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MATH162:INTRODUCTORY PURE MATHEMATICS II:COS:4048

MATH162:INTRODUCTORY PURE MATHEMATICS II:COS:4048

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Evaluate the integral $\displaystyle \int _0^{\frac {\pi }{2}} e^{\cos x} \sin x\, dx$ \displaystyle \int _0^{\frac {\pi }{2}} e^{\cos x} \sin x\, dx.

e-1

None of the other four choices.

$e^{\frac {\pi }{2}}$ e^{\frac {\pi }{2}}

$1-e^{\frac {\pi }{2}}$ 1-e^{\frac {\pi }{2}}

1-e

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Find the value of the integral $\displaystyle \int _0^{\frac {\pi }{4}} \tan x \,dx$ \displaystyle \int _0^{\frac {\pi }{4}} \tan x \,dx.

None of the other four choices.

$\frac {1}{2} \ln 2$ \frac {1}{2} \ln 2

$\ln \frac {1}{\sqrt {2}}$ \ln \frac {1}{\sqrt {2}}

$\frac {1}{2} \sqrt {2}$ \frac {1}{2} \sqrt {2}

$\frac {1}{2} \ln \sqrt {2}$ \frac {1}{2} \ln \sqrt {2}

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A cubic polynomial defined by f(x)=a x^3+b x^2+c x+d, where a, b, c and d are constants, has a minimum value of -4 when x=-1 and a maximum value of 4 when x=1. Evaluate $f^\prime (-2)$ f^\prime (-2).

-6

-18

None of the above choices.

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Evaluate the integral $\displaystyle \int \left (1+\tan ^2 x\right ) e^{\tan x} \,dx$ \displaystyle \int \left (1+\tan ^2 x\right ) e^{\tan x} \,dx.

$e^{\tan x}$ e^{\tan x}

None of the above choices.

$e^{\tan x}+c$ e^{\tan x}+c

$e^{\sec ^2 x}+c$ e^{\sec ^2 x}+c

$e^{\tan x}\sec ^2x+c$ e^{\tan x}\sec ^2x+c

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With the aid of the substitutions $e^x=\tan \theta$ e^x=\tan \theta and $\cosh x=\frac {1}{2}\left (e^x+e^{-x}\right )$ \cosh x=\frac {1}{2}\left (e^x+e^{-x}\right ), evaluate the integral $\displaystyle \int{sech} x \,dx$ \displaystyle \int{sech} x \,dx.

$2 \tan ^{-1} e^x+c$ 2 \tan ^{-1} e^x+c

$\tan ^{-1} e^x+c$ \tan ^{-1} e^x+c

None of the other four choices.

$2 \tan ^{-1} e^x$ 2 \tan ^{-1} e^x

$\tan ^{-1} e^x$ \tan ^{-1} e^x

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The curve y(x)=x^2-a x+b, where a and b are constants has a turning point at P(1,3). Find the value of b.

b=4

b=2

None of the other four choices.

b=3

b=-4

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Which of the choices below is the general solution of the differential equation $\dfrac {1}{y}\left (x^2+1\right ) \dfrac {d y}{d x}= \dfrac {2x}{y}$ \dfrac {1}{y}\left (x^2+1\right ) \dfrac {d y}{d x}= \dfrac {2x}{y} ?

$y = \ln c(x^2+1)$ y = \ln c(x^2+1)

$y = -\ln c(x^2+1)$ y = -\ln c(x^2+1)

$y=2\tan ^{-1}x^2 + c$ y=2\tan ^{-1}x^2 + c

None of the above choices.

$y= \frac {1}{2}\ln (x^2+1) +c$ y= \frac {1}{2}\ln (x^2+1) +c

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One side of a rectangle is three times the other. If the perimeter of the rectangle increases by $2 \%$ 2 \%, what is the percentage increase in the area of the rectangle?

$8 \%$ 8 \%

$4 \%$ 4 \%

$16 \%$ 16 \%

$2 \%$ 2 \%

None of the above choices.

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The angle at the point C of triangle A B C is always a right angle. If the sum of |C A| and |C B| is $6 \mathrm {~cm}$ 6 \mathrm {~cm}, find the maximum area of the triangle.

$4.5 \mathrm {~m}^2$ 4.5 \mathrm {~m}^2

None of the other four choices.

$4.5 \mathrm {~m}$ 4.5 \mathrm {~m}

$4.05 \mathrm {~cm}^2$ 4.05 \mathrm {~cm}^2

$4.5 \mathrm {~cm}$ 4.5 \mathrm {~cm}

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A function is defined parametrically by $x=\frac {1}{\sqrt {1+t^2}}$ x=\frac {1}{\sqrt {1+t^2}} , $y=\frac {t}{\sqrt {1+t^2}}$ y=\frac {t}{\sqrt {1+t^2}}. Find $\frac {dy}{dx}$ \frac {dy}{dx} in terms of t.