NCERT Solutions for class 13 Maths | Chapter 13 - Introduction to Limits and Derivatives

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Questions
1 The derivative of f(x) = |x| at x = 0 is ____.
A -1
B 1
C 0
D does not exist

Answer:does not exist
2 If y = 2 sin 3x cos x, then d20y/dx20 is ____.
A y20=−240cos4x−240cos2x
B y20=240cos4x−240cos2x
C y20=240cos4x+240cos2x
D y20=240sin4x+240sin2x

Answer:y20=240sin4x+240sin2x
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3 The displacement time relating to a speeding vehicle is given by D(t) = 5 t2- 3t + 30. Then, its speed at the end of 5 seconds is given by ____.
A 50 units/ sec
B 115 units/ sec
C 47 units/ sec
D 30 units/ sec

Answer:47 units/ sec
4 The point on the curve x2 + y2 - 2x + 1 = 0, at which the tangent is parallel to Y axis is given by ____.
A (0, -1)
B (0, 1)
C (1, 0)
D (-1, 0)

Answer:(1, 0)
5 The degree of a polynomial function is 5. Then, the degree of it's third derivative is ____.
A 2
B 3
C 0
D 5

Answer:2
6 The number of points in (1, 2) where f(x) = a[x2],a > 1 is not differentiable is ____.
A 0
B 2
C at least 5 points
D 3 points

Answer:2
7 The slope of the normal at the point θ=π/4 of the curve sin θ - cos θ is ____.
A 2
B -1/2
C 1/2
D -2

Answer:-1/2
8 If f(x) = ex log x, then f'(x) is ____.
A  xf'(x) - xf(x) = e-x
B ex f'(x) - ex f(x) = 1/x
C f1(x) = f(x) + ex/x
D f'(x) - f(x) = ex

Answer:f1(x) = f(x) + ex/x
9 The displacement of a particle describing a non - linear motion is given by s = (1/6) t3 - 8t. Then, the acceleration at the time when velocity vanishes is ____.
A 6 units/ s2
B 0
C 4 units/ s2
D 16 units/ s2

Answer:4 units/ s2
10 The mathematical expression x→a means_____
A x = a
B x ≠ a
C x is approaching a
D x is greater than a

Answer:x is approaching a
11 In the curve y2 = x3, the slope of the tangent at (4, 8) is ____.
A 4
B 8
C 1/2
D 3

Answer:3
12 If f(x) = emx + enx, then the second derivative f'(x) is ____.
A m2emx + n2enx
B memx + nenx
C emx + enx
D emx - enx

Answer:m2emx + n2enx
13 The deleted neighborhood of a is ____.
A (a - δ, a] i.e. left neighborhood
B [a, a + δ) i.e. right neighborhood
C a sphere of radius 'a' with the point 'a' deleted
D (a - δ, a + δ) - {a}

Answer:(a - δ, a + δ) - {a}
14 If y = tan x + sec x, then d2y/dx2 is ____.
A 0
B 1
C y
D (dy/dx)y

Answer:(dy/dx)y
15 If y = x2 sin 2x, then the first derivative is ____.
A 2(x cos 2x + sin 2x)
B (x cos 2x + sin 2x)
C 2x (x cos 2x + sin 2x)
D x2 (x cos 2x + sin 2x)

Answer:2x (x cos 2x + sin 2x)
16 The differential of (sin x + cos x)2is
A 0
B cos x - sin x
C 2 cos 2x
D cos2 x - sin2 x

Answer:2 cos 2x
17 In a real number line, the neighbourhood of positive real number 'a' with a very small radius δ>0 will be ____.
A (a - δ, a]
B [a, a + δ)
C (a - δ, a + δ)
D a sphere of radius a

Answer:(a - δ, a + δ)
18 If fx = 1 + x + x2/2+x3/3, then f' (x) is ____.
A 1/1 - x2
B 1+ x + x2
C 1 + x
D 1 - x

Answer:1+ x + x2
19 The ratio of the rate of change of area and the rate of change of radius with respect to given circle is ____.
A radius of the circle
B perimeter of the circle
C diameter of the circle
D area of the circle

Answer:perimeter of the circle
20 The limit of a function exists only if the left hand limit and the right hand limit coincide with each other.
A True
B False

Answer:True

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