Title Jee 2014 Booklet5 Hwt Differential Calculus 2 Analysis Tangent Monotonic Function Maxima And Minima 213.8 KB 11
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Page 1

Vidyamandir Classes

VMC/Differential Calculus-2 69 HWT/Mathematics

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-2 

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One
correct option.

1. The graph of  21 1y x   is :

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

2. The range of the function   1
2

f x
cos x

is :

(A)
1

1
3

,
 
 
 

(B) [0, 1] (C) 1 1,   (D)
1

0
3

,
 
 
 

3. The tangent at the point (5, 5) on the curve
3

2

10

x
y

x

meets the curve again at the point Q. The coordinates of Q are :

(A)
1

1
3

,
 

 
 

(B)
1

1
3

,
 
 
 

(C)  2 2,  (D)  2 1, 

4. For the curve defined as
2 2

1 1
2

1 1

2 1

x x
y cos ec sec

x x

         
      

the set of points at which the tangent is parallel to the x-axis is :

(A) [0, 1] (B) (0, 1) (C)    1 0 1, ,   (D) None of these

*5. Given  
2

2

16
f x tan x

   
 
 

and 0 1A R ,    

(A) Range of f (x) is A (B) Range of f (x) is A’

(C) Maximum value of f (x) =1 (D) Maximum value of    1 1f x  

6. The coordinates of a point on the curve 3 3 6x y xy  at which the tangent is parallel to the x-axis are :

(A)  4 3 5 32 2/ /,  (B)  4 3 5 32 2/ /, (C)  5 3 4 32 2/ /, (D) None of these

7. If  and  are the lengths of perpendiculars from the origin to the tangent and normal to the curve 2 3 2 3 2 35x y / / /

respectively then 2 24  is :

(A) 625 (B) 125 (C) 25 (D) 252/3

8. The curve 3 2 5y ax bx cx    touches x-axis at  2 0A , . The curve intersects the y-axis at a point B where its slope equals 3.
The value of ‘a’ is :

(A) 2 (B) 2 (C)
1

2

(D)

1

2

Y

X
O

Y

X
O

Y

X
O

Y

X
O

Page 2

Vidyamandir Classes

VMC/Differential Calculus-2 70 HWT/Mathematics

9. The function    34 242 80 32f x x x x    is :
(A) Monotonically increasing in    4 1 5, ,    (B) Monotonically increasing in    4 1 5, ,   
(C) Monotonically increasing in  4 5, (D) None of these

*10. If    2 4 2 4f x max x , | x |, | x |    then :
(A) f (x) is continuous for all x R (B) f (x) is differentiable except at

1 33

2
x
 

(C) f (x) has a critical point at x = 2 (D) f (x) has no maximum

Page 5

Vidyamandir Classes

VMC/Differential Calculus-2 73 HWT/Mathematics

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-2 

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1. Let f : a, b R   be a function such that for             0iv vc a, b , f c f c f c f c f c        , then :
(A) f has local extremum at x = c (B) f has neither local maximum nor local minimum at x = c
(C) f is necessarily a constant function (D) It is difficult to say whether (A) or (B)

2. The largest value of 3 22 3 12 10x x x   for 2 4x   occurs at x =
(A) 2 (B) 1 (C) 2 (D) 4

3. If the curves 2 16y x and 2 29 16x by  cut each other at right angles, then the value of b is :

(A) 2 (B) 4 (C) 9/2 (D) None

4. The curve 2
n n

x y

a b
   

    
   

touches the line 2
x y

a b
  at the point (a, b) for n =

(A) 1 (B) 2 (C) 4 (D) 0n 

5. The function   a sin x bcos xf x
c sin x d cos x

is monotonically decreasing in its domain if :

(A) 0ad bc  (B) 0ab cd  (C) ad bc (D) None of these

6. The complete set of values of x in which     22 2 4 1ef x log x x x     increases, is :

(A) (1, 2) (B) (2, 3) (C)
5

3
2

,
 
 
 

(D) (2, 4)

7. Segment of the tangent to the curve 2xy c at the point  x , y' which is contained between the co-ordinate axes, is bisected at the
point

(A)  x , y   (B)  y , x  (C)
2 2

x y
,
  

 
 

(D) None

8. The tangent and normal to the curve 2 2y sin x sin x  are drawn at
3

P x
 
 

 
. The area of the quadrilateral formed by the

tangent, the normal and co-ordinate axes is :

(A) 3 (B) 3 (C) 3 2/ (D) None

9. In 1 2,   the function   1f x x x   is :
(A) increasing (B) decreasing (C) constant (D) None of these

10. The number of solutions of the equations     0f xa g x  , where  0 0a , g x  and has minimum value 1/2, is :

(A) infinitely many (B) only one (C) two (D) zero

Page 6

Vidyamandir Classes

VMC/Differential Calculus-2 74 HWT/Mathematics

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-2 

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1. The curves 2 2 1ax by  and 2 2 1a x b y   intersect orthogonally if :

(A)
1 1 1 1

a b a b
  

 
(B)

1 1 1 1

a b a b
  

 
(C)

1 1 1 1

a a b b
  
 

(D) None of these

2. All the points on the curve y x sin x  at which the tangent is parallel to x-axis lie on a / an :
(A) straight line (B) circle (C) parabola (D) ellipse

3. If the function   a sin x cos xf x
sin x cos x

is increasing for all values of x, then :

(A) a > 1 (B) a < 1 (C) a < 2 (D) a > 2

4.  
6 : 1

7 : 1

x
f x

x x


 

 
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

5. The tangent to the curve    2 2x a cos cos , y a sin sin     , at the point
4

  is :

(A) parallel to x-axis (B) parallel to y-axis (C) parallel to line y = x(D) None of these

6. If curve 2y x bx c   touches the straight line y = x at the point (1, 1), then b and c are given by :
(A) 1 1, (B) 1 2, (C) 2, 1 (D) 1, 1

7.  
  
 

2 1
3

3

x x
f x x

x

 
  

. The minimum value of f (x) is equal to :

(A) 3 2 (B) 3 2 2 (C) 3 2 2 (D) 3 2 2

8. On [1, e], the least and greatest value of   2f x x ln x is :
(A) e, 1 (B) 1, e (C) 0, e2 (D) None

9. If 1 1[ ] [ ]sin x cos x  , where [.] denotes the greatest integer function, then complete set of values of x is :

(A) [cos 1, 1] (B) [sin 1, 1] (C) [cos 1, sin 1] (D) [0, 1]

10.    2 24 1 1f x max . , x , x x R     . Total number of points, where f (x) is not differentiable, is equal to :
(A) 2 (B) 4 (C) 6 (D) None

Page 10

Vidyamandir Classes

VMC/Differential Calculus-2 78 HWT/Mathematics

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-2 

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct. However, questions marked with '*' may have More than One
correct option.

1. If   xf x
sin x
 and   xg x

tan x
 where 0 1x  , then in this interval :

(A) Both f (x) and g (x) are increasing functions (B) Both f (x) and g (x) are decreasing functions
(C) f (x) is an increasing function (D) g (x) is an increasing function

2. The set of all x for which  1log x x  is :
(A)  0,  (B)  1,  (C)  1 0, (D) None of these

*3. The function     22 2 4 1f x log x x x     increases on the interval :

(A) (1, 2) (B) (2, 3) (C)
5

3
2

,
 
 
 

(D) (2, 4)

4. Let   3 2 25f x x ax bx sin x    be an increasing function on the set R. Then a and b satisfy :

(A) 2 3 15 0a b   (B) 2 3 15 0a b   (C) 2 3 15 0a b   (D) 0a  and 0b 

*5. The function   3 6 1ef x log x x     
is of the following types :

(A) even (B) odd (C) increasing (D) decreasing

6. If f is an increasing function and g is a decreasing function on an interval I such that fog exists, then :

(A) fog is an increasing function on I (B) fog is a decreasing function on I

(C) fog is neither increasing nor decreasing on I (D) None of these

*7. Which of the following functions are decreasing on 0
2

,
 

 
 

?

(A) cos x (B) 2cos x (C) 3cos x (D) tan x

8. The function 3 23 6 17y x x x    :

(A) increases everywhere (B) decreases everywhere

(C) increases for positive x and decreases for negative x

(D) increases for negative x and decreases for positive x

9. The interval in which the function x3 increases less rapidly than 26 15 5x x  , is :

(A)  1,  (B)  5 1, (C)  1 5, (D)  5, 

10. A condition for a function  y f x to have an inverse is that it should be :

(A) defined for all x (B) continuous everywhere

(C) strictly monotone and continuous in the domain (D) an even function

Page 11

Vidyamandir Classes

VMC/Differential Calculus-2 79 HWT/Mathematics

DATE : TIME : 30 Minutes MARKS : [ ___ /10] TEST CODE : DC-2 

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking.

Choose the correct alternative. Only one choice is correct.

1. The maximum value of
1

x

x
 
 
 

is :

(A) e (B) ee (C) 1 ee (D)
1

e

2. The value of a for which the function   1 3
3

f x a sin x sin x
 

   
 

has an extremum at
3

x

 is :

(A) 1 (B) 1 (C) 0 (D) 2

3. The number of critical points of  
2

1x
f x

x

 is :

(A) 1 (B) 2 (C) 3 (D) None of these

4. The greatest value of      1 3 1 31 1f x x x    on [0, 1] is :

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

3

5. The difference between the greatest and least values of the function   1 12 3
2 3

f x cos x cos x cos x   is :

(A)
2

3
(B)

8

7
(C)

9

4
(D)

3

8

6. On the interval [0, 1] the function  7525 1x x takes its maximum value at the point :

(A) 0 (B)
1

4
(C)

1

2
(D)

1

3

7. Let   2 4 20 1 2 nnP x a a x a x . . . a x     be a polynomial in a real variable x with 0 1 20 na a a . . . a     . The function
P(x) has :

(A) neither a maximum nor a minimum (B) only one maximum

(C) only one minimum (D) None of these

8. The maximum value of xy subject to 8x y  is :

(A) 8 (B) 16 (C) 20 (D) 24

9. The maximum area of the rectangle that can be inscribed in a circle of radius r is :

(A) 2r (B) 2r (C)
2

4

r
(D) 22r

10. If  
23 12 1 1 2

37 2 3

x x , x
f x

x , x

     
 

  
, then :

(A) f (x) is increasing in 1 2,   (B) f (x) is continuous in 1 3,  

(C) f (x) is maximum at x = 2 (D) All of these