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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.
| 1501. |
If \[f(x)=a+bx+c{{x}^{2}}\], where \[c>0\] and \[{{b}^{2}}-4ac |
| A. | \[\frac{1}{3}\{4f(1)+f(2)\}\] |
| B. | \[\frac{1}{2}\{f(0)+4f(1)+f(2)\}\] |
| C. | \[\frac{1}{2}\{f(0)+4f(1)\}\] |
| D. | \[\frac{1}{3}\{f(0)+4f(1)+f(2)\}\] |
| Answer» E. | |
| 1502. |
The value of a (a > 0) for which the area bounded by the curves \[y=\frac{x}{6}+\frac{1}{{{x}^{2}}},y=0,x=a\] and \[x=2a\]has the least value is |
| A. | 2 |
| B. | (b \[\sqrt{2}\] |
| C. | \[{{2}^{1/3}}\] |
| D. | 1 |
| Answer» E. | |
| 1503. |
The value of a (a > 0) for which the area bounded by the curves \[y=\frac{x}{6}+\frac{1}{{{x}^{2}}},y=0,x=a\] and \[x=2a\] has the least value is |
| A. | 2 |
| B. | \[\sqrt{2}\] |
| C. | \[{{2}^{1/3}}\] |
| D. | 1 |
| Answer» E. | |
| 1504. |
The area enclosed between the curves \[y={{\log }_{e}}(x+e),x={{\log }_{e}}\left( \frac{1}{y} \right)\] and the x-axis is |
| A. | 2 sq. units |
| B. | 1 sq. units |
| C. | 4 sq. units |
| D. | None of these |
| Answer» B. 1 sq. units | |
| 1505. |
The area bounded by the curve \[{{y}^{2}}(2a-x)={{x}^{3}}\]and the line \[x=2a\] is |
| A. | \[3\pi {{a}^{2}}\] sq. unit |
| B. | \[\frac{3\pi {{a}^{2}}}{2}\] sq. unit |
| C. | \[\frac{3\pi {{a}^{2}}}{4}\] sq. unit |
| D. | \[\frac{6\pi {{a}^{2}}}{5}\] sq. unit |
| Answer» C. \[\frac{3\pi {{a}^{2}}}{4}\] sq. unit | |
| 1506. |
The area of the region (in sq. units), in the first quadrant bounded by the parabola \[y=9{{x}^{2}}\] and the lines \[x=0,\,\,y=1\] and \[y=4\], is: |
| A. | 44446 |
| B. | 44269 |
| C. | 44262 |
| D. | 44453 |
| Answer» E. | |
| 1507. |
If the area enclosed by \[{{y}^{2}}=4\,ax\text{ }is\text{ }\frac{1}{3}sq.\] unit, then the roots of the equation \[{{x}^{2}}+2x=a,\] are |
| A. | -4 and 2 |
| B. | 2 and 4 |
| C. | -2 and -4 |
| D. | 8 and -8 |
| Answer» B. 2 and 4 | |
| 1508. |
A ball is dropped from a platform 19.6m high. Its position function is ? |
| A. | \[x=-4.9{{t}^{2}}+19.6(0\le t\le 1)\] |
| B. | \[x=-4.9{{t}^{2}}+19.6(0\le t\le 2)\] |
| C. | \[x=-9.8{{t}^{2}}+19.6(0\le t\le 2)\] |
| D. | \[x=-4.9{{t}^{2}}-19.6(0\le t\le 2)\] |
| Answer» C. \[x=-9.8{{t}^{2}}+19.6(0\le t\le 2)\] | |
| 1509. |
The equation of the tangent to the curve \[y={{e}^{-\left| x \right|}}\]at the point where the curve cuts the line \[x=1\] is |
| A. | \[e(x+y)=1\] |
| B. | \[y+ex=1\] |
| C. | \[y+x=e\] |
| D. | None of these |
| Answer» E. | |
| 1510. |
Find the minimum value of\[{{e}^{(2{{x}^{2}}-2x-1){{\sin }^{2}}x}}\]. |
| A. | 1 |
| B. | 2 |
| C. | 0 |
| D. | None of these |
| Answer» B. 2 | |
| 1511. |
Find the greatest value of the function \[f(x)=\frac{\sin \,\,2x}{\sin \left( x+\frac{\pi }{4} \right)}\] on the interval \[\left[ 0,\frac{\pi }{2} \right]\] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» B. 2 | |
| 1512. |
The velocity of telegraphic communication is given by\[v={{x}^{2}}\log (1/x)\], where x is the displacement. For maximum velocity, x equals to? |
| A. | \[{{e}^{1/2}}\] |
| B. | \[{{e}^{-1/2}}\] |
| C. | \[{{(2e)}^{-1}}\] |
| D. | \[2{{e}^{-1/2}}\] |
| Answer» C. \[{{(2e)}^{-1}}\] | |
| 1513. |
What is the value of p for which the function\[f(x)=p\,\,\sin x+\frac{\sin 3x}{3}\]has an extremum at\[x=\frac{\pi }{3}\]? |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | 2 |
| Answer» E. | |
| 1514. |
Two cyclists start from the junction of two perpendicular roads, their velocities being 3 v m/ minute and 4 v m/minute. The rate at which the two cyclists are separating is |
| A. | \[\frac{7}{2}v\] m/minute |
| B. | 5 v m/minute |
| C. | v m/minute |
| D. | None of these |
| Answer» C. v m/minute | |
| 1515. |
The equation of normal to the curve\[y={{(1+x)}^{y}}+{{\sin }^{-1}}({{\sin }^{2}}x)\] at \[x=0\] is |
| A. | \[x+y=1\] |
| B. | \[x-y=1\] |
| C. | \[x+y=-1\] |
| D. | \[x-y=-1\] |
| Answer» B. \[x-y=1\] | |
| 1516. |
If a circular plate is heated uniformly, its area expands 3c times as fast as its radius, then the value of c when the radius is 6 units, is |
| A. | \[4\pi \] |
| B. | \[2\pi \] |
| C. | \[6\pi \] |
| D. | \[3\pi \] |
| Answer» B. \[2\pi \] | |
| 1517. |
The point in the interval \[(0,2\pi )\] where \[f(x)={{e}^{x}}\sin x\] has maximum slope is |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\pi \] |
| D. | \[\frac{3\pi }{2}\] |
| Answer» B. \[\frac{\pi }{2}\] | |
| 1518. |
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Then the minimum length of the hypotenuse is |
| A. | \[{{\left( {{a}^{\frac{3}{2}}}+{{b}^{\frac{3}{2}}} \right)}^{\frac{2}{3}}}\] |
| B. | \[{{\left( {{a}^{\frac{2}{3}}}+{{b}^{\frac{2}{3}}} \right)}^{\frac{3}{2}}}\] |
| C. | \[{{\left( {{a}^{\frac{2}{3}}}+{{b}^{\frac{2}{3}}} \right)}^{3}}\] |
| D. | \[{{\left( {{a}^{\frac{3}{2}}}+{{b}^{\frac{3}{2}}} \right)}^{3}}\] |
| Answer» C. \[{{\left( {{a}^{\frac{2}{3}}}+{{b}^{\frac{2}{3}}} \right)}^{3}}\] | |
| 1519. |
What is the area of the largest rectangular field which can be enclosed with 200 m of fencing? |
| A. | \[1600\,{{m}^{2}}\] |
| B. | \[2100\,{{m}^{2}}\] |
| C. | \[2400\,{{m}^{2}}\] |
| D. | \[2500\,{{m}^{2}}\] |
| Answer» E. | |
| 1520. |
If the line joining the points (0, 3) and (5, -2) is a tangent to the curve \[y=\frac{c}{x+1},\] then the value of c is |
| A. | 1 |
| B. | -2 |
| C. | 4 |
| D. | None of these |
| Answer» D. None of these | |
| 1521. |
Let \[f'(x) |
| A. | \[f(g(x+1))>f(g(x+5))\] |
| B. | \[f(g(x))<f(g(f(x+2))\] |
| C. | \[g(f(x))<g(f(x+2))\] |
| D. | \[g(f(x))>g(f(x-2))\] |
| Answer» B. \[f(g(x))<f(g(f(x+2))\] | |
| 1522. |
A lamp of negligible height is placed on the ground \[{{l}_{1}}\] away from a wall. A man \[{{l}_{2}}\] m tall is walking at a speed of \[\frac{{{l}_{1}}}{10}\] m/s from the lamp to the nearest point on the wall. When he is midway between the lamp and the wall, the rate of change in the length of this shadow on the wall is |
| A. | \[-\frac{5{{l}_{2}}}{2}m/s\] |
| B. | \[-\frac{2{{l}_{2}}}{5}m/s\] |
| C. | \[-\frac{{{l}_{2}}}{2}m/s\] |
| D. | \[-\frac{{{l}_{2}}}{5}m/s\] |
| Answer» C. \[-\frac{{{l}_{2}}}{2}m/s\] | |
| 1523. |
If the relation between sub-normal SN and sub- tangent ST at any point S on the curve; \[b{{y}^{2}}={{(x+a)}^{3}}\] is \[p(SN)=q{{(ST)}^{2}},\] then the value of p/q is |
| A. | 8a/27 |
| B. | 27/8b |
| C. | 8b/27 |
| D. | 46600 |
| Answer» D. 46600 | |
| 1524. |
A man is moving away from a tower 41.6m high at a rate of 2 m/s. If the eye level of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower, is |
| A. | \[-\frac{4}{125}rad/s\] |
| B. | \[-\frac{2}{25}rad/s\] |
| C. | \[-\frac{1}{625}rad/s\] |
| D. | None of these |
| Answer» B. \[-\frac{2}{25}rad/s\] | |
| 1525. |
The function \[f:[0,3]\to [1,29],\] defined by \[f(x)=2{{x}^{3}}-15{{x}^{2}}+36x+1\], is |
| A. | One-one and onto |
| B. | Onto but not one-one |
| C. | One-one but not onto |
| D. | Neither one-one nor onto |
| Answer» C. One-one but not onto | |
| 1526. |
If at each point of the curve \[y={{x}^{3}}-a{{x}^{2}}+x+1,\]the tangent is inclined at an acute angle with the positive direction of the x-axis, then |
| A. | \[a>0\] |
| B. | \[a\le \sqrt{3}\] |
| C. | \[-\sqrt{3}\le a\le \sqrt{3}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1527. |
If f and g are two increasing functions such that fog is defined, then which one of the following is correct? |
| A. | Fog is always an increasing function |
| B. | Fog is always a decreasing function |
| C. | Fog is neither an increasing nor a decreasing function |
| D. | None of the above |
| Answer» B. Fog is always a decreasing function | |
| 1528. |
The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be |
| A. | 8 times the original |
| B. | 16 times the original |
| C. | 32 times the original |
| D. | 64 times the original |
| Answer» D. 64 times the original | |
| 1529. |
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle \[\theta \] with the horizontal. The value of \[\theta \] for which the height of G, the mid-point of the rod above the peg is minimum, is |
| A. | \[15{}^\circ \] |
| B. | \[30{}^\circ \] |
| C. | \[60{}^\circ \] |
| D. | \[75{}^\circ \] |
| Answer» D. \[75{}^\circ \] | |
| 1530. |
The difference between greatest and least value of \[f(x)=2\sin x+\sin 2x,x\in \left[ 0,\frac{3\pi }{2} \right]\] is- |
| A. | \[\frac{3\sqrt{3}}{2}\] |
| B. | \[\frac{3\sqrt{3}}{2}-2\] |
| C. | \[\frac{3\sqrt{3}}{2}+2\] |
| D. | None of these |
| Answer» D. None of these | |
| 1531. |
If OT is the perpendicular drawn from the origin to the tangent at any point t to the curve\[x=a\text{ }co{{s}^{3}}t,\text{ }y=\text{ }a\text{ }si{{n}^{3}}t\], then OT is equal to: |
| A. | a sin 2t |
| B. | \[\frac{a}{2}\sin 2t\] |
| C. | 2a sin 2t |
| D. | 2a |
| Answer» C. 2a sin 2t | |
| 1532. |
The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs Rs. 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to Rs. 300 per hour is |
| A. | 10 |
| B. | 20 |
| C. | 30 |
| D. | 40 |
| Answer» E. | |
| 1533. |
The two curves \[{{x}^{3}}-3x{{y}^{2}}+2=0\text{ }and\text{ }3{{x}^{2}}y-{{y}^{3}}=2\] |
| A. | Cuts at right angle |
| B. | Touch each other |
| C. | Cut at an angle \[\frac{\pi }{3}\] |
| D. | Cut at an angle \[\frac{\pi }{4}\] |
| Answer» B. Touch each other | |
| 1534. |
The motion of a particle is described as\[s=2-3t+4{{t}^{3}}\]. What is the acceleration of the particle at the point where its velocity is zero? |
| A. | 0 |
| B. | 4 unit |
| C. | 8 unit |
| D. | 12 unit |
| Answer» D. 12 unit | |
| 1535. |
Area of the triangle formed by the normal to the curve \[x={{e}^{\sin \,}}^{y}\] at (1, 0) with the coordinate axes is: |
| A. | ¼ |
| B. | ½ |
| C. | 44289 |
| D. | 1 |
| Answer» C. 44289 | |
| 1536. |
If the function \[y=\frac{ax+b}{(x-1)(x-4)}\] has turning point at \[P(2,-1)\], then |
| A. | \[a=b=1\] |
| B. | \[a=b=0\] |
| C. | \[a=1,b=0\] |
| D. | \[a=b=2\] |
| Answer» D. \[a=b=2\] | |
| 1537. |
Let \[g(x)=2f\left( \frac{x}{2} \right)+f(2-x)\] and \[f''(x) |
| A. | (1/2, 2) |
| B. | (4/3, 2) |
| C. | (0, 2) |
| D. | (0, 4/3) |
| Answer» E. | |
| 1538. |
A function g(x) is defined as \[g(x)=\frac{1}{4}f(2{{x}^{2}}-1)+\frac{1}{2}f(1-{{x}^{2}})\] and \[f(x)\] is an increasing function. Then g(x) is increasing in the interval |
| A. | \[(-1,1)\] |
| B. | \[\left( -\sqrt{\frac{2}{3},}0 \right)\cup \left( \sqrt{\frac{2}{3}},\infty \right)\] |
| C. | \[\left( -\sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}} \right)\] |
| D. | None of these |
| Answer» C. \[\left( -\sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}} \right)\] | |
| 1539. |
A curve is represented by the equation \[x=se{{c}^{2}}t\]and\[y=cot\text{ }t\], where t is a parameter. If the tangent at the point P on the curve where \[t=\pi /4\] meets the curve again at the point Q, then \[\left| PQ \right|\] is equal to |
| A. | \[\frac{5\sqrt{3}}{2}\] |
| B. | \[\frac{5\sqrt{5}}{2}\] |
| C. | \[\frac{2\sqrt{5}}{3}\] |
| D. | \[\frac{3\sqrt{5}}{2}\] |
| Answer» E. | |
| 1540. |
At what points of curve \[y=\frac{2}{3}{{x}^{3}}+\frac{1}{2}{{x}^{2}},\] the tangent makes equal angle with the axis? |
| 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 \[(-1,0)\] |
| 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 \[(-1,0)\] | |
| 1541. |
The total number of parallel tangents of\[{{f}_{1}}(x)={{x}^{2}}-x+1\] and \[{{f}_{2}}(x)={{x}^{3}}-{{x}^{2}}-2x+1\] is |
| A. | 2 |
| B. | 0 |
| C. | 1 |
| D. | Infinite |
| Answer» E. | |
| 1542. |
\[f(x)=\frac{\log (\pi +x)}{\log (e+x)}\] is |
| A. | Increasing in \[[0,\infty )\] |
| B. | Decreasing in \[[0,\infty )\] |
| C. | Decreasing in \[\left[ 0,\frac{\pi }{e} \right]\] & increasing in \[\left[ \frac{\pi }{e},\infty \right]\] |
| D. | Increasing in \[\left[ 0,\frac{\pi }{e} \right]\] & decreasing in \[\left[ \frac{\pi }{e},\infty \right)\] |
| Answer» C. Decreasing in \[\left[ 0,\frac{\pi }{e} \right]\] & increasing in \[\left[ \frac{\pi }{e},\infty \right]\] | |
| 1543. |
Find the angle between the tangent to the curve \[{{y}^{2}}=2ax\] at the points where x = a/2. |
| A. | \[180{}^\circ \] |
| B. | \[90{}^\circ \] |
| C. | \[0{}^\circ \] |
| D. | None of these |
| Answer» C. \[0{}^\circ \] | |
| 1544. |
Find the minimum value of the function\[\frac{40}{3{{x}^{4}}+8{{x}^{3}}-18{{x}^{2}}+60}\]. |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{2}{3}\] |
| C. | \[\frac{4}{3}\] |
| D. | \[\frac{5}{3}\] |
| Answer» C. \[\frac{4}{3}\] | |
| 1545. |
If the rate of change in volume of spherical soap bubble is uniform, then the rate of change of surface area varies as |
| A. | Square of radius |
| B. | Square root of radius |
| C. | Inversely proportional to radius |
| D. | Cube of the radius |
| Answer» D. Cube of the radius | |
| 1546. |
A stone thrown vertically upward satisfies the equation\[s=64t-16{{t}^{2}}\], where s is in meter and t is in second. What is the time required to reach the maximum height? |
| A. | 1s |
| B. | 2s |
| C. | 3s |
| D. | 4s |
| Answer» C. 3s | |
| 1547. |
If an equation of a tangent to the curve, \[y=\cos (x+y),\] \[-1 \le x\le 1 + \pi \], is \[x+2y=k\] then k is equal to: |
| A. | 1 |
| B. | 2 |
| C. | \[\frac{\pi }{4}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» E. | |
| 1548. |
A lamp is 50 ft. above the ground. A ball is dropped from the same height from a point 30 ft. away from the light pole. If ball falls a distance \[s=16{{t}^{2}}\] ft. in t seconds, then the speed of the shadow of the ball moving along the ground 1/2s later is |
| A. | -1500 ft/s |
| B. | 1500 ft/s |
| C. | -1600 ft/s |
| D. | 1600 ft/s |
| Answer» B. 1500 ft/s | |
| 1549. |
The velocity v of a particle at any instant t moving in a straight line is given by v = s + 1 where s metre is the distance travelled in t second. What is the time taken by the particle to cover a distance of 9m? |
| A. | 1 s |
| B. | \[(log\,\,10)s\] |
| C. | \[2\text{(}log\text{ }10)s\] |
| D. | \[10s\] |
| Answer» C. \[2\text{(}log\text{ }10)s\] | |
| 1550. |
What is the slope of the tangent to the curve\[x={{t}^{2}}+3t-8,y=2{{t}^{2}}-2t-5\,\,at\,\,t=2\]? |
| A. | 44354 |
| B. | 44383 |
| C. | 1 |
| D. | 44352 |
| Answer» C. 1 | |