PRE-BOARD EXAMINATION MATH

45/8

A point moves along the curve y = x³ - 3x + 5 so that x = (1/2)sqrt t + 3, where t is time. At what rate is y changing when t = 4?

x(t) = cos(4t) + 2sin(8t)

Which of the following is a period signal?

4π

Find the volume generated when the region bounded by y² = x³, y = 0, x = 2 is revolved about the x-axis.

0.05

Given that P(A) = 0.9, P(B) = 0.8 and P(A ∏ B) = 0.75. Find P(A' ∏ B'). A' means barred A and B' means barred B.

39.61 min, 70.5°F

A metal bar at a temperature of 100 deg F is placed in a room at a constant temperature of 0 deg F. If after 20 min the temperature of the bar is 50 deg F, find (a) the time it will take the bar to reach a temperature of 25 deg F and (b) the temperature of the bar after 10 minutes.

79.5°F

A body at a temperature of 50° F is placed outdoors where the temperature is 100° F. If after 5 minutes the temperature of the body is 60° F, find the temperature of the body after 20 minutes.

2/(s² + 1) + 3s/(s² + 4)

Find the Laplace transform of f(x) = 2sinx + 3cos2x.

1/4 - (1/4)cos 2x

Find Lˉ¹ 1/s(s² + 4).

Standard error

What is the standard deviation of a sampling distribution called?

a/(s² + k²)

Solve for the Laplace transform of sin ax.

(1/5)sin⁵x - (3/7)sin⁷x + (3/9)sin⁹x - (1/11)sin¹¹x + C

Evaluate the integral of sin⁴x cos⁷x dx.

9.26

Find the length of the indicated arc of the given curve y³ = 8x² from x = 1 to x = 8.

- 10 *

Find the slope of the tangent line to the following curves at the point x = 1 and y = 8 - 5x².

7

Let cos z = 2. Find cos 2z.

25 ft/hr

If the angle of elevation of the sum 45° and is decreasing by 1/4 radians per hour, how fast is the shadow cast on the ground a pole 50 ft tall lengthening?

7/20

My pencil holder contains 3 red pencils, 4 green pencils, and 3 blue pencils. My secretary has a pencil holder nearby that contains 4 blue pencils, 4 red pencils and 2 green pencils. What is the probability of my reaching out and grasping a blue pencil?

0.40

A committee of 5 persons is to be selected from a group of 5 men and 10 women. Find the probability that the committee consists of 2 men and 3 women.

y² = x²lnx² + kx²

Solve y' = (x² + y²)/xy.

bell-shaped

The shape of the normal curve is _______.

π/2 - arctan s/3

Find the Laplace transform of (sin3x)/x.

(y - 1)/(x - 2)

Find dy/dx of xy + x - 2y = 5.

2

Determine the initial value of the time-domain response of the following equation using the initial-value theorem. Y(s) = (2s + 1)/(s + 1 + j)(s + 1 - j).

1

Find the limit of [ln (2 + x)]/(x + 1) as x → - 1.

7

Given z₁ = 3 - 4i and z₂ = - 4 + 3i. Find |z₁ x z₂|.

7062

The population of a certain country is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000, estimate the number of people initially living in the country.

21°

Given that A = 10i + 11j - 2k and B = 5i + 12j. Find the angle between A and B.

Circle

Describe the locus represented by |z - i| = 2.

Y(s) = 2s/(s + 1)²(s + 2).y(t) = - 2te¯ᵗ + 4e¯ᵗ - 4e¯²ᵗ

Expand the following equation of Laplace transform in terms of its partial fraction and obtain the time domain response.

1t

Solve 100 d²N/dt² - 20 dN/dt + N = 0.N = C₁e^0.1t + C₂te^0.

1

The divergence of the vector field A = xa_x + ya_y + za_z is:

x³yˉ¹ = (2/3)y³ + C

Solve y' = 3yx²/(x³ + 2y⁴).

213.33

A solid has a base in the form of an ellipse with major axis 10 and minor axis 8. Find its volume if every section perpendicular to the major axis is an isosceles right triangle with one leg in the plane of the base.

non causal, linear and time-variant

The continuous time system described by y(t) = x(t²) is _____.

x⁴/12

Find f(x) * g(x) when f(x) = x and g(x) = x².

linear and time-varying

Consider a continuous-time system with input x(t) and output y(t) given by y(t) = x(t) cos (t). This system is ______.

differentiable

A function is said to be _________ at a point x_a if the derivative of the function exists at that point.

(1/3)sec³x - sec x + C

Evaluate the integral of tan³x sec x dx.

(s² + s + 2)/s²(s + 2)

Find the Laplace transform of eˉ²ᵗ + t.

The roots of the characteristic equation are on the left half of the s-plane and on the imaginary axis.

Which of the following statements with respect to the stability of a control system is NOT true?

20 min

A 50-gal tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find the amount of time required for overflow to occur.

Sqrt (1 + 36x²)

Find ds/dx at P(x, y) on the parabola y = 3x².

1/2

Evaluate the limit of [1/x - 1/(eˣ - 1)] as x → 0

T(t) = ce^-kt + 100

Solve dT/dt + kT = 100k, where k denotes a constant.

x² chi-square test

Which test we normally apply for qualitative data?

(3/7)i - (6/7)j + (2/7)k

Given that A = 3i - 6j + 2k. Find a unit vector in the direction of A.

h = 8(sqrt 2), r = 4(sqrt 2)

Determine the dimension of the right circular cylinder of maximum lateral surface area that can be inscribed in a sphere of radius 8.

All of the above statements are true

What does it mean when you calculate a 95% confidence interval?

20(sqrt2) x 15(sqrt 2)

A rectangle is inscribed in the ellipse x²/400 + y²/225 = 1 with its sides parallel to the axes of the ellipse. Find the dimensions of the rectangle with maximum area.

1200π

Oil from a leaking oil tanker radiates outward in the form of a circular film on the surface of the water. If the radius of the circle increases at the rate of 3 meters per minute, how fast is the area (in m²/s) of the circle increasing when the radius is 200 meters?

causal

A system is called _______ if its output at the present time depends on only the present and/or past values of the input.

6 V/m

Charge is distributed uniformly over the plane z = 10 cm with a density p_s = (1/3π) nC/m².

(1/2)eˉˣ - (1/2)cosx + (1/2)sinx

Find Lˉ¹ [1/(s + 1)(s² + 1).

t⁴/12

Find the convolution of f and g if f(t) = tu(t) and g(t) = t² u(t).

e³ˣ - e²ˣ

Find f(x) * g(x) when f(x) = e³ˣ and g(x) = e²ˣ.

694

A bacteria culture is known to grow at a rate proportional to the amount present. After one hour, 1000 strands of the bacteria are observed in the culture; and after four hours, 3000 strands. Find the approximate number of strands of the bacteria originally in the culture.

The convolution of two odd functions is an odd function.

Which of the following statements is not correct for convolution?

(1/3)x³ ln x - (1/9)x³ + C

Find the integral of x²lnx dx.

(1/3)tan³x - tan x + x + C

Find the integral of tan⁴xdx.

32π/3

Find the area of the surface of revolution generated by revolving about the x-axis the arc y² + 4x = 2 ln y from y = 1 to y = 3.

1

In Normal curve, the total area under the curve is _____.

(1/3)sin³x - (1/5)sin⁵x + C

Find the integral of sin²x cos³x dx.

Roots of the numerator of the closed loop transfer function

Zeroes are defined as:

5 μC

Find the charge in the volume defined by 0 ≤ x ≤ 1 m, 0 ≤ y ≤ 1 m, and 0 ≤ z ≤ 1 m if ρ = 30x²y (μC/m³). What change occurs for the limits - 1 ≤ y ≤ 0 m?

The probability of obtaining the observed sample data or more extreme results, assuming the null hypothesis is true.

The p-value in hypothesis testing represents:

y = C₁e²ˣ + C₂eˉˣ

Solve y'' - y' - 2y = 0.

4/3

Find the area enclosed by the curve y² = x² - x⁴.

1 + tan(t + C)

Solve dx/dt = x² - 2x + 2.

Ellipse

Describe the locus represented by |z + 2i| + |z - 2i| = 6.

< 50%

The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000 mm and 200 mm, respectively. What is the probability that the annual precipitation will be more than 1200 mm?

mean

Normal distribution is symmetric about _______.

17/6

Find the arc length of the curve 24xy = x⁴ + 48 from x = 2 to x = 4.

(y² - 2xy - 2x)/(x² - 2xy + 2y)

Find y', given x²y - xy² + x² + y² = 0.

F ratios

What do ANOVA calculate?

(1 + z¯¹)n

Find the Z-transform of {nCk}.

- 6i + 6k

Given points P(1, -1, 0), Q(2, 1, - 1), and R(-1, 1, 2). Find a vector perpendicular to the plane formed by P, Q, and R.

-4(1+ i)

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

- 6i + 12k

Given that A = - 8i - 2j - 4k and B = 2i + 2j + k. Find B x A.

2

Find the Wronskian of the set {eˣ, eˉˣ}

(1/4)(1 - cos2x)

Find Lˉ¹{1/s(s² + 4)} by convolutions.

0.304

A doctor's office sends half of its lab work to Lab X, one-fourth of its lab work to Lab Y, and the remainder to Lab Z. From Lab X, 1 report in 10 is late, from Lab Y, 1 in 8 is late, and from Lab Z, 1 in 12 is late. What is the probability that a late lab report came from Lab Y?

cos sqrt 6 x

Find the inverse Laplace transform of s/(s² + 6).

Feedback path

In signal flow path, what refers to the path from output node or a node near output node to a node near input node without repeating any of the nodes in between them?

0.34 min

A tank initially holds 100 gal of a brine solution containing 1 lb of salt. At t=0, another brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 3 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find the time at which the mixture in the tank contains 2 lbs of salt.

invertible

A system is called ________ if we can determine its input signal x uniquely by observing its output signa y.

1408π/5

Find the volume of the solid generated by revolving the region between the x-axis and the parabola y = 4x - x² about the line y = 6.

200 ft/s

A light is at the top of a pole 80 ft high. A ball is dropped at the same height from a point 20 ft from the light. Assuming that the ball is according to s = 16t², how fast is the shadow of the ball moving along the ground 1 second later.

- 12/(3 + 2x)²

Differentiate y = (3 - 2x)/(3 + 2x).

92/3

Find the area bounded by the parabola x = 8 + 2y - y², the y-axis, and the lines y = -1 and y = 3.

fundamental period

A continuous time signal is said to be periodic if x(t) = x(t + T₀). Here, T₀ is called ______.

π

What is the fundamental period of the following signal? (x)t = cos²t

R = (2/3)r

A right circular cylinder is inscribed in a right circular cone of radius r. Find the radius R of the cylinder if its volume is a maximum.

0.9990

Heather has a very important exam to take in the morning. Since she wants to be sure to that she will wake up in time, she sets two alarm clocks. One has a .95 probability that it will ring, and the other has a .98 probability that it will ring. She sets both clocks. What is the probability that at least one of the alarm clocks will wake her up?

12

A point moves in the plane according to the equations x = t² + 2t and y = 2t³ - 6t. Find dy/dx when t = 5.

19/33

Seven blue marbles and six red marbles are held in a single container. Marbles are randomly selected one at a time and not returned to the container. If the first two marbles selected are blue, what is the probability that at least two red marbles will be chosen in the next three selections?

11.55%

What constant interest rate is required if an initial deposit placed into an account that accrues interest compounded continuously is to double its value in six years?

y = C₁cos2x + C₂ sin2x

Solve y'' + 4y = 0.

0.72 N

Find the magnitude of the electrostatic force between a 100 mC charge at (-1, 1, -3) m and a 20 mC at (3, 1, 0) m.

x² + 2x + 1

Let f(x) = x² and g(x) = x + 1 Then (f ° g)(x) =