: (I) g is increasing in $\left(0,\frac{\pi }{4}\right)$ (II) g is decreasing in $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$ Then,

1. A.Both (I) and (II) are True
2. B.Neither (I) nor (II) is True
3. C.Only (I) is True
4. D.Only (II) is True

Let $f(x)=x^3+x^2{f}^{\prime }(1)+2x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3),x\in \mathbb{R}$. Then the value of $f^{\prime }(5)$ is :

1. A.$\frac{657}{5}$
2. B.$\frac{117}{5}$
3. C.$\frac{2}{5}$
4. D.$\frac{62}{5}$

Let the product of ${\omega }_1=(8+i)\sin \theta +(7+4i)\cos \theta$ and ${\omega }_2=(1+8i)\sin \theta +(4+7i)\cos \theta$ be $\alpha +i\beta$, $i=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha +\beta$ respectively. Then $\mathrm{p}+\mathrm{q}$ is equal to :

1. A.130
2. B.150
3. C.160
4. D.140

Let $\mathrm{A}(4,-2)$, $\mathrm{B}(1,1)$ and $\mathrm{C}(9,-3)$ be the vertices of a triangle ABC . Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and $AB$ of the triangle $ABC$ respectively, is ____.

Let $f(x)=a{x}^3+b{x}^2+cx+41$ be such that $f(1)=40$, $f^{\prime }(1)=2$ and $f^{\prime \prime }(1)=4$. Then $a^2+b^2+c^2$ is equal to:

1. A.54
2. B.51
3. C.73
4. D.62

Let $|\cos \theta \cos (60-\theta )\cos (60+\theta )|\le \frac{1}{8}$, $\theta \in [0,2\pi ]$. Then, the sum of all $\theta \in [0,2\pi ]$, where $\cos 3\theta$ attains its maximum value, is :

1. A.$6\pi$
2. B.$9\pi$
3. C.$18\pi$
4. D.$15\pi$

A variable line $\mathrm{L}$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $\mathrm{A}$ and $\mathrm{B}$. The minimum area of the triangle $\mathrm{OAB}$, where $\mathrm{O}$ is the origin, is :

1. A.35
2. B.25
3. C.30
4. D.40

Let $f(x)=4{\cos }^{3}x+3\sqrt{3}{\cos }^{2}x-10$. The number of points of local maxima of $f$ in interval $(0,2\pi )$ is

1. A.$1$
2. B.$3$
3. C.$4$
4. D.$2$

Let the maximum and minimum values of ${\left(\sqrt{8x-x^2-12}-4\right)}^2+(x-7{)}^2,x\in \mathbf{R}$ be M and m, respectively. Then ${\text{M}}^2-{\text{m}}^2$ is equal to _________.

Let $\mathrm{P}=\{\mathrm{z}\in \mathbb{C}:|z+2-3i|\le 1\}$ and $\mathrm{Q}=\{\mathrm{z}\in \mathbb{C}:z(1+i)+\bar{z}(1-i)\le -8\}$. Let in $\mathrm{P}\cap \mathrm{Q}$, $|z-3+2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If ${\left|z_1\right|}^2+2{\left|z_2\right|}^2=\alpha +\beta \sqrt{2}$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ equals ____.

Let $S=(-1,\infty )$ and $f:S\to \mathbb{R}$ be defined as $f(x)={\int }_{-1}^x{\left(e^t-1\right)}^{11}(2t-1{)}^5(t-2{)}^7(t-3{)}^{12}(2t-10{)}^{61}dt$, Let $\mathrm{p}=$ Sum of squares of the values of $x$, where $f(x)$ attains local maxima on $S$, and $\mathrm{q}=$ Sum of the values of $\mathrm{x}$, where $f(x)$ attains local minima on $S$. Then, the value of $p^2+2q$ is ____.

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _________.

Suppose $f(x)=\frac{(2^x+2^{-x})\tan x\sqrt{{\tan }^{-1}(x^2-x+1)}}{(7x^2+3x+1{)}^3}$. Then the value of $f^{\prime }(0)$ is equal to

1. A.$\pi$
2. B.$\sqrt{\pi }$
3. C.$0$
4. D.$\frac{\pi }{2}$

Let $f(x)=2^x-x^2,x\in \mathbb{R}$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f^{\prime }(x)$ intersect the $x$-axis, then the value of $m+n$ is _______.

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ passing through the origin. The tangent of an angle between the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ is :

1. A.$\frac{30}{41}$
2. B.$\frac{8}{5}$
3. C.$\frac{2}{5}$
4. D.$\frac{25}{41}$

Let $f(x)=x^3+x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)+f^{\prime \prime \prime }(3),x\in \mathbf{R}$. Then $f^{\prime }(10)$ is equal to ____.

Let $g(x)=3f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime }(x)>0$ for all $x\in (0,3)$. If $g$ is decreasing in $(0,\alpha )$ and increasing in $(\alpha ,3)$, then $8\alpha$ is :

1. A.$0$
2. B.$24$
3. C.$18$
4. D.$20$

Let $f(x)=\sum _{k=1}^{10}k{x}^k$, $x\in \mathbb{R}$. If $2f(2)+f^{\prime }(2)=119(2{)}^n+1$ then $n$ is equal to ___________

The area of the region enclosed by the curve $y=x^3$ and its tangent at the point $(-1,-1)$ is :

1. A.$\frac{23}{4}$
2. B.$\frac{19}{4}$
3. C.$\frac{27}{4}$
4. D.$\frac{31}{4}$

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5{b}^3{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

1. A.$110$
2. B.$108$
3. C.$90$
4. D.$55$

If $\mathrm{A}$ is the area in the first quadrant enclosed by the curve $\mathrm{C}:2x^2-y+1=0$, the tangent to $\mathrm{C}$ at the point $(1,3)$ and the line $x+y=1$, then the value of $60\mathrm{A}$ is _________.

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in ${\text{cm}}^2$) is equal to :

1. A.1025
2. B.900
3. C.800
4. D.675

The slope of tangent at any point (x, y) on a curve $y=y(x)$ is $\frac{x^2+y^2}{2xy}$, $x>0$. If $y(2)=0$, then a value of $y(8)$ is :

1. A.$-4\sqrt{2}$
2. B.$2\sqrt{3}$
3. C.$4\sqrt{3}$
4. D.$-2\sqrt{3}$

Let a common tangent to the curves $y^2=4x$ and $(x-4{)}^2+y^2=16$ touch the curves at the points P and Q. Then $(PQ{)}^2$ is equal to _________.

Let $g(x)=f(x)+f(1-x)$ and $f^{\prime \prime }(x)>0,x\in (0,1)$. If g is decreasing in the interval $(0,\alpha )$ and increasing in the interval $(\alpha ,1)$, then ${\tan }^{-1}(2\alpha )+{\tan }^{-1}\left(\frac{1}{\alpha }\right)+{\tan }^{-1}\left(\frac{\alpha +1}{\alpha }\right)$ is equal to :

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