MCQOPTIONS
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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 8651. |
The function \[\sin x(1+\cos x)\]at \[x=\frac{\pi }{3}\], is |
| A. | Maximum |
| B. | Minimum |
| C. | Neither maximum nor minimum |
| D. | None of these |
| Answer» B. Minimum | |
| 8652. |
The value of the function \[(x-1){{(x-2)}^{2}}\] at its maxima is |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | \[\frac{4}{27}\] |
| Answer» E. | |
| 8653. |
Let \[f(x)=\left\{ \begin{align} & {{x}^{\alpha }}\ln x,x>0 \\ & 0,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right\}\], Rolle?s theorem is applicable to f for \[x\in [0,1]\], if \[\alpha =\] [IIT Screening 2004] |
| A. | -2 |
| B. | -1 |
| C. | 0 |
| D. | \[\frac{1}{2}\] |
| Answer» E. | |
| 8654. |
If the function \[f(x)={{x}^{3}}-6a{{x}^{2}}+5x\]satisfies the conditions of Lagrange's mean value theorem for the interval [1, 2] and the tangent to the curve \[y=f(x)\]at \[x=\frac{7}{4}\]is parallel to the chord that joins the points of intersection of the curve with the ordinates \[x=1\] and \[x=2\]. Then the value of \[a\]is [MP PET 1998] |
| A. | \[\frac{35}{16}\] |
| B. | \[\frac{35}{48}\] |
| C. | \[\frac{7}{16}\] |
| D. | \[\frac{5}{16}\] |
| Answer» C. \[\frac{7}{16}\] | |
| 8655. |
In [0, 1] Lagrange's mean value theorem is NOT applicable to [IIT Screening 2003] |
| A. | \[f(x)=\left\{ \begin{align} & \frac{1}{2}-x,\,\,\,\,\,\,\,x<\frac{1}{2} \\ & {{\left( \frac{1}{2}-x \right)}^{2}},\,\,\,x\ge \frac{1}{2} \\ \end{align} \right.\] |
| B. | \[f(x)=\left\{ \begin{align} & \frac{\sin x}{x},\,\,\,x\ne 0 \\ & \,\,\,\,\,1\,\,\,,\,\,\,x=0 \\ \end{align} \right.\] |
| C. | \[f(x)=x|x|\] |
| D. | \[f(x)=|x|\] |
| Answer» B. \[f(x)=\left\{ \begin{align} & \frac{\sin x}{x},\,\,\,x\ne 0 \\ & \,\,\,\,\,1\,\,\,,\,\,\,x=0 \\ \end{align} \right.\] | |
| 8656. |
If the normal to the curve \[y=f(x)\] at the point \[(3,\,4)\] makes an angle \[\frac{3\pi }{4}\]with the positive x-axis then \[f'(3)\] is equal to [IIT Screening 2000; DCE 2001] |
| A. | \[-1\] |
| B. | \[-\frac{3}{4}\] |
| C. | \[\frac{4}{3}\] |
| D. | \[1\] |
| Answer» E. | |
| 8657. |
At what points of the curve \[y=\frac{2}{3}{{x}^{3}}+\frac{1}{2}{{x}^{2}},\]tangent makes the equal angle with axis [UPSEAT 1999] |
| A. | \[\left( \frac{1}{2},\,\frac{5}{24} \right)\] and \[\left( -1,\,-\frac{1}{6} \right)\] |
| B. | \[\left( \frac{1}{2},\,\frac{4}{9} \right)\] and \[\left( -1,\,0 \right)\] |
| C. | \[\left( \frac{1}{3},\,\frac{1}{7} \right)\] and \[\left( -3,\,\frac{1}{2} \right)\] |
| D. | \[\left( \frac{1}{3},\,\frac{4}{47} \right)\] and \[\left( -1,\,-\frac{1}{3} \right)\] |
| Answer» B. \[\left( \frac{1}{2},\,\frac{4}{9} \right)\] and \[\left( -1,\,0 \right)\] | |
| 8658. |
If a spherical balloon has a variable diameter \[3x+\frac{9}{2}\], then the rate of change of its volume with respect to x is |
| A. | \[27\pi {{(2x+3)}^{2}}\] |
| B. | \[\frac{27\pi }{16}{{(2x+3)}^{2}}\] |
| C. | \[\frac{27\pi }{8}{{(2x+3)}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8659. |
The values of ?a? for which the function \[(a+2){{x}^{3}}-3a{{x}^{2}}+9ax-1\] decreases monotonically throughout for all real x, are [Kurukshetra CEE 2002] |
| A. | \[a<-2\] |
| B. | \[a>-2\] |
| C. | \[-3<a<0\] |
| D. | \[-\infty <a\le -3\] |
| Answer» E. | |
| 8660. |
The derivative of \[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\]with respect to \[{{\tan }^{-1}}\left( \frac{2x\sqrt{1-{{x}^{2}}}}{1-2{{x}^{2}}} \right)\]at \[x=0\], is |
| A. | \[\frac{1}{8}\] |
| B. | \[\frac{1}{4}\] |
| C. | \[\frac{1}{2}\] |
| D. | 1 |
| Answer» C. \[\frac{1}{2}\] | |
| 8661. |
If \[y={{\sec }^{-1}}\frac{2x}{1+{{x}^{2}}}+{{\sin }^{-1}}\frac{x-1}{x+1}\],then \[\frac{dy}{dx}\]is equal to [Pb. CET 2000] |
| A. | 1 |
| B. | \[\frac{x-1}{x+1}\] |
| C. | Does not exist |
| D. | None of these |
| Answer» D. None of these | |
| 8662. |
If \[\sqrt{(1-{{x}^{6}})}+\sqrt{(1-{{y}^{6}})}={{a}^{3}}({{x}^{3}}-{{y}^{3}})\], then \[\frac{dy}{dx}=\] [Roorkee 1994] |
| A. | \[\frac{{{x}^{2}}}{{{y}^{2}}}\sqrt{\frac{1-{{x}^{6}}}{1-{{y}^{6}}}}\] |
| B. | \[\frac{{{y}^{2}}}{{{x}^{2}}}\sqrt{\frac{1-{{y}^{6}}}{1-{{x}^{6}}}}\] |
| C. | \[\frac{{{x}^{2}}}{{{y}^{2}}}\sqrt{\frac{1-{{y}^{6}}}{1-{{x}^{6}}}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 8663. |
If the path of a moving point is the curve \[x=at\], \[y=b\sin at\], then its acceleration at any instant [SCRA 1996] |
| A. | Is constant |
| B. | Varies as the distance from the axis of x |
| C. | Varies as the distance from the axis of y |
| D. | Varies as the distance of the point from the origin |
| Answer» D. Varies as the distance of the point from the origin | |
| 8664. |
A ball thrown vertically upwards falls back on the ground after 6 second. Assuming that the equation of motion is of the form \[s=ut-4.9{{t}^{2}}\], where s is in metre and t is in second, find the velocity at \[t=0\] |
| A. | \[0\,m/s\] |
| B. | 1 m/s |
| C. | 29.4 m/s |
| D. | None of these |
| Answer» D. None of these | |
| 8665. |
The function \[{{x}^{x}}\] is increasing, when [MP PET 2003] |
| A. | \[x>\frac{1}{e}\] |
| B. | \[x<\frac{1}{e}\] |
| C. | \[x<0\] |
| D. | For all real x |
| Answer» B. \[x<\frac{1}{e}\] | |
| 8666. |
The function \[f(x)=1-{{e}^{-{{x}^{2}}/2}}\] is [AMU 1999] |
| A. | Decreasing for all x |
| B. | Increasing for all x |
| C. | Decreasing for \[x<0\] and increasing for \[x>0\] |
| D. | Increasing for \[x<0\] and decreasing for \[x>0\] |
| Answer» D. Increasing for \[x<0\] and decreasing for \[x>0\] | |