Skip to content
Skip to main navigation
Skip to first column
Skip to second column

Test!

Test yourself!

Test

Test:	DELA - 2017 Fall - Final Exam-2
Number of Questions:	18
Time:

$\mbox{Let} \ W \ \mbox{be the set of singular} \ 2\times 2 \ \mbox{matrices under the usual operations. Then}$

	$W \ \mbox{is not a vector space, since it is not closed under addition.}$
	$\hspace{-6mm} W \ \mbox{is not a vector space, since it is not a subspace}\\ \mbox{of the vector space of} \ 2\times 2 \ \mbox{matrices.}$
	$\hspace{-6mm} W \ \mbox{is not a vector space, since it is not closed}\\ \mbox{under multiplication by a scalar.}$
	$W \ \mbox{is not a vector space, since} \ {\bf 0} \ \mbox{is not in} \ W.$
	$\mbox{None of the above}$

$\hspace{-6mm} \mbox{Let}\\[2mm] A=\left [ \begin{array}{cc} \sqrt{2}&\pi\\[2mm] e&3 \end{array} \right ]\\[2mm] \mbox{Find} \ A^{-1}.$

	$\dfrac{1}{\pi e-3\sqrt{2}}\left [ \begin{array}{cc} -\sqrt{2}&e\\[2mm] \pi&-3 \end{array} \right ]$
	$\dfrac{1}{3\sqrt{2}-\pi e}\left [ \begin{array}{cc} 3&-\pi\\[2mm] -e&\sqrt{2} \end{array} \right ]$
	$\dfrac{1}{3\sqrt{2}-\pi e}\left [ \begin{array}{cc} -\sqrt{2}&e\\[2mm] \pi&-3 \end{array} \right ]$
	$\mbox{None of the above}$
	$\dfrac{1}{\pi e-3\sqrt{2}}\left [ \begin{array}{cc} 3&-\pi\\[2mm] -e&\sqrt{2} \end{array} \right ]$

$\mbox{Vectors} \ {\bf v}_1,\ldots, {\bf v}_p \ \mbox{are said to be linearly dependent if}$

	$\mbox{there exist scalars} \ c_1, \ldots, c_p, \ \mbox{not all zero, such that} \ c_1{\bf v}_1+\ldots+c_p{\bf v}_p={\bf 0}.$
	$\mbox{None of the above}$
	$\mbox{there exist scalars} \ c_1, \ldots, c_p \ \mbox{such that} \ c_1{\bf v}_1+\ldots+c_p{\bf v}_p={\bf 0}.$
	$\mbox{there exist scalars} \ c_1, \ldots, c_p, \ \mbox{not all zero, such that} \ c_1{\bf v}_1+\ldots+c_p{\bf v}_p \not ={\bf 0}.$
	$\mbox{the vector equation} \ c_1{\bf v}_1+\ldots+c_p{\bf v}_p={\bf 0} \ \mbox{has only the trivial solution.}$
	$\mbox{the vector equation} \ c_1{\bf v}_1+\ldots+c_p{\bf v}_p={\bf 0} \ \mbox{has the trivial solution.}$
	$\mbox{there exist scalars} \ c_1, \ldots, c_p \ \mbox{such that} \ c_1{\bf v}_1+\ldots+c_p{\bf v}_p \not ={\bf 0}.$

$\mbox{Solve} \ (1+y^2)\dfrac{dy}{dx}=x\cos x.$

	$y^3+3y=-3(x\sin x-\cos x)+c$
	$\mbox{None of the above}$
	$y^3+3y=3(x\sin x+\cos x)+c$
	$y^3+3y=-3(x\sin x+\cos x)+c$
	$y^3+3y=3(x\sin x-\cos x)+c$

$\mbox{Solve} \ \dfrac{dy}{dx}-y=e^{2x}.$

	$y=e^x(e^x+c)$
	$\mbox{None of the above}$
	$y=e^{2x}(e^x+c)$
	$y=e^x(e^{2x}+c)$
	$y=e^{2x}(e^{2x}+c)$

$\mbox{Solve} \ [1+\ln(xy)]dx+xy^{-1}dy=0.$

	$xy\ln y=c$
	$x\ln (xy)=c$
	$y\ln (xy)=c$
	$xy\ln (xy)=c$
	$xy\ln x=c$
	$y\ln x=c$
	$x\ln y=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ ydx-(2x+y^4)dy=0.$

	$x^{-2}y-y^2/2=c$
	$xy^{-2}+y^2/2=c$
	$xy^{-2}-y^2/2=c$
	$\mbox{None of the above}$
	$x^{-2}y+y^2/2=c$

$\hspace{-6mm} \mbox{Compute the second Picard iteration for the initial-value problem}\\[2mm]\dfrac{dy}{dx}=4xy, \ y(0)=3.$

	$y_2(x)=3+4x^2+6x^4$
	$y_2(x)=3-4x^2+6x^4$
	$y_2(x)=3-4x^2-6x^4$
	$y_2(x)=3+4x^2-6x^4$
	$\mbox{None of the above}$

$\mbox{Solve} \ y^{\prime\prime}-y^{\prime}-2y=0.$

	$y=c_1e^{-x}+c_2xe^{2x}$
	$y=c_1e^{x}+c_2xe^{-2x}$
	$y=c_1e^{-x}+c_2xe^{-2x}$
	$y=c_1e^{x}+c_2xe^{2x}$
	$\mbox{None of the above}$

$\mbox{Find a particular solution of} \ y^{\prime\prime}+2y^{\prime}+y=50\sin 3t.$

	$y=4\sin 3t-3\cos 3t$
	$\mbox{None of the above}$
	$y=4\sin 3t+3\cos 3t$
	$y=-4\sin 3t-3\cos 3t$
	$y=-4\sin 3t+3\cos 3t$

$\mbox{Find a particular solution of} \ y^{\prime\prime}+xy=0.$

	$\mbox{None of the above}$
	$y(x)=x+\sum_{n=1}^{\infty}\dfrac{2\cdot 5\cdot 8\ldots (3n-1)(-1)^n}{(3n+1)!}x^{3n}$
	$y(x)=x+\sum_{n=1}^{\infty}\dfrac{1\cdot 3\cdot 5\ldots (2n-1)(-1)^n}{(2n+1)!}x^{2n+1}$
	$y(x)=x+\sum_{n=1}^{\infty}\dfrac{2\cdot 5\cdot 8\ldots (3n-1)(-1)^n}{(3n+1)!}x^{3n+1}$
	$y(x)=x+\sum_{n=1}^{\infty}\dfrac{1\cdot 3\cdot 5\ldots (2n-1)(-1)^n}{(2n+1)!}x^{2n}$

$\hspace{-6mm} \mbox{Find a basis of the vector space of all solutions of the system}\\[2mm] \left \{ \begin{aligned} x_1+x_2+x_3+x_4+x_5+x_6&=0\\[2mm] x_3+x_4+x_5+x_6&=0\\[2mm] x_4+x_5+x_6&=0 \end{aligned} \right .$

	$\left \{\left [\begin{array}{r}-1\\0\\1\\0\\0\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\0\\-1\\1\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\0\\-1\\0\\1\end{array}\right ]\right \}$
	$\left \{\left [\begin{array}{r}-1\\1\\0\\0\\0\\0\end{array}\right ], \ \left [\begin{array}{r}0\\1\\-1\\0\\0\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\0\\-1\\0\\1\end{array}\right ]\right \}$
	$\mbox{None of the above}$
	$\left \{\left [\begin{array}{r}-1\\1\\0\\0\\0\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\0\\-1\\1\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\0\\-1\\0\\1\end{array}\right ]\right \}$
	$\left \{\left [\begin{array}{r}-1\\1\\0\\0\\0\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\-1\\0\\1\\0\end{array}\right ], \ \left [\begin{array}{r}0\\0\\0\\-1\\0\\1\end{array}\right ]\right \}$

$\hspace{-6mm }\mbox{Solve the system} \ \dot{{\bf x}}=A{\bf x}, \ \mbox{where}\\[2mm] A=\left [ \begin{array}{rrr} 0&2&-3\\[2mm] -2&4&-3\\[2mm] -2&2&-1 \end{array} \right ]\\[2mm] \mbox{Hint: The eigenvalues of} \ A \ \mbox{are} \ \lambda_1=-1 \ \mbox{and} \ \lambda_2=2.$

	${\bf x}(t)=c_1e^{-t} \left [ \begin{array}{r} -1\\[2mm] 1\\[2mm] 1 \end{array} \right ]+ c_2e^{2t} \left [ \begin{array}{r} 1\\[2mm] -1\\[2mm] 0 \end{array} \right ]+ c_3e^{2t} \left [ \begin{array}{r} -3\\[2mm] 0\\[2mm] 2 \end{array} \right ]$
	${\bf x}(t)=c_1e^{-t} \left [ \begin{array}{r} 1\\[2mm] 1\\[2mm] 1 \end{array} \right ]+ c_2e^{2t} \left [ \begin{array}{r} 1\\[2mm] 1\\[2mm] 0 \end{array} \right ]+ c_3e^{2t} \left [ \begin{array}{r} 3\\[2mm] 0\\[2mm] 2 \end{array} \right ]$
	${\bf x}(t)=c_1e^{-t} \left [ \begin{array}{r} 1\\[2mm] 1\\[2mm] -1 \end{array} \right ]+ c_2e^{2t} \left [ \begin{array}{r} -1\\[2mm] 1\\[2mm] 0 \end{array} \right ]+ c_3e^{2t} \left [ \begin{array}{r} -3\\[2mm] 0\\[2mm] 2 \end{array} \right ]$
	$\mbox{None of the above}$
	${\bf x}(t)=c_1e^{-t} \left [ \begin{array}{r} 1\\[2mm] 1\\[2mm] 1 \end{array} \right ]+ c_2e^{2t} \left [ \begin{array}{r} 1\\[2mm] 1\\[2mm] 0 \end{array} \right ]+ c_3e^{2t} \left [ \begin{array}{r} -3\\[2mm] 0\\[2mm] 2 \end{array} \right ]$

$\hspace{-6mm} \mbox{Solve the system} \ \dot{{\bf x}}=A{\bf x}, \ \mbox{where}\\[2mm] A=\left [ \begin{array}{rr} 0&2\\[2mm] -2&0 \end{array} \right ]$

	$\mbox{None of the above}$
	${\bf x}(t)=c_1 \left [ \begin{array}{r} \sin 2t\\[2mm] \cos 2t \end{array} \right ]+ c_2 \left [ \begin{array}{r} \cos 2t\\[2mm] \sin 2t \end{array} \right ]$
	${\bf x}(t)=c_1 \left [ \begin{array}{r} \sin 2t\\[2mm] \cos 2t \end{array} \right ]+ c_2 \left [ \begin{array}{r} -\cos 2t\\[2mm] \sin 2t \end{array} \right ]$
	${\bf x}(t)=c_1 \left [ \begin{array}{r} -\sin 2t\\[2mm] \cos 2t \end{array} \right ]+ c_2 \left [ \begin{array}{r} \cos 2t\\[2mm] \sin 2t \end{array} \right ]$
	${\bf x}(t)=c_1 \left [ \begin{array}{r} -\sin 2t\\[2mm] \cos 2t \end{array} \right ]+ c_2 \left [ \begin{array}{r} -\cos 2t\\[2mm] \sin 2t \end{array} \right ]$

$\hspace{-6mm} \mbox{Solve the system} \ \dot{{\bf x}}=A{\bf x}, \ \mbox{where}\\[2mm] A=\left [ \begin{array}{rrr} 5&2&-1\\[2mm] 1&6&-1\\[2mm] 3&6&1 \end{array} \right ]\\[2mm] \mbox{Hint: The eigenvalues of} \ A \ \mbox{are} \ \lambda=4.$

	${\bf x}(t)=c_1e^{4t} \left [ \begin{array}{c} 1\\[2mm] 1\\[2mm] 3 \end{array} \right ]+c_2e^{4t} \left [ \begin{array}{c} 1+t\\[2mm] t\\[2mm] 3t \end{array} \right ]+c_3e^{4t} \left [ \begin{array}{r} -2\\[2mm] 1\\[2mm] 0 \end{array} \right ]$
	${\bf x}(t)=c_1e^{4t} \left [ \begin{array}{c} 1\\[2mm] 1\\[2mm] 3 \end{array} \right ]+c_2e^{4t} \left [ \begin{array}{c} 1-t\\[2mm] t\\[2mm] 3t \end{array} \right ]+c_3e^{4t} \left [ \begin{array}{r} -2\\[2mm] 1\\[2mm] 0 \end{array} \right ]$
	$\mbox{None of the above}$
	${\bf x}(t)=c_1e^{4t} \left [ \begin{array}{r} -1\\[2mm] 1\\[2mm] 3 \end{array} \right ]+c_2e^{4t} \left [ \begin{array}{c} 1-t\\[2mm] t\\[2mm] 3t \end{array} \right ]+c_3e^{4t} \left [ \begin{array}{r} 2\\[2mm] 1\\[2mm] 0 \end{array} \right ]$
	${\bf x}(t)=c_1e^{4t} \left [ \begin{array}{r} -1\\[2mm] 1\\[2mm] 3 \end{array} \right ]+c_2e^{4t} \left [ \begin{array}{c} 1+t\\[2mm] t\\[2mm] -3t \end{array} \right ]+c_3e^{4t} \left [ \begin{array}{r} -2\\[2mm] 1\\[2mm] 0 \end{array} \right ]$

$\mbox{Let} \ A=\left [ \begin{array}{rr} 6&-8\\[2mm] 2&-2 \end{array} \right ], \ \mbox{then} \ e^{At} \ \mbox{is}$

	$e^{2t}\left [ \begin{array}{cc} 1+4t&8t\\[2mm] 2t&1+4t \end{array} \right ]$
	$e^{2t}\left [ \begin{array}{cc} 1+4t&8t\\[2mm] -2t&1-4t \end{array} \right ]$
	$e^{2t}\left [ \begin{array}{cc} 1+4t&-8t\\[2mm] 2t&1-4t \end{array} \right ]$
	$\mbox{None of the above}$
	$e^{2t}\left [ \begin{array}{cc} 1-4t&8t\\[2mm] 2t&1-4t \end{array} \right ]$

$\hspace{-6mm} \mbox{Determine whether each solution} \ {\bf x}(t) \ \mbox{of the differential equation}\\[2mm] {\bf \dot{x}}= \left [ \begin{array}{rr} 1&-1\\[2mm] 1&2 \end{array} \right ]{\bf x}\\[2mm] \mbox{is stable or unstable.}$

	$\mbox{Stable, because at least one eigenvalue of} \ A \ \mbox{has positive real part.}$
	$\mbox{Unstable, because at least one eigenvalue of} \ A \ \mbox{has positive real part.}$
	$\mbox{None of the above}$
	$\mbox{Stable, because all the eigenvalues of} \ A \ \mbox{have negative real part.}$
	$\mbox{Unstable, because all the eigenvalues of} \ A \ \mbox{have negative real part.}$

$\hspace{-6mm} \mbox{Let}\\[2mm] f(x)= \begin{cases} 0&\mbox{for} \ \ 0 \,\,\textless\,\, x \le 2\\[3mm] 2&\mbox{for} \ \ 2 \,\,\textless\,\, x \,\,\textless\,\, 3 \end{cases}\\[2mm] \mbox{then} \ f(x)=\sum_{n=1}^{\infty}b_n\sin \dfrac{n\pi x}{3}, \ \mbox{where}$

	$b_n=\dfrac{4}{n\pi}\left (\cos \dfrac{2n\pi}{3}-(-1)^n\right )$
	$\mbox{None of the above}$
	$b_n=\dfrac{4}{n\pi}\left (\sin \dfrac{n\pi}{3}-(-1)^n\right )$
	$b_n=\dfrac{4}{n\pi}\left (\sin \dfrac{2n\pi}{3}-(-1)^n\right )$
	$b_n=\dfrac{4}{n\pi}\left (\cos \dfrac{n\pi}{3}-(-1)^n\right )$

Choose the answer

Powered by Valery Gromov

Main Menu

Home
Teaching
Calculus Website
Calculus Tests
Precalculus Website
Precalculus Tests
Linear Algebra Website
Linear Algebra Tests
Linear Algebra and Matrix Theory Website
Linear Algebra and Matrix Theory Tests
Differential Equations and Linear Algebra Website
Differential Equations and Linear Algebra Problems
Differential Equations and Linear Algebra Tests
All Tests