Differentiation & Applications · JEE Main previous year questions

2023 18 questions

1. Q252023 · 13 Apr 2023 · Shift 2· Numerical answer

   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 ___________

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   Answer: 10

2. Q112023 · 12 Apr 2023 · Shift 1

   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.$\frac{31}{4}$
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   Answer: C

3. Q12023 · 11 Apr 2023 · Shift 2

   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.$55$
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   Answer: C

4. Q242023 · 11 Apr 2023 · Shift 2· Numerical answer

   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 _________.

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   Answer: 16

5. Q92023 · 10 Apr 2023 · Shift 1

   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
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   Answer: C

6. Q192023 · 10 Apr 2023 · Shift 1

   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.$-2\sqrt{3}$
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   Answer: C

7. Q292023 · 10 Apr 2023 · Shift 1· Numerical answer

   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 _________.

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   Answer: 32

8. Q42023 · 10 Apr 2023 · Shift 2

   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.$\frac{3\pi }{2}$
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   Answer: B

9. Q262023 · 6 Apr 2023 · Shift 2· Numerical answer

   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 ________.

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   Answer: 18

10. Q292023 · 6 Apr 2023 · Shift 2· Numerical answer

    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 ____.

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    Answer: 5

11. Q102023 · 1 Feb 2023 · Shift 1

    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.${\alpha }^2+{\beta }^2=\frac{9}{2}$
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    Answer: B

12. Q292023 · 1 Feb 2023 · Shift 1· Numerical answer

    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 ________.

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    Answer: 14

13. Q42023 · 29 Jan 2023 · Shift 1

    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.$\frac{11}{8}$
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    Answer: C

14. Q302023 · 29 Jan 2023 · Shift 2· Numerical answer

    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 ____.

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    Answer: 4

15. Q62023 · 25 Jan 2023 · Shift 1

    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.$\frac{15}{44}$
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    Answer: A

16. Q202023 · 25 Jan 2023 · Shift 1

    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.$5\sqrt{3}$
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    Answer: B

17. Q292023 · 24 Jan 2023 · Shift 1· Numerical answer

    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 ____

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    Answer: 7

18. Q132023 · 24 Jan 2023 · Shift 2

    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.$3f(1)+f(2)=f(3)$
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    Answer: A