Teachoo MCQ Test

10 Question MCQ (including Assertion)

Chapter 12 Class 11 Limits and Derivatives | 10 questions

Timed test. Answer every question before time runs out. Correct answers and solutions appear after submission.
Time remaining 960
Question 1 of 10
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Question 1 of 10
The graph below shows the distance-time curve \( s = 4.9 t^2 \) for a free-falling body, along with secant chords \( AB_1 \) and \( AB_2 \) representing average velocities over time intervals starting at \( t = 2 \text{ s} \). As the time interval \( \Delta t \to 0 \), what geometric property of the curve at point \( A(2, 19.6) \) does the instantaneous velocity represent?
AB₂B₁C₂C₁22+t₂2+t₁Time t (s)Distance s (m)s = 4.9 t²
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Question 2 of 10
Assertion (A): The derivative of \( f(x) = x^n \) with respect to \( x \) is \( n x^{n-1} \) for any positive integer \( n \).
Reason (R): The derivative of \( g(x) = \sin x \) with respect to \( x \) is \( \cos x \).
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Question 3 of 10
Assertion (A): \( \lim_{x \to 0} \frac{1 - \cos 2x}{x^2} = 2 \).
Reason (R): The identity for \( 1 - \cos 2x \) is \( 1 - \cos 2x = \cos^2 x - \sin^2 x \).
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Question 4 of 10
Water is draining out of a conical reservoir. The volume of water remaining at time \( t \) minutes is given by \( V(t) = 100(10 - t)^2 \) liters for \( 0 \le t \le 10 \). At what rate is the water flowing out at \( t = 5 \) minutes?
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Question 5 of 10
Evaluate \( \lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1} \).
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Question 6 of 10
Assertion (A): For the function \( f(x) = \frac{x^2 - 4}{x - 2} \) (\( x \ne 2 \)), the limit \( \lim_{x \to 2} f(x) = 4 \).
Reason (R): The value of the function at \( x = 2 \) is equal to 4, i.e., \( f(2) = 4 \).
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Question 7 of 10
Evaluate \( \lim_{x \to 0} \frac{\tan x}{x} \).
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Question 8 of 10
Assertion (A): For the piecewise function \( f(x) = \begin{cases} x - 2 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ x + 2 & \text{if } x > 0 \end{cases} \), the limit \( \lim_{x \to 0} f(x) = 0 \).
Reason (R): The value of the function at \( x = 0 \) is defined as \( f(0) = 0 \).
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Question 9 of 10
Assertion (A): As the time interval \( \Delta t \to 0 \), the average velocity of a falling body with position \( s(t) = 4.9 t^2 \) over \( [t, t + \Delta t] \) approaches the instantaneous velocity \( 9.8 t \).
Reason (R): The instantaneous rate of change of displacement with respect to time is defined as the limiting value \( \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t} \).
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Question 10 of 10
Which key trigonometric identity is used in computing the derivative of \( f(x) = \cos x \) from first principles?
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