Free Printable Calculator Techniques

1.

In a class of 60 students, 43 like calculus, 38 like geometry, how many students like both subjects?

a)

20

b)

21

c)

22

d)

23

2.

In which interval will the following power series converge?

$\sum_{n=1}^{\infty}\frac{x^n}{\left(n^2+1\right)}$

a)

$\left(-\frac{4}{3},\ \frac{4}{3}\right)$

b)

$\left(-4,\ 1\right]$

c)

$\left[0,\ 2\right]$

d)

$\left[-1,\ 1\right]$

3.

Given the following functions:

$f\left(x\right)\ =\ 5x^2-4x+10$

$g\left(x\right)\ =\ 3x^3+18x^2-5x+7$

Find f(g(5)).

a)

5302732

b)

3253027

c)

4350278

d)

6534297

4.

Given the ff functions:
f(x) = 2x+3

g(x) = 3x+2

Evaluate: f(g(x))

a)

6x + 11

b)

6x + 7

c)

11x + 6

d)

7x + 6

5.

Find f^-1 (x)
$f\left(x\right)\ =\ \frac{2x+5}{4}$

a)

$f^{-1}\left(x\right)\ =\ \frac{4x-5}{4}$

b)

$f^{-1}\left(x\right)\ =\ \frac{4x+5}{4}$

c)

$f^{-1}\left(x\right)\ =\ \frac{2x-5}{4}$

d)

$f^{-1}\left(x\right)\ =\ \frac{2x+5}{4}$

6.

A class of students took examination in Math, Elecs and Comms. 50 passed Math, 48 passed Elecs and 42 passed Comms. 28 passed Math and Elecs, 20 passed Elecs and Comms, while 22 passed Math and Comms. If 15 passed all subjects how many students were there?

a)

80

b)

90

c)

85

d)

95

7.

Find $f^{-1}\left(5\right)$

$f\left(x\right)\ =\ \frac{2x+5}{4}$

a)

2.5

b)

5.5

c)

7.5

d)

9.5

8.

How many distinct elements/arrangements can be formed using all the letters of STATISTICS?

a)

30400

b)

40500

c)

50400

d)

60500

9.

In each of 4 races, the democrats have 60% chance of winning. Assuming that the races are independent of each other, what is the probability that the democrat won at least 2 races?

a)

0.7208

b)

0.8208

c)

0.9208

d)

0.6208

10.

Vehicles pass through a junction on a busy road at an average rate of 300 per hour. Find the probability that at most 5 cars passes in a given minute.

a)

0.3

b)

0.4

c)

0.5

d)

0.6

11.

Given the values: 74 85 94 100 115 and 130.

Find population standard deviation and variance.

a)

18.53, 343.55

b)

343.55, 18.53

c)

18.53, 340.55

d)

340.55, 18.53

12.

The annual salaries of employees in a company are approximately normally distributed with a mean of 2.5 million pesos and a standard deviation of 1 million pesos. What percent of people earn from between 2.25 and 3.25?

a)

0.753

b)

0.273

c)

0.573

d)

0.373

13.

Graduates from a certain business school plan to take an admission test. The scores of this test are roughly normally distributed with a mean of 530 and a standard deviation of 110. Which is the lowest score that belongs to the top 10

a)

780

b)

681

c)

781

d)

651

14.

What is the distance between (4,7) and (2,2)?

a)

5.39

b)

6.39

c)

7.39

d)

8.39

15.

Find the equation of the line passing through the points (-2, 5) and (4, -3)

a)

4x + 7y = 3

b)

3x + 4y = 7

c)

4x + 3y = 7

d)

7x + 4y = 3

16.

Find the equation of the line that passes through the point (-2 6) with a slope of 3.

a)

y = 12 - 3x

b)

y = 16 + 3x

c)

y = 16 - 3x

d)

y = 12 + 3x

17.

Find the midpoint of the line that passes through the point (-1, 2) and (3, -6)

a)

(1, -2)

b)

(2, -2)

c)

(-1, -2)

d)

(-1, -2)

18.

Find the equation of the line passing through the points (1, -2, 0) (3, 1, 4) and (0, -1, 2)

a)

2x + 8y + 5z = 18

b)

2x - 8y + 5z = 18

c)

2x - 8y - 5z = 18

d)

2x + 8y -5z = 18

19.

In a cartesian coordinate, the vertices of a triangle are the ff. points (13, 14) (16, 30) and (50, 10) What is the area?

a)

202

b)

402

c)

302

d)

502

20.

In a cartesian coordinate the vertices of a polygon are (4,1) (3,6) (-5,1) (-3,1) and (-3, -3). Find the area of the polygon.

a)

36.5

b)

38.5

c)

40.5

d)

42.5

21.

Find the coordinates of the centroid of the triangle formed by the ff. sets of 3 points: (7, 5) (-2, -5) (4, 6)

a)

(2, 3)

b)

(3, 3)

c)

(2, 2)

d)

(3, 2)

22.

Find the coordinates of the incenter of the triangle formed by following sets of 3 pts: (0,0) (0,14) (12, -5)

a)

(4,2)

b)

(6, -4)

c)

(4 -6)

d)

(2, -4)

23.

Find the indicated limit $\lim_{x,y\rightarrow1,1}\left(\frac{2x^2-xy\ -y^2}{x^2-y^2}\right)$

a)

1.5

b)

2.5

c)

3.5

d)

4.5

24.

Find the indicated limit:

$\lim_{x\rightarrow0}\ \frac{\cos\left(x\right)-1}{x^2}$

a)

-1

b)

1/2

c)

0

d)

-1/2

25.

Find the limit.

$\lim_{x\rightarrow+\infty}\ \frac{\sqrt[]{x^4+1}}{2x^2-5}$

a)

1/2

b)

-1/2

c)

does not exist

d)

0

26.

Find the indicated limit:

$\lim_{\left(x,\ y\right)\rightarrow\left(0,0\right)}\left(\frac{3xy}{x^2+y^2}\right)$

a)

3/2

b)

2/3

c)

-1

d)

limit does not exist

27.

Differentiate: $y=\cot^{-1}\left(3x\right)$

a)

$y'=\frac{3}{1+9x^2}$

b)

$y'=-\frac{3}{1+9x^2}$

c)

$y'=\frac{3}{1-9x^2}$

d)

$y'=-\frac{3}{1-9x^2}$

28.

Find the y'' of $y\ =\ \cot^{-1}\left(3x\right)$

a)

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

b)

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

c)

$-\frac{54x}{\left(1+9x^2\right)^2}$

d)

$\frac{54x}{\left(1+9x^2\right)^2}$

29.

Find y''' of $y\ =\cot^{-1}\left(3x\right)$

a)

$\frac{54+972x^2+13122x^4}{\left(1-9x^2\right)^4}$

b)

$\frac{54-972x^2-13122x^4}{\left(1+9x^2\right)^4}$

c)

$\frac{54+972x^2-13122x^4}{\left(1-9x^2\right)^4}$

d)

$\frac{54-972x^2+13122x^4}{\left(1+9x^2\right)^4}$

30.

Find the radius of curvature at x = pi/3 on $y\ =\ 2\log\left(\sin\left(\frac{x}{2}\right)\right)$

a)

0.26

b)

1.26

c)

3.26

d)

2.26

31.

Find the equation of the normal line to $y\ =\ \ x^3-2x^2+\ x-3$ at point (2, -1)

a)

y = 5x + 11

b)

y = 5x - 11

c)

y = 11x - 5

d)

y = 11x + 5

32.

Find the first order partial derivative of the given function with respect to y

$z\ =\ \sqrt[]{x^2+\ln\left(5x-3y^2\right)}$

a)

$-\frac{\frac{3y}{5x-3y^2}}{\sqrt[]{x^2+\ln\left(5x-3y^2\right)}}$

b)

$\frac{\frac{3y}{5x-3y^2}}{\sqrt[]{x^2+\ln\left(5x-3y^2\right)}}$

c)

$-\frac{\frac{6y}{5x-3y^2}}{\sqrt[]{x^2+\ln\left(5x-3y^2\right)}}$

d)

$\frac{\frac{6y}{5x-3y^2}}{\sqrt[]{x^2+\ln\left(5x-3y^2\right)}}$

33.

Differentiate

$10x^4-16xy^2+10y^3-48$

a)

$\frac{20y^3-9x^3}{3\left(5y^2-6xy\right)}$

b)

$\frac{20y^3-9x^3}{3\left(5y^2+6xy\right)}$

c)

$\frac{9y^3-20x^3}{3\left(5y^2-6xy\right)}$

d)

$\frac{9y^3+20x^3}{3\left(5y^2-6xy\right)}$

34.

Find the 2nd derivative of the function

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

a)

$\frac{4\left(x+y\right)}{\left(x+y+1\right)^3}$

b)

$\frac{4\left(x+y\right)}{\left(x+y-1\right)^3}$

c)

$\frac{4\left(x-y\right)}{\left(x+y+1\right)^3}$

d)

$\frac{4\left(x+y\right)}{\left(x-y-1\right)^3}$

35.

Find the slope of the curve $x^2+\ y^2-6x\ +10y\ +5\ =\ 0$ at point (1,0)

a)

1/5

b)

3/5

c)

4/5

d)

2/5

36.

In which interval is the function below increasing?

$f\left(x\right)\ =\ x^4-8x^2$

a)

$-\ \infty<x<-2$

b)

$0<x<2$

c)

$-2<x<0$

d)

$-2<x<+\infty$

37.

A sheet of metal is 12 x 10 is to be used to make an open box. Squares of equal sides x are cutout of each corner. If the sides are folded to make a box, What is the maximum volume possible?

a)

86.77

b)

96.77

c)

106.77

d)

206.77

38.

Determine the interval in which the concavity is upward

$f''\left(x\right)=\ -\sin\left(x\right)-\cos\left(x\right)$

a)

$\left[0,\frac{3pi}{4}\right]$

b)

$\left[\frac{3pi}{4},\frac{7pi}{4}\right]$

c)

$\left[\frac{3pi}{4},\ pi\right)$

d)

$\left(\frac{7pi}{4},\ 2pi\right)$

39.

The radius of a sphere is measured to be 5m. If the measurement is correct to within 0.05m. Estimate the propagated error and percent error in the volume of the sphere.

a)

$\pm30.708;\ \pm3.00\%$

b)

$\pm25.708;\ \pm3.00\%$

c)

$\pm15.708;\ \pm3.00\%$

d)

$\pm35.708;\ \pm3.00\%$

40.

Two poles, one 8 meters tall and another one 20 meters tall are 30 meters apart. A length of wire is attached to the top of each pole. Each wire is staked to the ground somewhere between the poles. How far from the shorter pole should the stake be to minimized the length of the wire used?

a)

60/7

b)

50/7

c)

40/7

d)

30/7

41.

A billboard 85 ft wide is perpendicular to a straight road and is 40 ft from the road. Find the point on the road at which the angle subtended by the billboard is a maximum.

a)

$30\sqrt[]{2}$

b)

$45.5$

c)

$66.7$

d)

$50\sqrt[]{2}$

42.

Three thousand feet from a radio tower, a man views a plane, which is flying away from him and towards the tower at an elevation of 4000ft. If the plane flies at a rate 600 ft/s at what rate does the distance between the man and the plane increase as the plane passes over the radio tower?

a)

360 ft/s

b)

380 ft/s

c)

400 ft/s

d)

450 ft/s

43.

What is the maximum volume of right circular cylinder of radius r and height h, which is inscribed in a right circular cone of radius 5 and height 20m?

a)

262.31 cu in.

b)

162.56 cu in.

c)

301.87 cu in.

d)

232.71 cu in.

44.

Evaluate: $\int_0^2xydy$

x = t³; y = 7t²

a)

1592

b)

1692

c)

1792

d)

1892

45.

Gas escapes from a spherical balloon at a rate of 2 cm³/min. Find the rate at which the surface area is decreasing in cm²/min when the radius is 8 cm.

a)

-0.25 cm²/min

b)

-0.5 cm²/min

c)

-1.5 cm²/min

d)

-1.0 cm²/min

46.

Water is draining from the bottom of a cone shaped funnel at the rate of 0.03 ft³/s. The height of the funnel is 2ft and the radius at the top of the funnel is 1 ft. At what rate is the height of the water in the funnel changing when the height of the water is 0.5 ft?

a)

-0.035

b)

-0.432

c)

-0.153

d)

-0.092

47.

Evaluate:
$\int_{ }^{ }\left(\frac{\cos^3\left(x\right)}{1-\sin\left(x\right)}\right)$

a)

$\cos\left(x\right)+\frac{1}{2}\cos^2\left(x\right)+C$

b)

$\sin\left(x\right)-\frac{1}{2}\sin^2\left(x\right)+C$

c)

$\sin\left(x\right)+\frac{1}{2}\sin^2\left(x\right)+C$

d)

$\sin\left(x\right)+\frac{1}{2}\cos^2\left(x\right)+C$

48.

Evaluate:

$\int_0^4\int_0^3\left(3xy^2+x^2y\right)dydx$

a)

412

b)

312

c)

512

d)

612

49.

Determine the orthogonal trajectories of the family of sinusoids:

$y=C\sin x$

a)

$\cos\left(x\right)=C_1e^{\frac{y^2}{2}}$

b)

$\sin\left(y\right)=C_1e^{\frac{x^2}{2}}$

c)

$\cos\left(x\right)=C_1e^{y^2}$

d)

$\sin\left(x\right)=C_1e^{y^2}$

50.

Find the area between $y\ =\ -x^3+x^2+16x$ and the $y\ =\ 4x$

a)

907/12

b)

937/12

c)

957/12

d)

967/12

51.

Evaluate

$\int_e^{e^2}\left(\frac{1}{x\ln\left(x\right)}dx\right)$

a)

ln(3)

b)

ln(2)

c)

ln(e)

d)

e

52.

Given the differential equation: $y'-2y=4\cos\left(x\right)-8\sin\left(x\right)$ ; y(0) = 3

Find the value of y(2).

a)

110.56

b)

121.72

c)

150.98

d)

98.99

53.

Evaluate:

$\int_0^{\frac{pi}{2}}\sin^5\beta\cos^5\beta$

a)

1/20

b)

1/60

c)

1/30

d)

1/50

54.

Solve for the D.E.

$y'''\ -\ y'\ =\sin\left(3x\right)$

a)

$y=C_1+C_2e^x+C_3e^{-x}+\frac{1}{30}\cos\left(3x\right)$

b)

$y=C_1+C_2e^{-2x}+C_3xe^{-x}+\frac{2}{30}\sin\left(3x\right)$

c)

$y=C_1e^{-x}+C_2xe^{-x}+C_3x^2e^{-x}+\frac{1}{30}\sin\left(3x\right)$

d)

$y=C_1+C_2e^{-x}+C_3xe^{-x}+\frac{2}{15}\cos\left(3x\right)$

55.

Find the area bounded by $r^2=\cos\theta$

a)

1

b)

4

c)

2

d)

3

56.

Evaluate: $\int_0^1\left(\frac{1}{\sqrt[]{y}}\right)$

a)

1

b)

3

c)

5

d)

2

57.

Find the general solution of the equation:

$y''-6y'+9y=0$

a)

$y=C_1xe^{3x}+C_2e^{3x}$

b)

$y=C_1e^{-3x}+C_2e^{3x}$

c)

$y=C_1xe^{3x}+C_2x^2e^{3x}$

d)

$y=C_1e^{-3x}+C_2e^{-3x}$

58.

Find the area between $y\ =\ x^{3\ }+\ 3x^{2\ }-4x$ and the x-axis.

a)

131/4

b)

125/4

c)

103/4

d)

151/4

59.

Solve for the general solution of the D.E.

$e^xdy+\left(ye^x+2x\right)dx=0$

a)

$xe^y+x^2=c$

b)

$xe^y+y^2=c$

c)

$ye^x+x^2=c$

d)

$ye^y+x^2=c$

60.

Derive the DE for the family of plane curves defined by:

$y^2=2cx$

a)

$2y-x^2y'\ =0$

b)

$x-2yy'=0$

c)

$x^2-xy'\ =0$

d)

$y-2xy'\ =0$

61.

Bismith-210 has a half-life of 5.0 days. Suppose that a sample originally has a mass of 800 mg. When will its mass be reduced to 1 mg.

a)

48.2

b)

56.2

c)

62.2

d)

38.2

62.

Form the D.E representing the family of parabolas with equation:

$y\ =\ C_1\left(x-C_2\right)^2$

a)

$yy''-x=2y'^{ }$

b)

$xyy''=y'^{ }$

c)

$2yy''=\left(y'\right)^2$

d)

$y^2y''=2y'$

63.

Solve for the DE of the following:

$y\ =\ C_1e^{-2x}+C_2xe^{-2x}+C_3e^{5x}$

a)

$y'''-y''-16y'-20y=0$

b)

$2y'''-3y''-16y'-2y=0$

c)

$3y'''-2y''-25y'-y=0$

d)

$5y'''-4y''-25y'-y=0$

64.

Solve for the given D.E

$y'\ =\ \frac{\left(x^2+y^2\right)}{xy}$ ; y(1) = -2

a)

$y=-2\sqrt[]{2x^2\ln\left(x^2\right)+4x^2}$

b)

$y=-2\sqrt[]{3x^2\ln\left(x^2\right)-4x^2}$

c)

$y=-\sqrt[]{x^2\ln\left(x^2\right)+4x^2}$

d)

$y=-\frac{1}{2}\sqrt[]{x^2\ln\left(x^2\right)+4x^2}$

65.

In an experiment, a population of insects grows at a rate proportional to its existing population. After the 3rd day of the experiment. There were 130 insects while after the 7th day, there were 380. How many insects were in the original population?

a)

48

b)

38

c)

58

d)

68

66.

A famous painting was purchased in 1941 for a price of 7000. The painting was sold in 1997 for 48.4 Million. What rate of interest compounded continuously did this earn

a)

16.8%

b)

19.8%

c)

18.8%

d)

15.8%

67.

A local restaurant has 1000 regular customers. On any given day, 25% of this regulars order a breakfast time of using eggs in the recipe. The latest shipment of eggs is contaminated with salmonella bacteria. How many people might get ill in 4 days?

a)

256

b)

586

c)

899

d)

724

68.

Suppose you need a pot of soup in a 75F room. Right when you take the soup from the stove, you measure its temperature to be 220F. Suppose after 20 min, the soup has cooled to 170F. Suppose you can eat the soup when it is 130F how long will it take to cool this temperature

a)

46 mins

b)

50 mins

c)

40 mins

d)

42 mins

69.

Evaluate and express in polar form:

$e^{\left(1+2i\right)}\overline{\left(\frac{5+2i}{4\cos\left(30\degree\right)+j\sin30\degree}\right)}$

a)

$3.66\angle2.143$

b)

$3.66\angle-2.143$

c)

$-3.66\angle2.143$

d)

$-3.66\angle-2.143$

70.

A pharmacist in a drugstore must check 1000 receipts per day. A new pharmacist was hired in the drugstore. In 1st week, the pharmacist was able to check 100 receipts per day. Estimate the number of receipts the pharmacist can check for the 2nd week.

a)

200

b)

190

c)

150

d)

220

71.

A lake is stocked with 100 fish. After 3 months, there are 250 fish. A study predicts that the lake can only support 1000 fish. Find the population of fish after another 3 months.

a)

300

b)

400

c)

500

d)

557

72.

A population of rabbits in a meadow is observed to be 200 rabbits at t = 0. After a month, the rabbit population is observed to have increased by 4%. If the maximum capacity of habitat is 750 rabbits, how many are there after a year?

a)

278

b)

302

c)

331

d)

435

73.

A tank has pure water flowing into it at 10L/min. The contents of the tank are kept thoroughly mixed and the contents flow out at 10L/min. Initially, the tank contains 10kg of salt in 100L in water. How much salt will there be in the tank after 30 mins?

a)

5.3 kg

b)

3.6 kg

c)

2.1 kg

d)

0.5 kg

74.

A tank contains 1,500 L of water and 20 kg of dissolved salt. Fresh water is entering the tank at 15 L/min (the solution stays perfectly mixed), and the solution drains at a rate of 10 L/min. How much salt is in the tank at t minutes and at 10 minutes?

a)

13.35 kg

b)

15.52 kg

c)

18.73 kg

d)

23.56 kg

75.

A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5L/min. The solution is kept mixed and is drained from the tank at a rate of 3L/min. Find the amount of salt after 20min.

a)

15.32 kg

b)

31.85 kg

c)

42.21 kg

d)

50.31 kg

76.

Express in rectangular form:

$e^{\left(4+2i\right)}$

a)

$-22.72+49.65i$

b)

$-22.72-49.65i$

c)

$22.72-49.65i$

d)

$22.72+49.65i$

77.

express in rectangular form:

$\frac{\left(4+7i\right)\left(e^{3.14j}\right)}{8\left(\cos60\degree+j\sin60\degree\right)}$

a)

1.008 + 0.00289i

b)

-1.008 - 0.00289i

c)

1.008 - 0.00289i

d)

-1.008 + 0.00289i

78.

Evaluate and express in exponential form

$i^{24}+i^{197}+i^{17892}-i^{614}$

a)

$\sqrt[]{13}e^{j0.588}$

b)

$\sqrt[]{-13}e^{j0.588}$

c)

$\sqrt[]{13}e^{-j0.588}$

d)

$-\sqrt[]{13}e^{j0.588}$

79.

Identify the principal root of $\left(3-6i\right)^{\frac{1}{3}}$

a)

$1.886\angle221.145\degree$

b)

$1.886\angle-141.145\degree$

c)

$1.886\angle98.855\degree$

d)

$1.886\angle-21.145\degree$

80.

Express in polar form:
$\left(2-9i\right)^{10}$

a)

$-4.437\ x\ 10^9\angle-54.71\degree$

b)

$4.437\ x\ 10^9\angle54.71\degree$

c)

$-4.437\ x\ 10^9\angle54.71\degree$

d)

$4.437\ x\ 10^9\angle-54.71\degree$

81.

Evaluate:

$\log_{\left(1+2i\right)}\left(3-4i\right)_{ }$

a)

$0.255-1.35i$

b)

$0.143+1.35i$

c)

$0.143-1.35i$

d)

$0.255+1.35i$

82.

Evaluate: $\left(3-4i\right)^{\left(1+2i\right)}$

a)

$-31.945e^{2.292j}$

b)

$32.699e^{2.292j}$

c)

$31.945e^{2.292j}$

d)

$-32.699e^{2.292j}$

83.

Evaluate:

$\tan^2\left(2+8i\right)$

a)

$1\angle-3.142$

b)

$-1\angle-3.142$

c)

$1\angle3.142$

d)

$-1\angle3.142$

84.

Evaluate:

$\coth^2\left(2+8i\right)$

Express in polar form.

a)

$-1.06\angle0.02$

b)

$1.06\angle0.02$

c)

$-0.93\angle0.02$

d)

$0.93\angle0.02$

85.

Solve for the determinant of the matrix

a)

-42

b)

-37

c)

-32

d)

-35

86.

Find the determinant of the matrix

a)

2101

b)

2202

c)

2303

d)

2404

87.

Find the determinant.

a)

11065

b)

14069

c)

-13068

d)

-12067

88.

Find the eigenvector of the given matrix

a)

b)

c)

d)

89.

Find the eigenvector of the given matrix if $\lambda=3$

a)

b)

c)

d)

90.

Find the eigenvalue of the given matrix

a)

5

b)

3

c)

2

d)

6

91.

Solve for the laplace transform of the equation

$f\left(t\right)\ =\ \frac{\sin\left(5t\right)}{e^{-2t}}-t^3$

a)

$\frac{25}{\left(s-2\right)^2+5}+\frac{36}{s^4}$

b)

$\frac{5}{\left(s-2\right)^2+25}-\frac{6}{s^4}$

c)

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

d)

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

92.

What is the laplace transform of δ(t – a)?

a)

$\frac{1}{s}$

b)

$e^{-as}$

c)

$e^{as}$

d)

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

93.

Evaluate the laplace transform of:

$\int_0^t\ e^{-3T}\cos\left(2T\right)dT$

a)

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

b)

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

c)

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

d)

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

94.

Solve for the laplace transform of:

$f\left(t\right)\ =\ e^{\left(3t-3\right)}u\left(t-1\right)$

a)

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

b)

$\frac{e^{-3s}}{s+3}$

c)

$\frac{6}{s^2+9}$

d)

$\frac{6}{s^2+9}$

95.

Find the fourier transform of $e^{-jnt}$

a)

$2\piδ\left(w+n\right)$

b)

$2\piδ\left(w-n\right)$

c)

$\piδ\left(w+n\right)$

d)

$\piδ\left(w-n\right)$

96.

Solve the inverse laplace of:

$F\left(s\right)=\frac{5}{\left(s-3\right)^4}$

a)

$f\left(t\right)=6e^{\left(-3t-3\right)}$

b)

$f\left(t\right)=5e^{\left(3t+3\right)}$

c)

$f\left(t\right)=\frac{5}{6}e^{\left(3t\right)t^3}$

d)

$f\left(t\right)=\frac{25}{36}e^{\left(-3t\right)t^3}$

97.

Solve for the b_ncoefficient of the fourier series of the function

a)

$b_n=\frac{3\left(1-\cos n\pi\right)}{n\pi}$

b)

$b_n=\frac{3\left(1-\cos n\pi\right)}{n\pi}$

c)

$b_n=\frac{3\left(1-\cos n\pi\right)}{2n\pi}$

d)

$b_n=\frac{3\left(1-\sin n\pi\right)}{2n\pi}$

98.

Solve for the fourier series coefficients of the function below with n = 2

a)

$A_2=0,\ B_2=-0.637$

b)

$A_2=0.637,\ B_2=-0.637$

c)

$A_2=0.637,\ B_2=0$

d)

$A_2=1.25,\ B_2=-0.637$

99.

Find the fourier series for the given function.

a)

$f\left(x\right)\ =\frac{4}{\pi}\sum_{k=0}^{\infty}\left(\frac{\cos\left(\left(2k+1\right)x\right)}{\left(2k+1\right)^2}\right)$

b)

$f\left(x\right)\ =\frac{2}{\pi}\sum_{k=0}^{\infty}\left(\frac{\cos\left(\left(2k-2\right)x\right)}{\left(2k+2\right)^2}\right)$

c)

$f\left(x\right)\ =\frac{2}{\pi}\sum_{k=0}^{\infty}\left(\frac{\cos\left(\left(2k+2\right)x\right)}{\left(2k+2\right)^2}\right)$

d)

$f\left(x\right)\ =\frac{4}{\pi}\sum_{k=0}^{\infty}\left(\frac{\cos\left(\left(2k-1\right)x\right)}{\left(2k+1\right)^2}\right)$

100.

Find the half-range sine series of the function.

a)

$f\left(x\right)=\sum_{n=1}^{\infty}\frac{-4}{n\pi}\left[2-\cos\left(n\pi\right)\right]\sin\left(nx\right)$

b)

$f\left(x\right)=\sum_{n=1}^{\infty}\frac{8}{n\pi}\left[1-\cos\left(n\pi\right)\right]\sin\left(nx\right)$

c)

$f\left(x\right)=\sum_{n=1}^{\infty}\frac{16}{n\pi}\left[2-\cos\left(n\pi\right)\right]\sin\left(nx\right)$

d)

$f\left(x\right)=\sum_{n=1}^{\infty}\frac{32}{n\pi}\left[1-\cos\left(n\pi\right)\right]\sin\left(nx\right)$

101.

Find the half range cosine series of a function

a)

$f\left(x\right)=2+\sum_{n=1}^{\infty}\frac{8}{n\pi}\sin\left(\frac{n\pi}{2}\right)\cos\left(nx\right)$

b)

$f\left(x\right)=2+\sum_{n=1}^{\infty}\frac{-8}{n\pi}\sin\left(\frac{n\pi}{2}\right)\cos\left(nx\right)$

c)

$f\left(x\right)=2+\sum_{n=1}^{\infty}\frac{4}{n\pi}\sin\left(\frac{n\pi}{2}\right)\cos\left(nx\right)$

d)

$f\left(x\right)=2+\sum_{n=1}^{\infty}\frac{-4}{n\pi}\sin\left(\frac{n\pi}{2}\right)\cos\left(nx\right)$

102.

What will be the fourier series of the function f(x) = 1−x²in the interval [−1, 1]?

a)

$\frac{2}{3}+\sum_{n=1}^{\infty}\frac{−4(−1)^n\cos(\pi nx)}{\pi^2n^4}$

b)

$\frac{2}{3}+\sum_{n=1}^{\infty}\frac{−16(−1)^n\cos(\pi nx)}{\pi^3n^2}$

c)

$\frac{2}{3}+\sum_{n=1}^{\infty}\frac{−4(−1)^n\cos(\pi nx)}{\pi^2n^2}$

d)

$\frac{2}{3}+\sum_{n=1}^{\infty}\frac{−16(−1)^n\cos(\pi nx)}{\pi^4n^2}$

103.

Solve for the Fourier Transform of the ff. function:

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

a)

$\frac{e^{-\left(2+j\omega\right)}}{2-\omega}$

b)

$\frac{e^{-3\left(2+j\omega\right)}}{2+j\omega}$

c)

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

d)

$\frac{e^{-\left(2-j\omega\right)}}{2+j\omega}$

104.

Find the fourier transform of:

a)

$\sqrt[]{\frac{2}{\pi}}\left(\frac{\cos\left(2\omega\right)}{\omega}\right)$

b)

$\sqrt[]{\frac{1}{2\pi}}\left(\frac{\cos\left(2\omega\right)}{\omega}\right)$

c)

$\sqrt[]{\frac{2}{\pi}}\left(\frac{\sin\left(2\omega\right)}{\omega}\right)$

d)

$\sqrt[]{\frac{1}{2\pi}}\left(\frac{\sin\left(2\omega\right)}{\omega}\right)$

105.

Find the fourier transform of the signal:

$x\left(t\right)=e^{-t}u\left(t\right)$

a)

$-\frac{1}{j\omega-1}$

b)

$\frac{\omega^2}{1+j\omega}$

c)

$\frac{1}{\left(j\omega-1\right)^2}$

d)

$\frac{1}{j\omega+1}$

106.

Find the inverse fourier transform of:
$x\left(\omega\right)=\frac{1}{3}\pi e^{-\left|\frac{\omega}{3}\right|}$

a)

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

b)

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

c)

$\frac{0.5}{1+\left(9t\right)^2}$

d)

$\frac{-2}{4+\left(3t\right)^2}$

107.

Find the z transform of the ff. sequence

$x\left(n\right)=10\sin\left(\frac{\pi n}{4}\right)u\left(n\right)$

a)

$x\left(z\right)=\frac{7.07z}{z^2-1.414z+1}$

b)

$x\left(z\right)=\frac{z}{2z^2-4z+1}$

c)

$x\left(z\right)=\frac{z^2}{z^2+1.414z-1}$

d)

$x\left(z\right)=\frac{3.33z}{z^2+2z-1}$

108.

$y\left(n\right)=\left(0.5\right)^{\left(n-5\right)}u\left(n-5\right)$

a)

$y\left(z\right)=\frac{z^{-4}}{z-0.5}$

b)

$y\left(z\right)=\frac{2z^{-4}}{z+0.5}$

c)

$y\left(z\right)=\frac{4}{z-0.25}$

d)

$y\left(z\right)=\frac{z^{-4}}{z-0.5}$

109.

Find the z transform of:

$x\left[n\right]=-a^nu\left(-n-1\right)$

a)

$x\left(z\right)=\frac{z^2}{z+a}$

b)

$x\left(z\right)=\frac{z+a}{z-a}$

c)

$x\left(z\right)=\frac{z}{z-a}$

d)

$x\left(z\right)=\frac{z}{z^2-a^2}$

110.

Find the convolution of

$f\left(t\right)=e^t$

$g\left(t\right)=e^{-2t}$

a)

$\frac{e^t-e^{-2t}}{3}$

b)

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

c)

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

111.

Which of the following is the inverse laplace of:

$\frac{1}{s^2-10s+21}$

a)

(1 - e^3t) * e^7t

b)

e^3t * e^7t

c)

e^3t * (1/3)e^7t

d)

(1 - e^3t) * 7t

112.

Find the taylor series for the function: $e^{-6x}$ about x = -4

a)

$\sum_{n=0}^{\infty}\left(\frac{\left(-6\right)^ne^{48}}{\left(n+1\right)!}\left(x+4\right)^n\right)$

b)

$\sum_{n=0}^{\infty}\left(\frac{\left(-6\right)^ne^{24}}{n!}\left(x+4\right)^n\right)$

c)

$\sum_{n=0}^{\infty}\left(\frac{\left(-6\right)^ne^{16}}{n!}\left(x+4\right)^n\right)$

d)

$\sum_{n=0}^{\infty}\left(\frac{\left(-6\right)^ne^{56}}{\left(n+1\right)!}\left(x+4\right)^n\right)$

113.

Find the Taylor Series for

$f(x)=x^4e^{-3x^2}$ about x=0

a)

$\sum_{n=0}^{\infty}\frac{(−3)^nx^{\left(2n+4\right)}}{n!}$

b)

$\sum_{n=0}^{\infty}\frac{(−6)^nx^{\left(3n+12\right)}}{n!}$

c)

$\sum_{n=0}^{\infty}\frac{(−9)^nx^{\left(3n+6\right)}}{n!}$

d)

$\sum_{n=0}^{\infty}\frac{(−12)^nx^{\left(3n+8\right)}}{n!}$

114.

Find the taylor series for:

$\frac{1}{x}$ at x = 1

a)

$1-\left(x-1\right)+\frac{1}{2}\left(x-1\right)^2-\frac{1}{6}\left(x-1\right)^3+\frac{1}{24}\left(x-1\right)^4+...$

b)

$1+\left(x-1\right)+\left(x-1\right)^2+\left(x-1\right)^3+\left(x-1\right)^4+...$

c)

$1-\left(x-1\right)+\left(x-1\right)^2-\left(x-1\right)^3+\left(x-1\right)^4+...$

d)

$1+\left(x-1\right)+\frac{1}{2}\left(x-1\right)^2+\frac{1}{6}\left(x-1\right)^3+\frac{1}{24}\left(x-1\right)^4+...$

115.

Determine the Taylor Series of the function:

$f\left(x\right)=\sqrt[]{\left(1+x\right)}$ $about\ x_0=0$

a)

$1+\frac{x}{2}-\frac{x^2}{8}+...$

b)

$1+\frac{x}{4}-\frac{x^2}{64}+...$

c)

$1+\frac{2x}{7}-\frac{3x^2}{11}+...$

d)

$3+\frac{x}{3!}-\frac{x^5}{5!}+...$

116.

Which of the following series is /are convergent?

a)

$\sum_{n=1}^{\infty}\frac{3^n}{2^n}$

b)

$\sum_{n=1}^{\infty}\frac{n^n}{n!}$

c)

Both of them

d)

None of these

117.

Solve for the radius of convergence of the given series:

$\sum_{n=0}^{\infty}\left(-1\right)^n\left(\frac{x^{2n}}{\left(2n\right)!}\right)$

a)

1

b)

2

c)

0

d)

+ infinity

118.

Solve for the radius of convergence of the given series:

$\sum_{n=1}^{\infty}\frac{3^nx^n}{n\left(4^n\right)}$

a)

1

b)

4/3

c)

1/2

d)

0.75

Calculator Techniques

Total questions: 118

Worksheet time: 10hrs 50mins

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