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BC MCQ Set 50 (CB Official 2013)
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8/20
Results by Question
1
MCQ
Correct
A curve is described by the parametric equations \(x = t^2 + 2 t\) and \(y = t^3 + t^2\). An equation of the line tangent to the curve at the point determined by \(t = 1\) is
A
\(2 x - 3 y = 0\)
B
\(4 x - 5 y = 2\)
C
\(4 x - y = 10\)
\(5 x - 4 y = 7\)
Correct Answer
E
\(5 x - y = 13\)
No explanation
2
MCQ
Correct
If \(3 x^2 + 2 x y + y^2 = 1\), then \(\dfrac{d y}{d x} =\)
A
\(-\dfrac{3 x + y}{y^2}\)
\(-\dfrac{3 x + y}{x + y}\)
Correct Answer
C
\(\dfrac{1 - 3 x - y}{x + y}\)
D
\(-\dfrac{3 x}{1 + y}\)
E
\(-\dfrac{3 x}{x + y}\)
No explanation
3
MCQ
Wrong
The table gives selected values for the derivative of a function \(g\) on \(-1 \leq x \leq 2\): \(g'(-1.0)=2\), \(g'(-0.5)=4\), \(g'(0)=3\), \(g'(0.5)=1\), \(g'(1.0)=0\), \(g'(1.5)=-3\), \(g'(2.0)=-6\). If \(g(-1) = -2\) and Euler's method with a step-size of \(1.5\) is used to approximate \(g(2)\), what is the resulting approximation?
A
\(-6.5\)
B
\(-1.5\)
C
\(1.5\)
\(2.5\)
Correct Answer
E
\(3\)
No explanation
4
MCQ
Correct
What are all values of \(x\) for which the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n 3^n}{x^n}\) converges?
A
All \(x\) except \(x = 0\)
B
\(|x| = 3\)
C
\(-3 \leq x \leq 3\)
\(|x| > 3\)
Correct Answer
E
The series diverges for all \(x\).
No explanation
5
MCQ
Correct
If \(\dfrac{d}{d x} f(x) = g(x)\) and if \(h(x) = x^2\), then \(\dfrac{d}{d x} f(h(x)) =\)
A
\(g(x^2)\)
B
\(2 x g(x)\)
C
\(g'(x)\)
\(2 x g(x^2)\)
Correct Answer
E
\(x^2 g(x^2)\)
No explanation
6
MCQ
Wrong
If \(F'\) is a continuous function for all real \(x\), then \(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{1}{h} \displaystyle\int_{a}^{a+h} F'(x) d x\) is
A
\(0\)
B
\(F(0)\)
My Answer
C
\(F(a)\)
D
\(F'(0)\)
\(F'(a)\)
Correct Answer
No explanation
7
MCQ
Correct
\(\displaystyle\int_{0}^{3} \dfrac{d x}{(1 - x)^2}\) is
A
\(-\dfrac{3}{2}\)
B
\(-\dfrac{1}{2}\)
C
\(\dfrac{1}{2}\)
D
\(\dfrac{3}{2}\)
divergent
Correct Answer
No explanation
8
MCQ
Wrong
Which of the following series converge to \(2\)? I. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{2 n}{n + 3}\) II. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{-8}{(-3)^n}\) III. \(\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{2^n}\)
A
I only
My Answer
B
II only
C
III only
D
I and III only
II and III only
Correct Answer
No explanation
9
MCQ
Correct
If the function \(f\) given by \(f(x) = x^3\) has an average value of \(9\) on the closed interval \([0, k]\), then \(k =\)
A
\(3\)
B
\(\sqrt{3}\)
C
\(18^{\dfrac{1}{3}}\)
D
\(36^{\dfrac{1}{4}}\)
\(36^{\dfrac{1}{3}}\)
Correct Answer
No explanation
10
MCQ
Correct
Which of the following integrals gives the length of the graph \(y = \sin(\sqrt{x})\) between \(x = a\) and \(x = b\), where \(0 < a < b\)?
A
\(\displaystyle\int_{a}^{b} \sqrt{x + \cos^2(sqrt(x))} d x\)
B
\(\displaystyle\int_{a}^{b} \sqrt{1 + \cos^2(sqrt(x))} d x\)
C
\(\displaystyle\int_{a}^{b} \sqrt{\sin^2(sqrt(x)) + \dfrac{1}{4 x} \cos^2(sqrt(x))} d x\)
\(\displaystyle\int_{a}^{b} \sqrt{1 + \dfrac{1}{4 x} \cos^2(sqrt(x))} d x\)
Correct Answer
E
\(\displaystyle\int_{a}^{b} \sqrt{\dfrac{1 + \cos^2(sqrt(x))}{4 x}} d x\)
No explanation
11
MCQ
Wrong
Which of the following integrals represents the area enclosed by the smaller loop of the graph of \(r = 1 + 2 \sin \theta\)?
\(\dfrac{1}{2} \displaystyle\int_{7 \dfrac{\pi}{6}}^{11 \dfrac{\pi}{6}} (1 + 2 \sin \theta)^2 d \theta\)
Correct Answer
B
\(\dfrac{1}{2} \displaystyle\int_{7 \dfrac{\pi}{6}}^{11 \dfrac{\pi}{6}} (1 + 2 \sin \theta) d \theta\)
C
\(\dfrac{1}{2} \displaystyle\int_{-\dfrac{\pi}{6}}^{7 \dfrac{\pi}{6}} (1 + 2 \sin \theta)^2 d \theta\)
D
\(\displaystyle\int_{-\dfrac{\pi}{6}}^{7 \dfrac{\pi}{6}} (1 + 2 \sin \theta)^2 d \theta\)
My Answer
E
\(\displaystyle\int_{7 \dfrac{\pi}{6}}^{-\dfrac{\pi}{6}} (1 + 2 \sin \theta) d \theta\)
No explanation
12
MCQ
Wrong
The third-degree Taylor polynomial about \(x = 0\) of \(\ln(1 - x)\) is
\(-x - \dfrac{x^2}{2} - \dfrac{x^3}{3}\)
Correct Answer
B
\(1 - x + \dfrac{x^2}{2}\)
C
\(x - \dfrac{x^2}{2} + \dfrac{x^3}{3}\)
D
\(-1 + x - \dfrac{x^2}{2}\)
E
\(-x + \dfrac{x^2}{2} - \dfrac{x^3}{3}\)
No explanation
13
MCQ
Wrong
If \(\dfrac{d y}{d x} = y \sec^2 x\) and \(y = 5\) when \(x = 0\), then \(y =\)
A
\(e^{\tan x} + 4\)
B
\(e^{\tan x} + 5\)
\(5 e^{\tan x}\)
Correct Answer
D
\(\tan x + 5\)
E
\(\tan x + 5 e^x\)
No explanation
14
MCQ
Correct
If \(f\) is differentiable at \(x = a\), which of the following could be false?
A
\(f\) is continuous at \(x = a\).
B
\(\operatorname*{lim}\limits_{x \rightarrow a} f(x)\) exists.
C
\(\operatorname*{lim}\limits_{x \rightarrow a} \dfrac{f(x) - f(a)}{x - a}\) exists.
D
\(f'(a)\) is defined.
\(f''(a)\) is defined.
Correct Answer
No explanation
15
MCQ
Wrong
A solid has a rectangular base that lies in the first quadrant and is bounded by the x- and y-axes and the lines \(x = 2\) and \(y = 1\). The height of the solid above the point \((x, y)\) is \(1 + 3 x\). Which of the following is a Riemann sum approximation for the volume of the solid?
A
\(\displaystyle\sum_{i=1}^n \dfrac{1}{n}\left(1 + \dfrac{3 i}{n}\right)\)
B
\(2 \displaystyle\sum_{i=1}^n \dfrac{1}{n}\left(1 + \dfrac{3 i}{n}\right)\)
C
\(2 \displaystyle\sum_{i=1}^n \dfrac{i}{n}\left(1 + \dfrac{3 i}{n}\right)\)
\(\displaystyle\sum_{i=1}^n \dfrac{2}{n}\left(1 + \dfrac{6 i}{n}\right)\)
Correct Answer
E
\(\displaystyle\sum_{i=1}^n \dfrac{2 i}{n}\left(1 + \dfrac{6 i}{n}\right)\)
No explanation
16
MCQ
Wrong
[Calculator] A particle moves along the x-axis so that at any time \(t \geq 0\) its velocity is given by \(v(t) = \ln(t + 1) - 2 t + 1\). The total distance traveled by the particle from \(t = 0\) to \(t = 2\) is
A
\(0.667\)
B
\(0.704\)
\(1.540\)
Correct Answer
D
\(2.667\)
E
\(2.901\)
No explanation
17
MCQ
Wrong
[Calculator] If the function \(f\) is defined by \(f(x) = \sqrt{x^3 + 2}\) and \(g\) is an antiderivative of \(f\) such that \(g(3) = 5\), then \(g(1) =\)
A
\(-3.268\)
\(-1.585\)
Correct Answer
C
\(1.732\)
D
\(6.585\)
E
\(11.585\)
No explanation
18
MCQ
Wrong
[Calculator] Let \(g\) be the function given by \(g(x) = \displaystyle\int_{1}^{x} 100(t^2 - 3 t + 2) e^{-t^2} d t\). Which of the following statements about \(g\) must be true? I. \(g\) is increasing on \((1, 2)\). II. \(g\) is increasing on \((2, 3)\). III. \(g(3) > 0\)
A
I only
II only
Correct Answer
C
III only
D
II and III only
E
I, II, and III
No explanation
19
MCQ
Wrong
For a series \(S\), let \(S = 1 - \dfrac{1}{9} + \dfrac{1}{2} - \dfrac{1}{25} + \dfrac{1}{4} - \dfrac{1}{49} + \dfrac{1}{8} - \dfrac{1}{81} + \dfrac{1}{16} - \dfrac{1}{121} + ... + a_n + ...\), where \(a_n = \begin{cases} \dfrac{1}{2^{(n-1) slash 2}} \text{if n is odd} \\ \dfrac{-1}{(n + 1)^2} \text{if n is even} \end{cases}\). Which of the following statements are true? I. \(S\) converges because the terms of \(S\) alternate and \(\operatorname*{lim}\limits_{n \rightarrow \infty} a_n = 0\). II. \(S\) diverges because it is not true that \(|a_{n+1}| < |a_n|\) for all \(n\). III. \(S\) converges although it is not true that \(|a_{n+1}| < |a_n|\) for all \(n\).
A
None
B
I only
C
II only
III only
Correct Answer
E
I and III only
No explanation
20
MCQ
Wrong
[Calculator] Let \(g\) be the function given by \(g(t) = 100 + 20 \sin\left(\dfrac{\pi t}{2}\right) + 10 \cos\left(\dfrac{\pi t}{6}\right)\). For \(0 \leq t \leq 8\), \(g\) is decreasing most rapidly when \(t =\)
A
\(0.949\)
\(2.017\)
Correct Answer
C
\(3.106\)
D
\(5.965\)
E
\(8.000\)
No explanation