Free Printable NU_CALCULUS II COMPETITION

1.

The effect of the limit process on the Riemann Sum (with regard to the area A, under a curve y = f(x) over [a,b] ) includes the following except:

a)

$\Delta x_k\ \rightarrow\ 0$

b)

$\sum_{k=1}^n\rightarrow\ \int_a^b$

c)

$f\left(x_k^{\cdot}\right)\ \rightarrow\ f\left(x\right)$

d)

$\Delta x_k\ \rightarrow\ dx$

2.

A solid of revolution is a . . . that is generated by revolving a . . . region about the axis of revolution.

a)

Plane, Solid

b)

Mass, Object

c)

Solid, Plane

d)

Solid, Shape

3.

Choose one of the options. The general equation of a conic section is written as:
$Ax^2$ + $Bxy$ + $Cy^2$ + $Dx$ + $Ey$ + $F$ $\ge$ 0

a)

False

b)

Not always

c)

True

4.

A pair of intersecting lines is an example of a degenerate conic section

a)

False

b)

Not always

c)

True

5.

Which of the following conic sections is associated with asymptotes?

a)

Parabola

b)

Circle

c)

Ellipse

d)

Hyperbola

6.

Work is done by applying a constant force W, on a body c, causing it to move from point a to point b on a flat surface. If b > a on the positive x-axis, the magnitude of this work done is given by:

a)

W * b

b)

W * c

c)

(b - a) * W

d)

a * W

7.

A homogeneous lamina has non-uniform composition.

a)

False

b)

True

c)

Not always

8.

This object in the image can be approximated to a lamina.

a)

True

b)

Iff it weighs below 5kg

c)

False

d)

Not always

9.

The method of integrating $\int_{ }^{ }f\left(x\right)g\left(x\right)\ dx$ as described in the image is called . . .

a)

Variable integration

b)

Tabular method

c)

Integration by parts

d)

Reduction formula

10.

For a given differential equation: $-4\frac{\text{d}^2x}{\text{d}y^2}\ +5\frac{\text{d}x}{\text{d}y}\ +8x\ +4\ =0$ . The order of the differential equation is:

a)

-4

b)

5

c)

2

d)

8

11.

Choose from any of the given options. This expression is a differential equation: $\frac{\text{d}^3y}{\text{d}x^3}\ -\ 2\frac{\text{d}y}{\text{d}x}\ +13\ >\ 0$

a)

False

b)

True

c)

Not always

12.

If the family of curves in the image represents the general solution of a differential equation, then each of the curves is called a/an . . .

(a)

13.

Which of the following is true. If the even-numbered terms of a sequence converges to L₁and the odd-numbered terms converge to L₂:

a)

Sequence converges to L₁ * L₂

b)

Sequence converges to L₂- L₁

c)

Sequence does not converge

d)

Sequence converges to L₂

14.

If there exists a unique point (x', y'), possibly in the region of a lamina, such that the effect of gravity on the lamina is equivalent to that of a single force acting at that point, assuming constant density, this point is called the. . .

a)

Centre of gravity

b)

Centroid

c)

Area under the lamina

d)

Radius of the lamina

15.

Consider the image of a weightless beam supporting 'n' masses. If we intend to calculate the moment about 'a' for the last mass on the beam and if the lever arm is positive, which of the following is false.

a)

The beam will rotate in a clockwise direction

b)

Moment about 'a' is negative

c)

Moment about 'a' is positive

16.

This formula compute the center of gravity of a lamina

a)

False

b)

Cannot say

c)

True

17.

Given the following problem: $\int_{ }^{ }\rho\left(x\right)f\left(x\right)\ dx$ , where $\rho\left(x\right)$ is a polynomial. The easiest method to solve this integral would be:

a)

Reduction method

b)

Tabular method

c)

Integration by parts

d)

Using trigonometric functions

18.

Answer True or False. A differential equation cannot contain more than 10 derivatives.

a)

False

b)

True

19.

A link is designed to support 500 connections on to an online event. We can describe 500 as the link's:

a)

Minimum capacity

b)

Number of participants

c)

Carrying capacity

d)

Reserve capacity

20.

If oil leaks from a tank at a rate of r(t) gallons per minute at time t, what does $\int_{0\ \ \ }^{180\ }r\left(t\right)\ dt\$ represent?

a)

Remaining gallons after 3hrs

b)

Gallons that leaked in the first 3hrs

c)

Rate of leakage after 3hrs

d)

Average leakage rate in the first 3hrs

21.

Answer True or False. A sequence always has infinite number of terms.

a)

False

b)

True

22.

The domain of a sequence is the set of . . .

a)

Real numbers

b)

Integers

c)

Rational numbers

d)

Complex numbers

23.

Answer True or False. It is possible to determine the convergence of a series without finding the sequence of partial sums.

a)

True

b)

False

24.

If the sequence of partial sums of a series is strictly increasing and unbounded above, the sequence and therefore the series is:

a)

Divergent

b)

Convergent

c)

Compliant

d)

Harmonic

25.

One can apply the divergence test to a finite series.

a)

True

b)

False

26.

The convergence or divergence of an infinite series is unchanged by adding or removing a finite number of terms.

a)

False

b)

True

c)

Not always

27.

Answer True of False. The Maclaurin's series (or polynomial) is a special case of the Taylor's series (or polynomial).

a)

Cannot say

b)

True

c)

False

28.

Every power series in x, is convergent for x = 0

a)

Cannot say

b)

True

c)

False

29.

In the convergence interval (-R,R) of a power series, R stands for

a)

Root of convergence

b)

Radius of convergence

c)

Random number

d)

Rational number

30.

You are asked to approximate a given function using n terms, if you use $\pi$ = 3.142, this is an example of . . .

a)

Truncation error

b)

Computation error

c)

Roundoff error

d)

Evaluation error

31.

If t is an unrestricted parameter of a set of parametric equations, then the domain of t is . . .

a)

$-\infty<t<+\infty$

b)

$-\infty\le t\le+\infty$

c)

$-\infty\ge t\ge+\infty$

d)

$-\infty>t>+\infty$

32.

The difference between a parametric curve and a normal curve (a set of points) is . . .

a)

Coordinate system

b)

Orientation

c)

Use of a parameter

d)

Number of variables

33.

In the rectangular coordinate system (i.e. in x and y), the tangent line to a parametric curve with parameter 'v' is horizontal when . . .

a)

$\frac{\text{d}y}{\text{d}v}=0$

b)

$\frac{\text{d}x}{\text{d}v}=0$

c)

$\frac{\text{d}v}{\text{d}y}=0$

d)

$\frac{\text{d}v}{\text{d}x}=0$

34.

Choose one of the options. Parametric equations of a given function are unique.

a)

Cannot say

b)

False

c)

True

35.

In the rectangular coordinate system (i.e. in x and y), the tangent line to a parametric curve with parameter 'f' is vertical when. . .

a)

$\frac{\text{d}x}{\text{d}f}=0$

b)

$\frac{\text{d}y}{\text{d}f}=0$

c)

$\frac{\text{d}f}{\text{d}x}=0$

d)

$\frac{\text{d}f}{\text{d}y}=0$

36.

In polar coordinates, r = a is . . .

a)

A circle of radius a, centered at the pole

b)

A circle of radius a, centered along the ray OP

c)

A "six-petal rose"

d)

A circle of radius a, centered at the origin

37.

Consider the family of curves given by the polar equations given below. How is the number of loops (or 'petals') related to n? Check all that apply.

a)

There are 4n loops when n is even

b)

There are 2n loops when n is odd

c)

There are n loops when n is odd

d)

There are 2n loops when n is even

e)

There are n loops when n is even

38.

At singular points for any given function, the general behaviour of its parametric equations. . .

a)

Depends on the given function

b)

Is unique

c)

Changes with periodicity

d)

Is undefined

39.

The error that results when an infinite series is approximated by a partial sum is called . . .

a)

Roundoff error

b)

Evaluation error

c)

Truncation error

d)

Computation error

40.

For the given power series, the set of x-values for which the series converges is called:

a)

Truth set

b)

Definition set

c)

Convergence set

d)

Valid set

41.

Find the coordinates at time t = 3 of a particle following the path:

$x=2t^3-9$

$y=2t^2-2$

Give your answer in the form (x,y)

(a)

42.

The decay constant of Actinium-89 is 0.032 inverse-years. What is its half-life? Give your answer in years, in 2 decimal points

(a)

43.

How much work is required to lift a 500-kg satellite to an altitude of $4\times10^6m$ above the surface of the Earth?
The gravitational force is $F\ =\ \frac{GMm}{r^2}$ , where

M is the mass of the Earth,

m is the mass of the satellite, and
r is the distance between the satellite and the Earth’s center.

The radius of the Earth is $6.4\ \times10^6m$ ,

its mass is $6\times10^{24\ kg}$ and gravitational constant, G is $6.67\times10^{-11}$ . Give your answer in the form a,b i.e. from $a\times10^b$ with a approx. to 3 decimal places
Hint: Work done = $\int_a^b\ F\ dr$

(a)

44.

The partial function decomposition of the function $\frac{-7x-67}{x^2+5x-14}\ is\ of\ the\ form\ \frac{A}{X}+\frac{B}{Y}$

(A,B) = ... Write you answer in the form A,B

(a)

45.

A 5-kg quantity of radioactive isotope decays to 1 kg after 2 years. Find the decay constant of the isotope. Give your answer to four decimal places

(a)

46.

Convert ( $2,\frac{\pi}{4}$ ) in polar coordinates to rectangular coordinates. Give your answer in the form (a,b) where both are a and b are in three decimal places

(a)

47.

Suppose a curve is traced by the parametric equations:

$x\ =\ 5\sin\left(t\right)$

$y\ =\ 46\ -15\cos^2\left(t\right)-30\sin\left(t\right)$

At what point (x,y) on this curve is the tangent line horizontal?

(a)

48.

Which is a parametric equation for the curve:
$49\ =\ \left(x-1\right)^2\ +\left(y-7\right)^2$
Hint: Use the parametric representation of a circle

a)

$c\left(t\right)\ =\ \left(7+7\cos\left(t\right),1+7\sin\left(t\right)\right)$

b)

$c\left(t\right)\ =\ \left(1+7\cos\left(t\right),7+7\sin\left(t\right)\right)$

c)

$c\left(t\right)\ =\ \left(7+49\cos\left(t\right),1+49\sin\left(t\right)\right)$

d)

$c\left(t\right)\ =\ \left(1+49\cos\left(t\right),7+49\sin\left(t\right)\right)$

49.

Find the coordinates at time t = 8 of a particle following the path:

$x=2t^3-9$

$y=2t^2-2$

Give your answer in the form (x,y)

(a)

50.

Find the exact length of the curve given by

$x=t^2$

$y=\frac{1}{3}t^3$ for $0\le t\le4$

Hint:

\int_a^b\ \sqrt{\left(\frac{dx}{dt}\right)^2+}\left(\frac{dy}{dt}\right)^2

Give your answer in 2 decimal places

(a)

NU_CALCULUS II COMPETITION

Total questions: 50

Worksheet time: 1hrs 10mins

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