Test Details

Application of Derivatives

Mar 31, 3:00 PM

120 minutes

Mar 31, 5:00 PM

50.0 marks

MCQ

25 questions

Manish Anand

I am B.Tech and M.Tech in mechanical engineering from NIT Patna. From past 20 years I am teaching for 11th, 12th ,IIT-JEE, MCA ENTRANCE EXAM, NET, GATE etc. Expertise in PURE AND APPLIED HIGHER ENGINEERING MATHEMATICS, DISCRETE MATHS, NUMERICAL METHODS, OPERATION RESEARCH, TOC,CO etc. I have worked with FIITJEE DELHI, BRILLIANT TUTORIAL, PATNA SUPER 30 etc as Senior Faculty of Maths. Presently I am running my own institute named VASISTHA MATHEMATICS in Patna and also working with RPSIT, NIT PAT

Class Details

Mathematics

12th Foundation With JEE

Mathematics

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Question Text

Question 1 :
What is the rate of change $\sqrt{x^2 + 16 }$ with respect to $x^2$ at x = 3 ?

Question 2 :
If $y = 6x -x^3$ and $x$ increases at the rate of $5$ units per second, the rate of change of slope when $x = 3$ is

Question 3 :
A particle moves along a straight line according to the law $s=16-2t+3t^3$, where $s$ metres is the distance of the particle from a fixed point at the end of $t$ second. The acceleration of the particle at the end of $2$ s is

Question 4 :
The sides of two squares are $x$ and $y$ respectively, such that $y = x + x^{2}$. The rate of change of area of second square with respect to area of first square is ________.

Question 5 :
The side of a square sheet is increasing at the rate of $4 cm$ per minute. The rate by which the area increasing when the side is $8 cm$ long is.

Question 6 :
Consider the following statements:<br/>1. $\dfrac {dy}{dx}$ at a point on the curve gives slope of the tangent at that point.<br/>2. If $a(t)$ denotes acceleration of a particle, then $\displaystyle \int a(t) dt + c$ give velocity of the particle.<br/>3. If $s(t)$ gives displacement of a particle at time $t$, then $\dfrac {ds}{dt}$ gives its acceleration at that instant.<br/>Which of the above statements is/ are correct?

Question 7 :
The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+15$ is

Question 8 :
The area of an equilateral triangle of side $'a'$ feet is increasing at the rate of $4 sq.ft./sec$. The rate at which the perimeter is increasing is

Question 9 :
If the distance $s$ travelled by a particle in time $t$ is given by $s=t^2-2t+5$, then its acceleration is

Question 11 :
Sand is pouring from a pipe at the rate of $12$ cc/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always $\dfrac {1}{6}^{th}$ of the radius of the base How fast is the height of the sand cone increasing when the height is $4$ cm.

Question 12 :
The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate, then the ratio of the change of their areas is

Question 13 :
The rate of change of the volume of a cone withrespect to the radius of its base is.

Question 14 :
Side of an equilateral triangle expands at the rate of $2 cm/sec$. The rate of increase of its area when each side is $10cm$ is

Question 15 :
If the rate of change of area of a square plate is equal to that of the rate of change of its perimeter, then length of the side is?

Question 16 :
The population $p(t)$at time t of a certain mouse species satisfies the differential equation $\dfrac { dp\left( t \right) }{ dt } =0.5p\left( t \right) -450. $ If $p\left( 0 \right) =850,$ then the time at which the population becomes zero is:

Question 17 :
If $s=e^t(\sin t -\cos t)$ is the equation of motion of a moving particle then acceleration at time $t$ is given by

Question 18 :
Find an angle $\theta, 0 < \theta < \dfrac{\pi}{2}$, which increases twice as fast as its sine.

Question 19 :
A spherical balloon is being inflated so that its volume increases uniformly at the rate of $ 40 cm^3/min. $ At $r=8 cm$, its surface area increases at the rate

Question 20 :
The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is $8m/{s}^{2}$. The time taken by the particle to move the second metre is

Question 21 :
The rate at which ice-ball melts is proportional to the amount of ice in it. If half of it melts in $20$ minutes, the amount of ice left after $40$ minutes compared to it original amount is

Question 22 :
A balloon is pumped at the rate of a cm$^{3}$/minute. The rate of increase of its surface area when the radius is b cm, is

Question 23 :
The surface area of sphere when the volume is increasing at the same rate as its radius is:

Question 24 :
Initially, equation of ellipse was $3x^2+4y^2=12$. Keeping major axis constant, ellipse is bulge to form circle (with major axis as diameter). Its eccentricity changes at a rate $0.1/$sec. Time taken to form this circle is

Question 25 :
A variable triangle $ABC$ is inscribed in a circle of diameter $x$ units. If at a particular instant the rate of change of side, '$a$' is $x/2$ times the rate of change of the opposite angle $A$, then $A$ =

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