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Final step AOD By ABHIJIT KUMAR JHA

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ABJSIR

LEVEL�I1. A circular metal plate expands under heating so that its radius increase by 2%. Find the approximate

increase in the area of the plate, if the radius of the plate before heating is 10cm.

2. The length x of a rectangle is decreasing at the rate of 2 cm/sec and width y is increasing at the rateof 2 cm/sec. When x = 12 cm and y = 5 cm, find the rate of change of(i) the perimeter and (ii) the area of the rectangle

3. A ladder 16 cm 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 cm/sec. How fast is its height on the wall decreasing when thefoot of the ladder is 4 cm away from the wall ?

4. Prove that the tangent to 4x2 � 9y2 = 36 which is perpendicular to the straight 5x + 2y � 10 = 0does not exist.

5. If , are the intercepts made on the axes by the tangent at any point of the curve

x = a y bcos , sin3 3 , prove that 2

2

2

21

a b .

6. Show that the normal to the rectangular hyperbola xy = c2 at the point t1 meets the curve again at

the point t2 such that t

13 t

2 = �1.

7. A(0, 1), B2

1,FHGIKJ are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists

a point P on the curve between A and B such that tangent at P is parallel to AB. Find the coordinatesof P.

8. If f(x) differentiable in [1, 5], then show that )b(f).a(f8)1(f)5(f 22 , where ]5,1[b,a .

9. Let the function f be a continuous in [a, b] and derivable in (a, b) show that there exist a number.

)b,a(c such that 2c [f(a) � f(b)] = ]ba)[c(f 22 .

10. If a function f is continuous in [a, b] and differentiable in (a, b), where ab > 0, then prove that there

exists at least one c (a, b) for which )c(fc

a

1

b

1)a(f)b(f 2

.

11. Find the condition for the polynomial equation f(x) = 0 to have a repeated real roots by using

Rolle�s theorem. Hence or otherwise prove that x

r

r

r

n

!

0

0 can not have a repeated root.

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ABJSIR

12. If a, b, c be non-zero real numbers such that :

( cos )( )1 8 2

0

1 z x ax bx c dx = ( cos )( )1 08 2

0

2 z x ax bx c dx , then equation ax2 + bx

+ c = 0 will have one root between 0 and 1 other root between 1 and 2.13. Prove that if (n � 1)a

12 � 2na

2 < 0 then the roots of the equation

xn + a1xn � 1 + ....+ a

n�1 x + an = 0 cannot be all real.

14. Given that

)1n(

0kk 0a . Where 1n,.......,2,1,0kRa k . Show that

0ax2.a..........x).1n(ax.n.a1n2n10 ii

2ni

1ni

has at least one real root in (�3, 3)

for any permutation )a,.........a,aofa,.........a,a( 1n10iii 1n10 .

15. Determine the intervals in which the function f(x) = x2/(x-1), x 1 increasing or decreasing.

16. Find the set of all values of �a� for which the function, f(x) = a

a

FHG

IKJ

4

11 x5 - 3x + log 5 decreases

for all real �x�.17. If f(x) = 2ex - a e-x + (2a + 1)x - 3 monotonically increases for every x R then find the range of

values of �a�.

18. Find the values of �a� for which the function f(x) = sin sin sinx a x x ax 21

33 2 increase

throughout the number line.19. The interval to which b may belong so that the function,

f(x) =

1b

bb4211

2

x3 + 5x + 16 , increases for all x.

20. Let g (x) = 2 f

2

x + f (2 � x) and f (x) < 0 x (0, 2). Find the intervals of increase

and decrease of g(x).

21. Prove the inequality, tan

tan

x

x

x

x2

1

2

1

for 021 2 x x

.

Prove the following :

22. 2 3sin tanx x x for 02

x

.

23. 1 � x < e-x < 1 - x + x2

2 for all x 0 .

24. x x x x x for x2 1 2 4 1 2 1 log log ( , )b g .

25. 21

12 1

1

3 10 1

2

2x

x

xx

x

xfor x

FHG

IKJ log . ( , ) .

26. e x x ex xtan

1 2 21e j for x > 0.




ABJSIR

27. cos(sin x) > sin(cos x) for 0 < x < 2

.

28. Show that sin cosp q attains a maximum, when tan 1 p

q.

29. Find the least value of f(x) = a2 sec2x + b2cosec2x, given ab 0.

30. Show that f(x) = | x |m | x - 1|n, x R m n N , , , m, n > 1, has a point of maxima at which the value

is m n

m n

m n

m n( )( ) .

31. Find the value of �p� for which f(x) = x3 + 6(p - 3)x2 + 3(p2 - 4)x + 10 has a positive point ofmaximum.

32. Find the polynomial f(x) of degree 6, which satisfies Limf x

xx

x

FHGIKJ0 3

1

1( )

/

= e2 and has local maximum

at x = 1 and local minimum at x = 0 and 2.33. Show that the surface area of a closed cuboid with square base and given volume is minimum,

when it is a cube.34. A box of maximum volume with top open is to be made by cutting out four equal squares

from four corners of a square tin sheet of side length a ft, and then folding up the flaps.Find the side of the square cut off.

35. Show that the height of the closed cylinder of given surface and maximum volume, is equal to thediameter of its base.

36. Find the volume of the largest cylinder that can be inscribed in a sphere of radius �r� cm.37. A cone is circumscribed about a sphere of radius �r�. Show that the volume of the cone is minimum

when its semi-vertical angle is, sin FHGIKJ

1 1

3.

38. If the sum of the lengths of the hypotenuse and another side of a right angled triangle is given, show

that the area of the triangle is a maximum when the angle between these sides is / 3 .39. Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is

isosceles.40. In constructing an A-C transformer it is important to insert into the coil a cross-shaped iron core of

greatest possible surface area. Shown in the figure the cross-section of the core with appropriatedimensions. Find the most suitable x and y if the radius of the coil is equal to a




ABJSIR

LEVEL�II

1. A particle moves so that the space described in time t is square root of a quadratic function of t.

Then, prove that the acceleration of the particle varies as 3

1

s.

2. A light shines from the top of a pole 50 ft high. A ball is drooped from the same height from a point30 ft away from the light. How fast is the shadow of the ball moving along the ground 1/2 sec.later? [Assume the ball falls a distance s = 16 t2 ft in �t� sec.

3. An air force plane is ascending vertically at the rate of 100 km/h. If the radius of the earth is R Km,how fast the area of the earth, visible from the plane increasing at 3 min after it started ascending.

Take visible area A = 2 2R h

R h where h is height of the plane in kms above the earth.

4. If x1 and y

1 be the intercepts on the axes of �x� and �y� cut off by the tangent to the curve,

x

a

y

b

n nFHGIKJ FHGIKJ 1 , then show that

a

x

b

y

n n n n

1

1

1

1

1FHGIKJ

FHGIKJ

/( ) /( )

.

5. Prove that the curves y = f(x), (f(x) > 0) and y = f(x) sin x, where f(x) is differentiable function,have common tangents at common points.

6. A curve is given by the equations x = at2 and y = at3. A variable pair of perpendicular lines throughthe origin �O� meet the curve at P and Q. Show that the locus of the point of intersection of thetangents at P and Q is 4y2 = 3ax - a2.

7. Show that the condition, that the curves x2/3 + y2/3 = c2/3 and x

a

y

b

2

2

2

21 may touch, if c = a + b,

where a, b and Rc .

8. For the function F(x) = 20

| |t dtxz , find the tangent lines which are parallel to the bisector of the angle

in the first quadrant.

9. The tangent at a variable point P of the curve y = x2 - x3 meets it again at Q. Show that the locus ofthe middle point of PQ is y = 1 - 9x + 28 x2 - 28 x3.

10. If a = -1, b 1 and f(x) = 1/| x |, show that the conditions of Lagrange�s mean value theorem are

not satisfied in the interval [a, b], but the conclusion of the theorem is true if and only if b 1 2 .

11. Let y = f(x) be differentiable in the closed interval [2002, 2004] and f(2002) = f(2004) = 0. Showthat there exist a point on the curve y = f(x) at which the length of the subtangent is 2003.




ABJSIR

12. Let f(x) be a differentiable function on [�1, 1]. If f(1) = 0 and f(x) > 0 for all x in (�1, 1), prove that

the equation )x(f)x(fs)x(f)x(f.r has a solution in )Rs&r()1,1( .

13. Let f(x) and g(x) be differentiable functions such that )x(g).x(f)x(g).x(f for any real x. Provethat between any two real solution of f(x) = 0, there is at least one real solution of g(x) = 0.Give one example of such that a pair of solution.

14. Suppose f is continuous on [a, b] and differentiable on (a, b). Assume further that f(b) � f(a) = b � a.Prove that for every positive integer n, there exist distinct points c

1, c

2,........,c

n in (a, b) such that

n)c(f.......)c(f)c(f n21 .

15. Prove that f(x) = 11

FHGIKJx

x

is always an increasing function for every x > 0.

16. Prove that ( )a b a bn n n for all a, b > 0 and 0 < n < 1.

17. Show that the function f(x) = x + cosx � a is an increasing function and hence deduce that theequation x + cosx = a has no positive root for a <1 and has one positive root for a > 1.

18. Let ( )x = f t dta

x

( )z and f(x) satisfies the following conditions

f(x + y) = f(x) + f(y) + 2xy - 1 x y R, and f a a( ) ( )0 3 2 where a is constant and

a FHG

IKJ

1 13

2

1 13

2, , then prove that ( )x is entirely increasing.

19. Let f be differentiable at every value of x and suppose that f(1) = 1, that 0)x(f on )1,( and

that 0)x(f on ),1( , show that 1)x(f for all Rx .

20. Find the range of parameter b, for which the function f(x) is entirely increasing or decreasing for all

values of x where, f (x) = x

0

2 dt)tcosbbt( .

21. Let f (x) = [� c2 + (b � 1) c � 2] x + (sin2 x + cos4 x) dx. If f (x) be an increasing function of

x x R then find all possible values of b (if c R).

22. Find all the values of the parameter �a� for which the function ; f(x) = 8ax - a sin 6x - 7x - sin 5xincreases and has no critical points for all x R .




ABJSIR

23. Find the set of all values of the parameter �a� for which the functionf(x) = sin 2x - 8(a + 1)sin x + (4a2 + 8a - 14)x increases for all x R and has no critical points forall x R .

24. Prove that following inequality 1x,0x,)xx(

1x

3

1

1x

xn33

.

25. Show that the volume of the greatest cylinder which can be inscribed in a cone of height �h� and

semi - vertical angle is 4

273 2 h tan .

26. A light hangs above the centre of a table of radius r ft. The illumination at any point on the table isdirectly proportional to the cosine of the angle of incidence. (i.e. the angle a ray of light makes withthe normal )and is inversely proportional to the square of the distance from the light. How far thelight be above the table in order to give the strongest illumination at the edges of the table.

27. A closed rectangular box with a square base is to be made to contain 1000 cubic feet. The cost ofthe material per square foot for the bottom is 15 paise, for the top 25 paise and for the sides 20paise. The labor charges for making the box are Rs. 3/-. Find the dimensions of the box when thecost is minimum.

28. Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for thegiven constant length L of the median drawn to its lateral side.

29. A segment of a line bisects a triangle ABC with sides a, b, c into two equal areas. Find the length

of the shortest segment.

30. Show that the altitude of the greatest equilateral triangle that can be circumscribed about a given

triangle ABC with its sides a, b, c is a b ab C2 2

1 2

23

FHGIKJ

RSTUVWcos

/

.

31. The three sides of a trapezium are equal each being 6 cms long, find the area of the trapezium whenit is maximum.

32. One corner of a long rectangular sheet of paper of width 1 unit is folded over so as to reach theopposite edge of the sheet. Find the minimum length of the crease.

33. A ladder is to be carried in a horizontal position round a corner formed by two streets, a feet andb feet wide meeting at right angles. Prove that the length of the longest ladder that will pass roundthe corner without jamming is, (a2/3 + b2/3)3/2.

34. Two towns located on the same side of the river agree to construct a pumping station and filtrationplant at the river�s edge, to be used jointly to supply the towns with water. If the distance of the twofrom the river are �a� and �b� and the distance between them is �c� show that the pipe lines joiningthem to the pumping station is atleast as great as c ab2 4 .




ABJSIR

35. A circle of radius 1 unit touches positive x-axis and positive y-axis at P and Q respectively.A variable line 1 passing through origin intersects circle C in two points M and N. Find the equationof the line for which area of triangle MNQ is maximum.

36. A perpendicular is drawn from the centre to a tangent to an ellipse x

a

y

b

2

2

2

21 . Find the greatest

value of the intercept between the point of contact and the foot of the perpendicular.

37 In the graph of the function y = 2

3 x log x, where x [e�1.5, find the point P (x, y) such that the

segment of the tangent to the graph of the function at the point, intercepted between the point P andy-axis, is shortest.

38. A figure is bounded by the curves, y = x2 + 1, y = 0, x = 0 and x = 1. At what point (a, b), a tangentshould be drawn to the curve, y = x2 + 1 for it to cut off a trapezium of the greatest area from thefigure.

39. A private telephone company serving a small community makes a profit of Rs. 12.00 per subscriber,if it has 725 subscribers. It decides to reduce the rate by a fixed sum for each subscriber over 725,thereby reducing the profit by one paise per subscriber. Thus there will be profit of Rs. 11.99 oneach the 726 subscriber, Rs. 11.98 on each of 727 subscribers etc. What is the number of subscriberswhich will give the company the maximum profit ?

40. Let f(x) = 4 3 3 2 3

18 3

3 2x x b b x

x xe

RSTlog ( ),

, . Find all possible real values of b such that f(x)

has the smallest value at x = 3.

41. Find the minimum value of x x xx1 2

2

12

2

2

29

FHG

IKJb g where x1 0 2 ,d i and x R2

.

42. Let f be a function defined on an interval [a, b] what conditions could you place on f to guarantee

that min )x(fmax)ab(

)a(f)b(f)x(f

, where )x(fmin refer to the minimum and maximum

values of )x(f on [a, b]. Give reason for your answer..

43. Find the greatest and the least values of the function f(x) defined as below. f(x) = minimum of

3 8 6 24 1 1 24 3 2t t t t t x x ; ,m r maximum of 31

42 1 2 42t t t x x RST

UVW sin ; , .




ABJSIR

SET�I1. Angle of intersection of x2 + y2 - 6x - 2y - 10 = 0 and y = 2x - 5 is

(A) 4

(B)

6

(C)

3

(D)

2

2. Total number of parallel tangents of f1(x) = x2 - x + 1 and f

2(x) = x3 - x2 - 2x + 1 is equal to

(A) 2 (B) 3 (C) 4 (D) none of these

3. Tangents are drawn to y = cos x from the point P(0, 0). Points of contact of these tangents willalways lie on

(A) 1y

1

x

122 (B) 1

y

1

x

122 (C) x2 + y2 = 1 (D) x2 - y2 = 1

4. The curve x2 - 4y2 + c = 0 and y2 = 4x will intersect orthogonally for

(A) )16,0(c (B) )4,3(c (C) )4,3(c (D) none of these

5. If the line joining the points (0, 3) and (5, -2) is a tangent to the curve y = x1

ax

, then

(A) 31a (B) a (C) 31a (D) 322a

6. If the line ax + by + c = 0 is a tangent to the curve xy + 2 = 0 then(A) a > 0, b > 0 (B) a > 0, b < 0 (C) a > 0, c > 0 (D) a > 0, c < 0

7. 1 and 2 are the side lengths of two variable squares S1 and S

2 respectively. If 63

221

then rate of change of the area of S2 with respect to rate of change of the area of S

1 when 12

is equal to

(A) 4

3(B)

3

4(C)

2

3(D)

32

1

8. Total number of values of �x� where the function f(x) = x2cosxcos attains its maximum value is(A) 1 (B) 2 (C) 4 (D) none of these

9. If A + B = 3

2 where A, B > 0, then minimum value of sec A + sec B is equal to

(A) 4 (B) 8 (C) 6 (D) none of these

10. If x = a is the point of local maxima for y = f(x), then which of the following is always true

(A) 0)A(f (B) 0)A(f,0)A(f

(C) 0)A(f,0)A(f (D) none of these

11. Let f(x) = {x}, where {.} denotes the fractional part. For f(x), x = 5 is(A) a point of local maxima (B) a point of local minima(C) neither a point of local minima nor maxima (D) a stationary point




ABJSIR

12. f(x) =

1x,x7

1x,6 then for f(x), x = 1 is

(A) a point of local maxima (B) a point of local minima(C) neither a point of local minima nor maxima (D) a stationary point

13. f(x) =

0x,ax

0x,2

xcos

. Then x = 0 will be point of local maxima for f(x) if

(A) )1,1(a (B) )1,0(a (C) 0a (D) 1a

14. f(x) = 0x,x

1x , then

(A) f(x) has no point of local maxima (B) f(x) has no point local minima(C) f(x) has exactly one point of local minima (D) f(x) has exactly two points of local minima

15. If f(x) = x3 + ax2 + bx + c attains its local minima at certain negative real number then(A) a2 - 3b > 0, a < 0, b < 0 (B) a2 - 3b > 0, a < 0, b > 0(C) a2 - 3b > 0, a > 0, b < 0 (D) a2 - 3b > 0, a > 0, b > 0

16. Let f(x) = ax3 + bx2 + cx + d, 0a . If x1 and x

2 are the real and distinct roots of 0)x(f then

f(x) = 0 will have three real and distinct roots if(A) x

1 . x

2 < 0 (B) f(x

1) . f(x

2) > 0 (C) f(x

1) . f(x

2) < 0 (D) x

1 . x

2 > 0

17. A rectangle is inscribed in an equilateral triangle of side length �2a� units. Maximum area of thisrectangle can be

(A) 2a3 (B) 4

a3 2

(C) a2 (D) 2

a3 2

18. If the equation 3ax2 + 2bx + c = 0 has its coefficients such that a + b + c = 0 where a, b, c Rthen the equation has at least one real root in the interval

(A) (�1, 1) (B) (1, 2) (C) 1 3

,2 2

(D) none of these

19. If Rx0)x(f and g (x) = f(x2 � 2) + f(6 � x2) then

(A) g(x) is an increasing in [�2, 0] (B) g(x) is an increasing in ),2[ (C) g(x) has a local minima at x = � 2 (D) g(x) has a local maxima at x = 2

20. f(x) = x

0

432 dt)))at(cos(log(loglog . If f(x) is increasing for all real values of x then

(A) )1,1(a (B) )5,1(a (C) ),1(a (D) ),5(a

21. Let �P� be a point on x2 = 4y that is nearest to the point A(0, 4) then coordinates of �P� are

(A) (4, 4) (B) (0, 0) (C) 2,8 (D) (2, 1)




ABJSIR

22. Let Rx0)x(f and g(x) = f(2 - x) + f(4 + x). Then g(x) is increasing in

(A) )1,( (B) )0,( (C) ),1( (D) none of these

23. Let ),0[),0[:gand),0[),0[:f be non-increasing and non-decreasing functions,h(x) = g(f(x)). If �f� and �g� are differentiable for all points in their respective domains and h(0) = 0then h(x) will always be(A) an increasing function (B) a decreasing function(C) identically zero (D) none of these

24. If xy = 10 then minimum value of 12x2 + 13y2 is equal to

(A) 15 (B) 3940 (C) 133 (D) 1330

25. If 9 - x2 >| x � a| has atleast one negative solution, where then complete set of values of a is

(A) 25

, 92

(B) 35

, 94

(C) 37

, 92

(D) 37

, 94

26. f(x) be a differentiable function such that )))ax(cos(log(log

1)x(f

4/13 . If f(x) is increasing for

all values of x then

(A) ),5(a (B)

4

5,1a (C)

5,

4

5a (D) none of these

27. Let f(x) be a function such that ))ax(sin(loglog)x(f 33/1 . If f(x) is decreasing for all real

values of x then

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

28. Tangents are drawn to x2 + y2 = 16 from the point P(0, h). These tangents meet the x-axis at A andB. If the area of triangle PAB is minimum then

(A) 212h (B) 26h (C) 28h (D) 24h

29. Tangents PA and PB are drawn to y = x2 - x + 1 from the point

h,2

1P . If the area of triangle

PAB is maximum then

(A) 4

1h (B)

2

1h (C) h = - 2 (D) none of these

30. The curve C1 : y = 1 - cos x, ),0(x and C

2 : y = a|x|

2

3 will touch each other if

(A) 32

3a

(B)

322

3a

(C)

32

1a

(D)

34

3a




ABJSIR

SET�II1. The parabolas y2 = 4ax and x2 = 4by intersect orthogonally at point P(x

1, y

1) where 0y.x 11

provided(A) b = a2 (B) b = a3 (C) b3 = a2 (D) none of these

2. Two variable curves C1 : y2 = 4a(x - b

1) and C

2 : x2 = 4a(y - b

2) where �a� is a given positive real

number and b1 and b

2 are variables, touch each other. Locus of the point of contact is

(A) xy = a2 (B) xy = 2a2 (C) xy = 4a2 (D) none of these

3. Point on y2 = 4x that is nearest to the circle x2 + (y - 12)2 = 1, is(A) (4, -4) (B) (4, 4) (C) (9, 6) (D) (9, -6)

4. The function f(x) = 4x

2x2

2

has

(A) no point of local minima (B) no point of local maxima(C) exactly one point of local minima (D) exactly one point of local maxima

5. The function f(x) = x(x2 - 4)n (x2 - x + 1), Nn assumes a local minima at x = 2 then(A) �n� can be any odd number (B) �n� can only be an odd prime number(C) �n� can be any even number (D) �n� can only be a multiple of four

6.1tan x, | x |

4f (x)| x |, | x |

2 4

, then

(A) f(x) has no point of local maxima (B) f(x) has no point of local minima(C) f(x) has exactly one point of local maxima (D) f(x) has exactly two points of local minima

7. f(x) = ex.cos x, ]2,0[x . The slope of tangent of the function is minimum for

(A) x (B) 4

x

(C) 4

3x

(D)

2

3x

8. If f(x) = xbx|x|na 2 has extremes at x = 1 and x = 3 then

(A) 8

1b,

4

3a (B)

8

1b,

4

3a (C)

8

1b,

4

3a (D)

8

1b,

4

3a

9. Total number of critical points of f(x) = 2x

|x2| are equal to

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

10. x

0

2 )x(f).2,0(x,dttcot)1t()x(f attains local maximum value at




ABJSIR

(A) 2

x

(B) x = 1 (D) 2

3x

(D) none of these

11. x

0

t dttsin)tcost(sin)1t()1e()x(f ,

2,

2x then f(x) is

(A) Decreasing in , 02

, Decreasing in , 14

, Decreasing in ,4

(B) Decreasing in ,2 4

, Decreasing in 1, , Decreasing in 5

, 24

(C) Decreasing in , 14

, Decreasing in 5

,4

(D) Decreasing in 0,4

, Decreasing in 1, , Decreasing in 5

, 24

12. RR:f be a differentiable function Rx . If tangent drawn to the curve at any point

)b,a(x always lie below the curve then

(A) )b,a(x0)x(f,0)x(f (B) )b,a(x0)x(f,0)x(f

(C) )b,a(x0)x(f,0)x(f (D) none of these

13. A spherical balloon is pumped at the constant rate of 3 m3/min. The rate of increase of its surfacearea at certain instant is found to be 5 m2/min. At this instant its radius is equal to

(A) m5

1(B) m

5

3(C) m

5

6(D) m

5

2

14. The abscissa of points P and Q on the curve y = ex + e�x such that tangents at P and Q make600 with x-axis

(A)

2

73n and

2

53n (B)

2

73n

(C) 7 3

n2

(D)

2

73n

15. A lamp of negligible height, is placed on the ground � 1 � m away from a wall. A man � 2 � m tall is

walking at a speed of sec/m10

1 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 his shadow on thewall is




ABJSIR

(A) sec/m2

5 2 (B) sec/m5

2 2 (C) sec/m2

2 (D) sec/m5

2

16. Let f(x) and g(x) be real valued functions such that f(x) . g(x) = 1 Ry,x . If )x(f and

)x(g exist for all values of x, and )x(f and )x(g are never zero, then)x(g

)x(g

)x(f

)x(f

is equal to

(A) )x(f

)x(g2 (B) )x(g

)x(g2 (C) )x(g

)x(f2 (D) )x(f

)x(f2

17. Consider the parabola y2 = 4x. )4,4(A and )6,9(B be two fixed points on the parabola.Let �C� be a moving point on the parabola between A and B such that the area of triangle ABC ismaximum, then coordinate of �C� is

(A)

1,

4

1(B) (4, 4) (C) 32,3 (D) 32,3

18. The equation x3 - 3x + a = 0 will have exactly one real root if(A) (0, 2) (B) (-2, 2)(C) ),2()2,( (D) (-2, 0)

19. The inequality x2 - 4x > cot-1 x is true in(A) [0, 4] (B) (4, 5)(C) ),5[]1,( (D) (-1, 4)

20. Total number of critical points of f(x) = max. {sin x, cos x} )2,2(x is equal to(A) 5 (B) 7 (C) 4 (D) 3

21. The equation x3 - 3x + [a] = 0, where [.] denotes the greatest integer function, will have real anddistinct roots if(A) )2,(a (B) )2,0(a(C) ),0()2,(a (D) )2,1[a

22. y = f(x) is parabola, having its axis parallel to y-axis. If the line y = x touches this parabola at x = 1,then(A) 1)0(f)1(f (B) 1)1(f)0(f (C) 1)0(f)1(f (D) 1)1(f)0(f

23. Let 0)x(g and Rx0)x(f then(A) g ( f (x + 1) ) > g ( f (x - 1) ) (B) f ( g (x - 1) ) > f ( g (x + 1) )(C) g ( f (x + 1) ) < g ( f (x - 1) ) (D) g ( g (x + 1) ) < g ( g (x - 1) )

24. If Rx0)x(f then for any two real numbers x1 and x

2, )xx( 21




ABJSIR

(A) 2

)x(f)x(f

2

xxf 2121

(B) 2

)x(f)x(f

2

xxf 2121

(C) 2

)x(f)x(f

2

xxf 2121

(D)

2

)x(f)x(f

2

xxf 2121

25. Let 0)x(sinf and

2

,0x0)x(sinf and g(x) = f(sin x) + f(cos x), then g(x) is

decreasing in

(A)

2,

4(B)

4,0 (C)

2,0 (D)

2,

6

26. f(x) = 2 sin3 x - 3 sin2 x + 12 sin x + 5

2

,0x then

(A) �f � is increasing in

2,0

(B) �f � is decreasing in

2,0

(C) �f � is increasing in

4,0 and decreasing in

2,

4

(D) �f � is decreasing in

4,0 and increasing in

2,

4

27. Let f(x) =

3x,18x

3x0),3a3a(nxx4 23 . Complete set of �a� such that f(x) has a local

minima at x = 3, is

(A) [-1, 2] (B) ),2()1,( (C) [1, 2] (D) ),2()1,(

28. The equation x + cos x = a has exactly one positive root. Complete set of values of �a� is(A) (0, 1) (B) )1,( (C) (-1, 1) (D) ),1(

29. If the function f(x) = 2

x

e).4x(x

has its local maxima at x = a then

(A) a = 22 (B) a = � 22 (C) a = 4 (D) a = � 4

30. If the curve 14

y

a

x 2

2

2

and y2 = 16 x intersect at right angle then

(A) 1a (B) 3a (C) 3

1a (D) 2a




ABJSIR

SET�III

1. If f is twice differentiable at x = a ; then which of the following is True

(A) If f(a) is an extreme value of f(x), then 0)a(f

(B) If 0)a(f , then f(a) is an extreme value of f(x)

(C) If 0)a(f and f (a) 0 then function has a local minima at x = a(D) none of these

2. The line ax � by + c = 0 is normal to the curve xy = � 1 then which one of the following is/are is nottrue(A) a > 0, b > 0 (B) a < 0, b < 0 (C) a > 0, b < 0 (D) a < 1, b > 1

3. Let p(x) = a0 + a

1x2 + a

2x4 + .....................................+ a

nx2n be a polynomial in a real variable

x with 0 < a0 < a

1 < a

2 <........<a

n. The function p(x) has

(A) Neither a maximum nor a minimum (B) only one maximum(C) only one minimum (D) none of these

4. At x = a, there is minimum for a given function f (x), then

(A) x alim

f (x) = x alim

f (x) (B) x alim

f (x) > 0 , x alim

f (x) < 0

(C) x alim

f (x) < 0, x alim

f (x) < 0 (D) nothing can be said

5. Let f be a twice differentiable function satisfying f(1) = e; f(2) = e2 ; f(3) = e3 , then which of thefollowing is/are false

(A) f(x) = ]3,1[xex

(B) xe)x(f has atleast three solution in [1, 3]

(C) xe)x(f has atleast two solution in [1, 3]

(D) xe)x(f has a solution in [0, 4]

6. Let f(x) and g(x) are defined and differentiable for x x0 and f(x

0) = g(x

0) , f (x) > g (x) for x >

x0 , then which of the following is/are not true

(A) f(x) > g(x) for some x > x0

(B) f(x) = g(x) for some x > x0

(C) f(x) > g(x) for all x > x0

(D) f(x) > g(x) for no x > x0

7. If the function f(x) increases in the interval (a, b) then the function (x) = [f(x)]2

(A) increases in (a, b)(B) decreases in (a, b)(C) we cannot say that (x) increases or decreases in (a, b)(D) none of these




ABJSIR

8. Let f(x) = x

2sin , 0 x 1

3 2 x , x 1

, then

(A) f(x) has local maxima at x = 1(B) f(x) has local minima at x = 1(C) f(x) does not have any local extrema at x = 1(D) f(x) has a global minima at x = 1

9. Among the following statements which one is/are true

(A) ),0(inx)x1(n (B) x n(1 x) in (0, )

(C) )2/,0(inxxtan (D) )2/,0(inxxtan

10. If a < b < c < d and x R then the least value of the function,f(x) = x a + x b + x c + x d is(A) a + c b d (B) a + b + c + d (C) c + d a b (D) a + b c d

11. Let f(x) be a differentiable function upto any order such that f (x).f (x) 0 x R . If and

be the two consecutive real roots of f(x) = 0, then

(A) )x(f must be equal to zero for atleast one ),(x

(B) )x(f must be equal to zero for atleast one ),(x

(C) 0)x(f ),(x (D) none of these

12. Among the following statements which one is/are true(A) The cubic equation x3 + 2x2 + x + 5 = 0 has three real roots.(B) The cubic equation x3 + 2x2 + x + 5 = 0 has only one real root.(C) The cubic equation x3 + 2x2 + x + 5 = 0 has only real root , such that [ ] = �3.

(D) The cubic equation x3 + 2x2 + x + 5 = 0 has three real roots ,, , such that

1][,2][,3][ , (where [.] denotes the greatest integer function)

13. Let f(x) = | x 1| a, x 1

2x 3, x 1

. If f(x) has a local minima at x = 1, then a is not

(A ) less than 5 (B) greater than or equal to 5(C) less than or equal to 5 (D) none of these

14. A car is driven at speed of x km/hr., where )120,20(x and its mileage is given by

)x(g

))x(g(n)x(f

, where g(x) = 1x50

1e

, then the best economical speed is

(A) 70 km./hr. (B) 49 + e km./hr.(C) 50 km./hr. (D) 59 + e km./hr.




ABJSIR

W I Read the following passage and give the answer of question 15 to 17If a function f(x) is :(a) continuous is closed interval [a, b],(b) differentiable in open interval (a, b), then exists at least one c between a and b such

that ab

)a(f)b(f)c(f

.

15. Suppose f(x) =

0x2,xx2

2x0,xx22

2

, then in the interval [�2, 2]

(A) both LMVT and Rolle's theorem can be applied(B) only LMVT can be applied(C) only Rolle's theorem can be applied(D) neither Rolle's theorem nor LMVT can be applied

16. By Lagrange's Mean Value Theorem, which of the following is true for x > 1

(A) 1 + x n x < x < 1 + n x (B) 1 + n x < x < 1 + x n x

(C) x < 1 + x n x < 1 + n x (D) 1 + n x < 1 + x n x < x

17. If f(x) and g(x) satisfy the conditions of Mean Value Theorem on the interval [a, b], then whichof the following function satisfies the conditions of Rolle's Theorem on [a, b](A) g(a) f(x) + g(b) g(x) (B) (g (a) + g(b)) f(x) + (f(a) + f(b)) g(x)(C) (g(b) � g(a)) f(x) + (f(a) � f(b)) g(x) (D) none of these

W II Read the following passage and answer the question 18 to question 21A conical vessel is to be prepared out of a circular sheet of copper of unit radius as shown in thefigure where be the angle of the sector removed (i.e. AOB ), then

18. The volume of the vessel. ( If )

(A) 24

(B)

6

3 2(C)

24

3 (D) none of these

19. The value of �r� for which volume is maximum (when is variable)

(A) 3

2(B)

3

2(C)

3

2(D) none of these

20. The value of � � for which volume is maximum (when is variable)

(A) 2 2

3

(B)

3

222 (C) 2 (D) none of these




ABJSIR

21. The sectorial area is to be removed from the sheet so that vessel has the maximum volume, is

(A) 23 (B)

3

23(C)

3

2 (D) none of these

W III Read the following passage and give the answer of question 22 to 24Let f be continuous and differentiable on an interval I. Then f is increasing or decreasing on I if and

only if 0)( xf or 0)( xf respectively for all x in I. Answer the following questions from 5 to 8.

22. Let xxxf cos)( . Then the equation 0)( xf has

(A) Unique solution in )6/,0( (B) Unique solution in )3/,6/( (C) infinitely many solutions in )4/,0( (D) infinitely many solutions in )2/,0(

23. Let f be continuous and differentiable function such that f (x) and )(xf have opposite signseverywnere. Then(A) f is increasing (B) f is decreasing(C) | f | is increasing and decreasing (D) | f | is decreasing

24. Let dxxxexf x )2)(1()( . Then f decreases in the interval :

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

W IV Consider the following function and answer the question 25 to question 29f(x) = 2x3 � 3(a � 3)x2 + 6ax + a + 2

25. The value of �a� for which f(x) has exactly one point of local maxima and one point of local minima

(A) ),9()1,( (B) ( , 1] [9, ) (C) [1, 9] (D) (1, 9)

26. The value of �a� for which f(x) has local minima at some negative real x

(A) ),9()1,( (B) ( , 1] [9, ) (C) (0, 1) (D) (1, 9)

27. The values of �a� for which f(x) has local maxima at some negative and local minima atpositive real x

(A) ( , 0] [9, ) (B) (9, )

(C) (0, 1] (D) )0,(

28. The values of �a� for which f(x) has no point of extrema

(A) [1, 9] (B) (C) )0,( (D) (1, 9)

29. The values of �a� for which f(x) is increasing in ),2[

(A) [1, 9] (B) (1, 9) (C) ( , 9] (D) ( , 1]




ABJSIR

W V Consider the following function and answer the question 30 to question 33Consider the curve x = 1 � 3t2, y = t � 3t3 . If tangent at point (1 � 3t2, t � 3t3 ) inclined at an angle

to positive x�axis and tangent at point P(�2, 2) cuts the curve again at Q.

30. The curve is symmetrical about(A) y � x = 0 (B) y + x = 0 (C) y = 0 (D) x = 0

31. sectan is equal to(A) t (B) 3t (C) t + 3t2 (D) none of these

32. The point Q will be

(A)

9

2,

3

1(B) (1, �2) (C) (�2, 1) (D) none of these

33. The angle between the tangents at P and Q will be

(A) 6

(B)

4

(C)

3

(D)

2

W VI Read the following passage and give the answer of question 34 to 38

A circular arc PQ of radius 1 subtends an angle of x radian at its centre O, ( 0 < x < ) as shownin the figure. The point R is the intersection of the two tangents at points P and Q of the arc. Let usdefine the following functionS(x) = area of the sector OPQT(x) = area of the triangle PQRU(x) = area of the shaded region

34.

4S has value

(A) 4

(B)

8

(C)

2

1(D) none of these

35. The expression for T(x) is

(A) xsin2

1(B)

2

xsin

2

xtan

(C)

2

xsin

2

xtan

2

1 2(D) none of these




ABJSIR

36. If x0 , then the function U(x) is(A) always increasing(B) always decreasing

(C) increases in

2,0 and decrease in

,2

(D) decreases in

2,0 and increases in

,2

37. For the domain x0 , the root of the equation )x(T)x(U2

x is

(A) 4

(B)

3

(C)

2

(D) none of these

38. The value of the limit x 0

U(x)Lim

T(x), is equal to

(A) 1 (B) 2

3(C)

3

2(D) none of these

W VII Consider the following function and answer the question 39 to question 41Suppose f

(x) is continuous on an interval I, and a and b are two points of I. Then if y

0 is a number

between f (a) and f

(b), there exists a number c between a and b such that f

(c) = y

0 . In particular

if f (a) and f

(b) possess opposite signs

, then there exists atleast one solution of the equation

f (x) = 0 in the open interval (a , b) .

39. Let 52x

1

3x

1

1x

1)x(f

, then f(x) = 0 has

(A) no real roots (B) at most one real roots(C) exactly 3 real roots (D) exactly 2 real roots

40. f(x) = ax2 + bx + c and 3a + b + 3c = 0, then f(x) = 0 has(A) two real and distinct roots(B) two real and equal roots(C) non real roots(D) real roots as well as non real roots depending upon a, b and c

41. Let R),0[:f be a continuous function such that 23f(1) + 10f(2) + 2005f(3) = 0 , thenf (x) = 0 has always at least one real root in(A) [2, 3] (B) [1, 2] (C) (1, 2) (D) (0, 4)




ABJSIR

LEVEL�I ANSWER

1. 8 sq. cm 2. 0 cm /sec,14cm2/sec 3. sec/cm15

2

7. 6

3

2,FHGIKJ 15. I in 0, , D in [0, 1), I in ,2 , D in (1, 2]

16. L

NMOQP

43 21

21, ( , ) 17. a 0 18. [ , )1

19. b (� 7, � 1) (2, 3) 20. g (x) is I in (0,4/3] and D in [4/3,2)

29. (a + b)2 31. p < -2, 2 < p < 4 - 8

332. 2x4 - 12/5 x5 + 2/3 x6

34. max. at a/6 36. 4

3 3

3r

40. x = a sin , y = a cos , where = 0.5 tan-12

LEVEL�II

2. -1500 ft/sec 3. 200

5

3

2

R

R( ) km2/h 8. 4

3xy and

4

1xy

26. r 2

2 feet 27. side 10 height 10 28. cos A = 0.8

29. ( )( )c a b c a b

231. 27 3 sq. cms 38.

1

2

5

4,FHGIKJ

32. 4

33 units 35. y = x

3

136. |a - b|

37. when x = e�4/3 39. 962 or 963 41. 2 2

43. greatest = 14, least = 8

SET�I

1. D 2. D 3. B 4. D 5. B6. B 7. D 8. A 9. A 10. D




ABJSIR

11. B 12. C 13. D 14. C 15. D

16. C 17. D 18. A 19. D 20. D

21. C 22. C 23. C 24. B 25. D

26. D 27. B 28. D 29. D 30. A

SET�II1. D 2. C 3. B 4. D 5. C

6. C 7. A 8. C 9. D 10. A

11. C 12. C 13. C 14. D 15. B

16. D 17. A 18. C 19. C 20. B

21. D 22. C 23. C 24. B 25. B

26. A 27. C 28. D 29. A 30. D

SET�III1. AC 2. ABD 3. C 4. D 5. ABC

6. ABD 7. C 8. A 9. AC 10. C

11. B 12. BC 13. AC 14. C 15. A

16. B 17. C 18. C 19. C 20. B

21. B 22. B 23. D 24. C 25. A

26. C 27. D 28. A 29. C 30. C

31. B 32. A 33. D 34. C 35. B

36. A 37. C 38. C 39. C 40. A

41. D


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