Differentiation & Applications · JEE Main previous year questions

2021 7 questions

1. Q62021 · 1 Sept 2021 · Shift 2

   The function $f(x)=x^3-6x^2+ax+b$ is such that $f(2)=f(4)=0$. Consider two statements : Statement 1 : there exists $x_1,x_2\in (2,4)$, $x_1<x_2$, such that $f^{\prime }(x_1)=-1$ and $f^{\prime }(x_2)=0$. Statement 2 : there exists $x_3,x_4\in (2,4)$, $x_3<x_4$, such that f is decreasing in $(2,x_4)$, increasing in $(x_4,4)$ and $2f^{\prime }(x_3)=\sqrt{3}f(x_4)$. Then

   1. A.both Statement 1 and Statement 2 are true
   2. B.Statement 1 is false and Statement 2 is true
   3. C.both Statement 1 and Statement 2 are false
   4. D.Statement 1 is true and Statement 2 is false
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   Answer: A

2. Q292021 · 1 Sept 2021 · Shift 2· Numerical answer

   A man starts walking from the point $\mathrm{P}(-3,4)$, touches the x-axis at R, and then turns to reach at the point $\mathrm{Q}(0,2)$. The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $50\left((\mathrm{PR}{)}^2+(\mathrm{RQ}{)}^2\right)$ is equal to ___________.

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   Answer: 1250

3. Q302021 · 20 Jul 2021 · Shift 1· Numerical answer

   Let $y=mx+c$, $m>0$ be the focal chord of $y^2=-64x$, which is tangent to $(x+10{)}^2+y^2=4$. Then, the value of $4\sqrt{2}(m+c)$ is equal to ________.

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   Answer: 34

4. Q272021 · 18 Mar 2021 · Shift 2· Numerical answer

   Let P(x) be a real polynomial of degree 3 which vanishes at $x=-3$. Let P(x) have local minima at $x=1$, local maxima at $x=-1$ and ${\int }_{-1}^{1}P(x)dx=18$, then the sum of all the coefficients of the polynomial P(x) is equal to ____.

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   Answer: 8

5. Q122021 · 26 Feb 2021 · Shift 1

   The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point :

   1. A.$\left(3,\frac{21}{2}\right)$
   2. B.$(0,0)$
   3. C.$(2,9)$
   4. D.$(2,2)$
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   Answer: D

6. Q172021 · 26 Feb 2021 · Shift 2

   Let slope of the tangent line to a curve at any point $P(x,y)$ be given by $\frac{x{y}^2+y}{x}$. If the curve intersects the line $x+2y=4$ at $x=-2$, then the value of y, for which the point $(3,y)$ lies on the curve, is :

   1. A.$-\frac{18}{19}$
   2. B.$-\frac{4}{3}$
   3. C.$\frac{18}{35}$
   4. D.$-\frac{18}{11}$
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   Answer: A

7. Q102021 · 25 Feb 2021 · Shift 2

   The minimum value of $f(x)=a^{a^x}+a^{1-a^x}$, where $a,x\in \mathbb{R}$ and $a>0$, is equal to :

   1. A.$a+\frac{1}{a}$
   2. B.$2a$
   3. C.$a+1$
   4. D.$2\sqrt{a}$
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   Answer: D