MCQOPTIONS
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MCQOPTIONS
This section includes 26 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
If \[y={{x}^{{{x}^{x......\infty }}}}\], then \[\frac{dy}{dx}=\] [UPSEAT 2004; DCE 2000] |
| A. | \[\frac{{{y}^{2}}}{x(1+y\log x)}\] |
| B. | \[\frac{{{y}^{2}}}{x(1-y\log x)}\] |
| C. | \[\frac{y}{x(1+y\log x)}\] |
| D. | \[\frac{y}{x(1-y\log x)}\] |
| Answer» C. \[\frac{y}{x(1+y\log x)}\] | |
| 2. |
If \[y={{(x\log x)}^{\log \,\log x}}\], then \[\frac{dy}{dx}=\] [Roorkee 1981] |
| A. | \[{{(x\log x)}^{\log \log x}}\left\{ \frac{1}{x\log x}(\log x+\log \log x)+(\log \,\,\log x)\text{ }\left( \frac{1}{x}+\frac{1}{x\log x} \right)\text{ } \right\}\] |
| B. | \[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\] |
| C. | \[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\] |
| D. | None of these |
| Answer» B. \[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\] | |
| 3. |
The function \[f(x)=\frac{\text{ln}(\pi +x)}{\text{ln}(e+x)}\] is [IIT 1995] |
| A. | Increasing on \[\left[ 0,\,\infty\right)\] |
| B. | Decreasing on \[\left[ 0,\,\infty\right)\] |
| C. | Decreasing on \[\left[ 0,\frac{\pi }{e} \right)\]and increasing on \[\left[ \frac{\pi }{e},\infty\right)\] |
| D. | Increasing on \[\left[ 0,\frac{\pi }{e} \right)\] and decreasing on \[\left[ \frac{\pi }{e},\infty\right)\] |
| Answer» C. Decreasing on \[\left[ 0,\frac{\pi }{e} \right)\]and increasing on \[\left[ \frac{\pi }{e},\infty\right)\] | |
| 4. |
If \[x=\sin t\] and \[y=\sin pt\], then the value of\[\left( 1-{{x}^{2}} \right)\frac{{{d}^{2}}y}{d{{x}^{2}}}-x\frac{dy}{dx}+{{p}^{2}}y\]is equal to [Pb. CET 2002] |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | \[\sqrt{2}\] |
| Answer» B. 1 | |
| 5. |
If \[{{I}_{n}}=\frac{{{d}^{n}}}{d{{x}^{n}}}({{x}^{n}}\log x),\]then \[{{I}_{n}}-n{{I}_{n-1}}=\] [EAMCET 2003] |
| A. | \[n\] |
| B. | \[n-1\] |
| C. | \[n!\] |
| D. | \[(n-1)!\] |
| Answer» E. | |
| 6. |
If a spherical balloon has a variable diameter \[3x+\frac{9}{2}\], then therate 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 | |
| 7. |
The rate of change of \[\sqrt{({{x}^{2}}+16)}\] with respect to \[\frac{x}{x-1}\] at \[x=3\] is [AMU 2001; MP PET 1987] |
| A. | 2 |
| B. | \[\frac{11}{5}\] |
| C. | \[-\frac{12}{5}\] |
| D. | \[-3\] |
| Answer» D. \[-3\] | |
| 8. |
If \[t=\frac{{{v}^{2}}}{2}\],then \[\left( -\frac{df}{dt} \right)\]is equal to, (where f is acceleration) [MP PET 1991] |
| A. | \[{{f}^{2}}\] |
| B. | \[{{f}^{3}}\] |
| C. | \[-{{f}^{3}}\] |
| D. | \[-{{f}^{2}}\] |
| Answer» C. \[-{{f}^{3}}\] | |
| 9. |
If \[x=\sec \theta -\cos \theta \]and \[y={{\sec }^{n}}\theta -{{\cos }^{n}}\theta \], then [IIT 1989] |
| A. | \[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}={{n}^{2}}({{y}^{2}}+4)\] |
| B. | \[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}={{x}^{2}}({{y}^{2}}+4)\] |
| C. | \[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}=({{y}^{2}}+4)\] |
| D. | None of these |
| Answer» B. \[({{x}^{2}}+4)\text{ }{{\left( \frac{dy}{dx} \right)}^{2}}={{x}^{2}}({{y}^{2}}+4)\] | |
| 10. |
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. | |
| 11. |
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. \[f(x)=\left\{ \begin{align} & \frac{\sin x}{x},\,\,\,x\ne 0 \\ & \,\,\,\,\,1\,\,\,,\,\,\,x=0 \\ \end{align} \right.\] | |
| 12. |
If the function \[f(x)=2{{x}^{3}}-9a{{x}^{2}}\] \[+12{{a}^{2}}x+1,\]where \[a>0\] attains its maximum and minimum at p and q respectively such that \[{{p}^{2}}=q\], then aequals [AIEEE 2003] |
| A. | 3 |
| B. | 1 |
| C. | 2 |
| D. | \[\frac{1}{2}\] |
| Answer» D. \[\frac{1}{2}\] | |
| 13. |
The maximum value of exp \[(2+\sqrt{3}\cos x+\sin x)\] is [AMU 1999] |
| A. | \[\exp (2)\] |
| B. | \[\exp (2-\sqrt{3})\] |
| C. | \[\exp (4)\] |
| D. | 1 |
| Answer» D. 1 | |
| 14. |
The point(s) on the curve \[{{y}^{3}}+3{{x}^{2}}=12y\] where the tangent is vertical (parallel to y-axis), is (are) [IIT Screening 2002] |
| A. | \[\left( \pm \frac{4}{\sqrt{3}},-2 \right)\] |
| B. | \[\left( \pm \frac{\sqrt{11}}{3},1 \right)\] |
| C. | \[(0,\,0)\] |
| D. | \[\left( \pm \frac{4}{\sqrt{3}},2 \right)\] |
| Answer» E. | |
| 15. |
The radius of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is [AMU 1999] |
| A. | \[\frac{2}{3}R\] |
| B. | \[\sqrt{\frac{2}{3}}R\] |
| C. | \[\frac{3}{4}R\] |
| D. | \[\sqrt{\frac{3}{4}}R\] |
| Answer» C. \[\frac{3}{4}R\] | |
| 16. |
A man of height 1.8 metre is moving away from a lamp post at the rate of 1.2 \[m/\sec .\] If the height of the lamp post be 4.5 metre, then the rate at which the shadow of the man is lengthening is [MP PET 1989] |
| A. | \[0.4\,\,m/\sec \] |
| B. | \[0.8\,\,m/\sec \] |
| C. | \[1.2\,\,m/\sec \] |
| D. | None of these |
| Answer» C. \[1.2\,\,m/\sec \] | |
| 17. |
The volume of a spherical balloon is increasing at the rate of 40 cubic centrimetre per minute. The rate of change of the surface of the balloon at the instant when its radius is 8 centimetre, is [Roorkee 1983] |
| A. | \[\frac{5}{2}\] sq cm/min |
| B. | 5 sq cm/min |
| C. | 10 sq cm/min |
| D. | 20 sq cm/min |
| Answer» D. 20 sq cm/min | |
| 18. |
Let \[f:(0,\,+\infty )\to R\] and \[F(x)=\int_{0}^{x}{f(t)\,dt}\]. If \[F({{x}^{2}})={{x}^{2}}(1+x)\], then \[f(4)\] equals [IIT Screening 2001] |
| A. | \[\frac{5}{4}\] |
| B. | 7 |
| C. | 4 |
| D. | 2 |
| Answer» D. 2 | |
| 19. |
A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 cm/sec. The height of the upper end while it is descending at the rate of 4 cm/sec is[Kerala(Engg.) 2005] |
| A. | \[4\sqrt{3}\]m |
| B. | \[5\sqrt{3}\]m |
| C. | \[5\sqrt{2}\,m\] |
| D. | 8 m |
| E. | 6 m |
| Answer» C. \[5\sqrt{2}\,m\] | |
| 20. |
A spherical iron ball 10 cmin radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is [AIEEE 2005] |
| A. | \[\frac{1}{54\pi }\]cm/min |
| B. | \[\frac{5}{6\pi }\] cm/min |
| C. | \[\frac{1}{36\pi }\] cm/min |
| D. | \[\frac{1}{18\pi }\] cm/min |
| Answer» E. | |
| 21. |
If the distance ?s? metre traversed by a particle in t seconds is given by \[s={{t}^{3}}-3{{t}^{2}}\], then the velocity of the particle when the acceleration is zero, in metre/sec is [Karnataka CET 2004] |
| A. | 3 |
| B. | ? 2 |
| C. | ? 3 |
| D. | 2 |
| Answer» D. 2 | |
| 22. |
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when the side is 10 cm is [Kerala (Engg.) 2002] |
| A. | \[\sqrt{3}\] sq. unit/sec |
| B. | 10 sq. unit/sec |
| C. | \[10\sqrt{3}\] sq. unit/sec |
| D. | \[\frac{10}{\sqrt{3}}\] sq. unit/sec |
| Answer» D. \[\frac{10}{\sqrt{3}}\] sq. unit/sec | |
| 23. |
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 | |
| 24. |
A man 2metre high walks at a uniform speed 5 metre/hour away from a lamp post 6 metre high. The rate at which the length of his shadow increases is |
| A. | 5 m/h |
| B. | \[\frac{5}{2}\]m/h |
| C. | \[\frac{5}{3}\]m/h |
| D. | \[\frac{5}{4}\]m/h |
| Answer» C. \[\frac{5}{3}\]m/h | |
| 25. |
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 | |
| 26. |
A particle moves in a straight line in such a way that its velocity at any point is given by \[{{v}^{2}}=2-3x\], where x is measured from a fixed point. The acceleration is [MP PET 1992] |
| A. | Uniform |
| B. | Zero |
| C. | Non-uniform |
| D. | Indeterminate |
| Answer» B. Zero | |