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

(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) = elog 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 y= 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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