Calculus PYQ Quiz

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Question 1

Consider the function f(x) = sin(x) in the interval [π/4, 7π/4]. The number and location(s) of the local minima of this function are

  • One, at π/2

  • One, at 3π/2

  • Two, at π/2 and 3π/2

  • Two, at π/4 and 3π/2

Question 2

Which one of the following functions is continuous at x = 3?

  • [Tex]f(x)=\left\{\begin{array}{lll}2, & \text { if } & x=3 \\ x-1, & \text { if } & x>3 \\ \frac{x+3}{3}, & \text { if } & x<3\end{array}\right.[/Tex]

  • [Tex]f(x)=\left\{\begin{array}{lll}4, & \text { if } & x=3 \\ 8-x & \text { if } & x \neq 3\end{array}\right.[/Tex]

  • [Tex]f(x)=\left\{\begin{array}{lll}x+3, & \text { if } & x \leq 3 \\ x-4, & \text { if } & x>3\end{array}\right.[/Tex]

  • [Tex]f(x)=\frac{1}{x^3-27}, [/Tex]if x ≠ 3

Question 3

Function f is known at the following points: 

 

gatecs201310



 

  • 9.003 
     

  • 9.017 
     

  • 8.983 
     

  • 9.045
     

Question 4

Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?
 

  • 7
     

  • 1
     

  • 1/8
     

  • 8
     

Question 5

Consider a function f(x) = 1 – |x| on –1 ≤ x ≤ 1. The value of x at which the function attains a maximum and the maximum value of the function are:

  • 0, –1

  • –1, 0

  • 0, 1

  • –1, 2

Question 6

The trapezoidal method is used to evaluate the numerical value of [Tex]\int_0^1 e^x\,dx [/Tex]. Consider the following values for the step size h.

i. 10-2

ii. 10-3

iii. 10-4

iv. 10-5

For which of these values of the step size h, is the computed value guaranteed to be correct to seven decimal places. Assume that there are no round-off errors in the computation.

  • (iv) only

  • (iii) and (iv) only

  • (ii), (iii) and (iv) only

  • (i), (ii), (iii) and (iv)

Question 7

Suppose that f : R→R is a continuous function on the interval [−3,3] and a differentiable function in the interval (−3,3) such that for every x in the interval, f′(x)≤2. If f(−3)=7, then f(3) is at most __________ .

  • 19

  • 17

  • 22

  • 10

Question 8

[Tex]\lim_{x \to 4} \frac{\sin(x - 4)}{x-4} [/Tex]= ____.

Note: This question was asked as a Numerical Answer Type.

  • 0

  • 1

  • 2

  • 3

Question 9

Let f(x) = x

–(1/3)

and A denote the area of the region bounded by f(x) and the X-axis, when x varies from –1 to 1. Which of the following statements is/are True? 

1. f is continuous in [–1, 1]
2. f is not bounded in [–1, 1]
3. A is nonzero and finite

  • 2 only

  • 3 only

  • 2 and 3 only

  • 1, 2 and 3

Question 10

The minimum number of equal length subintervals needed to approximate 

[Tex]\int_{1}^{2}xe^xdx[/Tex]

 to an accuracy of at least 

[Tex]\frac{1}{3} * 10^{-6}[/Tex]

 using the trapezoidal rule is
 

  • 1000 e
     

  • 1000
     

  • 100 e 

  • 100
     

There are 30 questions to complete.

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