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question_answer1)
The distance of the point on \[y={{x}^{4}}+3{{x}^{2}}+2x\] which is nearest to the line \[y=2x-1\] is
A)
\[\frac{2}{\sqrt{5}}\] done
clear
B)
\[\sqrt{5}\] done
clear
C)
\[\frac{1}{\sqrt{5}}\] done
clear
D)
\[5\sqrt{5}\] done
clear
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question_answer2)
The maximum area of a right angled triangle with hypotenuse h is:
A)
\[\frac{{{h}^{2}}}{2\sqrt{2}}\] done
clear
B)
\[\frac{{{h}^{2}}}{2}\] done
clear
C)
\[\frac{{{h}^{2}}}{\sqrt{2}}\] done
clear
D)
\[\frac{{{h}^{2}}}{4}\] done
clear
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question_answer3)
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 done
clear
B)
a = b = 1 done
clear
C)
a = b = 2 done
clear
D)
a = b = -1 done
clear
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question_answer4)
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 done
clear
B)
50 done
clear
C)
60 done
clear
D)
40 done
clear
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question_answer5)
The function \[f(x)=\frac{{{x}^{2}}}{{{e}^{x}}}\] monotonically increasing if
A)
x < 0 only done
clear
B)
x > 2 only done
clear
C)
0 < x < 2 done
clear
D)
\[x\in (-\infty ,0)\cup (2,\infty )\] done
clear
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question_answer6)
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 done
clear
B)
4 unit done
clear
C)
8 unit done
clear
D)
12 unit done
clear
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question_answer7)
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 done
clear
B)
Fog is always a decreasing function done
clear
C)
Fog is neither an increasing nor a decreasing function done
clear
D)
None of the above done
clear
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question_answer8)
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)
\[{{\left( {{a}^{\frac{3}{2}}}+{{b}^{\frac{3}{2}}} \right)}^{\frac{2}{3}}}\] done
clear
B)
\[{{\left( {{a}^{\frac{2}{3}}}+{{b}^{\frac{2}{3}}} \right)}^{\frac{3}{2}}}\] done
clear
C)
\[{{\left( {{a}^{\frac{2}{3}}}+{{b}^{\frac{2}{3}}} \right)}^{3}}\] done
clear
D)
\[{{\left( {{a}^{\frac{3}{2}}}+{{b}^{\frac{3}{2}}} \right)}^{3}}\] done
clear
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question_answer9)
The number of tangents to the curve \[{{x}^{3/2}}+{{y}^{3/2}}=2{{a}^{3/2}},\,\,\,a>0,\] which are equally inclined to the axes, is
A)
2 done
clear
B)
1 done
clear
C)
0 done
clear
D)
4 done
clear
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question_answer10)
The curve \[y=x{{e}^{x}}\] has minimum value equal to
A)
\[-\frac{1}{e}\] done
clear
B)
\[\frac{1}{e}\] done
clear
C)
\[-e\] done
clear
D)
e done
clear
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question_answer11)
What is the minimum value of \[px+qy\] \[(p>0,q>0)\] when\[xy={{r}^{2}}\]?
A)
\[2r\sqrt{pq}\] done
clear
B)
\[2pq\sqrt{r}\] done
clear
C)
\[-2r\sqrt{pq}\] done
clear
D)
\[2rpq\] done
clear
View Solution play_arrow
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question_answer12)
The number of solutions of the equation \[3\tan x+{{x}^{3}}=2\,\,in\left( 0,\frac{\pi }{4} \right).\] is
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
Infinite done
clear
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question_answer13)
The straight line \[\frac{x}{a}+\frac{y}{b}=2\] touches the curve \[{{\left( \frac{x}{a} \right)}^{n}}+{{\left( \frac{y}{b} \right)}^{n}}=2\] at the point (a, b) for
A)
n = 1, 2 done
clear
B)
n = 3, 4, -5 done
clear
C)
n = 1, 2, 3 done
clear
D)
Any value of n done
clear
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question_answer14)
The function \[f(x)=1+x(\sin x)[\cos x],0<x\le \frac{\pi }{2}\](where [.] is G.I.F.)
A)
Is continuous on \[\left( 0,\frac{\pi }{2} \right)\] done
clear
B)
Is strictly increasing in \[\left( 0,\frac{\pi }{2} \right)\] done
clear
C)
Is strictly decreasing in \[\left( 0,\frac{\pi }{2} \right)\] done
clear
D)
Has global maximum value 2 done
clear
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question_answer15)
The function \[f(x)=2\,\,\log (x-2)-{{x}^{2}}+4x+1\]increases on the interval
A)
(1, 2) done
clear
B)
(2, 3) done
clear
C)
(1/2, 3) done
clear
D)
(2, 4) done
clear
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question_answer16)
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 done
clear
B)
2 only done
clear
C)
Both 1 and 2 done
clear
D)
Neither 1 nor 2 done
clear
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question_answer17)
The radius of a circle is uniformly increasing at the rate of 3 cm/s. What is the rate of increase in area, when the radius is 10 cm?
A)
\[6\pi \,c{{m}^{2}}/s\] done
clear
B)
\[10\pi \,c{{m}^{2}}/s\] done
clear
C)
\[30\pi ;c{{m}^{2}}/s\] done
clear
D)
\[60\,\pi \,c{{m}^{2}}/s\] done
clear
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question_answer18)
If \[x\text{ }cos\theta +y\text{ }sin\text{ }\theta =2\] is perpendicular to the line\[x-y=3\], then what is one of the value of\[\theta \]?
A)
\[\pi /6\] done
clear
B)
\[\pi /4\] done
clear
C)
\[\pi /2\] done
clear
D)
\[\pi /3\] done
clear
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question_answer19)
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 done
clear
B)
Only one maximum done
clear
C)
Only one minimum done
clear
D)
Only one maximum and only one minimum done
clear
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question_answer20)
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)
\[162\,\,c{{m}^{3}}/s\] done
clear
B)
\[172\,\,c{{m}^{3}}/s\] done
clear
C)
\[182\,\,c{{m}^{3}}/s\] done
clear
D)
\[192\,\,c{{m}^{3}}/s\] done
clear
View Solution play_arrow
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question_answer21)
What is the interval in which the function \[f(x)=\sqrt{9-{{x}^{2}}}\] is increasing? \[(f(x)>0)\]
A)
\[0<x<3\] done
clear
B)
\[-3<x<0\] done
clear
C)
\[0<x<9\] done
clear
D)
\[-3<x<3\] done
clear
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question_answer22)
If at any instant t, for a sphere, r denotes the radius, S denotes the surface area and V denotes the volume, then what is \[\frac{dV}{dt}\] equal to?
A)
\[\frac{1}{2}S\frac{dr}{dt}\] done
clear
B)
\[\frac{1}{2}r\frac{dS}{dt}\] done
clear
C)
\[r\frac{dS}{dt}\] done
clear
D)
\[\frac{1}{2}{{r}^{2}}\frac{dS}{dt}\] done
clear
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question_answer23)
The largest area of a trapezium inscribed in a semi-circle of radius R, if the lower base is on the diameter, is
A)
\[\frac{3\sqrt{3}}{4}{{R}^{2}}\] done
clear
B)
\[\frac{\sqrt{3}}{2}{{R}^{2}}\] done
clear
C)
\[\frac{3\sqrt{3}}{8}{{R}^{2}}\] done
clear
D)
\[{{R}^{2}}\] done
clear
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question_answer24)
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)
\[f\{g(x)\}\ge f\{g(0)\}\] done
clear
B)
\[g\{f(x)\}\le g\{f(0)\}\] done
clear
C)
\[f\{g(2)\}=7\] done
clear
D)
None of these done
clear
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question_answer25)
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)
\[x+2y=1\] done
clear
B)
\[x+2y=\pi /2\] done
clear
C)
\[x+2y=\pi /4\] done
clear
D)
None of these done
clear
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question_answer26)
The range of the function\[f(x)=2\sqrt{x-2}+\sqrt{4-x}\] is
A)
\[\left( \sqrt{2},\sqrt{10} \right)\] done
clear
B)
\[\left[ \sqrt{2},\sqrt{10} \right)\] done
clear
C)
\[\left( \sqrt{2},\sqrt{10} \right]\] done
clear
D)
\[\left[ \sqrt{2},\sqrt{10} \right]\] done
clear
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question_answer27)
What is the slope of the tangent to the curve\[y={{\sin }^{-1}}({{\sin }^{2}}x)at\,\,x=0\]?
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
None of these done
clear
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question_answer28)
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)
7/6 done
clear
B)
6/7 done
clear
C)
1 done
clear
D)
5/6 done
clear
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question_answer29)
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 done
clear
B)
\[(log\,\,10)s\] done
clear
C)
\[2\text{(}log\text{ }10)s\] done
clear
D)
\[10s\] done
clear
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question_answer30)
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 done
clear
B)
1500 ft/s done
clear
C)
-1600 ft/s done
clear
D)
1600 ft/s done
clear
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question_answer31)
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 done
clear
B)
2 done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[\frac{\pi }{2}\] done
clear
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question_answer32)
The approximate value of \[{{(0.007)}^{1/3}}\]
A)
\[\frac{23}{120}\] done
clear
B)
\[\frac{27}{120}\] done
clear
C)
\[\frac{19}{120}\] done
clear
D)
\[\frac{17}{120}\] done
clear
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question_answer33)
The equation of the normal to the curve \[y=\left| {{x}^{2}}-\left| x \right| \right|\] at\[x=-2\].
A)
\[3y=x+8\] done
clear
B)
\[x=3y+4\] done
clear
C)
\[y=2x+8\] done
clear
D)
\[y=3x\] done
clear
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question_answer34)
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)
\[\frac{1}{4}\] done
clear
B)
1 done
clear
C)
\[\frac{1}{3}\] done
clear
D)
\[\frac{1}{2}\] done
clear
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question_answer35)
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 done
clear
B)
2s done
clear
C)
3s done
clear
D)
4s done
clear
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question_answer36)
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 done
clear
B)
Square root of radius done
clear
C)
Inversely proportional to radius done
clear
D)
Cube of the radius done
clear
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question_answer37)
Find the minimum value of the function\[\frac{40}{3{{x}^{4}}+8{{x}^{3}}-18{{x}^{2}}+60}\].
A)
\[\frac{1}{3}\] done
clear
B)
\[\frac{2}{3}\] done
clear
C)
\[\frac{4}{3}\] done
clear
D)
\[\frac{5}{3}\] done
clear
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question_answer38)
Find the angle between the tangent to the curve \[{{y}^{2}}=2ax\] at the points where x = a/2.
A)
\[180{}^\circ \] done
clear
B)
\[90{}^\circ \] done
clear
C)
\[0{}^\circ \] done
clear
D)
None of these done
clear
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question_answer39)
\[f(x)=\frac{\log (\pi +x)}{\log (e+x)}\] is
A)
Increasing in \[[0,\infty )\] done
clear
B)
Decreasing in \[[0,\infty )\] done
clear
C)
Decreasing in \[\left[ 0,\frac{\pi }{e} \right]\] & increasing in \[\left[ \frac{\pi }{e},\infty \right]\] done
clear
D)
Increasing in \[\left[ 0,\frac{\pi }{e} \right]\] & decreasing in \[\left[ \frac{\pi }{e},\infty \right)\] done
clear
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question_answer40)
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 done
clear
B)
0 done
clear
C)
1 done
clear
D)
Infinite done
clear
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question_answer41)
How many tangents are parallel to x-axis for the curve\[y={{x}^{2}}-4x+3\]?
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
No tangent is parallel to x-axis done
clear
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question_answer42)
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)
\[68\,\,c{{m}^{2}}\] done
clear
B)
\[70\,\,c{{m}^{2}}\] done
clear
C)
\[71.25\,\,c{{m}^{2}}\] done
clear
D)
\[72.25\,\,c{{m}^{2}}\] done
clear
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question_answer43)
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)
\[k<3\] done
clear
B)
\[k\le 3\] done
clear
C)
\[k>3\] done
clear
D)
\[k\ge 3\] done
clear
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question_answer44)
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 \[\left( -1,-\frac{1}{6} \right)\] done
clear
B)
\[\left( \frac{1}{2},\frac{4}{9} \right)\] and \[(-1,0)\] done
clear
C)
\[\left( \frac{1}{3},\frac{1}{7} \right)\] and \[\left( -3,\frac{1}{2} \right)\] done
clear
D)
\[\left( \frac{1}{3},\frac{4}{47} \right)\] and \[\left( -1,-\frac{1}{3} \right)\] done
clear
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question_answer45)
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)
\[\frac{5\sqrt{3}}{2}\] done
clear
B)
\[\frac{5\sqrt{5}}{2}\] done
clear
C)
\[\frac{2\sqrt{5}}{3}\] done
clear
D)
\[\frac{3\sqrt{5}}{2}\] done
clear
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question_answer46)
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)
\[(-1,1)\] done
clear
B)
\[\left( -\sqrt{\frac{2}{3},}0 \right)\cup \left( \sqrt{\frac{2}{3}},\infty \right)\] done
clear
C)
\[\left( -\sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}} \right)\] done
clear
D)
None of these done
clear
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question_answer47)
Let \[g(x)=2f\left( \frac{x}{2} \right)+f(2-x)\] and \[f''(x)<0\forall x\in (0,2)\]. Then g(x) increases in
A)
(1/2, 2) done
clear
B)
(4/3, 2) done
clear
C)
(0, 2) done
clear
D)
(0, 4/3) done
clear
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question_answer48)
If the function \[y=\frac{ax+b}{(x-1)(x-4)}\] has turning point at \[P(2,-1)\], then
A)
\[a=b=1\] done
clear
B)
\[a=b=0\] done
clear
C)
\[a=1,b=0\] done
clear
D)
\[a=b=2\] done
clear
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question_answer49)
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 done
clear
B)
½ done
clear
C)
3/4 done
clear
D)
1 done
clear
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question_answer50)
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)
\[162\,\,c{{m}^{3}}/s\] done
clear
B)
\[172\,\,c{{m}^{3}}/s\] done
clear
C)
\[182\,\,c{{m}^{3}}/s\] done
clear
D)
\[192\,\,c{{m}^{3}}/s\] done
clear
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question_answer51)
If\[f(x)=x\ell nx\], then \[f(x)\] attains minimum value at which one of the following points?
A)
\[x={{e}^{-2}}\] done
clear
B)
\[x=e\] done
clear
C)
\[x={{e}^{-1}}\] done
clear
D)
\[x=2{{e}^{-1}}\] done
clear
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question_answer52)
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 done
clear
B)
300 done
clear
C)
200 done
clear
D)
100 done
clear
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question_answer53)
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 done
clear
B)
Touch each other done
clear
C)
Cut at an angle \[\frac{\pi }{3}\] done
clear
D)
Cut at an angle \[\frac{\pi }{4}\] done
clear
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question_answer54)
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 done
clear
B)
20 done
clear
C)
30 done
clear
D)
40 done
clear
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question_answer55)
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 done
clear
B)
\[\frac{a}{2}\sin 2t\] done
clear
C)
2a sin 2t done
clear
D)
2a done
clear
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question_answer56)
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)
\[\frac{3\sqrt{3}}{2}\] done
clear
B)
\[\frac{3\sqrt{3}}{2}-2\] done
clear
C)
\[\frac{3\sqrt{3}}{2}+2\] done
clear
D)
None of these done
clear
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question_answer57)
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)
\[15{}^\circ \] done
clear
B)
\[30{}^\circ \] done
clear
C)
\[60{}^\circ \] done
clear
D)
\[75{}^\circ \] done
clear
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question_answer58)
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 done
clear
B)
16 times the original done
clear
C)
32 times the original done
clear
D)
64 times the original done
clear
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question_answer59)
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 done
clear
B)
91 done
clear
C)
84 done
clear
D)
96 done
clear
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question_answer60)
If \[A>0,\text{ }B>0\] and \[A+B=\pi /3,\] then the maximum value of tan A tan B is
A)
\[\frac{1}{\sqrt{3}}\] done
clear
B)
\[\frac{1}{3}\] done
clear
C)
\[3\] done
clear
D)
\[\sqrt{3}\] done
clear
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question_answer61)
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 done
clear
B)
f has neither local maximum nor minimum at x = c done
clear
C)
f is necessarily a constant function done
clear
D)
it is difficult to say whether or (b) done
clear
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question_answer62)
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)
\[a>0\] done
clear
B)
\[a\le \sqrt{3}\] done
clear
C)
\[-\sqrt{3}\le a\le \sqrt{3}\] done
clear
D)
None of these done
clear
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question_answer63)
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 done
clear
B)
Onto but not one-one done
clear
C)
One-one but not onto done
clear
D)
Neither one-one nor onto done
clear
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question_answer64)
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)
\[-\frac{4}{125}rad/s\] done
clear
B)
\[-\frac{2}{25}rad/s\] done
clear
C)
\[-\frac{1}{625}rad/s\] done
clear
D)
None of these done
clear
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question_answer65)
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 done
clear
B)
27/8b done
clear
C)
8b/27 done
clear
D)
8/27 done
clear
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question_answer66)
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)
\[-\frac{5{{l}_{2}}}{2}m/s\] done
clear
B)
\[-\frac{2{{l}_{2}}}{5}m/s\] done
clear
C)
\[-\frac{{{l}_{2}}}{2}m/s\] done
clear
D)
\[-\frac{{{l}_{2}}}{5}m/s\] done
clear
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question_answer67)
Let \[f'(x)<0\] and \[g'(x)>0\] for all real x, then
A)
\[f(g(x+1))>f(g(x+5))\] done
clear
B)
\[f(g(x))<f(g(f(x+2))\] done
clear
C)
\[g(f(x))<g(f(x+2))\] done
clear
D)
\[g(f(x))>g(f(x-2))\] done
clear
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question_answer68)
Let \[f(x)\] be a function defined as follows: \[f(x)=\sin ({{x}^{2}}-3x),x\le 0;\] and \[6x+5{{x}^{2}},x>0\] Then at \[x=0,f(x)\]
A)
Has a local maximum done
clear
B)
done
clear
C)
Is discontinuous done
clear
D)
None of these done
clear
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question_answer69)
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 done
clear
B)
-2 done
clear
C)
4 done
clear
D)
None of these done
clear
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question_answer70)
What is the area of the largest rectangular field which can be enclosed with 200 m of fencing?
A)
\[1600\,{{m}^{2}}\] done
clear
B)
\[2100\,{{m}^{2}}\] done
clear
C)
\[2400\,{{m}^{2}}\] done
clear
D)
\[2500\,{{m}^{2}}\] done
clear
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question_answer71)
The point in the interval \[(0,2\pi )\] where \[f(x)={{e}^{x}}\sin x\] has maximum slope is
A)
\[\frac{\pi }{4}\] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\pi \] done
clear
D)
\[\frac{3\pi }{2}\] done
clear
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question_answer72)
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)
\[4\pi \] done
clear
B)
\[2\pi \] done
clear
C)
\[6\pi \] done
clear
D)
\[3\pi \] done
clear
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question_answer73)
The equation of normal to the curve\[y={{(1+x)}^{y}}+{{\sin }^{-1}}({{\sin }^{2}}x)\] at \[x=0\] is
A)
\[x+y=1\] done
clear
B)
\[x-y=1\] done
clear
C)
\[x+y=-1\] done
clear
D)
\[x-y=-1\] done
clear
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question_answer74)
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 done
clear
B)
5 v m/minute done
clear
C)
v m/minute done
clear
D)
None of these done
clear
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question_answer75)
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 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
2 done
clear
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question_answer76)
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)
\[{{e}^{1/2}}\] done
clear
B)
\[{{e}^{-1/2}}\] done
clear
C)
\[{{(2e)}^{-1}}\] done
clear
D)
\[2{{e}^{-1/2}}\] done
clear
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question_answer77)
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 done
clear
B)
2 done
clear
C)
3 done
clear
D)
None of these done
clear
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question_answer78)
Find the minimum value of\[{{e}^{(2{{x}^{2}}-2x-1){{\sin }^{2}}x}}\].
A)
1 done
clear
B)
2 done
clear
C)
0 done
clear
D)
None of these done
clear
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question_answer79)
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)
\[e(x+y)=1\] done
clear
B)
\[y+ex=1\] done
clear
C)
\[y+x=e\] done
clear
D)
None of these done
clear
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question_answer80)
A ball is dropped from a platform 19.6m high. Its position function is ?
A)
\[x=-4.9{{t}^{2}}+19.6(0\le t\le 1)\] done
clear
B)
\[x=-4.9{{t}^{2}}+19.6(0\le t\le 2)\] done
clear
C)
\[x=-9.8{{t}^{2}}+19.6(0\le t\le 2)\] done
clear
D)
\[x=-4.9{{t}^{2}}-19.6(0\le t\le 2)\] done
clear
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