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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:
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