application of derivative wa

Mathamtics

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Question bank on Application of DerivativeSelect the correct alternative : (Only one is correct)

Q.1 Suppose x1 & x2 are the point of maximum and the point of minimum respectively of the function

f(x) = 2x3 9 ax2 + 12 a2x + 1 respectively, then for the equality 21x = x2 to be true the value of 'a' must be

(A) 0 (B) 2 (C) 1 (D) 1/4

Q.2 Point 'A' lies on the curve 2xey and has the coordinate (x,

2xe ) where x > 0. Point B has thecoordinates (x, 0). If 'O' is the origin then the maximum area of the triangle AOB is

(A) e2

1(B)

e4

1(C)

e

1(D)

e8

1

Q.3 The angle at which the curve y = KeKx intersects the y-axis is :

(A) tan 1 k2 (B) cot 1 (k2) (C) sec 1 1 4

k (D) none

Q.4 {a1, a2, ....., a4, ......} is a progression where an = n

n

2

3 200 . The largest term of this progression is :

(A) a6 (B) a7 (C) a8 (D) none

Q.5 The angle between the tangent lines to the graph of the function f (x) = x

2

dt)5t2( at the points where

the graph cuts the x-axis is(A) /6 (B) /4 (C) /3 (D) /2

Q.6 The minimum value of the polynomial x(x + 1) (x + 2) (x + 3) is :(A) 0 (B) 9/16 (C) 1 (D) 3/2

Q.7 The minimum value of tan

tan

x

x

6

is :

(A) 0 (B) 1/2 (C) 1 (D) 3

Q.8 The difference between the greatest and the least values of the function, f (x) = sin2x – x on

2,

2

(A) (B) 0 (C) 32

3 (D) 3

2

2

3

Q.9 The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at therate of 0.2 cm/min. The rate of change of the volume of the cylinder, in cm3/min, when the radius is 2 cmand the height is 3 cm is

(A) – 2 (B) – 5

8(C) –

5

3(D)

5

2

Q.10 If a variable tangent to the curve x2y = c3 makes intercepts a, b on x and y axis respectively, then thevalue of a2b is

(A) 27 c3 (B)3c

27

4(C)

3c4

27(D)

3c9

4

Q.11 Difference between the greatest and the least values of the functionf (x) = x(ln x – 2) on [1, e2] is

(A) 2 (B) e (C) e2 (D) 1

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Q.12 Let f (x) =

n2

0r

r

n

xtan

xtan, n N, where x

2,0

(A) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains exactly one integral point.(B) f (x) is bounded and it takes both of it's bounds and the range of f (x) contains more than one integral point.(C) f (x) is bounded but minimum and maximum does not exists.(D) f (x) is not bounded as the upper bound does not exist.

Q.13 If f (x) = x3 + 7x – 1 then f (x) has a zero between x = 0 and x = 1. The theorem which best describesthis, is(A) Squeeze play theorem (B) Mean value theorem(C) Maximum-Minimum value theorem (D) Intermediate value theorem

Q.14 Consider the function f (x) =

0xfor0

0xforx

sinx

then the number of points in (0, 1) where the

derivative f (x) vanishes , is(A) 0 (B) 1 (C) 2 (D) infinite

Q.15 The sum of lengths of the hypotenuse and another side of a right angled triangle is given. The area of thetriangle will be maximum if the angle between them is :

(A) 6

(B) 4

(C) 3

(D) 5

12

Q.16 In which of the following functions Rolle’s theorem is applicable?

(A) f(x) =

1x,0

1x0,x on [0, 1] (B) f(x) =

0x,0

0x,x

xsin

on [–, 0]

(C) f(x) = 1x

6xx2

on [–2,3] (D) f(x) =

1xif6

]3,2[on,1xif1x

6x5x2x 23

Q.17 Suppose that f (0) = – 3 and f ' (x) 5 for all values of x. Then the largest value which f (2) can attain is(A) 7 (B) – 7 (C) 13 (D) 8

Q.18 The tangent to the graph of the function y = f(x) at the point with abscissa x = a forms with the x-axisan angle of /3 and at the point with abscissa x = b at an angle of /4, then the value of the integral,

a

b

f (x) . f (x) dx is equal to

(A) 1 (B) 0 (C) 3 (D) –1[ assume f (x) to be continuous ]

Q.19 Let C be the curve y = x3 (where x takes all real values). The tangent at A meets the curve again at B. Ifthe gradient at B is K times the gradient at A then K is equal to(A) 4 (B) 2 (C) – 2 (D) 1/4

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Q.20 The vertices of a triangle are (0, 0), (x, cos x) and (sin3x, 0) where 0 < x <2

. The maximum area for

such a triangle in sq. units, is

(A) 32

33(B)

32

3(C)

32

4(D)

32

36

Q.21 The subnormal at any point on the curve xyn = an + 1 is constant for :(A) n = 0 (B) n = 1 (C) n = 2 (D) no value of n

Q.22 Equation of the line through the point (1/2, 2) and tangent to the parabola y = x2

2 + 2 and secant to

the curve y = 4 2 x is :

(A) 2x + 2y 5 = 0 (B) 2x + 2y 3 = 0 (C) y 2 = 0 (D) none

Q.23 The lines y = 3

2x and y =

2

5x intersect the curve 3x2

+ 4xy + 5y2 4 = 0 at the points P and Q respectively.The tangents drawn to the curve at P and Q(A) intersect each other at angle of 45º(B) are parallel to each other(C) are perpendicular to each other(D) none of these

Q.24 The least value of 'a' for which the equation,

4 1

1sin sinx x

= a has atleast one solution on the interval (0, /2) is :

(A) 3 (B) 5 (C) 7 (D) 9

Q.25 If f(x) = 4x3 x2 2x + 1 and g(x) = [ Min f t t x x

x x

( ) : ;

;

0 0 1

3 1 2

then

g1

4

+ g

3

4

+ g

5

4

has the value equal to :

(A) 7

4(B)

9

4(C)

13

4(D)

5

2

Q.26 Given : f (x) =

3/2

x2

14

g (x) =

0x,1

0x,x

]x[nta

h (x) = {x} k (x) = )3x(log25

then in [0, 1] Lagranges Mean Value Theorem is NOT applicable to(A) f, g, h (B) h, k (C) f, g (D) g, h, k

Q.27 Two curves C1 : y = x2 – 3 and C2 : y = kx2 , Rk intersect each other at two different points. The

tangent drawn to C2 at one of the points of intersection A (a,y1) , (a > 0) meets C1 again at B(1,y2)

21 yy . The value of ‘a’ is

(A) 4 (B) 3 (C) 2 (D) 1

Q.28 f (x) = dxx1

1

x1

12

22

then f is

(A) increasing in (0, ) and decreasing in (– , 0) (B) increasing in (– , 0) and decreasing in (0, (C) increasing in (– , (D) decreasing in (– ,

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Q.29 The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. Thefraction of width folded over if the area of the folded part is minimum is :(A) 5/8 (B) 2/3 (C) 3/4 (D) 4/5

Q.30 A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy planebounded by the lines y = 0, y = 3x, and y = 30 – 2x. The largest area of such a rectangle is

(A) 8

135(B) 45 (C)

2

135(D) 90

Q.31 Which of the following statement is true for the function

0xx43

x

1x0x

1xx

)x(f

3

3

(A) It is monotonic increasing Rx (B) f (x) fails to exist for 3 distinct real values of x

(C) f (x) changes its sign twice as x varies from (– , )(D) function attains its extreme values at x1 & x2 , such that x1, x2 > 0

Q.32 A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when thedepth of the liquid in it is x cm, the volume of the liquid in it is, x2 (15 x) cu. cm. The length EF is:(A) 7.5 cm (B) 8 cm (C) 10 cm (D) 12 cm

Q.33 Coffee is draining from a conical filter, height and diameter both 15 cms into a cylinderical coffee potdiameter 15 cm. The rate at which coffee drains from the filter into the pot is 100 cu cm /min.The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 10 cm, is

(A) 16

9(B) 9

25(C) 3

5(D) 9

16

Q.34 Let f (x) and g (x) be two differentiable function in R and f (2) = 8, g (2) = 0, f (4) = 10 and g (4) = 8then(A) g ' (x) > 4 f ' (x) x (2, 4) (B) 3g ' (x) = 4 f ' (x) for at least one x (2, 4)(C) g (x) > f (x) x (2, 4) (D) g ' (x) = 4 f ' (x) for at least one x (2, 4)

Q.35 Let m and n be odd integers such that o < m < n. If f(x) = xm

n for x R, then(A) f(x) is differentiable every where (B) f (0) exists(C) f increases on (0, ) and decreases on (– , 0) (D) f increases on R

Q.36 A horse runs along a circle with a speed of 20 km/hr . A lantern is at the centre of the circle . A fence isalong the tangent to the circle at the point at which the horse starts . The speed with which the shadow ofthe horse move along the fence at the moment when it covers 1/8 of the circle in km/hr is(A) 20 (B)40 (C) 30 (D) 60

Q.37 Give the correct order of initials T or F for following statements. Use T if statement is true and F if it isfalse.

Statement-1: If f : R R and c R is such that f is increasing in (c – , c) and f is decreasing in(c, c + ) then f has a local maximum at c. Where is a sufficiently small positive quantity.Statement-2 : Let f : (a, b) R, c (a, b). Then f can not have both a local maximum and a point ofinflection at x = c.Statement-3 : The function f (x) = x2 | x | is twice differentiable at x = 0.Statement-4 : Let f : [c – 1, c + 1] [a, b] be bijective map such that f is differentiable at c then f–1

is also differentiable at f (c).(A) FFTF (B) TTFT (C) FTTF (D) TTTF

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Q.38 Let f : [–1, 2] R be differentiable such that 0 f ' (t) 1 for t [–1, 0] and – 1 f ' (t) 0 fort [0, 2]. Then(A) – 2 f (2) – f (–1) 1 (B) 1 f (2) – f (–1) 2(C) – 3 f (2) – f (–1) 0 (D) – 2 f (2) – f (–1) 0

Q.39 A curve is represented by the equations, x = sec2 t and y = cot t where t is a parameter. If the tangent

at the point P on the curve where t = /4 meets the curve again at the point Q then PQ is equal to:

(A) 5 3

2(B)

5 5

2(C)

2 5

3(D)

3 5

2

Q.40 For all a, b R the function f (x) = 3x4 4x3 + 6x2 + ax + b has :(A) no extremum (B) exactly one extremum(C) exactly two extremum (D) three extremum .

Q.41 The set of values of p for which the equation ln x px = 0 possess three distinct roots is

(A)

e

1,0 (B) (0, 1) (C) (1,e) (D) (0,e)

Q.42 The sum of the terms of an infinitely decreasing geometric progression is equal to the greatest value ofthe function f (x) = x3 + 3x – 9 on the interval [– 2, 3]. If the difference between the first and the secondterm of the progression is equal to f ' (0) then the common ratio of the G.P. is(A) 1/3 (B) 1/2 (C) 2/3 (D) 3/4

Q.43 The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height mustbe equal to :

(A) 1

3(B)

2

3(C)

1

3(D) 1

Q.44 The lateral edge of a regular rectangular pyramid is 'a' cm long . The lateral edge makes an angle withthe plane of the base. The value of for which the volume of the pyramid is greatest, is :

(A) 4

(B) sin 1 2

3(C) cot 1 2 (D)

3

Q.45 In a regular triangular prism the distance from the centre of one base to one of the vertices of the otherbase is l. The altitude of the prism for which the volume is greatest :

(A)

2(B)

3(C)

3(D)

4

Q.46 Let f (x) =

1xif)2x(

1xifx

3

53

then the number of critical points on the graph of the function is(A) 1 (B) 2 (C) 3 (D) 4

Q.47 The curve y exy + x = 0 has a vertical tangent at :(A) (1, 1) (B) (0, 1) (C) (1, 0) (D) no point

Q.48 Number of roots of the equation x2 . e2 x = 1 is :(A) 2 (B) 4 (C) 6 (D) zero

Q.49 The point(s) at each of which the tangents to the curve y = x3 3x2 7x + 6 cut off on the positivesemi axis OX a line segment half that on the negative semi axis OY then the co-ordinates the point(s) is/are given by :(A) ( 1, 9) (B) (3, 15) (C) (1, 3) (D) none

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Q.50 A curve with equation of the form y = ax4 + bx3 + cx + d has zero gradient at the point (0, 1) and alsotouches the x-axis at the point ( 1, 0) then the values of x for which the curve has a negative gradient are(A) x > 1 (B) x < 1 (C) x < 1 (D) 1 x 1

Q.51 Number of solution(s) satisfying the equation, 3x2 2x3 = log2 (x2 + 1) log2 x is :

(A) 1 (B) 2 (C) 3 (D) none

Q.52 Consider the functionf (x) = x cos x – sin x, then identify the statement which is correct .

(A) f is neither odd nor even (B) f is monotonic decreasing at x = 0(C) f has a maxima at x = (D) f has a minima at x = –

Q.53 Consider the two graphs y = 2x and x2 – xy + 2y2 = 28. The absolute value of the tangent of the anglebetween the two curves at the points where they meet, is(A) 0 (B) 1/2 (C) 2 (D) 1

Q.54 The x-intercept of the tangent at any arbitrary point of the curve 22 y

b

x

a = 1 is proportional to:

(A) square of the abscissa of the point of tangency(B) square root of the abscissa of the point of tangency(C) cube of the abscissa of the point of tangency(D) cube root of the abscissa of the point of tangency.

Q.55 For the cubic, f (x) = 2x3 + 9x2 + 12x + 1 which one of the following statement, does not hold good?(A) f (x) is non monotonic(B) increasing in (– , – 2) (–1, ) and decreasing is (–2, –1)(C) f : R R is bijective(D) Inflection point occurs at x = – 3/2

Q.56 The function 'f' is defined by f(x) = xp (1 x)q for all x R, where p,q are positive integers, has amaximum value, for x equal to :

(A) pq

p q(B) 1 (C) 0 (D)

p

p q

Q.57 Let h be a twice continuously differentiable positive function on an open interval J. Let

g(x) = ln )x(h for each x J

Suppose 2)x('h > h''(x) h(x) for each x J. Then

(A) g is increasing on J (B) g is decreasing on J(C) g is concave up on J (D) g is concave down on J

Q.58 Let f (x) =

2

1xif0

2

1xif

1x2

)1x6)(1x(

then at x = 2

1

(A) f has a local maxima (B) f has a local minima(C) f has an inflection point (D) f has a removable discontinuity

Q.59 Let f (x) and g (x) be two continuous functions defined from R R, such that f (x1) > f (x2) and

g (x1) < g (x2), x1 > x2 , then solution set of )2(gf 2 > )43(gf is

(A) R (B) (C) (1, 4) (D) R – [1, 4]

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Q.60 If f(x) = x

x2

(t 1) dt , 1 x 2, then global maximum value of f(x) is :

(A) 1 (B) 2 (C) 4 (D) 5

Q.61 A right triangle is drawn in a semicircle of radius 1/2 with one of its legs along the diameter. The maximumarea of the triangle is

(A) 4

1(B)

32

33(C)

16

33(D)

8

1

Q.62 At any two points of the curve represented parametrically by x = a (2 cos t cos 2t) ;y = a (2 sin t sin 2t) the tangents are parallel to the axis of x corresponding to the values of theparameter t differing from each other by :(A) 2/3 (B) 3/4 (C) /2 (D) /3

Q.63 Let x be the length of one of the equal sides of an isosceles triangle, and let be the angle between them.If x is increasing at the rate (1/12) m/hr, and is increasing at the rate of /180 radians/hr then the ratein m2/hr at which the area of the triangle is increasing when x = 12 m and = /4

(A) 21/2

5

21 (B)

2

73 · 21/2 (C)

52

3 21 (D) 21/2

52

1

Q.64 If the function f (x) = 4x

xx3t 2

, where 't' is a parameter has a minimum and a maximum then the

range of values of 't' is(A) (0, 4) (B) (0, ) (C) (– , 4) (D) (4, )

Q.65 The least area of a circle circumscribing any right triangle of area S is :

(A) S (B) 2 S (C) 2 S (D) 4 S

Q.66 A point is moving along the curve y3 = 27x. The interval in which the abscissa changes at slower rate thanordinate, is(A) (–3 , 3) (B) (– , ) (C) (–1, 1) (D) (– , –3) (3, )

Q.67 Let f (x) and g (x) are two function which are defined and differentiable for all x x0. If f (x0) = g (x0) andf ' (x) > g ' (x) for all x > x0 then(A) f (x) < g (x) for some x > x0 (B) f (x) = g (x) for some x > x0

(C) f (x) > g (x) only for some x > x0 (D) f (x) > g (x) for all x > x0

Q.68 P and Q are two points on a circle of centre C and radius , the angle PCQ being 2 then the radius ofthe circle inscribed in the triangle CPQ is maximum when

(A) 22

13sin

(B)2

15sin

(C) 2

15sin

(D) 4

15sin

Q.69 The line which is parallel to x-axis and crosses the curve y = x at an angle of 4

is

(A) y = 1/2 (B) x = 1/2 (C) y = 1/4 (D) y = 1/2

Q.70 The function S(x) =

x

0

2

dt2

tsin has two critical points in the interval [1, 2.4]. One of the critical

points is a local minimum and the other is a local maximum. The local minimum occurs at x =

(A) 1 (B) 2 (C) 2 (D) 2

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Q.71 For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in km). If the speedof the current is steady at C km/hr then the most economical speed of the steamer going against thecurrent will be(A) 1.25 C (B) 1.5 C (C) 1.75C (D) 2 C

Q.72 Let f and g be increasing and decreasing functions, respectively from [0 , ) to [0 , ). Leth (x) = f [g (x)] . If h (0) = 0, then h (x) h (1) is :(A) always zero (B) strictly increasing (C) always negative (D) always positive

Q.73 The set of value(s) of 'a' for which the function f (x) = a x3

3 + (a + 2) x2 + (a 1) x + 2 possess a

negative point of inflection .(A) ( , 2) (0, ) (B) { 4/5 }(C) ( 2, 0) (D) empty set

Q.74 A function y = f (x) is given by x = 1

1 2 t & y =

1

1 2t t( ) for all t > 0 then f is :

(A) increasing in (0, 3/2) & decreasing in (3/2, )(B) increasing in (0, 1)(C) increasing in (0, )(D) decreasing in (0, 1)

Q.75 The set of all values of ' a ' for which the function ,

f (x) = (a2 3 a + 2) cos sin2 2

4 4

x x

+ (a 1) x + sin 1 does not possess critical points is:

(A) [1, ) (B) (0, 1) (1, 4) (C) ( 2, 4) (D) (1, 3) (3, 5)

Q.76 Read the following mathematical statements carefully:I. Adifferentiable function ' f ' with maximum at x = c f ''(c) < 0.II. Antiderivative of a periodic function is also a periodic function.

III. If f has a period T then for any a R. T

0

dx)x(f = T

0

dx)ax(f

IV. If f (x) has a maxima at x = c , then 'f ' is increasing in (c – h, c) and decreasing in (c, c + h)as h 0 for h > 0.

Now indicate the correct alternative.(A) exactly one statement is correct. (B) exactly two statements are correct.(C) exactly three statements are correct. (D) All the four statements are correct.

Q.77 If the point of minima of the function, f(x) = 1 + a2x – x3 satisfy the inequality

x x

x x

2

2

2

5 6

< 0, then 'a' must lie in the interval:

(A) 3 3 3 3, (B) 2 3 3 3,

(C) 2 3 3 3, (D) 3 3 2 3 2 3 3 3, ,

Q.78 The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of theradius and increases three times as fast as radius. When the radius is 1cm the altitude is 6 cm. When theradius is 6cm, the volume is increasing at the rate of 1Cu cm/sec. When the radius is 36cm, the volumeis increasing at a rate of n cu. cm/sec. The value of 'n' is equal to:(A) 12 (B) 22 (C) 30 (D) 33

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Q.79 Two sides of a triangle are to have lengths 'a' cm & 'b' cm. If the triangle is to have the maximum area,then the length of the median from the vertex containing the sides 'a' and 'b' is

(A) 1

22 2a b (B)

2

3

a b(C)

a b2 2

2

(D)

a b 23

Q.80 Let a > 0 and f be continuous in [– a, a]. Suppose that f ' (x) exists and f ' (x) 1 for all x (– a, a). Iff (a) = a and f (– a) = – a then f (0)(A) equals 0 (B) equals 1/2 (C) equals 1 (D) is not possible to determine

Q.81 The lines tangent to the curves y3 – x2y + 5y – 2x = 0 and x4 – x3y2 + 5x + 2y = 0 at the origin intersectat an angle equal to(A) /6 (B) /4 (C) /3 (D) /2

Q.82 The cost function at American Gadget is C(x) = x3 – 6x2 + 15x (x in thousands of units and x > 0)The production level at which average cost is minimum is(A) 2 (B) 3 (C) 5 (D) none

Q.83 A rectangle has one side on the positive y-axis and one side on the positive x - axis. The upper right hand

vertex on the curve y = nx

x2. The maximum area of the rectangle is

(A) e–1 (B) e – ½ (C) 1 (D) e½

Q.84 A particle moves along the curve y = x3/2 in the first quadrant in such a way that its distance from theorigin increases at the rate of 11 units per second. The value of dx/dt when x = 3 is

(A) 4 (B) 9

2(C)

3 3

2(D) none

Q.85 Number of solution of the equation 3tanx + x3 = 2 in

4,0 is

(A) 0 (B) 1 (C) 2 (D) 3

Q.86 Let f (x) = ax2 – b | x |, where a and b are constants. Then at x = 0, f (x) has(A) a maxima whenever a > 0, b > 0 (B) a maxima whenever a > 0, b < 0(C) minima whenever a > 0, b > 0 (D) neither a maxima nor minima whenever a > 0, b < 0

Q.87 Let f (x) =

x

1

dtt

tn)t(nt

ll (x > 1) then

(A) f (x) has one point of maxima and no point of minima.(B) f ' (x) has two distinct roots(C) f (x) has one point of minima and no point of maxima(D) f (x) is monotonic

Q.88 Consider f (x) = | 1 – x | 1 x 2 and

g (x) = f (x) + b sin2

x, 1 < x < 2

then which of the following is correct?(A) Rolles theorem is applicable to both f, g and b = 3/2(B) LMVT is not applicable to f and Rolles theorem if applicable to g with b = 1/2(C) LMVT is applicable to f and Rolles theorem is applicable to g with b = 1(D) Rolles theorem is not applicable to both f, g for any real b.

Mathamtics


Q.89 Consider f (x) =

x

0

dtt

1t and g (x) = f (x) for x

3,

2

1

If P is a point on the curve y = g(x) such that the tangent to this curve at P is parallel to a chord joining

the points

2

1g,

2

1 and (3, g(3) ) of the curve, then the coordinates of the point P

(A) can't be found out (B)

28

65,

4

7(C) (1, 2) (D)

6

5,

2

3

Q.90 The co-ordinates of the point on the curve 9y2 = x3 where the normal to the curve makes equalintercepts with the axes is

(A)

3

1,1 (B) 3,3 (C)

3

8,4 (D)

5

6

5

2,

5

6

Q.91 The angle made by the tangent of the curve x = a (t + sint cost) ; y = a (1 + sint)2 with the x-axis at anypoint on it is

(A) t24

1 (B) tcos

tsin1 (C) t2

4

1(D)

t2cos

tsin1

Q.92 If f (x) = 1 + x + x

1

2 dtnt2tn ll , then f (x) increases in

(A) (0, ) (B) (0, e–2) (1, ) (C) no value (D) (1, )

Q.93 The function f (x) =)xe(n

)x(n

l

l is :

(A) increasing on [0, ) (B) decreasing on [0, )(C) increasing on [0, /e) & decreasing on [/e, ) (D) decreasing on [0, /e) & increasing on [/e, )

Directions for Q.94 to Q.96Suppose you do not know the function f (x), however some information about f (x) is listed below. Readthe following carefully before attempting the questions(i) f (x) is continuous and defined for all real numbers(ii) f '(–5) = 0 ; f '(2) is not defined and f '(4) = 0(iii) (–5, 12) is a point which lies on the graph of f (x)(iv) f ''(2) is undefined, but f ''(x) is negative everywhere else.(v) the signs of f '(x) is given below

Q.94 On the possible graph of y = f (x) we have(A) x = – 5 is a point of relative minima.(B) x = 2 is a point of relative maxima.(C) x = 4 is a point of relative minima.(D) graph of y = f (x) must have a geometrical sharp corner.

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Q.95 From the possible graph of y = f (x), we can say that(A) There is exactly one point of inflection on the curve.(B) f (x) increases on – 5 < x < 2 and x > 4 and decreases on – < x < – 5 and 2 < x < 4.(C) The curve is always concave down.(D) Curve always concave up.

Q.96 Possible graph of y = f (x) is

(A) (B)

(C) (D)

Directions for Q.97 to Q.100

Consider the function f (x) =

x

x

11

then

Q.97 Domain of f (x) is(A) (–1, 0) (0, ) (B) R – { 0 } (C) (– , –1) (0, ) (D) (0, )

Q.98 Which one of the following limits tends to unity?

(A) )x(Limx

f (B) )x(Lim

0xf

(C) )x(Lim

1xf

(D) )x(Lim

xf

Q.99 The function f (x)(A) has a maxima but no minima (B) has a minima but no maxima(C) has exactly one maxima and one minima (D) has neither a maxima nor a minima

Q.100 Range of the function f (x) is(A) (0, ) (B) (– , e) (C) (1, ) (D) (1, e) (e, )

Q.101 A cube of ice melts without changing shape at the uniform rate of 4 cm3/min. The rate of change of thesurface area of the cube, in cm2/min, when the volume of the cube is 125 cm3, is(A) – 4 (B) – 16/5 (C) – 16/6 (D) – 8/15

Q.102 Let f (1) = – 2 and f ' (x) 4.2 for 1 x 6. The smallest possible value of f (6), is(A) 9 (B) 12 (C) 15 (D) 19

Q.103 Which of the following six statements are true about the cubic polynomialP(x) = 2x3 + x2 + 3x – 2?

(i) It has exactly one positive real root.(ii) It has either one or three negative roots.(iii) It has a root between 0 and 1.(iv) It must have exactly two real roots.(v) It has a negative root between – 2 and –1.(vi) It has no complex roots.(A) only (i), (iii) and (vi) (B) only (ii), (iii) and (iv)(C) only (i) and (iii) (D) only (iii), (iv) and (v)

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Q.104 Given that f (x) is continuously differentiable on a x b where a < b, f (a) < 0 and f (b) > 0, which ofthe following are always true?(i) f (x) is bounded on a x b.(ii) The equation f (x) = 0 has at least one solution in a < x < b.(iii) The maximum and minimum values of f (x) on a x b occur at points where f ' (c) = 0.(iv) There is at least one point c with a < c < b where f ' (c) > 0.(v) There is at least one point d with a < d < b where f ' (c) < 0.(A) only (ii) and (iv) are true (B) all but (iii) are true(C) all but (v) are true (D) only (i), (ii) and (iv) are true

Q.105 Consider the function f (x) = 8x2 – 7x + 5 on the interval [–6, 6]. The value of c that satisfies theconclusion of the mean value theorem, is(A) – 7/8 (B) – 4 (C) 7/8 (D) 0

Q.106 Consider the curve represented parametrically by the equationx = t3 – 4t2 – 3t andy = 2t2 + 3t – 5 where t R.

If H denotes the number of point on the curve where the tangent is horizontal and V the number of pointwhere the tangent is vertical then(A) H = 2 and V = 1 (B) H = 1 and V = 2(C) H = 2 and V = 2 (D) H = 1 and V = 1

Q.107 At the point P(a, an) on the graph of y = xn (n N) in the first quadrant a normal is drawn. The normal

intersects the y-axis at the point (0, b). If 2

1bLim

0a

, then n equals

(A) 1 (B) 3 (C) 2 (D) 4

Q.108 Suppose that f is a polynomial of degree 3 and that f ''(x) 0 at any of the stationary point. Then(A) f has exactly one stationary point. (B) f must have no stationary point.(C) f must have exactly 2 stationary points. (D) f has either 0 or 2 stationary points.

Q.109 Let f (x) =

0xfor8x

0xforx

2

2

. Then x intercept of the line that is tangent to the graph of f (x) is

(A) zero (B) – 1 (C) –2 (D) – 4

Q.110 Suppose that f is differentiable for all x and that f '(x) 2 for all x. If f (1) = 2 and f (4) = 8 then f (2)has the value equal to(A) 3 (B) 4 (C) 6 (D) 8

Q.111 There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree plantedin the orchard, the output per additional tree drops by 10 apples. Number of trees that should be addedto the existing orchard for maximising the output of the trees, is(A) 5 (B) 10 (C) 15 (D) 20

Q.112 The ordinate of all points on the curve y = xcos3xsin2

122

where the tangent is horizontal, is

(A) always equal to 1/2(B) always equal to 1/3(C) 1/2 or 1/3 according as n is an even or an odd integer.(D) 1/2 or 1/3 according as n is an odd or an even integer.

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Select the correct alternatives : (More than one are correct)

Q.113 The equation of the tangent to the curve x

a

y

b

n n = 2 (n N) at the point with abscissa equal to

'a' can be :

(A) x

a

y

b

= 2 (B) x

a

y

b

= 2 (C) x

a

y

b

= 0 (D) x

a

y

b

= 0

Q.114 The function y = 2 1

2

x

x

(x 2) :

(A) is its own inverse (B) decreases for all values of x(C) has a graph entirely above x-axis (D) is bound for all x.

Q.115 If x

a

y

b = 1 is a tangent to the curve x = Kt, y =

K

t, K > 0 then :

(A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0

Q.116 The extremum values of the function f(x) = 1

4

1

4sin cosx x , where x R is :

(A) 4

8 2(B)

2 2

8 2(C)

2 2

4 2 1(D)

4 2

8 2

Q.117 The function f(x) = 1 4

0

tx

dt is such that :

(A) it is defined on the interval [ 1, 1] (B) it is an increasing function(C) it is an odd function (D) the point (0, 0) is the point of inflection

Q.118 Let g(x) = 2 f (x/2) + f (1 x) and f (x) < 0 in 0 x 1 then g(x) :(A) decreases in [0, 2/3) (B) decreases in (2/3, 1](C) increases in [0, 2/3) (D) increases in (2/3, 1]

Q.119 The abscissa of the point on the curve xy = a + x, the tangent at which cuts off equal intersects from

the co-ordinate axes is : ( a > 0)

(A) a

2(B)

a

2(C) a 2 (D) a 2

Q.120 The function sin ( )

sin ( )

x a

x b

has no maxima or minima if :

(A) b a = n , n I (B) b a = (2n + 1) , n I(C) b a = 2n , n I (D) none of these .

Q.121 The co-ordinates of the point P on the graph of the function y = e–|x| where the portion of the tangentintercepted between the co-ordinate axes has the greatest area, is

(A) 11,e

(B)

11,e

(C) (e, e–e) (D) none

Q.122 Let f(x) = (x2 1)n (x2 + x + 1) then f(x) has local extremum at x = 1 when :(A) n = 2 (B) n = 3 (C) 4 (D) n = 6

Q.123 For the function f(x) = x4 (12 ln x 7)(A) the point (1, 7) is the point of inflection (B) x = e1/3 is the point of minima(C) the graph is concave downwards in (0, 1) (D) the graph is concave upwards in (1, )

Q.124 The parabola y = x2 + px + q cuts the straight line y = 2x 3 at a point with abscissa 1. If the distancebetween the vertex of the parabola and the x axis is least then :(A) p = 0 & q = 2(B) p = 2 & q = 0(C) least distance between the parabola and x axis is 2(D) least distance between the parabola and x axis is 1

Q.125 The co-ordinates of the point(s) on the graph of the function, f(x) = x x3 2

3

5

2 + 7x - 4 where the

tangent drawn cut off intercepts from the co-ordinate axes which are equal in magnitude but opposite insign, is(A) (2, 8/3) (B) (3, 7/2) (C) (1, 5/6) (D) none

Q.126 On which of the following intervals, the function x100 + sin x 1 is strictly increasing.(A) ( 1, 1) (B) (0, 1) (C) (/2, ) (D) (0, /2)

Q.127 Let f(x) = 8x3 – 6x2 – 2x + 1, then(A) f(x) = 0 has no root in (0,1) (B) f(x) = 0 has at least one root in (0,1)

(C) f (c) vanishes for some )1,0(c (D) none

Q.128 Equation of a tangent to the curve y cot x = y3 tan x at the point where the abscissa is 4

is :

(A) 4x + 2y = + 2 (B) 4x 2y = + 2 (C) x = 0 (D) y = 0

Q.129 Let h (x) = f (x) {f (x)}2 + {f (x)}3 for every real number ' x ' , then :(A) ' h ' is increasing whenever ' f ' is increasing(B) ' h ' is increasing whenever ' f ' is decreasing(C) ' h ' is decreasing whenever ' f ' is decreasing(D) nothing can be said in general.

Q.130 If the side of a triangle vary slightly in such a way that its circum radius remains constant, then,

da

A

db

B

dc

Ccos cos cos is equal to:

(A) 6 R (B) 2 R (C) 0 (D) 2R(dA + dB + dC)

Q.131 In which of the following graphs x = c is the point of inflection .

(A) (B) (C) (D)

Q.132 An extremum value of y = 0

x

(t 1) (t 2) dt is :

(A) 5/6 (B) 2/3 (C) 1 (D) 2

Q.1 B Q.2 D Q.3 B Q.4 B Q.5 D Q.6 C Q.7 D

Q.8 A Q.9 D Q.10 C Q.11 B Q.12 A Q.13 D Q.14 D

Q.15 C Q.16 D Q.17 A Q.18 D Q.19 A Q.20 A Q.21 C

Q.22 A Q.23 C Q.24 D Q.25 D Q.26 A Q.27 B Q.28 C

Q.29 B Q.30 C Q.31 C Q.32 C Q.33 D Q.34 D Q.35 D

Q.36 B Q.37 A Q.38 A Q.39 D Q.40 B Q.41 A Q.42 C

Q.43 C Q.44 C Q.45 B Q.46 C Q.47 C Q.48 B Q.49 B

Q.50 C Q.51 A Q.52 B Q.53 C Q.54 C Q.55 C Q.56 D

Q.57 D Q.58 C Q.59 C Q.60 C Q.61 B Q.62 A Q.63 D

Q.64 C Q.65 A Q.66 C Q.67 D Q.68 B Q.69 D Q.70 C

Q.71 B Q.72 A Q.73 A Q.74 B Q.75 B Q.76 A Q.77 D

Q.78 D Q.79 A Q.80 A Q.81 D Q.82 B Q.83 A Q.84 A

Q.85 B Q.86 A Q.87 D Q.88 C Q.89 D Q.90 C Q.91 A

Q.92 A Q.93 B Q.94 D Q.95 C Q.96 C Q.97 C Q.98 B

Q.99 D Q.100 D Q.101 B Q.102 D Q.103 C Q.104 D Q.105 D

Q.106 B Q.107 C Q.108 D Q.109 B Q.110 B Q.111 C Q.112 D

Q.113 A,B Q.114 A,B Q.115 A,D Q.116 A,C Q.117 A,B,C,D Q.118 B,C

Q.119 A,B Q.120 A,B,C Q.121 A,B Q.122 A,C,D Q.123 A,B,C,D Q.124 B,D

Q.125 A,B Q.126 B,C,D Q.127 B,C Q.128 A,B,D Q.129 A,C Q.130 C,D

Q.131 A,B,D Q.132 A,B

ANSWER KEY

application of derivative wa

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