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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.
| 8601. |
If \[\frac{{{(1-3x)}^{1/2}}+{{(1-x)}^{5/3}}}{\sqrt{4-x}}\]is approximately equal to \[a+bx\]for small values of x, then \[(a,b)\]= |
| A. | \[\left( 1,\frac{35}{24} \right)\] |
| B. | \[\left( 1,-\frac{35}{24} \right)\] |
| C. | \[\left( 2,\frac{35}{12} \right)\] |
| D. | \[\left( 2,-\frac{35}{12} \right)\] |
| Answer» C. \[\left( 2,\frac{35}{12} \right)\] | |
| 8602. |
In the expansion of \[{{\left( 2{{x}^{2}}-\frac{1}{x} \right)}^{12}}\], the term independent of x is [MP PET 2001] |
| A. | 10th |
| B. | 9th |
| C. | 8th |
| D. | 7th |
| Answer» C. 8th | |
| 8603. |
The ratio of the coefficient of terms \[{{x}^{n-r}}{{a}^{r}}\]and \[{{x}^{r}}{{a}^{n-r}}\] in the binomial expansion of \[{{(x+a)}^{n}}\]will be |
| A. | \[x:a\] |
| B. | \[n:r\] |
| C. | \[x:n\] |
| D. | None of these |
| Answer» E. | |
| 8604. |
The angle between the curves \[y=\sin x\] and \[y=\cos x\] is [EAMCET 2003] |
| A. | \[{{\tan }^{-1}}(2\sqrt{2})\] |
| B. | \[{{\tan }^{-1}}(3\sqrt{2})\] |
| C. | \[{{\tan }^{-1}}(3\sqrt{3})\] |
| D. | \[{{\tan }^{-1}}(5\sqrt{2})\] |
| Answer» B. \[{{\tan }^{-1}}(3\sqrt{2})\] | |
| 8605. |
The angle of intersection of the curves \[y={{x}^{2}}\] and \[x={{y}^{2}}\] at (1, 1) is [Roorkee 2000; Karnataka CET 2001] |
| A. | \[{{\tan }^{-1}}\left( \frac{4}{3} \right)\] |
| B. | \[{{\tan }^{-1}}(1)\] |
| C. | \[{{90}^{o}}\] |
| D. | \[{{\tan }^{-1}}\left( \frac{3}{4} \right)\] |
| Answer» E. | |
| 8606. |
Let \[f:[a,b]\to R\] be a function such that for \[c\in (a,b),f'(c)=f'(c)=f'''(c)={{f}^{iv}}(c)={{f}^{v}}(c)=0\]. Then |
| A. | f has a local extremum at x = c |
| B. | f has neither local maximum nor minimum at x = c |
| C. | f is necessarily a constant function |
| D. | it is difficult to say whether or (b) |
| Answer» E. | |
| 8607. |
What is the product of two parts of 20, such that the product of one part and the cube of the other is maximum? |
| A. | 75 |
| B. | 91 |
| C. | 84 |
| D. | 96 |
| Answer» B. 91 | |
| 8608. |
If \[A>0,\text{ }B>0\] and \[A+B=\pi /3,\] then the maximum value of tan A tan B is |
| A. | \[\frac{1}{\sqrt{3}}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[3\] |
| D. | \[\sqrt{3}\] |
| Answer» C. \[3\] | |
| 8609. |
The profit function, in rupees, of a firm selling x items \[(x\ge 0)\] per week is given by\[P(x)=-3500+(400-x)x\]. How many items should the firm sell so that the firm has maximum profit? |
| A. | 400 |
| B. | 300 |
| C. | 200 |
| D. | 100 |
| Answer» D. 100 | |
| 8610. |
If\[f(x)=x\ell nx\], then \[f(x)\] attains minimum value at which one of the following points? |
| A. | \[x={{e}^{-2}}\] |
| B. | \[x=e\] |
| C. | \[x={{e}^{-1}}\] |
| D. | \[x=2{{e}^{-1}}\] |
| Answer» D. \[x=2{{e}^{-1}}\] | |
| 8611. |
A wire 34 cm long is to be bent in the form of a quadrilateral of which each angle is \[90{}^\circ \]. What is the maximum area which can be enclosed inside the quadrilateral? |
| A. | \[68\,\,c{{m}^{2}}\] |
| B. | \[70\,\,c{{m}^{2}}\] |
| C. | \[71.25\,\,c{{m}^{2}}\] |
| D. | \[72.25\,\,c{{m}^{2}}\] |
| Answer» E. | |
| 8612. |
If \[f(x)=k{{x}^{3}}-9{{x}^{2}}+9x+3\] is monotonically increasing in every interval, then which one of the following is correct? |
| A. | \[k<3\] |
| B. | \[k\le 3\] |
| C. | \[k>3\] |
| D. | \[k\ge 3\] |
| Answer» D. \[k\ge 3\] | |
| 8613. |
How many tangents are parallel to x-axis for the curve\[y={{x}^{2}}-4x+3\]? |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | No tangent is parallel to x-axis |
| Answer» B. 2 | |
| 8614. |
The function \[f(x)=\frac{{{x}^{2}}}{{{e}^{x}}}\] monotonically increasing if |
| A. | x < 0 only |
| B. | x > 2 only |
| C. | 0 < x < 2 |
| D. | \[x\in (-\infty ,0)\cup (2,\infty )\] |
| Answer» D. \[x\in (-\infty ,0)\cup (2,\infty )\] | |
| 8615. |
The cost of running a bus from A to B, is Rs. \[\left( av+\frac{b}{v} \right),\] where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/ h) of the bus is: |
| A. | 45 |
| B. | 50 |
| C. | 60 |
| D. | 40 |
| Answer» D. 40 | |
| 8616. |
If the curve \[y=a{{x}^{2}}-6x+b\] passes through (0, 2) and has its tangent parallel to the x-axis at \[x=\frac{3}{2},\] then |
| A. | a = b = 0 |
| B. | a = b = 1 |
| C. | a = b = 2 |
| D. | a = b = -1 |
| Answer» D. a = b = -1 | |
| 8617. |
If the sub-normal at any point on \[y={{a}^{1-n}}{{x}^{n}}\] is of constant length, then the value of n is |
| A. | \[\frac{1}{4}\] |
| B. | 1 |
| C. | \[\frac{1}{3}\] |
| D. | \[\frac{1}{2}\] |
| Answer» E. | |
| 8618. |
The equation of the normal to the curve \[y=\left| {{x}^{2}}-\left| x \right| \right|\] at\[x=-2\]. |
| A. | \[3y=x+8\] |
| B. | \[x=3y+4\] |
| C. | \[y=2x+8\] |
| D. | \[y=3x\] |
| Answer» B. \[x=3y+4\] | |
| 8619. |
The approximate value of \[{{(0.007)}^{1/3}}\] |
| A. | \[\frac{23}{120}\] |
| B. | \[\frac{27}{120}\] |
| C. | \[\frac{19}{120}\] |
| D. | \[\frac{17}{120}\] |
| Answer» B. \[\frac{27}{120}\] | |
| 8620. |
The equation of one of the tangents to the curve \[y=\cos (x+y),-2\pi \le x\le 2\pi \] that is parallel to the line \[x+2y=0,\] is |
| A. | \[x+2y=1\] |
| B. | \[x+2y=\pi /2\] |
| C. | \[x+2y=\pi /4\] |
| D. | None of these |
| Answer» C. \[x+2y=\pi /4\] | |
| 8621. |
Let f and g be functions from the interval \[[0,\infty )\] to the interval\[[0,\infty )\], f being an increasing and g being a decreasing function. If \[f\{g(0)\}=0\] then |
| A. | \[f\{g(x)\}\ge f\{g(0)\}\] |
| B. | \[g\{f(x)\}\le g\{f(0)\}\] |
| C. | \[f\{g(2)\}=7\] |
| D. | None of these |
| Answer» C. \[f\{g(2)\}=7\] | |
| 8622. |
The largest area of a trapezium inscribed in a semi-circle of radius R, if the lower base is on the diameter, is |
| A. | \[\frac{3\sqrt{3}}{4}{{R}^{2}}\] |
| B. | \[\frac{\sqrt{3}}{2}{{R}^{2}}\] |
| C. | \[\frac{3\sqrt{3}}{8}{{R}^{2}}\] |
| D. | \[{{R}^{2}}\] |
| Answer» B. \[\frac{\sqrt{3}}{2}{{R}^{2}}\] | |
| 8623. |
If water is poured into an inverted hollow cone whose semi-vertical angle is \[30{}^\circ \]. Its depth (measured along the axis) increases at the rate of 1 cm/s. The rate at which the volume of water increases when the depth is 24 cm is |
| A. | \[162\,\,c{{m}^{3}}/s\] |
| B. | \[172\,\,c{{m}^{3}}/s\] |
| C. | \[182\,\,c{{m}^{3}}/s\] |
| D. | \[192\,\,c{{m}^{3}}/s\] |
| Answer» E. | |
| 8624. |
What is the interval in which the function \[f(x)=\sqrt{9-{{x}^{2}}}\] is increasing? \[(f(x)>0)\] |
| A. | \[0<x<3\] |
| B. | \[-3<x<0\] |
| C. | \[0<x<9\] |
| D. | \[-3<x<3\] |
| Answer» C. \[0<x<9\] | |
| 8625. |
Let \[P(x)={{a}_{0}}+{{a}_{1}}{{x}^{2}}+{{a}_{2}}{{x}^{4}}+.....+{{a}_{n}}{{x}^{2n}}\] be a polynomial in a real variable x with\[0<{{a}_{0}}<{{a}_{1}}<{{a}_{2}}<....<{{a}_{n}}\]. The function P(x) has |
| A. | Neither a maximum nor a minimum |
| B. | Only one maximum |
| C. | Only one minimum |
| D. | Only one maximum and only one minimum |
| Answer» D. Only one maximum and only one minimum | |
| 8626. |
Consider the following statements: 1. \[f(x)=\] ln x is an increasing function on \[\left( 0,\infty \right).\] 2. \[f(x)={{e}^{x}}-x(ln\,\,x)\] is an increasing function on \[\left( 1,\,\infty \right)\]. Which of the above statements is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» D. Neither 1 nor 2 | |
| 8627. |
The function \[f(x)=2\,\,\log (x-2)-{{x}^{2}}+4x+1\]increases on the interval |
| A. | (1, 2) |
| B. | (2, 3) |
| C. | (1/2, 3) |
| D. | (2, 4) |
| Answer» C. (1/2, 3) | |
| 8628. |
The function \[f(x)=1+x(\sin x)[\cos x],0 |
| A. | Is continuous on \[\left( 0,\frac{\pi }{2} \right)\] |
| B. | Is strictly increasing in \[\left( 0,\frac{\pi }{2} \right)\] |
| C. | Is strictly decreasing in \[\left( 0,\frac{\pi }{2} \right)\] |
| D. | Has global maximum value 2 |
| Answer» B. Is strictly increasing in \[\left( 0,\frac{\pi }{2} \right)\] | |
| 8629. |
The number of solutions of the equation \[3\tan x+{{x}^{3}}=2\,\,in\left( 0,\frac{\pi }{4} \right).\] is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | Infinite |
| Answer» B. 2 | |
| 8630. |
What is the minimum value of \[px+qy\] \[(p>0,q>0)\] when\[xy={{r}^{2}}\]? |
| A. | \[2r\sqrt{pq}\] |
| B. | \[2pq\sqrt{r}\] |
| C. | \[-2r\sqrt{pq}\] |
| D. | \[2rpq\] |
| Answer» B. \[2pq\sqrt{r}\] | |
| 8631. |
The curve \[y=x{{e}^{x}}\] has minimum value equal to |
| A. | \[-\frac{1}{e}\] |
| B. | \[\frac{1}{e}\] |
| C. | \[-e\] |
| D. | e |
| Answer» B. \[\frac{1}{e}\] | |
| 8632. |
If from mean value theorem, \[f'({{x}_{1}})=\frac{f(b)-f(a)}{b-a}\], then [MP PET 1999] |
| A. | \[a<{{x}_{1}}\le b\] |
| B. | \[a\le {{x}_{1}}<b\] |
| C. | \[a<{{x}_{1}}<b\] |
| D. | \[a\le {{x}_{1}}\le b\] |
| Answer» D. \[a\le {{x}_{1}}\le b\] | |
| 8633. |
If \[f(x)=\cos x,0\le x\le \frac{\pi }{2}\], then the real number ?c? of the mean value theorem is [MP PET 1994] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[{{\sin }^{-1}}\left( \frac{2}{\pi } \right)\] |
| D. | \[{{\cos }^{-1}}\left( \frac{2}{\pi } \right)\] |
| Answer» D. \[{{\cos }^{-1}}\left( \frac{2}{\pi } \right)\] | |
| 8634. |
In the Mean-Value theorem \[\frac{f(b)-f(a)}{b-a}=f'(c),\] if \[a=0,b=\frac{1}{2}\] and \[f(x)=x(x-1)(x-2),\]the value of c is [MP PET 2003] |
| A. | \[1-\frac{\sqrt{15}}{6}\] |
| B. | \[1+\sqrt{15}\] |
| C. | \[1-\frac{\sqrt{21}}{6}\] |
| D. | \[1+\sqrt{21}\] |
| Answer» D. \[1+\sqrt{21}\] | |
| 8635. |
The function \[f(x)={{x}^{3}}-6{{x}^{2}}+ax+b\] satisfy the conditions of Rolle's theorem in [1, 3]. The values of a and b are |
| A. | 11, ? 6 |
| B. | ? 6, 11 |
| C. | ?11, 6 |
| D. | 6, ?11 |
| Answer» B. ? 6, 11 | |
| 8636. |
If the length of sub-normal is equal to the length of sub-tangent at my point (3, 4) on the curve y=f(x) and the tangent at (3, 4) to y=f(x) meets the coordinate axes at A and B, then the maximum area of the triangle OAB, where O is origin, is |
| A. | 45/2 |
| B. | 49/2 |
| C. | 44252 |
| D. | 81/2 |
| Answer» C. 44252 | |
| 8637. |
A point on the parabola \[{{y}^{2}}=18x\] at which the ordinate increases at twice the rate of the abscissa is |
| A. | (2, 4) |
| B. | (2, -4) |
| C. | \[\left( \frac{-9}{8},\frac{9}{2} \right)\] |
| D. | \[\left( \frac{9}{8},\frac{9}{2} \right)\] |
| Answer» E. | |
| 8638. |
If the function \[f(x)=a{{x}^{3}}+b{{x}^{2}}+11x-6\] satisfies condition of Rolle's theorem in [1, 3] for \[x=2+\frac{1}{\sqrt{3}}\], then values of a and b , respectively, are |
| A. | \[-\,3,\text{ }2\] |
| B. | \[2,\,\,-4\] |
| C. | 1, 6 |
| D. | none of these |
| Answer» E. | |
| 8639. |
The point \[(0,\,5)\]is closest to the curve \[{{x}^{2}}=2y\] at [MNR 1983] |
| A. | \[(2\sqrt{2},0)\] |
| B. | (0, 0) |
| C. | \[(2,\,2)\] |
| D. | None of these |
| Answer» E. | |
| 8640. |
If \[f(x)=x+\frac{1}{x},\] x > 0, then its greatest value is [RPET 2002] |
| A. | ? 2 |
| B. | 0 |
| C. | 3 |
| D. | None of these |
| Answer» E. | |
| 8641. |
If \[{{a}^{2}}{{x}^{4}}+{{b}^{2}}{{y}^{4}}={{c}^{6}},\] then maximum value of xy is [RPET 2001] |
| A. | \[\frac{{{c}^{2}}}{\sqrt{ab}}\] |
| B. | \[\frac{{{c}^{3}}}{ab}\] |
| C. | \[\frac{{{c}^{3}}}{\sqrt{2ab}}\] |
| D. | \[\frac{{{c}^{3}}}{2ab}\] |
| Answer» D. \[\frac{{{c}^{3}}}{2ab}\] | |
| 8642. |
The function \[f(x)=x+\sin x\] has [AMU 2000] |
| A. | A minimum but no maximum |
| B. | A maximum but no minimum |
| C. | Neither maximum nor minimum |
| D. | Both maximum and minimum |
| Answer» D. Both maximum and minimum | |
| 8643. |
The denominator of a fraction number is greater than 16 of the square of numerator, then least value of the number is [RPET 2000] |
| A. | \[-1/4\] |
| B. | \[-1/8\] |
| C. | \[1/12\] |
| D. | \[1/16\] |
| Answer» C. \[1/12\] | |
| 8644. |
The maximum value of the function \[{{x}^{3}}+{{x}^{2}}+x-4\] is |
| A. | 127 |
| B. | 4 |
| C. | Does not have a maximum value |
| D. | None of these |
| Answer» D. None of these | |
| 8645. |
The maximum and minimum values of \[{{x}^{3}}-18{{x}^{2}}+96x\] in interval (0, 9) are [RPET 1999] |
| A. | 160, 0 |
| B. | 60, 0 |
| C. | 160, 128 |
| D. | 120, 28 |
| Answer» D. 120, 28 | |
| 8646. |
The number of values of x where the function \[f(x)=\cos x+\cos (\sqrt{2}x)\] attains its maximum is [IIT 1998; DCE 2001, 05] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | Infinite |
| Answer» C. 2 | |
| 8647. |
One maximum point of \[{{\sin }^{p}}x{{\cos }^{q}}x\]is [RPET 1997; AMU 2000] |
| A. | \[x={{\tan }^{-1}}\sqrt{(p/q)}\] |
| B. | \[x={{\tan }^{-1}}\sqrt{(q/p)}\] |
| C. | \[x={{\tan }^{-1}}(p/q)\] |
| D. | \[x={{\tan }^{-1}}(q/p)\] |
| Answer» B. \[x={{\tan }^{-1}}\sqrt{(q/p)}\] | |
| 8648. |
A minimum value of \[\int_{0}^{x}{t{{e}^{-{{t}^{2}}}}}\]dt is [EAMCET 2003] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 0 |
| Answer» E. | |
| 8649. |
If from a wire of length 36 metre a rectangle of greatest area is made, then its two adjacent sides in metre are [MP PET 1998] |
| A. | 6, 12 |
| B. | 9, 9 |
| C. | 10, 8 |
| D. | 13, 5 |
| Answer» C. 10, 8 | |
| 8650. |
\[{{x}^{x}}\] has a stationary point at [Karnataka CET 1993] |
| A. | \[x=e\] |
| B. | \[x=\frac{1}{e}\] |
| C. | \[x=1\] |
| D. | \[x=\sqrt{e}\] |
| Answer» C. \[x=1\] | |