Quiz about PDE, Vector Spaces, Laplace Transform

Question 1

The solution of the equation z = px + qy + pq is____

Accepted Answer

z = ax + by + ab

Answer

z = ax + by + pq

Answer

z = bx + ay + ab

Answer

z = px + qy + ab

Question 2

$$The\ solution\ of\ \frac{\partial^2z}{\partial y^2}=0$$

Accepted Answer

z = yf(x)+g(x)

Answer

z = yf(y)+g(x)

Answer

z = xf(x)+g(x)

Answer

z = xf(y)+g(x)

Question 3

The solution of simultaneous linear differential equations is of the form___

Accepted Answer

$$\Phi\left(u,v\right)=0$$

Answer

$$\Phi\left(u\right)=0$$

Answer

$$\Phi\left(v\right)=0$$

Answer

$$\Phi\left(x,y\right)=0$$

Question 4

Find the roots of $$m^3-m^2+m-1=0$$

Accepted Answer

1,i,-i

Answer

1,-1,i

Answer

i,-i,0

Answer

1,-1,0

Question 5

Find the complementary function of $$\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial x\partial y}-2\frac{\partial^2z}{\partial y^2}=0$$

Accepted Answer

$$Z=f_1\left(y+x\right)+f_2\left(y-2x\right)$$

Answer

$$Z=f_1\left(y-x\right)+f_2\left(y+2x\right)$$

Answer

$$Z=f_1\left(y\right)+f_2\left(y+2x\right)$$

Answer

$$Z=f_1\left(y+x\right)+xf_2\left(y-2x\right)$$

Question 6

Find the complementary function of $$\left(D^3-7DD$$

Accepted Answer

$$Z=f_1\left(y-x\right)+f_2\left(y-2x\right)+f_3\left(y+3x\right)$$

Answer

$$Z=f_1\left(y+x\right)+f_2\left(y+2x\right)+f_3\left(y-3x\right)$$

Answer

$$Z=f_1\left(y+x\right)+f_2\left(y-2x\right)+xf_3\left(y+3x\right)$$

Answer

$$Z=f_1\left(y-x\right)+f_2\left(y\right)+f_3\left(y+3x\right)$$

Question 7

Solve $$\left(D^3-DD$$

Accepted Answer

$$Z=f_1\left(y+x\right)+f_2\left(y-x\right)+xf_3\left(y-x\right)$$

Answer

$$Z=f_1\left(y-2x\right)+xf_2\left(y-2x\right)+f_3\left(y+x\right)$$

Answer

$$Z=f_1\left(y\right)+xf_2\left(y\right)+f_3\left(y-x\right)$$

Answer

$$Z=f_1\left(y+x\right)+f_2\left(y-x\right)+f_3\left(y-x\right)$$

Question 8

The subsidiary equation for Lagrange's linear equation

Accepted Answer

$$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$$

Answer

$$\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{z}$$

Answer

$$\frac{dx}{p}=\frac{dy}{q}=\frac{dz}{r}$$

Question 9

The multipliers for $$x\left(y-z\right)p+y\left(z-x\right)q=z\left(x-y\right)$$ is

Accepted Answer

$$1,1,1\ and\ \frac{1}{x},\frac{1}{y},\frac{1}{z}$$

Answer

$$1,1,1\ and\ x,y,z$$

Answer

$$0,0,0\ and\ 1,1,1$$

Question 10

Which of the following is clairaut's form

Accepted Answer

$$z=px+qy+\sqrt{pq}$$

Answer

$$p+q=1$$

Answer

$$p+q=x+y$$

Question 11

$$.$$ Find the P.I of $$\left(D^3+D^2D$$

Accepted Answer

$$0$$

Answer

$$1$$

Answer

$$e^x$$

Answer

$$none$$

Question 12

$$.$$ Find the P.I of $$\left(D^2-5DD$$

Accepted Answer

$$\frac{1}{3}e^{\left(x+y\right)}$$

Answer

$$e^{\left(x+y\right)}$$

Answer

$$\frac{1}{2}e^{\left(x+y\right)}$$

Answer

$$\frac{1}{4}e^{\left(x+y\right)}$$

Question 13

$$\ .$$ Find the P.I of $$\left(D^3-3D^2D$$

Accepted Answer

$$\ \frac{1}{27}e^{\left(x+2y\right)}$$

Answer

$$\ \frac{1}{6}e^{\left(x+2y\right)}$$

Answer

$$\ \frac{1}{27}e^{\left(2x+y\right)}$$

Answer

$$\ \frac{1}{6}e^{\left(2x+y\right)}$$

Question 14

$$.$$ Find the P.I of $$\left(D^2-2DD$$

Accepted Answer

$$-\frac{1}{16}\cos(x-3y)$$

Answer

$$-\frac{1}{16}\sin⁡(x-3y)$$

Answer

$$\frac{1}{16}\cos⁡(x-3y)$$

Answer

$$\frac{1}{16}\sin⁡(x-3y)$$

Question 15

$$.$$ Find the P.I of $$\left(D^2+3DD$$

Accepted Answer

$$\frac{1}{4}\sin⁡y$$

Answer

$$\frac{1}{4}\cos⁡y$$

Answer

$$\frac{1}{4}\cos⁡x$$

Answer

$$\frac{1}{4}\sin x$$

Question 16

The set { (1,-2,6), (5,-10,30) } is

Accepted Answer

Linearly dependent

Answer

Linearly independent

Answer

Basis

Answer

None of the above

Question 17

u = (-1, 1/4) and v = (4, -1/8). find $$\parallel u\parallel$$ .

Accepted Answer

$$\sqrt{\frac{17}{16}}$$

Answer

$$\sqrt{\frac{1025}{64}}$$

Answer

$$\frac{\sqrt{577}}{4}$$

Answer

$$\sqrt{\frac{17}{4}}$$

Question 18

Find the distance between two vertors u and v.

u = (1, 2, 0) and v = (-1, 4, 1)

Accepted Answer

3

Answer

$$\sqrt{3}$$

Answer

6

Answer

9

Question 19

Find the dot product $$u\ \circ\ v$$
u = (3, 4) and v = (2, -3)

Accepted Answer

-6

Answer

-30

Answer

13

Answer

25

Question 20

Find the dot product $$v\ \circ\ v$$
u = (3, 4) and v = (2, -3)

Accepted Answer

13

Answer

-6

Answer

-30

Answer

25

Question 21

If S = {v1 ,v2 , . . ,vn } is a set of vectors in a finite dimensional vector space V, then S is called a basis for V if:

Accepted Answer

both A and B

Answer

S spans V

Answer

S is linearly independent

Answer

either A or B

Question 22

If W is subspace of vector space v then

Accepted Answer

dim w $$\le$$ dim V

Answer

dim W = dim V

Answer

dim w $$<$$ dim V

Answer

dim w $$>$$ dim V

Question 23

The subset { (1,-2), (2,9), (-4,3 } of $$R^2$$ is

Accepted Answer

Linearly dependent

Answer

Linearly independent

Answer

Basis

Answer

None of the above

Question 24

Let V be a vector space, and let S be a subset of V. What does it mean when we say that S spans V?

Accepted Answer

Every vector in V can be expressed as a linear combination of vectors in S.

Answer

The elements of S are all distinct from each other.

Answer

Every vector in V has exactly one representation as a linear combination of vectors in S.

Answer

S is a basis for V.

Question 25

Find the distance between two vertors u and v.

u = (1, 2, 0) and v = (-1, 4, 1)

Accepted Answer

3

Answer

$$\sqrt{3}$$

Answer

6

Answer

9

Question 26

Find $$L^{-1}\left(\frac{s}{s^2+25}\right)$$

Accepted Answer

cos 5t

Answer

cos 25t

Answer

sin 5t

Question 27

Find $$L^{-1}\left(\frac{1}{s^2+1}\right)$$

Accepted Answer

sin t

Answer

sin 5t

Answer

cos t

Question 28

$$L^{-1}\left(\frac{6}{\left(s\right)}-\frac{4s}{s^2+25}\right)$$

Accepted Answer

$$6-4\cos5t$$

Answer

$$1-4\cos5t$$

Answer

$$6-\cos5t$$

Answer

$$6t-4\cos5t$$

Question 29

$$L^{-1\ }\left(\frac{4s}{s^2+25}+\frac{2}{3\left(s^2+1\right)}\right)$$

Accepted Answer

$$4\ \cos\ 5t\ +\frac{2}{3}\sin\ t$$

Answer

$$4\ \cos\ 25t+\frac{2}{3}\sin\ t$$

Answer

$$\cos\ 5t+\ \frac{3}{2}\sin\ t$$

Question 30

$$\frac{1}{s^9}$$

Inverse Laplace Transform of

Accepted Answer

t8/8!

Answer

t9 /8!

Answer

t9/9!

Answer

t8/9!

Question 31

$$L^{-1}\left(\frac{3}{s^2+36}\right)$$

Accepted Answer

$$\frac{1}{2\ }\sin\ 6t$$

Answer

$$\frac{3}{5}\sin\ 5t$$

Answer

$$\cosh2t$$

Question 32

$$L^{-1}\left(\frac{3s}{\left(s+2\right)^4}\right)$$

Accepted Answer

$$e^{-2t}\left(\frac{3}{2}t^2-t^{\left(3\right)}\right)$$

Answer

$$\frac{1}{2}e^{-t}\left(3t^2-t^{\left(3\right)}\right)$$

Answer

$$e^{-t}\left(\frac{3}{2}t^2-t^{\left(3\right)}\right)$$

Question 33

$$s^2+2s+5$$ Complete the square

Accepted Answer

$$\left(s+1\right)^2+4$$

Answer

$$\left(s+4\right)^2+2$$

Answer

$$\left(s+2\right)^2+4$$

Answer

$$\left(s+2\right)^2+16$$

Question 34

Find the partial fraction for $$\frac{\left(2s-8\right)}{\left(s-2\right)\left(s-3\right)}$$

Accepted Answer

$$\frac{4}{s-2}-\frac{2}{s-3}$$

Answer

$$\frac{2}{s-2}-\frac{4}{s-3}$$

Answer

$$\frac{3}{s-2}-\frac{2}{s-3}$$

Question 35

Laplace Transforms of e^at

Accepted Answer

1/s-a

Answer

1/s+a

Question 36

Laplace Transform of tⁿ

Accepted Answer

n!/sⁿ⁺¹

Answer

n!/sn

Answer

(n-1)!/sⁿ⁺¹

Question 37

Laplace Transform of cosh at

Accepted Answer

s/s2-a2

Answer

a/s2+a2

Answer

s/s2+a2

Question 38

Which one is the correct definition of Laplace Transform?

Accepted Answer

$$F\left(s\right)=\int_0^{\infty}e^{-st}f\left(t\right)dt$$

Answer

$$F\left(s\right)=\int_0^tf\left(t\right)g\left(t-u\right)$$

Question 39

What is the Laplace Transform for $$f\left(t\right)=t^3$$ ?

Accepted Answer

$$\frac{6}{s^4}$$

Answer

$$\frac{1}{s^3}$$

Answer

$$\frac{2}{s^3}$$

Answer

$$\frac{3}{s^3}$$

Question 40

What is Laplace Transform for $$\sin\ 2t$$ ?

Accepted Answer

$$\frac{2}{s^2+4}$$

Answer

$$\frac{2}{s^2+2}$$

Answer

$$\frac{s}{s^2+2}$$

Answer

$$\frac{s}{s^2+4}$$

Question 41

What is Laplace Transform for $$6\ \sin\ 2t$$ ?

Accepted Answer

$$\frac{12}{s^2+4}$$

Answer

$$\frac{2}{s^2+4}$$

Answer

$$\frac{s}{s^2+4}$$

Answer

$$\frac{6}{s^2+4}$$

Question 42

What is the Laplace Transform for $$f\left(t\right)=t^3$$ ?

Accepted Answer

$$\frac{6}{s^4}$$

Answer

$$\frac{1}{s^3}$$

Answer

$$\frac{2}{s^3}$$

Answer

$$\frac{3}{s^3}$$

Question 43

Find $$F\left(s\right),\ given\ f\left(t\right)=e^{2t}\sin\ 2t$$

Accepted Answer

$$\frac{s}{\left(s-2\right)^2+4}$$

Answer

$$\frac{1}{\left(s-1\right)^2+1^{ }}$$

Answer

$$\frac{s}{\left(s-4\right)^2+1}$$

Question 44

What is Laplace Transform for $$6\ \sin\ 2t$$ ?

Accepted Answer

$$\frac{12}{s^2+4}$$

Answer

$$\frac{2}{s^2+4}$$

Answer

$$\frac{s}{s^2+4}$$

Answer

$$\frac{6}{s^2+4}$$

Question 45

Find $$L\left(\cos\ t\right)$$

Accepted Answer

$$\frac{s}{s^2+1}$$

Answer

$$\frac{1}{s^2+1^{ }}$$

Answer

$$\frac{1}{s^2+4}$$

Question 46

Find $$L\left(t^3-3t^2+5t\right)$$

Accepted Answer

$$\frac{6}{s^4}-\left(\frac{6}{s^3}\right)+\frac{5}{s^2}$$

Answer

$$\frac{3}{s^3}-\left(\frac{1}{s^{\left(2\right)}}\right)+\frac{1}{s}$$

Answer

$$\frac{2}{s^3}-\left(\frac{3}{s^{\left(2\right)}}\right)+\frac{5}{s}$$

Question 47

$$L\left\{e^{dt}\right\}$$

Accepted Answer

$$\frac{1}{s-d}$$

Answer

$$\frac{1}{s-a}$$

Answer

$$\frac{1}{s-b}$$

Answer

$$\frac{1}{s-c}$$

Question 48

Define this property: $$L\left(f\left(t\right)e^{at}\right)=F\left(s-a\right)$$

Accepted Answer

First Shifting Property

Answer

Linearity Property

Answer

Convolution Theorem

Answer

Second Shifting Property

Create a free account and access millions of resources

Similar Resources on Wayground

Popular Resources on Wayground

Discover more resources for Mathematics