Differentiation & Applications · JEE Main previous year questions

2024 13 questions

1. Q32024 · 9 Apr 2024 · Shift 1

   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
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   Answer: B

2. Q62024 · 9 Apr 2024 · Shift 1

   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.$15\pi$
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   Answer: A

3. Q152024 · 9 Apr 2024 · Shift 1

   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
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   Answer: C

4. Q32024 · 8 Apr 2024 · Shift 1

   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.$2$
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   Answer: D

5. Q232024 · 5 Apr 2024 · Shift 2· Numerical answer

   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 _________.

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   Answer: 1600

6. Q272024 · 1 Feb 2024 · Shift 1· Numerical answer

   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 ____.

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   Answer: 36

7. Q252024 · 31 Jan 2024 · Shift 1· Numerical answer

   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 ____.

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   Answer: 27

8. Q212024 · 30 Jan 2024 · Shift 1· Numerical answer

   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 _________.

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   Answer: 10

9. Q12024 · 29 Jan 2024 · Shift 1

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

10. Q292024 · 29 Jan 2024 · Shift 1· Numerical answer

    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 _______.

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    Answer: 5

11. Q142024 · 27 Jan 2024 · Shift 1

    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.$\frac{25}{41}$
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    Answer: A

12. Q232024 · 27 Jan 2024 · Shift 1· Numerical answer

    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 ____.

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    Answer: 202

13. Q142024 · 27 Jan 2024 · Shift 2

    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.$20$
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    Answer: C