Free Printable MODEL EXAM-MA8352-LAPDE

1.

Let (V,+,.) be a vector space. Then, which one is true?

a)

There exist an element y in

V, Such that x+y=x for all x in V

b)

There exist an element y in

V, Such that x.y=x for all x in V

c)

There exist an element y in

V, Such that x+y=y for all x in V

d)

There exist an element y in

V, Such that x.y=y for all x in V

2.

Which of the following are vector spaces under usual addition and scalar multiplication?

a)

M₂(R) over R

b)

Q over R

c)

Z over R

d)

R over C.

3.

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 ?

a)

The elements of S are all distinct from each other

b)

S has at least as many elements as the dimension V

c)

S is a basis for V

d)

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

4.

When will $W=\left\{\left(x,y,z\right)\right\}$ be a subspace of $R^{3\ }$ ?

a)

x+y+z=1

b)

x-y-z=2

c)

2x+3y=0

d)

2x+3=4z

5.

The set $S=\left\{1,x+x^2,4x^2\right\}$ is

a)

Linearly independent

b)

Linearly dependent

6.

What is the linear span L(S) of the set $S=\left\{e_1,e_2,\frac{e_1+e_2}{2}\right\}$ in $R^3$ ?

a)

xy - plane

b)

x – axis

c)

y – axis

d)

yz- plane

7.

Let A and B be two subspaces of a vector space V. Then A U B is a subspace of V if and only if

a)

$A\cap B=\left\{0\right\}$

b)

$A\cup B=B$

c)

$A\cap B=\phi$

d)

$A\cap B=\left\{0,1\right\}$

8.

Which among the following statements is true?

a)

A subset of linearly dependent is linear dependent.

b)

The union of two subspaces is a subspace.

c)

A spanning set should be a basis for a vector space.

d)

A subset of a linearly independent is linearly independent.

9.

Dim M_4x3(R) is

a)

12

b)

43

c)

64

d)

81

10.

If S={(1,0,0),(2,0,0),(3,0,0)} is a subset of R³, the dim(L(S))=

a)

1

b)

2

c)

3

d)

4

11.

Which of the following is not a linear transformation?

a)

$T\left(x,y\right)=\left(x,-y\right)$

b)

$T\left(x,y\right)=\left(-x,y\right)$

c)

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

d)

$T\left(x,y\right)=\left(x+2,y+4\right)$

12.

Suppose that $T:R^2\longrightarrow R^2$ is linear and T(1,0)=(1,4) and T(1,1)=(2,5). What is T(2,3)

a)

(2,5)

b)

(5,11)

c)

(5,2)

d)

(8,5)

13.

If T is a linear transformation from V to W, which of the following is correct?

a)

R(T) is a subspace of V

b)

R(T) is a subspace of W

c)

N(T) is a subspace of W

d)

None of the others

14.

Find N(T) if $T:R^3\longrightarrow R^2$ is a linear transformation defined by $T\left(x,y,z\right)=\left(x-y,2z\right)$

a)

$N\left(T\right)=\left\{\left(a,0,0\right):a\text{ }\in R\right\}$

b)

$N\left(T\right)=\left\{0\text{ }\right\}$

c)

$N\left(T\right)=R^2$

d)

$N\left(T\right)=\left\{\left(a,a,0\right):a\in R\text{ }\right\}$

15.

Let $I$ be the identity transformation on the vector space V . Then

a)

$I$ is 1-1

b)

$I$ is onto

c)

Both $I$ is 1-1 and onto

d)

neither $I$ is 1-1 nor onto

16.

$If\ I\ \ is\ \ n\times n\ \ identity\ \ matrix\ \ then\ \ \rho\left(I\right)\ =$

a)

$n-1$

b)

$n$

c)

$n+1$

d)

$doesnot\ \ exists$

17.

$Let\ \ A\ \ be\ \ n\times n\ \ non\sin gular\ \ then\ \ \rho\left(A\right)\ =$

a)

$1$

b)

$2$

c)

$n$

d)

$n^2$

18.

A is a $3\ \times3$ upper triangular matrix with the diagonal entries are 3, 4 and 5, then the eigen values are

a)

$3,4,5$

b)

$7,9,8$

c)

$2,3,4$

d)

Need more information

19.

$If\ \ A\ \ is\ \ 1\times n\ \ matrix\ \ and\ \ B\ \ is\ \ n\times1\ \ matrix\ \ then\ \ \rho\left(AB\right)\ =$

a)

$1$

b)

$0$

c)

$n$

d)

$doesnot\ \ exists$

20.

$.$ $If\ I\ \ is\ \ n\times n\ \ identity\ \ matrix\ \ then\ \ \rho\left(I\right)\ =$

a)

$n-1$

b)

$n$

c)

$n+1$

d)

$doesnot\ \ exists$

21.

Pick out the even function.

a)

$\sin x$

b)

$\tan x$

c)

$x^2$

d)

$x^3$

22.

The period of $\sin x$ is

a)

$2π$

b)

$π$

c)

$π/2$

d)

$3π/2$

23.

The value of $a_0$ in the Fourier series of $f(x)=x$ in $(0,2π)$ is

a)

$2π$

b)

$π$

c)

$0$

d)

$-π$

24.

if $f\left(x\right)$ is an even function, then the value of $b_n$ in the Fourier series for $f\left(x\right)$ in $(-π,π)$ is

a)

$\left(2/π\right)∫_0^πf(x)\sin xdx$

b)

$π$

c)

$\left(1/π\right)\int_{-\pi}^{\pi}f(x)\sin xdx$

d)

$0$

25.

Which of the following function doesnot have a Fourier series in $(0,2π)$ ?

a)

$\sin x$

b)

$\tan x$

c)

$\cos x$

d)

$x^2$

26.

Pick out one of the conditions of Dirichlet’s condition.

a)

f\left(x\right)

is periodic, single-valued and finite

b)

$f\left(x\right)$ has infinite number of infinite discontinuous in any one period

c)

$f\left(x\right)$ is infinite valued function

d)

$f\left(x\right)$ has infinite number of maxima and minima

27.

if $|f(x)|$ is even whenever $f(x)$ is

a)

Even only

b)

Odd only

c)

Neither even nor odd

d)

Either even or odd

28.

Which of the following function is neither even nor odd?

a)

$\sin x$

b)

$x^2+x$

c)

$\cos x$

d)

$x^2$

29.

if $f\left(x\right)$ is an odd function, then the value of $a_n$ in the Fourier series for $f\left(x\right)$ in $(-π,π)$ is

a)

$\left(2/π\right)∫_0^πf(x)\sin xdx$

b)

$π$

c)

$\left(1/π\right)\int_{-\pi}^{\pi}f(x)\sin xdx$

d)

$0$

30.

if $f\left(x\right)$ is an even function, then $a_n$ in the Fourier series for $f\left(x\right)$ in $(-π,π)$ is given by

a)

$\left(2/π\right)∫_0^πf(x)\sin xdx$

b)

$π$

c)

$\left(2/π\right)∫_0^πf(x)\cos xdx$

d)

$0$

31.

The nature of the one-dimensional wave equation is

a)

Hyperbolic

b)

Parabolic

c)

Elliptic

d)

None of These

32.

The nature of the one-dimensional heat equation is

a)

Circular

b)

Elliptic

c)

Parabolic

d)

Hyperbolic

33.

The nature of PDE 4u_xx+3 u_xy+3 u_yy=0

a)

Parabolic

b)

Hyperbolic

c)

Elliptic

d)

Laplace

34.

The PDE u_xx+ u_yy= 0, is known as

a)

1-D heat equation

b)

1-D wave equation

c)

Laplace equation

d)

None of these

35.

Let PDE u_xx+ u_yy= u_t, By method separation, we consider the solution

a)

u(x,y)=X(x)Y(y)

b)

u(x,y,t)=X(x)Y(y)T(t)

c)

u(x,t)=X(x)T(t)

d)

None of these

36.

Let PDE c²(u_xx+ u_yy)= u_t, By is known as

a)

2-D heat equation

b)

2-D wave equation

c)

Laplace equation

d)

None of these

37.

Let PDE c²(u_xx+ u_yy)= u_tt, By is known as

a)

2-D heat equation

b)

2-D wave equation

c)

Laplace equation

d)

None of these

38.

The nature of PDE u_xx+4 u_xy+3 u_yy,=0

a)

Circular

b)

Elliptic

c)

Hyperbolic

d)

Parabolic

39.

The nature of PDE 4u_xx+3 u_xy=0

a)

Circular

b)

Elliptic

c)

Hyperbolic

d)

Parabolic

40.

A rod of length 10 m has temperature 30⁰C and 40⁰C at end points. What is the temperature gradient

a)

1⁰C per cm

b)

3⁰C per cm

c)

2⁰C per cm

d)

None of these

41.

If x = (1+i, 4) and y = (2-3i, 4+5i) then

< 𝒙, 𝒚 > =

a)

15-15i

b)

15+15i

c)

1+i

d)

1-i

42.

The real inner product space is also known as

(a)

43.

Every inner product Space is a Normed Linear Space . TRUE or FALSE?

a)

TRUE

b)

FALSE

44.

The vector Orthogonal to ( 1 , 2 , 1 ) and ( 3 , 1 , 0 ) is

a)

( 1 , -3 , -5 )

b)

( -1 , -3 , 5 )

c)

( 1 , -3 , 5 )

d)

( -1 , -3 , -5 )

45.

The Distance between the vectors ( 7,1 ) and (3,2 ) in

R^2

a)

$\sqrt{14}$

b)

$\sqrt{15}$

c)

$\sqrt{16}$

d)

$\sqrt{17}$

46.

The triangle inequality only holds in finite dimensional inner product space. TRUE or FALSE?

a)

TRUE

b)

FALSE

47.

An orthonormal basis must be an ordered basis. TRUE or FALSE?

a)

TRUE

b)

FALSE

48.

An inner product is a scalar valued function on the set of

ordered pairs of vectors. TRUE or FALSE?

a)

TRUE

b)

FALSE

49.

There is exactly one inner product on the vector space

R^n

. TRUE or FALSE?

a)

TRUE

b)

FALSE

50.

If < 𝑥, 𝑦 > = 0 for all x in an inner product space then y = 0. TRUE or FALSE?

a)

TRUE

b)

FALSE

MODEL EXAM-MA8352-LAPDE

Total questions: 50

Worksheet time: 4hrs 10mins

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