Engineering Mathematics – I

Unit – I [Differential Calculus]

Tangent, Normal and Maxima, Minima

Video Explanation: https://youtu.be/G1D7MdH_rOE

1

The equation of the tangent to the curve 1

yx

at 3x is

A. 9 1y x

B. 9 6x y

C. 9 0x y

D. 3 1y x

ANSWER:

2

The slope of the tangent to the curve 2 2y x x at 3x is

A. 1

2 3

B. 1

2 3

C. 1

3

D. 2

3

ANSWER:

3

The equation of normal line to the curve 4 2 xy x e at the point (0,2) is

A. 2 4x y

B. 2 2 0x y

C. 3 4x y

D. 3 2x y

ANSWER:

4

The equation of the tangent line to the curve 3y x at the point (2,8) is

A. 3 16x y

B. 3 16x y

C. 12 16x y

D. 12 16x y

ANSWER:

5

If ( ) 0f x , what does that mean with respect to the tangent line?

A. Tangent line is vertical

B. Tangent line has a positive slope

C. Tangent line is horizontal

D. Tangent has a negative slope

ANSWER:

6

Condition for vertical tangent of the curve ( )y f x is

A. 0dy

dx

B. dy

dx

C. 0dy

dx

D. 0dy

dx

ANSWER:

7

Condition for tangent parallel to x -axis for the curve ( )y f x is

A. 0dy

dx

B. dy

dx

C. 0dy

dx

D. 0dy

dx

ANSWER:

8

The equation of the tangent to the curve 2

4y x

x , that is parallel to the x -axis, is

A. 0y

B. 1y

C. 2y

D. 3y

ANSWER:

9

The tangent to the curve 2y ax bx at (2, 8) is parallel to x -axis, then

A. 2, 2a b

B. 2, 4a b

C. 2, 8a b

D. 4, 4a b

ANSWER:

10

The points at which the tangents to the curve 4 22 2y x x are parallel to x -axis are

A. (0,2), ( 1,1), (1,1)

B. (0,2), ( 1,1), (1, 1)

C. (0,2), (1,1), (1, 1)

D. (0, 2), ( 1,1), (1,1)

ANSWER:

11

If normal to the curve ( )y f x is parallel to x -axis, then which of the following is true?

A. 0dy

dx

B. 1dy

dx

C. 0dx

dy

D. 1dx

dy

ANSWER:

12

The sum of intercepts on co-ordinate axes made by tangent to the curve x y a

is

A. a

B. 2a

C. 2 a

D. a

ANSWER:

13

The points on the curve 312y x x at which the gradient is zero are

A. (2, 16) and (2, 0)

B. (2, 16) and ( 2, 16)

C. (2, 16) and ( 2, 16)

D. (0, 16) and (2, 16)

ANSWER:

14

The slope of the normal to the curve 4 2 xy x e at the point (0,2) is

A. 1

2

B. 1

2

C. 1

3

D. 1

3

ANSWER:

15

The equation of the normal line to the curve sin2

xy

at (1, 1) is

A. 1y

B. 1x

C. y x

D. 2

1 1y x

ANSWER:

16

The equation of the normal line to the curve siny x at 3

x

is

A. 3 1

2 2 3y x

B. 3 1

2 2 3y x

C. 3

22 3

y x

D. 3

22 3

y x

ANSWER:

17

The normal to the curve (1 cos )x a , siny a at always pass through the

fixed point

A. ( , )a a

B. (0, )a

C. (0,0)

D. ( ,0)a

ANSWER:

18

If 2x t and 2y t , then the equation of the normal at 1t is

A. 3 0x y

B. 1 0x y

C. 1 0x y

D. 3 0x y

ANSWER:

19

The slope of the tangent to the curve 2 3 8x t t ,

22 2 5y t t at the point

(2, 1) is

A. 22

7

B. 6

7

C. 6

D. None of these

ANSWER:

20

The point of the curve 2 2( 3)y x at which the normal is parallel to the line

2 1 0y x is

A. (5, 2)

B. 1

, 22

C. (5, 2)

D. 3

, 22

ANSWER:

21

The line 2x y is tangent to the curve 2 3 2x y at its point

A. (1, 1)

B. 1, 1

C. 1, 1

D. 1, 1

ANSWER:

22

The tangent drawn at the point (0,1) on the curve 2xy e meet x -axis at the point

A. 1

, 02

B. 1

, 02

C. 2, 0

D. 0, 0

ANSWER:

23

Angle between the tangents to the curve 2 5 6y x x at the points (2, 0) and

(3, 0) is

A. 2

B. 6

C. 4

D. 3

ANSWER:

24

Angle between the tangents to the curves 2y x and

2x y at the point (1, 1) is

A. 1tan (1)

B. 2

C. 1 4

tan3

D. 1 3

tan4

ANSWER:

25

Which of the following is monotonic

A. 2y x

B. y x

C. siny x

D. cosy x

ANSWER:

26

Which of the following statement(s) is/are true?

A. The graph of an increasing function rises from left to right.

B. The graph of a decreasing function falls from left to right.

C. The monotonicity of a function refers to if the function is increasing or decreasing.

D. All of these statements are true.

ANSWER:

27

For all x in the given interval, if ( ) 0f x , then ( )f x is

A. increasing in the interval

B. decreasing in the interval

C. constant in the interval

D. Undefined in the interval

ANSWER:

28

For all x in the given interval, if ( ) 0f x , then ( )f x is

A. increasing in the interval

B. decreasing in the interval

C. constant in the interval

D. Undefined in the interval

ANSWER:

29

The critical numbers of 35 6y x x are

A. 3

2

B. 2

3

C. 2

5

D. 5

2

ANSWER:

30

The function 2( )f x x is decreasing in

A. ,

B. ,0

C. 0,

D. 2,

ANSWER:

31

The function 2

( ) , ( 1)1

xf x x

x

is increasing on the interval

A. ,1

B. 0,

C.

D. None of these

ANSWER:

32

In the interval 0,1 , the function 2 1x x is

A. Increasing

B. Decreasing

C. Neither increasing nor decreasing

D. None of these

ANSWER:

33

On the interval 0,2

, the function ( ) logsinf x x is

A. Increasing

B. Decreasing

C. Neither increasing nor decreasing

D. None of these

ANSWER:

34

The function 3 2( ) 2 9 12 1f x x x x is decreasing in the interval_______.

A. 2,

B. 2, 1

C. , 1

D. , 2 and 1,

ANSWER:

35

The function 3( ) 27 5f x x x is an increasing function, when

A. 3x

B. 3x

C. 3x

D. 3x

ANSWER:

36

For every value of x on R , the function ( ) xf x e is

A. Decreasing

B. Increasing

C. Neither increasing nor decreasing

D. None of these

ANSWER:

37

For every value of x on R , the function 1

( )3x

f x is

A. Decreasing

B. Increasing

C. Neither increasing nor decreasing

D. None of these

ANSWER:

38

The function ( ) tanf x x x is

A. monotonically increasing

B. monotonically deceasing

C. neither increasing nor decreasing

D. both increasing and decreasing

ANSWER:

39

The function 1

( ) , ( 0)f x x xx

is a non-increasing function in the interval

A. 1,1

B. 0,1

C. 1,0

D. 2,1

ANSWER:

40

The function ( )f x x is increasing in

A. ,

B. ,0

C. 0,

D. 2,

ANSWER:

41

Find the range of values of x for which the function is increasing 25 8 2y x x

A. 3x

B. 2x

C. 2x

D. 2x

ANSWER:

42

The minimum value of the function 2( ) 5 8 14f x x x is

A. 32

B. 54

5

C. 54

5

D. 32

ANSWER:

43

The maximum value of the function 2( ) 3 5 21f x x x is

A. 232

3

B. 254

5

C. 277

12

D. 235

3

ANSWER:

44

The maximum value of the function 3 2( ) 2 3 36f x x x x is

A. 81

B. 61

C. 51

D. 31

ANSWER:

45

The minimum value of the function 4 3( ) 4f x x x is

A. 28

B. 3

C. 27

D. 16

ANSWER:

46

The function 2

2

x

x has a local minimum at

A. 2x

B. 0x

C. 1x

D. 2x

ANSWER:

47

The maximum value of 3 3x x in the interval [0, 2] is

A. 1

B. 2

C. 0

D. 2

ANSWER:

48

The function 23 2 5y x x has

A. 31

max at8

B. 31

min at8

C. 49

min at8

D. 49

max at8

ANSWER:

49

Does the graph of the following function have a maximum or minimum? What is it?

A. min at 2

B. max at 0

C. max at 2

D. min at 0

ANSWER:

50

For what value of x , the function 3 2( ) 2 3f x x x

attain local minimum

A. 1

B. 0

C. 1.5

D. there is no minimum

ANSWER:

51

Which of the following curves is concave down?

A. 2y x

B. 2y x

C. xy e

D. 2 2 3y x x

ANSWER:

52

The domain of concavity of the function 22y x is

A. ( , 0)

B. everywhere concave upward

C. everywhere concave downward

D. (0, )

ANSWER:

53

The curve 3 2y ax bx cx d has a point of inflection at 1x then

A. 0a b

B. 3 0a b

C. 3 0a b

D. 3 1a b

ANSWER:

54

The point of inflection of the curve 3 22 3 36y x x x is

A. 1 37

,3 2

B. 1 37

,2 2

C. 1 37

,2 3

D. 1 37

,3 3

ANSWER:

55

The points of inflection of the curve 2 42 2y x x is

A. 1 17 1 17

, and ,9 92 2

B. 1 17 1 17

, and ,9 93 3

C. 1 23 1 23

, and ,9 92 2

D. 1 23 1 23

, and ,9 93 3

ANSWER:

56

The points of inflection of the function 4 3( ) 12 6 9f x x x x on the interval

2 10x is at

A. 0, 6x

B. 0, 6x

C. 6x

D. 12x

ANSWER: