Advanced Mathematics

2√5

Vectors u and v are given by u = (2 , 0) and v = (-3 , 1). What is the length of vector w given by w = -u - 2v?

-44

Evaluate the following determinant: |7 8 9 4|

4

The following equation involves two determinants: |3 x 2 2| = |2 - 1x - 3|, the value of x is?

-46

Evaluate the following determinant: |1 5 - 2 2 1 - 3 3 - 2 1|

-28

Compute the value of x from the following: x = |4 - 1 2 3 2 0 2 1 1 0 3 0 1 1 4 2 4 5|

-5

Evaluate the following determinant: D = |1 4 2 - 1 2 - 1 0 - 3 - 2 3 1 2 0 2 1 4|

|-1 -1 0 3|

Given matrix A = |1 3 - 2 1| and matrix B = |-1 -2 -11|. Find A + 2B.

|118 - 20 - 5|

Elements of matrix B = |1 2 0 -5|; Elements of matrix C = |3 6 4 1|

Find the elements of the product of the two matrices, matrix BC.

x = -2; y = 6

Solve for x and y from the given relationship: |1 1 3 2||xy| = 20

5

In the complex number 3+4i, the absolute value is:

345.96°

In the complex number 8-2i, the amplitude is:

24 cis 150°

(6 cis 120°)(4 cis 30°) is equal to:

3cis30°

(20 cis 80°)/(10 cis 50°) is equal to:

5

The value of x + y in the complex equation 3 + xi = y + 2i is:

18+i

Multiply (3-2i)(4+3i).

1+2i

Divide (4+3i)/(2-i).

i

Find the value of i⁹.

0

Simplify i¹⁹⁹⁷+i¹⁹⁹⁹, where I is an imaginary number.

-46+9i

Expand (2+√-9)³

5∟143.13°

Write -4+3i in polar form.

-1+i

Simplify: i³⁰-2i²⁵+3i¹⁷.

-√70

Evaluate the value of √-10 x √-7.

-21i

Perform the indicated operation: √-9 x ∛-343.

-8+4i

What is the quotient when 4+8i is divided by i³?

5e^i36.87°

What is the exponential form of the complex number 4+3i?

5+12i

What is the algebraic form of the complex number 13e^i67.38°?

±3i

Solve for x that satisfy the equation x²+36 = 9-2x².

2.565+1.176i

Evaluate ln (5+12i).

(1, -2)

Add the given vectors: (-4, 7) + (5, -9).

√21

Find the length of vector (2, 1,1).

6.00

Find the length of the vector (2, 4, 4).

44

Given 2 x 2 matrix [9 4 7 8], find its determinant.

|(ag+bi+ck) (ah+bj+cl) (dg+ei+fk) (dh+ej+fl)|

Find the product MN of the following matrices: M = |a b c d e f| and N = |g g i j k l|

sine

The following Fourier series equation represents a periodic ____wave i(x)= i + i cos x + i² cos 2x+ i³ cos 3x +…+i sin x + i² sin 2x+ i³ sin 3x+…

-101

Evaluate the value of the determinant |1 6 0 4 2 7 0 5 3|

cos st

The inverse laplace transform of s/[(square) + (w square)] is:

s/[(square) + (w square)]

The laplace transform of cos wt is:

cos kt

K divided by [(s square) + (k square)] is inverse laplace transform of:

[2 e (exp – t) – 4e (exp – 3t)]

Find the inverse transform of [2/(s+1)] – [(4/(s+3)] is equal to:

1/ (s + 4)

What is the laplace transform of e^(-4t)?

I(S) = 2e^(-25t) sin100t

Determine the inverse laplace transform of I(S) = 200 / [(s^2) + 50s + 10625]

e^(-at) sin wt

Determine the inverse laplace transform of (s+a) / [(s+a) ^2 + w^2]

10e^(-10t) – 10e^(-20t)

Determine the inverse laplace transform of 100/ [(S+10) (S+20)]

3, 7

If i = √-1, solve for x and y if x+2+4i = 5+(y-3)i.

3cis30°

(20 cis 80°)/(10 cis 50°) is equal to:

5

The value of x + y in the complex equation 3 + xi = y + 2i is:

18+i

Multiply (3-2i)(4+3i).

1+2i

Divide (4+3i)/(2-i).

i

Find the value of i⁹.

-1

If (x-2)i = y-3i, solve for x.

4

Evaluate i¹¹³ + 4i⁸⁴ + i³.

-2i

If z = 2 + i and w = i – 2, find (z-w)/(z+w).

3

Given 

1	2	6
X	-2	1
0	1	-1

= 25, find x. 

(1+i)/2

Rationalize (2+i)/(3-i)

-1

Determine the value of the given determinant

1	1	0	0
1	1	1	0
0	1	1	1
0	0	1	1

5+i or -5-i

If z² = 24 + 10i, find z.

5+7i

Express 3i + 5 + (square root of -16) in the standard form.

-1+i

Write (square root of 2) cis 135 degrees in rectangular form.

2-5i

Give the conjugate of 2 + (square root of -25) in the standard form.

8 cis 240°

Express -4 - 4√3 i   in trigonometric form.

6.47 cis 3°

Find the quotient of (50 + j35)/(8 + j5).

-√3 + i

The rationalized value of ( 4  - 4√3 i ) / ( -2√3 + 2i ) is:

-4(√3 – i)

Find the product of (4cis120°)(2cis30°) in rectangular form.

√2 ( 1 + i )

Evaluate  [6( cos80° + isin80° ) / 3( cos35° + isin35° )]

-8

Evaluate   [2(cos60° + isin60°)]³

16

Evaluate  (1 + i)^8

cis 300°

Express 1/2 (1 - √3 i ) in trigonometric form.

0

Simplify the expression i^1997 + i^1999, where i is an imaginary number.

3i

Simplify: i^29 + i^21 + i

2i-1

Write in the form a + bi the expression i^3217 – i^427 + i^18

7

The expression 3 + 4i is a complex number. Compute its absolute value.

5 cis 53.1°

Write the polar form of the vector 3 + j4.

-sqrt(70)

Evaluate the value of sqrt(-10) x sqrt(-7).

-3+i

Simplify (3 – i)^2 – 7(3 – i) + 10

30.8 cis 45°

If A = 40 e^(j120°), B = 2 cis -40°, C = 26.46 + j0, solve for A + B + C.

8i

What is 4i cube times 2i square?

12.5 + j21.65

What is the simplified complex expression of (4.33 + j2.5) square?

-4(1+i)

Find the value of (1 + i)^5, where I is an imaginary number.

1.9 + j1.1

Find the principal 5^th root of [ 50(cos 150° + jsin 150°) ].

-8+4i

What is the quotient when 4 + 8i is divided by i^3?

(-18+i)/25

If A = -2 – 3i, and B = 3 + 4i, what is A/B?

1+2i

Rationalize (4 + 3i) / (2 - i).

(-7+17i)/13

Simplify [(2 + 3i)(5 - i)] / [(3 – 2i)^2].

1.12 - j0.66

What is the simplified expression of the complex number (6 + j2.5)/(3 + j4)?

square root of 3+i

Perform the operation: 4 (cos 60° + i sin 60°) divided by 2 (cos 30° + i sin 30°) in rectangular coordinates.

-10+j10

A/B = 2 at 180°, 

A + C = -5 + j15, 

C = conjugate of B. Find A.

0.707

Evaluate cosh(jπ/4).

j1.732

Evaluate tanh(jπ/3).

1.28+j0.98

Evaluate ln (2 + j3).

4

Evaluate the terms of a Fourier series 2 e^(j10πt) + e^(- j10πt) at t = 1.

e^(ix) = cos x + i sin x

Given the following series:

sin x = x – x^3/3! + x^5/5! + …

cos x 1 – x^2/2! + x^4/4! + …

e^x = 1 + x + x^2/2! + x^3/3! + …

What relation can you draw from these series?

10 e^(j40πt)

One term of a Fourier series in cosine form is 10 cos 40πt. Write it in exponential form.

5

Evaluate the determinant:

1	2	3
-2	-1	-2
3	1	4

-101

Evaluate the determinant:

1	6	0
4	2	7
0	5	3

326

Evaluate the determinant:

2	14	3	1
1	5	-1	3
1	-2	2	-3
3	-4	-3	-4

-28

Compute the value of x by determinant. 

x =	4	-1	2	3
	2	0	2	1
	10	3	0	1
	14	2	4	5

3

Given the equations:

x + y + z = 2

3x – y – 2z = 4

5x – 2y + 3z = -7

Solve for y by determinants.

(1, 2, -1)

Solve the equations by Cramer’s Rule:

2x – y + 3z = -3

3x + 3y – z = 10

-x – y + z = -4

(1/4) e^t sinht

Determine the inverse laplace transform of 1 / (4s^2 – 8s).

2 cos 3x – 6 sin 3x

Find the inverse laplace transform of (2s - 18) / (s^2 + 9) as a function of x.