Differentiation & Applications · JEE Main previous year questions

2020 9 questions

1. Q72020 · 6 Sept 2020 · Shift 1

   Let $m$ and $M$ be respectively the minimum and maximum values of $$\left|\begin{array}{ccc}{\cos }^{2}x& 1+{\sin }^{2}x& \sin 2x\\ 1+{\cos }^{2}x& {\sin }^{2}x& \sin 2x\\ {\cos }^{2}x& {\sin }^{2}x& 1+\sin 2x\end{array}\right|$$ Then the ordered pair $(m,M)$ is equal to :

   1. A.$(-3,-1)$
   2. B.$(-4,-1)$
   3. C.$(1,3)$
   4. D.$(-3,3)$
   Show answer

   Answer: A

2. Q242020 · 6 Sept 2020 · Shift 1· Numerical answer

   Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If $\mathrm{AD}=8\text{ m}$, $\mathrm{BC}=11\text{ m}$ and $\mathrm{AB}=10\text{ m}$; then the distance (in meters) of a point M on AB from the point A such that ${\mathrm{MD}}^2+{\mathrm{MC}}^2$ is minimum is ______ .

   Show answer

   Answer: 5

3. Q192020 · 6 Sept 2020 · Shift 2

   If the tangent to the curve, $y=f(x)=x{\log }_{e}x$, $(x>0)$ at a point $(c,f(c))$ is parallel to the line-segment joining the points $(1,0)$ and $(e,e)$, then c is equal to :

   1. A.$\frac{e-1}{e}$
   2. B.$e^{\left(\frac{1}{1-e}\right)}$
   3. C.$e^{\left(\frac{1}{e-1}\right)}$
   4. D.$\frac{1}{e-1}$
   Show answer

   Answer: C

4. Q52020 · 3 Sept 2020 · Shift 2

   Suppose f(x) is a polynomial of degree four, having critical points at $-1,0,1$. If $\mathrm{T}=\{x\in \mathbb{R}\mid f(x)=f(0)\}$, then the sum of squares of all the elements of T is :

   1. A.6
   2. B.2
   3. C.8
   4. D.4
   Show answer

   Answer: D

5. Q132020 · 3 Sept 2020 · Shift 2

   If the surface area of a cube is increasing at a rate of $3.6\mathrm{c}{\mathrm{m}}^2/sec$, retaining its shape; then the rate of change of its volume (in $\mathrm{c}{\mathrm{m}}^3/sec$), when the length of a side of the cube is 10 cm, is :

   1. A.9
   2. B.10
   3. C.18
   4. D.20
   Show answer

   Answer: A

6. Q142020 · 2 Sept 2020 · Shift 1

   If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :

   1. A.6
   2. B.12
   3. C.-12
   4. D.-24
   Show answer

   Answer: C

7. Q182020 · 2 Sept 2020 · Shift 1

   Let P(h, k) be a point on the curve $y=x^2+7x+2$, nearest to the line, $y=3x-3$. Then the equation of the normal to the curve at P is :

   1. A.$x-3y-11=0$
   2. B.$x-3y+22=0$
   3. C.$x+3y-62=0$
   4. D.$x+3y+26=0$
   Show answer

   Answer: D

8. Q222020 · 8 Jan 2020 · Shift 2· Numerical answer

   Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. Then ƒ(x) has a local minima at x = _______.

   Show answer

   Answer: 3

9. Q132020 · 7 Jan 2020 · Shift 2

   Let $f(x)$ be a polynomial of degree 5 such that $x=\pm 1$ are its critical points. If ${\lim }_{x\to 0}\left(2+\frac{f(x)}{x^3}\right)=4$, then which one of the following is not true?

   1. A.$f(1)-4f(-1)=4$.
   2. B.$x=1$ is a point of minima and $x=-1$ is a point of maxima of f.
   3. C.$x=1$ is a point of maxima and $x=-1$ is a point of minimum of f.
   4. D.f is an odd function.
   Show answer

   Answer: B