Application of Derivatives

Q60 The curves $y = 4{x^2} + 2x - 8$ and $y = {x^3} - x + 13$ touch each other at the point ……….. FillBlank Q61 The equation of normal to the curve $y = \tan x$ at (0,0) is…………… FillBlank Q62 The values of $a$ for which the function $f(x) = \sin x - ax + b$ increases on $R$ are ……... FillBlank Q63 The function $f(x) = \frac{{2{x^2} - 1}}{{{x^4}}}$, (where, $\left. {x > 0} \right)$ decreases in the interval…………….. FillBlank Q64 The least value of function $f(x) = ax + \frac{b}{x}($ where $,a > 0,b > 0,x > 0)$ is…………… FillBlank Q25 If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, then show that the area of triangle is maximum, when the angle between them is $\frac{\pi }{3}$. LA Q26 Find the points of local maxima, local minima and the points of inflection of the function $f(x) = {x^5} - 5{x^4} + 5{x^3} - 1$. Also, find the corresponding local maximum and local minimum values. LA Q27 A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs.300 per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs.1 per one subscriber will discontinue the service. Find what increase will bring maximum profit? LA Q28 If the straight line $x\cos \alpha + y\sin \alpha = p$ touches the curve $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,$ then prove that ${a^2}{\cos ^2}\alpha + {b^2}{\sin ^2}\alpha = {p^2}$. LA Q29 If an open box with square base is to be made of a given quantity of card board of area ${c^2}$, then show that the maximum volume of the box is $\frac{{{c^3}}}{{6\sqrt 3 }}{\rm{cu}}$ units. LA Q30 Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume. LA Q31 I the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum? LA Q32 If $AB$ is a diameter of a circle and $C$ is any point on the circle, then show that the area of $\Delta ABC$ is maximum, when it is isosceles. LA Q33 A metal box with a square base and vertical sides is to contain $1024\;{\rm{c}}{{\rm{m}}^3}$. If the material for the top and bottom costs Rs.5 per ${\rm{c}}{{\rm{m}}^2}$ and the material for the sides costs Rs.2.50 per ${\rm{c}}{{\rm{m}}^2}$. Then, find the least cost of the box. LA Q34 The sum of surface areas of a rectangular parallelopiped with sides $x$, $2x$ and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if $x$ is equal to three times the radius of the sphere. Also, find the minimum value of the sum of their volumes. LA Q1 A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is propotional to the surface. Prove that the radius is decreasing at a constant rate. SA Q2 If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius. SA Q3 A kite is moving horizontally at a height of 151.5 m. If the speed of kite is $10\;{\rm{m}}/{\rm{s}}$, how fast is the string being let out, when the kite is $250\;{\rm{m}}$ away from the boy who is flying the kite, if the height of boy is $1.5\;{\rm{m}}$ ? SA Q4 Two men $A$ and $B$ start with velocities $v$ at the same time from the junction of two roads inclined at ${45^\circ }$ to each other. If they travel by different roads, then find the rate at which they are being separated. SA Q5 Find an angle $\theta$, where $0 < \theta < \frac{\pi }{2}$, which increases twice as fast as its sine. SA Q6
Find the approximate value of ${(1.999)^5}$. SA Q7 Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively. SA Q8 A man, $2\;{\rm{m}}$ tall, walks at the rate of $1\frac{2}{3}\;{\rm{m}}/{\rm{s}}$ towards a street light which is $5\frac{1}{2}\;{\rm{m}}$ above the ground. At what rate is the tip of his shadow moving and at what rate is the length of the shadow changing when he is $3\frac{1}{3}\;{\rm{m}}$ from the base of the light? SA Q9 A swimming pool is to be drained for cleaning. If $L$ represents the number of litres of water in the pool $t$ seconds after the pool has been plugged off to drain and $L = 200{(10 - t)^2}$. How fast is the water running out at the end of $5\;{\rm{s}}$ and what is the average rate at which the water flows out during the first $5\;{\rm{s}}$ ? SA Q10 The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side. SA Q11 If $x$ and $y$ are the sides of two squares such that $y = x - {x^2}$, then find the rate of change of the area of second square with respect to the area of first square. SA Q12 Find the condition that curves $2x = {y^2}$ and $2xy = k$ intersect orthogonally. SA Q13 Prove that the curves $xy = 4$ and ${x^2} + {y^2} = 8$ touch each other. SA Q14 Find the coordinates of the point on the curve $\sqrt x + \sqrt y = 4$ at which tangent is equally inclined to the axes. SA Q15 Find the angle of intersection of the curves $y = 4 - {x^2}$ and $y = {x^2}$. SA Q16 Prove that the curves ${y^2} = 4x$ and ${x^2} + {y^2} - 6x + 1 = 0$ touch each other at the point (1,2) . SA Q17 Find the equation of the normal lines to the curve $3{x^2} - {y^2} = 8$ which are parallel to the line $x + 3y = 4$. SA Q18 At what points on the curve ${x^2} + {y^2} - 2x - 4y + 1 = 0$, the tangents are parallel to the Y-axis? SA Q19 Show that the line $\frac{x}{a} + \frac{y}{b} = 1$, touches the curve $y = b \cdot {e^{ - x/a}}$ at the point, where the curve intersects the axis of $Y$. SA Q20 Show that $f(x) = 2x + {\cot ^{ - 1}}x + \log \left( {\sqrt {1 + {x^2}} - x} \right)$ is increasing in $R$. SA Q21 Show that for $a \ge 1,f(x) = \sqrt 3 \sin x - \cos x - 2ax + b$ is decreasing in $R$. SA Q22 Show that $f(x) = {\tan ^{ - 1}}(\sin x + \cos x)$ is an increasing function in $\left( {0,\frac{\pi }{4}} \right)$. SA Q23 At what point, the slope of the curve $y = - {x^3} + 3{x^2} + 9x - 27$ is maximum? Also, find the maximum slope. SA Q24 Prove that $f(x) = \sin x + \sqrt 3 \cos x$ has maximum value at $x = \frac{\pi }{6}$. SA

Q1 Find the rate of change of the area of a circle with respect to its radius $r$ when

(a) $r = 3$ cm

(b) $r = 4$ cm SA Q2 The volume of a cube is increasing at the rate of $8{\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{/s}}$. How fast is the surface area increasing when the length of an edge is $12{\rm{ cm}}$? SA Q3 The radius of a circle is increasing uniformly at the rate of $3{\rm{cm/s}}$. Find the rate at which the area of the circle is increasing when the radius is $10{\rm{ cm}}$. SA Q4 An edge of a variable cube is increasing al the rate of $3{\rm{cm/s}}$. How fast is the volume of the cube increasing when the edge is $10{\rm{ cm}}$long? SA Q5 A stone is dropped into a quiet lake and waves move in circles at the speed of $5{\rm{cm/s}}$. At the instant when the radius of the circular wave is $8{\rm{ cm}}$, how fast is the enclosed area increasing? SA Q6 The radius of a circle is increasing at the rate of $0.7{\rm{cm/s}}$ . What is the rate of increase of its circumference? SA Q7 The length $x$ of a rectangle is decreasing at the rate of $5{\rm{ cm/minute}}$ and the width $y$ is increasing at the rate of $4{\rm{ cm/minute}}$. When $x = 8{\rm{cm}}$ and $y = 6{\rm{cm}}$, find the rates of change of

(a) perimeter, and

(b) the area of the rectangle. SA Q8 A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15$ cm. SA Q9 A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. SA Q10 A ladder $5$m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2{\rm{cm/s}}$. How fast is its height on the wall decreasing when the foot of the ladder is $4{\rm{m}}$ away from the wall ? SA Q11 A particle moves along the curve $6y = {x^3} + 2$. Find the points on the curve at which the $y -$ coordinate is changing $8$ times as the $x -$coordinate. SA Q12 The radius of an air bubble is increasing at the rate of $\cfrac{1}{2}{\rm{cm/s}}$. At what rate is the volume of the bubble increasing when the radius is $1{\rm{cm}}$? SA Q13 A balloon, which always remains spherical, has a variable diameter $\cfrac{3}{2}(2x + 1)$. Find the rate of change of its volume with respect to $x$. SA Q14 Sand is pouring from a pipe at the rate of $12{\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{/s}}$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is $4{\rm{ cm}}$? SA Q15 The total cost $C(x)$ in rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007{x^3} - 0.003{x^2} + 15x + 4000$.Find the marginal cost when $17$ units are produced. SA Q16 The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 13{x^2} + 2x + 15$. Find the marginal revenue when $x = 7$. SA Q17 The rate of change Of the area of a circle with respect to its radius $r$ at $r = 6$ cm is

(A) $10\pi$

(B) $12\pi$

(C) $8\pi$

(D) $11\pi$ SA Q18 The total revenue in Rupees received from the sale of $x$ units of a product is given by$R\left( x \right) = 3{x^2} + 36x + 5$. The marginal revenue, when $x = 15$ is

(A) $116$

(B) $96$

(C) $90$

(D) $126$ SA

Q1 Show that the function given by $f(x) = 3x + 17$ is strictly increasing on $R$. SA Q2 Show that the function given by $f\left( x \right) = {e^{2x}}$ is strictly increasing on $R$. SA Q3 Show that the function given by $f\left( x \right) = \sin x$ is

(a) strictly increasing in $\left( {0,\cfrac{\pi }{2}} \right)$

(b) strictly decreasing in $\left( {\cfrac{\pi }{2},\;\pi } \right)$

(c) neither increasing nor decreasing in $\left( {0,\pi } \right)$ SA Q4 Find the intervals in which the function $f$ given by $f(x) = 2{x^2} - 3x$ is

(a) strictly increasing

(b) strictly decreasing SA Q5 Find the intervals in which the function $f$ given by $f(x) = 2{x^3} - 3{x^2} - 36x + 7$ is

(a) strictly increasing

(b) strictly decreasing SA Q6 Find the intervals in which the following functions are strictly increasing or decreasing:

(a) ${x^2} + 2x - 5$

(b) $10 - 6x - 2{x^2}$

(c) $- 2{x^3} - 9{x^2} - 12x + 1$

(d) $6 - 9x - {x^2}$

(e) ${\left( {x + 1} \right)^3}{\left( {x - 3} \right)^3}$ SA Q7 Show that $y = \log \left( {1 + x} \right) - \cfrac{{2x}}{{2 + x}},x > - 1$, is an increasing function of $x$ throughout its domain. SA Q8 Find the values of $x$ for which $y = {\left[ {x\left( {x - 2} \right)} \right]^2}$ is an increasing function. SA Q9 Prove that $y = \cfrac{{4\sin \theta }}{{\left( {2 + \cos \theta } \right)}} - \theta$ is an increasing function of $\theta$ in $\left[ {0,\cfrac{\pi }{2}} \right]$. SA Q10 Prove that the logarithmic function is strictly increasing on $(0,\infty )$ SA Q11 Prove that the function $f$ given by $f\left( x \right) = {x^2} - x + 1$ is neither strictly increasing nor strictly decreasing on $( - 1,1)$ . SA Q12 Which of the following functions are strictly decreasing on $\left( {0,\cfrac{\pi }{2}} \right)$?

(A) $\cos x$

(B) $\cos 2x$

(C) $\cos 3x$

(D) $\tan x$ SA Q13 On which of the following intervals is the function $f$ given by $f(x) = {x^{100}} + \sin x - 1$ strictly decreasing?

(A) $\left( {0,1} \right)$

(B) $\left( {\cfrac{\pi }{2},\pi } \right)$

(C) $\left( {0,\cfrac{\pi }{2}} \right)$

(D) None of these SA Q14 Find the least value of $a$ such that the function $f$ given by $f(x) = {x^2} + ax + 1$ is strictly increasing on $(1,\;2)$. SA Q15 Let $I$ be any interval disjoint from $\left( { - 1,1} \right)$. Prove that the function $f$ given by $f(x) = x + \cfrac{1}{x}$ is strictly increasing on $I$. SA Q16 Prove that the function $f$ given by $f\left( x \right) = \log \sin x$ is strictly increasing on $\left( {0,\cfrac{\pi }{2}} \right)$ and strictly decreasing on $\left( {\cfrac{\pi }{2},\pi } \right)$. SA Q17 Prove that the function $f$ given by $f\left( x \right) = \log \cos x$ is strictly decreasing on $\left( {0,\cfrac{\pi }{2}} \right)$ and strictly increasing on $\left( {\cfrac{\pi }{2},\pi } \right)$. SA Q18 Prove that the function given by $f(x) = {x^3} - 3{x^2} + 3x - 100$ is increasing in $R$. SA Q19 The interval in which $y = {x^2}{e^{ - x}}$ is increasing, is

(A) $\left( { - \infty ,\infty } \right)$

(B) $( - 2,0)$

(C) $(2,\infty )$

(D) $(0,2)$ SA

Q1 Find the slope of the tangent to the curve $y = 3{x^4} - 4x$ at $x = 4.$ SA Q2 Find the slope of the tangent to the curve $y = \cfrac{{x - 1}}{{x - 2}},x \ne 2$ at $x = 10.$ SA Q3 Find the slope of the tangent to curve $y = {x^3} - x + 1$ at the point whose $x -$coordinate is $2$. SA Q4 Find the slope of the tangent to the curve $y = {x^3} - 3x + 2$ at the point whose $x -$coordinate is $3$. SA Q5 Find the slope of the normal to the curve $x = a{\cos ^3}\theta ,y = a{\sin ^3}\theta$at $\theta = \cfrac{\pi }{4}$. SA Q6 Find the slope of the normal to the curve $x = 1 - a\sin \theta ,\,y = b{\cos ^2}\theta$ at $\theta = \cfrac{\pi }{2}$. SA Q7 Find points at which the tangent to the curve $y = {x^3} - 3{x^2} - 9x + 7$ is parallel to the $x -$axis. SA Q8 Find a point on the curve $y = {\left( {x - 2} \right)^2}$ at which the tangent is parallel to the chord joining the points $\left( {2,0} \right)$ and $\left( {4,4} \right)$. SA Q9 Find the point on the curve $y = {x^3} - 11x + 5$ at which the tangent is $y = x - 11$. SA Q10 Find the equations of all lines having slope $- 1$ that are tangents to the curve $y = \cfrac{1}{{x - 1}},\,\,x \ne 1$. SA Q11 Find the equations of all lines having slope $2$ which are tangents to the curve $y = \cfrac{1}{{x - 3}},x \ne 3.$ SA Q12 Find the equations of all lines having slope $0$ which are tangents to the curve $y = \cfrac{1}{{{x^2} - 2x + 3}}.$ SA Q13 Find points on the curve $\cfrac{{{x^2}}}{9} + \cfrac{{{y^2}}}{{16}} = 1$ at which the tangents are

(i) parallel to $x$-axis

(ii) parallel to $y$-axis. SA Q14 Find the equations of the tangent and normal to the given curves at the indicated points:

(i) $y = {x^4} - 6{x^3} + 13{x^2} - 10x + 5$ at $(0,5)$

(ii) $y = {x^4} - d + 13{x^2} - 10x + 5\;$at $(1,\;3)$

(iii) $y = {x^3}$ at $\left( {1,1} \right)$

(iv) $y = {x^2}\;$at $(0,0)$

(v) $x = \cos t,y = \sin t$ at $t = \cfrac{\pi }{4}.$ SA Q15 Find the equation of the tangent line to the curve $y = {x^2} - 2x + 7$, which is

(a) parallel to the line $2x - y + 9 = 0$

(b) perpendicular to the line $5y - 15x = 13$. SA Q16 Show that the tangents to the curve $y = 7{x^3} + 11$ at the points where $x = 2$ and $x = - 2$ are parallel. SA Q17 Find the points on the curve $y = {x^3}$ at which the slope of the tangent is equal to the $y$-coordinate of the point. SA Q18 For the curve$y = 4{x^3} - 2{x^5}$, find all the points at which the tangent passes through the origin. SA Q19 Find the points on the curve ${x^2} + {y^2} - 2x - 3 = 0$ at which the tangents are parallel to the $x -$ axis. SA Q20 Find the equation of the normal at the point $\left( {a{m^2},\,a{m^3}} \right)$ for the curve $a{y^2} = {x^3}$. SA Q21 Find the equation of the normals to the curve $y = {x^3} + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$. SA Q22 Find the equations of the tangent and normal to the parabola ${y^2} = 4ax$ at the point $(a{t^2},2at)$ SA Q23 Prove that the curves $x = {y^2}and\,\,xy = k$ cut at right angles, if $8{k^2} = 1.$ SA Q24 Find the equations of the tangent and normal to the hyperbola $\cfrac{{{x^2}}}{{{a^2}}} - \cfrac{{{y^2}}}{{{b^2}}} = 1$ at the point $({x_0},{y_0}).$ SA Q25 Find the equation of the tangent to the curve $y = \sqrt {3x - 2}$ which is parallel to the line $4x - 2y + 5 = 0.$ SA Q26 The slope of the normal to the curve $y = 2{x^2} + 3\sin x$ at $x = 0$ is

(A) $3$

(B) $\cfrac{1}{3}$

(C) $- 3$

(D) $- \cfrac{1}{3}$ SA Q27 The line $y = x + 1$ is a tangent to the curve ${y^2} = 4x$ at the point

(A) $(1,\;2)$

(B) $(2,1)$

(C) $(1,\; - 2)$

(D) $( - 1,2)$ SA

Q1 Using differentials, find the approximate value of each of the following up to 3 places of decimal.

(i) $\sqrt {25.3}$

(ii) $\sqrt {49.5}$

(iii) $\sqrt {0.6}$

(iv) ${(0.009)^{1/3}}$

(v) ${(0.999)^{1/10}}$

(vi) ${(15)^{114}}$

(vii) ${(26)^{1/3}}$

(viii) ${(255)^{1/4}}$

(ix) ${(82)^{1/4}}$

(x) ${(401)^{1/2}}$

(xi) ${(0.0037)^{1/2}}$

(xii) ${(26.57)^{1/3}}$

(xiii) ${(81.5)^{1/4}}$

(xiv) ${(3.968)^{3/2}}$

(xv) ${(32.15)^{1/5}}$ SA Q2 Find the approximate value of $f\left( {2.01} \right)$, where $f\left( x \right) = 4{x^2} + 5x + 2$. SA Q3 Find the approximate value of $f\left( {5.001} \right)$, where $f\left( x \right) = {x^3} - 7{x^2} + 15$. SA Q4 Find the approximate change in the volume $V$ of a cube of side $x$ metres caused by increasing the side by $1\%$. SA Q5 Find the approximate change in the surface area of a cube of side $x$ metres caused by decreasing the side by $1\%$. SA Q6 If the radius of a sphere is measured as $7$m with an error of $0.02$m, then find the approximate error in calculating its volume. SA Q7 If the radius of a sphere is measured as $9{\rm{ m}}$ with an error of $0.03{\rm{ m}}$, then find the approximate error in calculating its surface area. SA Q8 If $f\left( x \right) = 3{x^2} + 15x + 5$, then the approximate value of $f\left( {3.02} \right)$ is

(A) $47.66$

(B) $57.66$

(C) $67.66$

(D) $77.66$ SA Q9 The approximate change in the volume of a cube of side $x$ metres caused by increasing the side by $3\%$ is

(A) $0.06{\rm{ }}{x^3}{{\rm{m}}^3}$

(B) $0.6{x^3}{{\rm{m}}^3}$

(C) $0.09{x^3}{{\rm{m}}^3}$

(D) $0.9{x^3}{{\rm{m}}^3}$ SA

Q1 Find the maximum and minimum values, if any, of the following functions given by

(i) $f(x) = {\left( {2x - 1} \right)^2} + 3$

(ii) $f\left( x \right) = 9{x^2} + 12x + 2$

(iii) $f\left( x \right) = - {\left( {x - {\rm{1}}} \right)^2} + 10$

(iv) $g\left( x \right) = {x^3} + 1$ SA Q2 Find-the maximum and minimum values, if any, of the following functions given by

(i) $f(x\rangle = |x + 2| - 1$

(ii) $g(x) = - |x + 1| + 3$

(iii) $h(x) = \sin (2x) + 5$

(iv) $f(x\rangle = |\sin 4x + 3|$

(v) $h(x) = x + 1,x \in ( - 1,1)$ SA Q3 Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

(i) $f(x) = {x^2}$

(ii) $g(x) = {x^3} - 3x$

(iii) $h(x) = \sin x + \cos x,0 < x < \cfrac{\pi }{2}$

(iv) $f(x) = \sin x - \cos x,0 < x < 2\pi$

(v) $f(x) = {x^3} - 6{x^2} + 9x + 15$

(vi) $g(x) = \cfrac{x}{2} + \cfrac{2}{x},x > 0$

(vii) $g(x) = \cfrac{1}{{{x^2} + 2}}$

(viii) $f(x) = x\sqrt {1 - x} ,x > 0$ SA Q4 Prove that the following functions do not have maxima or minima:

(i) $f(x) = {e^x}$

(ii) $g(x) = \log x$

(iii) $h(x) = {x^3} + {x^2} + x + 1$ SA Q5 Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i) $f\left( x \right) = {x^3},x \in [ - 2,2]$

(ii) $f\left( x \right) = \sin x + \cos x,x \in [0,\pi ]$

(iii) $f\left( x \right) = 4x - \cfrac{1}{2}{x^2},x \in \left[ { - 2,\cfrac{9}{2}} \right]$

(iv) $f\left( x \right) = {(x - 1)^2} + 3,x \in [ - 3,1]$ SA Q6 Find the maximum profit that a company can make , if the profit function is given by $p\left( x \right) = 41 - 72x - 18{x^2}$ . SA Q7 Find both the maximum value and the minimum value of $3{x^4} - 8{x^3} + 12{x^2} - 48x + 25$ on the interval $\left[ {0,3} \right]$. SA Q8 At what points in the interval $\left[ {0,2\pi } \right]$, does the function $\sin 2x$ attain its maximum value? SA Q9 What is the maximum value of the function $\sin x + \cos x$ ? SA Q10 Find the maximum value of $2{x^3} - 24x + 107$ in the interval $\left[ {1,{\rm{ }}3} \right]$. Find the maximum value of the same function in $[ - 3,\; - 1]$. SA Q11 It is given that at $x = 1$, the function ${x^4} - 63{x^2} + ax + 9$ attains its maximum value, on the interval $\left[ {0,2} \right]$. Find the value of $a$. SA Q12 Find the maximum and minimum values of $x + \sin 2x$ on $[0,2\pi ]$. SA Q13 Find two numbers whose sum is $24$ and whose product is as large as possible. SA Q14 Find two positive numbers $x$ and $y$ such that $x + y = 60$ and $x{y^3}$ is maximum. SA Q15 Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product ${x^2}{y^5}$ is maximum. SA Q16 Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum. SA Q17 A square piece of tin of side $18{\rm{ cm}}$ is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? SA Q18 A rectangular sheet of tin $45{\rm{ cm by }}24{\rm{ cm}}$ is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum? SA Q19 Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area. SA Q20 Show that the right circular cylinder of given surface and maximum volume is such that height is equal to the diameter of the base. SA Q21 Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, of the dimensions of the can which has the minimum surface area. SA Q22 A wire of length $28{\rm{ m}}$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? SA Q23 Prove that the volume of the largest cone that can be inscribed in a sphere of radius $R$ is $8/27$ of the volume of the sphere. SA Q24 Show that the right circular cone of least curved surface and given volume has an altitude equal to $\sqrt 2$ times the radius of the base. SA Q25 Show that the semi-vertical angle of the cone of maximum volume and of given slant height is ${\tan ^{ - 1}}\sqrt 2$. SA Q26 Show that the semi-vertical angle of right circular cone of given surface area and maximum volume is ${\sin ^{ - 1}}\left( {\cfrac{1}{3}} \right)$

. SA Q27 The point on the curve ${x^2} = 2y$ which is nearest to the point $(0,5)$ is

(A) $(2\sqrt 2 ,4)$

(B) $(2\sqrt 2 ,0)$

(C) $(0,0)$

(D) $(2,2)$ SA Q28 For all real values of$x$, the minimum value of $\cfrac{{1 - x + {x^2}}}{{1 + x + {x^2}}}$ is

(A) $0$

(B) $1$

(C) $3$

(D) $(1/3)$ SA Q29 The maximum value of ${[x(x - 1) + 1]^{1/3}},0 \le x \le 1$ is

(A) ${\left( {\cfrac{1}{3}} \right)^{1/3}}$

(B) $\cfrac{1}{2}$

(C) $1$

(D) $0$ SA

Q1 Using differentials, find the approximate value of each of the following :

(a) ${\left( {\cfrac{{17}}{{81}}} \right)^{1/4}}$

( b) ${(33)^{1/5}}$ SA Q2 Show that the function given by $f(x) = \cfrac{{\log x}}{x}$ has maximum at $x = e$. SA Q3 The two equal sides of an isosceles triangle with fixed base $b$ are decreasing at rate of $3$ cm per second. How fast is the area decreasing when the two equal sides are equal to the base? SA Q4 Find the equation of the normal to curve ${y^2} = 4x$ at the point $(1,2)$. SA Q5 Show that the normal at any point $\theta$ to the curve ` $x = a\cos \theta + a\theta \sin \theta ,y = a\sin \theta - a\theta \cos \theta$ is at a constant distance from the origin. SA Q6 Find the intervals in which the function $f$ given by $f(x) = \cfrac{{4\sin x - 2x - x\cos x}}{{2 + \cos x}}$ is

(i) increasing

(ii) decreasing SA Q7 Find the intervals in which the function $f$ given by
$f(x) = {x^3} + \cfrac{1}{{{x^3}}},x \ne 0$ is

(i) increasing

(ii) decreasing. SA Q8 Find the maximum area of an isosceles triangle inscribed in the ellipse $\cfrac{{{x^2}}}{{{a^2}}} + \cfrac{{{y^2}}}{{{b^2}}} = 1$ with its vertex at one end of the major axis. SA Q9 A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2m$ and volume is $8{m^3}$. If building of tank costs $Rs.70$ per sq. metre for the base and $Rs.45$ per square metre for sides. What is the cost of least expensive tank? SA Q10 The sum of the perimeter of a circle and square is $k$, where $k$ is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. SA Q11 A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10m$. Find the dimensions of the window to admit maximum light through the whole opening. SA Q12 A point on the hypotenuse of a right triangle is at distances $a$ and $b$ from the sides of the triangle. Show that the minimum length of the hypotenuse is ${({a^{2/3}} + {b^{2/3}})^{3/2}}.$ SA Q13 Find the points at which the function $f$ given by $f(x) = {(x - 2)^4}{(x + 1)^3}$ has

(i) local maxima

(ii) local minima

(iii) point of inflexion. SA Q14 Find the absolute maximum and minimum values of the function $f$ given by $f(x) = {\cos ^2}x + \sin x,x \in [0,\;\pi ].$ SA Q15 Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$is $\cfrac{{4r}}{3}$. SA Q16 . Lel $f$ be a function defined on $[a,\;b]$ such that $f'(x) > 0,$ for all $x \in (a,b).$ Then, prove that $f$ is an increasing function on $(a,b).$ SA Q17 Show that the height of the cylinder of maximum voIume that can be inscribed in a sphere of radius $R$ is $\cfrac{{2R}}{{\sqrt 3 }}$. Also, find the maximum volume. SA Q18 Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi vertical, angle $\alpha$ is one-third that of the cone and the greatest volume of cylinder is $\cfrac{4}{{27}}\pi {h^3}{\tan ^2}a.$ SA Q19 A cylindrical tank of radius $10$m is being filled with wheat at the rate of $314$ cubic metres per hour. Then the depth of the wheat is increasing at the rate of

(A) $1\,\,m/h$

(B) $0.1\,\,m/h$

(C) $1.1\,\,m/h$

(D) $0.5\,\,m/h$ SA Q20 The slope of the tangent to the curve $x = {t^2} + 3t - 8,y = 2{t^2} - 2t - 5$ at the point $(2, - 1)$ is

(A) $\cfrac{{22}}{7}$

(B) $\cfrac{6}{7}$

(C)$\cfrac{7}{6}$

(D)$\cfrac{{ - 6}}{7}$ SA Q21 The line $y = mx + 1$ is a tangent to the curve ${y^2} = 4x$ if the value of $m$ is

(A) $1$

(B) $2$

(C) $3$

(D) $112$ SA Q22 The normal at the point $(1,\;1)$ on the curve $2y + {x^2} = 3$ is

(A) $x + y = 0$

(B) $x - y = 0$

(C) $x + y + 1 = 0$

(D) $- x + y + 2 = 0$ SA Q23 The normal to the curve ${x^2} = 4y$ passing $(1,2)$ is

(A) $x + y = 3$

(B) $x - y = 3$

(C) $x + y = 1$

(D) $x - y = 1$ SA Q24 The points of the curve $9{y^2} = {x^3}$, where the normal to the curve makes equal intercepts with the axes are

(A) $\left( {4,\; \pm \cfrac{8}{3}} \right)$

(B) $\left( {4,\;\cfrac{{ - 8}}{3}} \right)$

(C) $\left( {4,\; \pm \cfrac{3}{8}} \right)$

(D) $\left( { \pm 4,\;\cfrac{3}{8}} \right)$ SA