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Mathematics · Textbook solutions

Application of Derivatives

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 161 questions

  1. Find the equations of tangent and normal to the curve at the given point on it.
    2.1.2 SolvedEx.1 (i)
    y=2x3−x2+2y = 2x^3 - x^2 + 2 at (12,2)\left(\frac{1}{2}, 2\right)
  2. 2.1.2 SolvedEx.1 (ii)
    x3+2x2y−9xy=−2x^3 + 2x^2 y - 9xy = -2 at (2,1)(2, 1)
  3. 2.1.2 SolvedEx.1 (iii)
    x=2sin⁡3θx = 2 \sin^3 \theta, y=3cos⁡3θy = 3 \cos^3 \theta at θ=π4\theta = \frac{\pi}{4}
  4. 2.1.2 SolvedEx.2
    Find points on the curve given by y=x3−6x2+x+3y = x^3 - 6x^2 + x + 3 where the tangents are parallel to the line y=x+5y = x + 5.
  1. 2.1.3 SolvedEx.1
    A stone is dropped in to a quiet lake and waves in the form of circles are generated, radius of the circular wave increases at the rate of 5 cm/ sec. At the instant when the radius of the circular wave is 8 cm, how fast the area enclosed is increasing ?
  2. 2.1.3 SolvedEx.2
    The volume of the spherical ball is increasing at the rate of 4π4\pi cc/sec. Find the rate at which the radius and the surface area are changing when the volume is 288π288\pi cc.
  3. 2.1.3 SolvedEx.3
    Water is being poured at the rate of 36 m3^{3}/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising.
  4. 2.1.3 SolvedEx.4
    A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which (i) his shadow is lengthening. (ii) the tip of the shadow is moving.
  1. 2.1.4 SolvedEx.1
    A car is moving in such a way that the distance it covers, is given by the equation s=4t2+3ts = 4t^2 + 3t where ss is in meters and tt is in seconds. What would be the velocity and the acceleration of the car at time t=20t = 20 second ?
  2. 2.1.4 SolvedEx.2
    The displacement of a particle at time tt is given by s=2t3−5t2+4t−3s = 2t^3 - 5t^2 + 4t - 3. Find the time when the acceleration is 14 ft/ sec2^2, the velocity and the displacement at that time.
  1. Find the equations of tangents and normals to the curve at the point on it.
    Ex 2.1 Q.1 (i)
    y=x2+2ex+2y = x^2 + 2e^x + 2 at (0,4)(0, 4)
  2. Ex 2.1 Q.1 (ii)
    x3+y3−9xy=0x^3 + y^3 - 9xy = 0 at (2,4)(2, 4)
  3. Ex 2.1 Q.1 (iii)
    x2−3xy+2y2=5x^2 - \sqrt{3}xy + 2y^2 = 5 at (3,2)(\sqrt{3}, 2)
  4. Ex 2.1 Q.1 (iv)
    2xy+πsin⁡y=2π2xy + \pi \sin y = 2\pi at (1,π2)\left(1, \dfrac{\pi}{2}\right)
  5. Ex 2.1 Q.1 (v)
    xsin⁡2y=ycos⁡2xx \sin 2y = y \cos 2x at (π4,π2)\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)
  6. Ex 2.1 Q.1 (vi)
    x=sin⁡θx = \sin \theta and y=cos⁡2θy = \cos 2\theta at θ=π6\theta = \dfrac{\pi}{6}
  7. Ex 2.1 Q.1 (vii)
    x=tx = \sqrt{t}, y=t−1ty = t - \dfrac{1}{\sqrt{t}} at t=4t = 4.
  8. Ex 2.1 Q.2
    Find the point on the curve y=x−3y = \sqrt{x - 3} where the tangent is perpendicular to the line 6x+3y−5=06x + 3y - 5 = 0.
  9. Ex 2.1 Q.3
    Find the points on the curve y=x3−2x2−xy = x^3 - 2x^2 - x where the tangents are parallel to 3x−y+1=03x - y + 1 = 0.
  10. Ex 2.1 Q.4
    Find the equations of the tangents to the curve x2+y2−2x−4y+1=0x^2 + y^2 - 2x - 4y + 1 = 0 which are parallel to the X-axis.
  11. Ex 2.1 Q.5
    Find the equations of the normals to the curve 3x2−y2=83x^2 - y^2 = 8, which are parallel to the line x+3y=4x + 3y = 4.
  12. Ex 2.1 Q.6
    If the line y=4x−5y = 4x - 5 touches the curve y2=ax3+by^2 = ax^3 + b at the point (2,3)(2, 3) find aa and bb.
  13. Ex 2.1 Q.7
    A particle moves along the curve 6y=x3+26y = x^3 + 2 Find the points on the curve at which y-coordinate is changing 8 times as fast as the X-coordinate.
  14. Ex 2.1 Q.8
    A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.020.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?
  15. Ex 2.1 Q.9
    The surface area of a spherical balloon is increasing at the rate of 2 cm2^2/ sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?
  16. Ex 2.1 Q.10
    If each side of an equilateral triangle increases at the rate of 2\sqrt{2} cm/ sec, find the rate of increase of its area when its side of length 3 cm .
  17. Ex 2.1 Q.11
    The volume of a sphere increase at the rate of 20 cm3^3/ sec. Find the rate of change of its surface area when its radius is 5 cm.
  18. Ex 2.1 Q.12
    The edge of a cube is decreasing at the rate of 0.60.6 cm/sec. Find the rate at which its volume is decreasing when the edge of the cube is 2 cm.
  19. Ex 2.1 Q.13
    A man of height 2 meters walks at a uniform speed of 6 km/hr away from a lamp post of 6 meters high. Find the rate at which the length of the shadow is increasing.
  20. Ex 2.1 Q.14
    A man of height 1.51.5 meters walks toward a lamp post of height 4.54.5 meters, at the rate of (34)\left(\dfrac{3}{4}\right) meter/sec. Find the rate at which (i) his shadow is shortening. (ii) the tip of the shadow is moving.
  21. Ex 2.1 Q.15
    A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.21.2 meters per second, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall.
  22. Ex 2.1 Q.16
    If water is poured into an inverted hollow cone whose semi-vertical angel is 30∘30^\circ, so that its depth (measured along the axis) increases at the rate of 1 cm/ sec. Find the rate at which the volume of water increasing when the depth is 2 cm.
  1. 2.2.1 SolvedEx.1
    Find the approximate value of 64.1\sqrt{64.1}.
  2. 2.2.1 SolvedEx.2
    Find the approximate value of (3.98)3(3.98)^3.
  3. 2.2.1 SolvedEx.3
    Find the approximate value of sin⁡(30∘ 30′)\sin(30^\circ\, 30'). Given that 1∘=0.0175c1^\circ = 0.0175^c and cos⁡30∘=0.866\cos 30^\circ = 0.866.
  4. 2.2.1 SolvedEx.4
    Find the approximate value of tan⁡−1(0.99)\tan^{-1}(0.99), Given that π≈3.1416\pi \approx 3.1416.
  5. 2.2.1 SolvedEx.5
    Find the approximate value of e1.005e^{1.005}. Given that e=2.7183e = 2.7183.
  6. 2.2.1 SolvedEx.6
    Find the approximate value of log⁡10(998)\log_{10}(998). Given that log⁡10e=0.4343\log_{10} e = 0.4343.
  7. 2.2.1 SolvedEx.7
    Find the approximate value of f(x)=x3+5x2−2x+3f(x) = x^3 + 5x^2 - 2x + 3 at x=1.98x = 1.98.
  1. Find the approximate value of given functions, at required points.
    Ex 2.2 Q.1 (i)
    8.95\sqrt{8.95}
  2. Ex 2.2 Q.1 (ii)
    283\sqrt[3]{28}
  3. Ex 2.2 Q.1 (iii)
    31.985\sqrt[5]{31.98}
  4. Ex 2.2 Q.1 (iv)
    (3.97)4(3.97)^4
  5. Ex 2.2 Q.1 (v)
    (4.01)3(4.01)^3
  6. Find the approximate value of
    Ex 2.2 Q.2 (i)
    sin⁡(61∘)\sin(61^\circ) given that 1∘=0.0175c1^\circ = 0.0175^c, 3=1.732\sqrt{3} = 1.732
  7. Ex 2.2 Q.2 (ii)
    sin⁡(29∘ 30′)\sin(29^\circ\,30') given that 1∘=0.0175c1^\circ = 0.0175^c, 3=1.732\sqrt{3} = 1.732
  8. Ex 2.2 Q.2 (iii)
    cos⁡(60∘ 30′)\cos(60^\circ\,30') given that 1∘=0.0175c1^\circ = 0.0175^c, 3=1.732\sqrt{3} = 1.732
  9. Ex 2.2 Q.2 (iv)
    tan⁡(45∘ 40′)\tan(45^\circ\,40') given that 1∘=0.0175c1^\circ = 0.0175^c.
  10. Find the approximate value of
    Ex 2.2 Q.3 (i)
    tan⁡−1(0.999)\tan^{-1}(0.999)
  11. Ex 2.2 Q.3 (ii)
    cot⁡−1(0.999)\cot^{-1}(0.999)
  12. Ex 2.2 Q.3 (iii)
    tan⁡−1(1.001)\tan^{-1}(1.001)
  13. Find the approximate value of
    Ex 2.2 Q.4 (i)
    e0.995e^{0.995}
  14. Ex 2.2 Q.4 (ii)
    e2.1e^{2.1} given that e2=7.389e^2 = 7.389
  15. Ex 2.2 Q.4 (iii)
    32.013^{2.01} given that log⁡3=1.0986\log 3 = 1.0986
  16. Find the approximate value of
    Ex 2.2 Q.5 (i)
    log⁡e(101)\log_e(101) given that log⁡e10=2.3026\log_e 10 = 2.3026
  17. Ex 2.2 Q.5 (ii)
    log⁡e(9.01)\log_e(9.01) given that log⁡3=1.0986\log 3 = 1.0986
  18. Ex 2.2 Q.5 (iii)
    log⁡10(1016)\log_{10}(1016) given that log⁡10e=0.4343\log_{10} e = 0.4343
  19. Find the approximate value of
    Ex 2.2 Q.6 (i)
    f(x)=x3−3x+5f(x) = x^3 - 3x + 5 at x=1.99x = 1.99
  20. Ex 2.2 Q.6 (ii)
    f(x)=x3+5x2−7x+10f(x) = x^3 + 5x^2 - 7x + 10 at x=1.12x = 1.12
  1. Check whether conditions of Rolle's theorem are satisfied by the following functions.
    2.3.1 SolvedEx.1 (i)
    f(x)=2x3−5x2+3x+2f(x) = 2x^3 - 5x^2 + 3x + 2, x∈[0,32]x \in \left[0, \dfrac{3}{2}\right]
  2. 2.3.1 SolvedEx.1 (ii)
    f(x)=x2−2x+3f(x) = x^2 - 2x + 3, x∈[1,4]x \in [1, 4]
  3. 2.3.1 SolvedEx.2
    Verify Rolle's theorem for the function f(x)=x2−4x+10f(x) = x^2 - 4x + 10 on [0,4][0, 4].
  4. 2.3.1 SolvedEx.3
    Given an interval [a,b][a, b] that satisfies hypothesis of Rolle's theorem for the function f(x)=x3−2x2+3f(x) = x^3 - 2x^2 + 3. It is known that a=0a = 0. Find the value of bb.
  5. 2.3.1 SolvedEx.4
    Verify Rolle's theorem for the function f(x)=ex(sin⁡x−cos⁡x)f(x) = e^x(\sin x - \cos x) on [π4,5π4]\left[\dfrac{\pi}{4}, \dfrac{5\pi}{4}\right].
  1. 2.3.2 SolvedEx.1
    Verify Lagrange's mean value theorem for the function f(x)=x+4f(x) = \sqrt{x + 4} on the interval [0,5][0, 5].
  2. 2.3.2 SolvedEx.2
    Verify Lagrange's mean value theorem for the function f(x)=x+1xf(x) = x + \dfrac{1}{x} on the interval [1,3][1, 3].
  1. Check the validity of the Rolle's theorem for the following functions.
    Ex 2.3 Q.1 (i)
    f(x)=x2−4x+3f(x) = x^2 - 4x + 3, x∈[1,3]x \in [1, 3]
  2. Ex 2.3 Q.1 (ii)
    f(x)=e−xsin⁡xf(x) = e^{-x} \sin x, x∈[0,π]x \in [0, \pi]
  3. Ex 2.3 Q.1 (iii)
    f(x)=2x2−5x+3f(x) = 2x^2 - 5x + 3, x∈[1,3]x \in [1, 3]
  4. Ex 2.3 Q.1 (iv)
    f(x)=sin⁡x−cos⁡x+3f(x) = \sin x - \cos x + 3, x∈[0,2π]x \in [0, 2\pi]
  5. Ex 2.3 Q.1 (v)
    f(x)=x2f(x) = x^2 if 0≤x≤20 \leq x \leq 2 =6−x= 6 - x if 2≤x≤62 \leq x \leq 6
  6. Ex 2.3 Q.1 (vi)
    f(x)=x23f(x) = x^{\frac{2}{3}}, x∈[−1,1]x \in [-1, 1]
  7. Ex 2.3 Q.2
    Given an interval [a,b][a, b] that satisfies hypothesis of Rolle's thorem for the function f(x)=x4+x2−2f(x) = x^4 + x^2 - 2. It is known that a=−1a = -1. Find the value of bb.
  8. Verify Rolle's theorem for the following functions.
    Ex 2.3 Q.3 (i)
    f(x)=sin⁡x+cos⁡x+7f(x) = \sin x + \cos x + 7, x∈[0,2π]x \in [0, 2\pi]
  9. Ex 2.3 Q.3 (ii)
    f(x)=sin⁡(x2)f(x) = \sin \left( \frac{x}{2} \right), x∈[0,2π]x \in [0, 2\pi]
  10. Ex 2.3 Q.3 (iii)
    f(x)=x2−5x+9f(x) = x^2 - 5x + 9, x∈[1,4]x \in [1, 4]
  11. Ex 2.3 Q.4
    If Rolle's theorem holds for the function f(x)=x3+px2+qx+5f(x) = x^3 + px^2 + qx + 5, x∈[1,3]x \in [1, 3] with c=2+13c = 2 + \frac{1}{\sqrt{3}}, find the values of pp and qq.
  12. Ex 2.3 Q.5
    Rolle's theorem holds for the function f(x)=(x−2)log⁡xf(x) = (x - 2) \log x, x∈[1,2]x \in [1, 2], show that the equation xlog⁡x=2−xx \log x = 2 - x is satisfied by at least one value of xx in (1,2)(1, 2).
  13. Ex 2.3 Q.6
    The function f(x)=x(x+3)e−x2f(x) = x (x + 3) e^{-\frac{x}{2}} satisfies all the conditions of Rolle's theorem on [−3,0][-3, 0]. Find the value of cc such that f′(c)=0f'(c) = 0.
  14. Verify Lagrange's mean value theorem for the following functions.
    Ex 2.3 Q.7 (i)
    f(x)=log⁡xf(x) = \log x, on [1,e][1, e]
  15. Ex 2.3 Q.7 (ii)
    f(x)=(x−1)(x−2)(x−3)f(x) = (x - 1) (x - 2) (x - 3) on [0,4][0, 4]
  16. Ex 2.3 Q.7 (iii)
    f(x)=x2−3x−1f(x) = x^2 - 3x - 1, x∈[−117,137]x \in \left[ -\frac{11}{7}, \frac{13}{7} \right]
  17. Ex 2.3 Q.7 (iv)
    f(x)=2x−x2f(x) = 2x - x^2, x∈[0,1]x \in [0, 1]
  18. Ex 2.3 Q.7 (v)
    f(x)=x−1x−3f(x) = \frac{x - 1}{x - 3} on [4,5][4, 5]
  1. 2.4.1 SolvedEx.1
    Show that the function f(x)=x3+10x+7f(x) = x^3 + 10x + 7 for x∈Rx \in R is strictly increasing.
  2. 2.4.1 SolvedEx.2
    Test whether the function f(x)=x3+6x2+12x−5f(x) = x^3 + 6x^2 + 12x - 5 is increasing or decreasing for all x∈Rx \in R.
  3. 2.4.1 SolvedEx.3
    Find the values of xx, for which the funciton f(x)=x3+12x2+36x+6f(x) = x^3 + 12x^2 + 36x + 6 is (i) monotonically increasing. (ii) monotonically decreasing.
  1. 2.4.3 SolvedEx.1
    Find the local maxima or local minima of f(x)=x3−3xf(x) = x^3 - 3x.
  1. 2.4.4 SolvedEx.1
    Find the local maximum and local minimum value of f(x)=x3−3x2−24x+5f(x) = x^3 - 3x^2 - 24x + 5.
  2. 2.4.4 SolvedEx.2
    A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
  3. 2.4.4 SolvedEx.3
    A Rectangular sheet of paper has it area 24 sq. meters. The margin at the top and the bottom are 75 cm each and at the sides 50 cm each. What are the dimensions of the paper, if the area of the printed space is maximum ?
  4. 2.4.4 SolvedEx.4
    An open box is to be cut out of piece of square card of side 18 cm by cutting of equal squares from the corners and turning up the sides. Find the maximum volume of the box.
  5. 2.4.4 SolvedEx.5
    Two sides of a triangle are given, find the angle between them such that the area of the triangle is maximum.
  6. 2.4.4 SolvedEx.6
    The slant side of a right circular cone is ll. Show that the semi-vertical angle of the cone of maximum volume is tan⁡−1(2)\tan^{-1}(\sqrt{2}).
  7. 2.4.4 SolvedEx.7
    Find the height of a covered box of fixed volume so that the total surface area of the box is minimum whose base is a rectangle with one side three times as long as the other.
  1. Test whether the following functions are increasing or decreasing.
    Ex 2.4 Q.1 (i)
    f(x)=x3−6x2+12x−16f(x) = x^3 - 6x^2 + 12x - 16, x∈Rx \in \mathbb{R}
  2. Ex 2.4 Q.1 (ii)
    f(x)=2−3x+3x2−x3f(x) = 2 - 3x + 3x^2 - x^3, x∈Rx \in \mathbb{R}
  3. Ex 2.4 Q.1 (iii)
    f(x)=x−1xf(x) = x - \dfrac{1}{x}, x∈Rx \in \mathbb{R} and x≠0x \neq 0
  4. Find the values of xx for which the following functions are strictly increasing -
    Ex 2.4 Q.2 (i)
    f(x)=2x3−3x2−12x+6f(x) = 2x^3 - 3x^2 - 12x + 6
  5. Ex 2.4 Q.2 (ii)
    f(x)=3+3x−3x2+x3f(x) = 3 + 3x - 3x^2 + x^3
  6. Ex 2.4 Q.2 (iii)
    f(x)=x3−6x2−36x+7f(x) = x^3 - 6x^2 - 36x + 7
  7. Find the values of xx for which the following functions are strictly decreasing -
    Ex 2.4 Q.3 (i)
    f(x)=2x3−3x2−12x+6f(x) = 2x^3 - 3x^2 - 12x + 6
  8. Ex 2.4 Q.3 (ii)
    f(x)=x+25xf(x) = x + \dfrac{25}{x}
  9. Ex 2.4 Q.3 (iii)
    f(x)=x3−9x2+24x+12f(x) = x^3 - 9x^2 + 24x + 12
  10. Ex 2.4 Q.4
    Find the values of xx for which the function f(x)=x3−12x2−144x+13f(x) = x^3 - 12x^2 - 144x + 13 (a) Increasing (b) Decreasing
  11. Ex 2.4 Q.5
    Find the values of xx for which f(x)=2x3−15x2−144x−7f(x) = 2x^3 - 15x^2 - 144x - 7 is (a) strictly increasing (b) strictly decreasing
  12. Ex 2.4 Q.6
    Find the values of xx for which f(x)=xx2+1f(x) = \dfrac{x}{x^2 + 1} is (a) strictly increasing (b) strictly decreasing
  13. Ex 2.4 Q.7
    Show that f(x)=3x+13xf(x) = 3x + \dfrac{1}{3x} increasing in (13,1)\left(\dfrac{1}{3}, 1\right) and decreasing in (19,13)\left(\dfrac{1}{9}, \dfrac{1}{3}\right).
  14. Ex 2.4 Q.8
    Show that f(x)=x−cos⁡xf(x) = x - \cos x is increasing for all xx.
  15. Find the maximum and minimum of the following functions -
    Ex 2.4 Q.9 (i)
    y=5x3+2x2−3xy = 5x^3 + 2x^2 - 3x
  16. Ex 2.4 Q.9 (ii)
    f(x)=2x3−21x2+36x−20f(x) = 2x^3 - 21x^2 + 36x - 20
  17. Ex 2.4 Q.9 (iii)
    f(x)=x3−9x2+24xf(x) = x^3 - 9x^2 + 24x
  18. Ex 2.4 Q.9 (iv)
    f(x)=x2+16x2f(x) = x^2 + \dfrac{16}{x^2}
  19. Ex 2.4 Q.9 (v)
    f(x)=xlog⁡xf(x) = x \log x
  20. Ex 2.4 Q.9 (vi)
    f(x)=log⁡xxf(x) = \dfrac{\log x}{x}
  21. Ex 2.4 Q.10
    Divide the number 30 in to two parts such that their product is maximum.
  22. Ex 2.4 Q.11
    Divide the number 20 in to two parts such that sum of their squares is minimum.
  23. Ex 2.4 Q.12
    A wire of length 36 meter is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
  24. Ex 2.4 Q.13
    A ball is thrown in the air. Its height at any time t is given by h=3+14t−5t2h = 3 + 14t - 5t^2. Find the maximum height it can reach.
  25. Ex 2.4 Q.14
    Find the largest size of a rectangle that can be inscribed in a semi circle of radius 1 unit, So that two vertices lie on the diameter.
  26. Ex 2.4 Q.15
    An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of πa3\pi a^3 cu. cm of water. Find the dimensions so that sheet required is minimum.
  27. Ex 2.4 Q.16
    The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area ?
  28. Ex 2.4 Q.17
    A box with a square base is to have an open top. The surface area of the box is 192 sq.cm. What should be its dimensions in order that the volume is largest ?
  29. Ex 2.4 Q.18
    The profit function P(x)P(x) of a firm, selling xx items per day is given by P(x)=(150−x)x−1625P(x) = (150 - x)x - 1625. Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
  30. Ex 2.4 Q.19
    Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
  31. Ex 2.4 Q.20
    Show that among rectangles of given area, the square has the least perimeter.
  32. Ex 2.4 Q.21
    Show that the height of a closed right circular cylinder, of a given volume and least surface area, is equal to its diameter.
  33. Ex 2.4 Q.22
    Find the volume of the largest cylinder that can be inscribed in a sphere of radius 'r' cm.
  34. Ex 2.4 Q.23
    Show that y=log⁡(1+x)−2x2+xy = \log(1 + x) - \dfrac{2x}{2 + x}, x>−1x > -1 is an increasing function on its domain.
  35. Ex 2.4 Q.24
    Prove that y=4sin⁡θ2+cos⁡θ−θy = \dfrac{4 \sin \theta}{2 + \cos \theta} - \theta is an increasing function of θ∈[0,π2]\theta \in \left[0, \dfrac{\pi}{2}\right].
  1. Misc I Q.1
    If the function f(x)=ax3+bx2+11x−6f(x) = ax^3 + bx^2 + 11x - 6 satisfies conditions of Rolle's theorem in [1,3][1, 3] and f′(2+13)=0f'\left(2 + \frac{1}{\sqrt{3}}\right) = 0, then values of aa and bb are respectively.
    1. A.
      1,−61, -6
    2. B.
      −2,1-2, 1
    3. C.
      −1,−6-1, -6
    4. D.
      −1,6-1, 6
  2. Misc I Q.2
    If f(x)=x2−1x2+1f(x) = \frac{x^2 - 1}{x^2 + 1}, for every real xx, then the minimum value of ff is -
    1. A.
      11
    2. B.
      00
    3. C.
      −1-1
    4. D.
      22
  3. Misc I Q.3
    A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 1.5 m/ sec. The length of the higher point of ladder when the foot of the ladder is 4.0 m away from the wall decreases at the rate of
    1. A.
      11
    2. B.
      22
    3. C.
      2.52.5
    4. D.
      33
  4. Misc I Q.4
    Let f(x)f(x) and g(x)g(x) be differentiable for 0<x<10 < x < 1 such f(0)=0f(0) = 0, g(0)=0g(0) = 0, f(1)=6f(1) = 6. Let there exist a real number c in (0,1)(0, 1) such that f′(c)=2g′(c)f'(c) = 2g'(c), then the value of g(1)g(1) must be
    1. A.
      11
    2. B.
      33
    3. C.
      2.52.5
    4. D.
      −1-1
  5. Misc I Q.5
    If f(x)=x3−6x2+9x+18f(x) = x^3 - 6x^2 + 9x + 18, then f(x)f(x) is strictly decreasing in -
    1. A.
      (−∞,1)(-\infty, 1)
    2. B.
      [3,∞)[3, \infty)
    3. C.
      (−∞,1]∪[3,∞)(-\infty, 1] \cup [3, \infty)
    4. D.
      (1,3)(1, 3)
  6. Misc I Q.6
    If x=−1x = -1 and x=2x = 2 are the extreme points of y=αlog⁡x+βx2+xy = \alpha \log x + \beta x^2 + x then
    1. A.
      α=−6,β=12\alpha = -6, \beta = \frac{1}{2}
    2. B.
      α=−6,β=−12\alpha = -6, \beta = -\frac{1}{2}
    3. C.
      α=2,β=−12\alpha = 2, \beta = -\frac{1}{2}
    4. D.
      α=2,β=12\alpha = 2, \beta = \frac{1}{2}
  7. Misc I Q.7
    The normal to the curve x2+2xy−3y2=0x^2 + 2xy - 3y^2 = 0 at (1,1)(1, 1)
    1. A.
      Meets the curve again in second quadrant.
    2. B.
      Does not meet the curve again.
    3. C.
      Meets the curve again in third quadrant.
    4. D.
      Meets the curve again in fourth quadrant.
  8. Misc I Q.8
    The equation of the tangent to the curve y=1−ex2y = 1 - e^{\frac{x}{2}} at the point of intersection with Y-axis is
    1. A.
      x+2y=0x + 2y = 0
    2. B.
      2x+y=02x + y = 0
    3. C.
      x−y=2x - y = 2
    4. D.
      x+y=2x + y = 2
  9. Misc I Q.9
    If the tangent at (1,1)(1, 1) on y2=x(2−x)2y^2 = x(2 - x)^2 meets the curve again at P then P is
    1. A.
      (4,4)(4, 4)
    2. B.
      (−1,2)(-1, 2)
    3. C.
      (3,6)(3, 6)
    4. D.
      (94,38)\left(\frac{9}{4}, \frac{3}{8}\right)
  10. Misc I Q.10
    The appoximate value of tan⁡(44∘ 30′)\tan(44^\circ\ 30') given that 1∘=0.01751^\circ = 0.0175.
    1. A.
      0.89520.8952
    2. B.
      0.95280.9528
    3. C.
      0.92850.9285
    4. D.
      0.98250.9825
  1. Misc II Q.1
    If the curves ax2+by2=1ax^2 + by^2 = 1 and a′x2+b′y2=1a'x^2 + b'y^2 = 1 intersect orthogonally, then prove that 1a−1b=1a′−1b′\frac{1}{a} - \frac{1}{b} = \frac{1}{a'} - \frac{1}{b'}.
  2. Misc II Q.2
    Determine the area of the triangle formed by the tangent to the graph of the function y=3−x2y = 3 - x^2 drawn at the point (1,2)(1, 2) and the cordinate axes.
  3. Misc II Q.3
    Find the equation of the tangent and normal drawn to the curve y4−4x4−6xy=0y^4 - 4x^4 - 6xy = 0 at the point M(1,2)M(1, 2).
  4. Misc II Q.4
    A water tank in the form of an inverted cone is being emptied at the rate of 2 cubic feet per second. The height of the cone is 8 feet and the radius is 4 feet. Find the rate of change of the water level when the depth is 6 feet.
  5. Misc II Q.5
    Find all points on the ellipse 9x2+16y2=4009x^2 + 16y^2 = 400, at which the y-coordinate is decreasing and the x-coordinate is increasing at the same rate.
  6. Misc II Q.6
    Verify Rolle's theorem for the function f(x)=2ex+e−xf(x) = \frac{2}{e^x + e^{-x}} on [−1,1][-1, 1].
  7. Misc II Q.7
    The position of a particle is given by the function s(t)=2t2+3t−4s(t) = 2t^2 + 3t - 4. Find the time t=ct = c in the interval 0≤t≤40 \leq t \leq 4 when the instantaneous velocity of the particle equals to its average velocity in this interval.
  8. Misc II Q.8
    Find the approximate value of the function f(x)=x2+3xf(x) = \sqrt{x^2 + 3x} at x=1.02x = 1.02.
  9. Misc II Q.9
    Find the approximate value of cos⁡−1(0.51)\cos^{-1}(0.51) given π=3.1416\pi = 3.1416, 23=1.1547\frac{2}{\sqrt{3}} = 1.1547.
  10. Misc II Q.10
    Find the intervals on which the function y=xxy = x^x, (x>0)(x > 0) is increasing and decreasing.
  11. Misc II Q.11
    Find the intervals on the which the function f(x)=xlog⁡xf(x) = \frac{x}{\log x}, is increasing and decreasing.
  12. Misc II Q.12
    An open box with a square base is to be made out of a given quantity of sheet of area a2a^2, Show the maximum volume of the box is a363\frac{a^3}{6\sqrt{3}}.
  13. Misc II Q.13
    Show that of all rectangles inscribed in a given circle, the square has the maximum area.
  14. Misc II Q.14
    Show that a closed right circular cyclinder of given surface area has maximum volume if its height equals the diameter of its base.
  15. Misc II Q.15
    A window is in the form of a rectangle surmounted by a semi-circle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
  16. Misc II Q.16
    Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
  17. Misc II Q.17
    A wire of length ll is cut in to two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is least, if the radius of the circle is half the side of the square.
  18. Misc II Q.18
    A rectangular sheet of paper of fixed perimeter with the sides having their length in the ratio 8:158 : 15 converted in to an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum valume. Find the lengths of the sides of rectangular sheet of paper.
  19. Misc II Q.19
    Show that the altitude of the right circular cone of maximum volume that can be inscribed in a shpere of radius rr is 4r3\frac{4r}{3}.
  20. Misc II Q.20
    Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R3\frac{2R}{\sqrt{3}}. Also find the maximum volume.
  21. Misc II Q.21
    Find the maximum and minimum values of the function f(x)=cos⁡2x+sin⁡xf(x) = \cos^2 x + \sin x.