Number and algebra
John invests $8000 in an account that pays compound interest at an annual rate of 5%, compounded annually. How much will his investment be worth after 3 years?
9261
The population of a town is modeled by the exponential function P(t)=5000×(1.04)t, where t is the number of years since 2020. According to this model, what will be the population in 2030? (Round to the nearest whole number).
Approximately 7401
A circle has a radius of 5 cm. Find the length of an arc that subtends an angle of 120∘ at the center.
( 120/ 360 )× 2π(5)=(1/3) ×10π= 310π
≈10.5 cm
Explain the difference between discrete and continuous data, providing an example of each.
Discrete data can only take specific numerical values (e.g., number of students in a class), while continuous data can take any value within a given range (e.g., height of a student).
What is the gradient of the tangent to the curve y=x2 at the point where x=3
y′=2x, so at x=3, the gradient is 2(3)=6.
Write the number 0.00045 in the form a×10k, where 1≤a<10 and k is an integer.
4.5×10−4
The height of a projectile is modeled by the quadratic function h(t)=−5t2+20t+10, where h is in meters and t is in seconds. Find the time at which the projectile reaches its maximum height.
t=2 seconds
From a point on the ground 20 meters away from the base of a building, the angle of elevation to the top of the building is 75∘. Calculate the height of the building.
Approximately 74.6 meters
(using tan75∘= height/ 20 )
The following data set represents the scores of 7 students on a test: 6, 7, 7, 8, 9, 10, 10. Find the median and the mode of this data set.
Median = 8, Mode = 7 and 10
Find the equation of the tangent to the curve y=x2−4x+3 at the point where x=1.
When x=1, y=12−4(1)+3=0. The point is (1,0). The derivative is y′=2x−4. At x=1, the gradient is 2(1)−4=−2. The equation of the tangent is y−0=−2(x−1), which simplifies to y=−2x+2.
Solve the following system of linear equations using algebraic methods:
2x+y−z=5
x−3y+2z=−1
3x+2y+z=8
x=2,y=1,z=0
A sinusoidal function is given by f(x)=3sin(2x)+1. State the amplitude and the equation of the principal axis of this function.
Amplitude = 3, Principal axis: y=1
A right cone has a base radius of 3 cm and a slant height of 5 cm. Calculate its total surface area
Curved surface area = πrl=π(3)(5)=15π cm2.
Base area = πr2=π(3)2=9π cm2.
Total surface area = 15π+9π=24π
≈75.40 cm2
The heights of a group of students are normally distributed with a mean of 168 cm and a standard deviation of 6 cm. Approximately what percentage of students have heights between 162 cm and 174 cm?
Approximately 68%
A curve has a gradient function given by dxdy=3x2−4x. If the curve passes through the point (2,1), find the equation of the curve.
y=x3−2x2+C. Using the point (2,1):
1=(2)3−2(2)2+C ⟹1=8−8+C ⟹C=1.
The equation of the curve is y=x3−2x2+1.
The population of a town is modeled by the function P(t)=15000×(1.02)t, where t is the number of years since 2020. In what year will the population reach 20,000?
Approximately the year 2034
Determine the coordinates of the point(s) of intersection between the line y=x+2 and the parabola y=x2
The coordinates of the points of intersection are (−1,1) and (2,4).
Find the equation of the perpendicular bisector of the line segment joining the points P(2,3) and Q(6,1)
The midpoint of PQ is (4,2). The gradient of PQ is =-0.5. The gradient of the perpendicular bisector is 2. The equation is y−2=2(x−4), which simplifies to y=2x−6.
A bag contains 4 blue tokens and 6 yellow tokens. Two tokens are drawn at random without replacement. What is the probability that the first token is blue and the second token is yellow?
4 / 10 × 6 / 9 = 24/ 90
= 4 / 15 .
0.267
The rate of flow of water into a tank is given by R(t)=20e−0.1t liters per minute, where t is the time in minutes since the start. Find the total amount of water that flows into the tank during the first 10 minutes.
The total amount is given by ∫ 0 to 10 of 20e−0.1t dt
=(−200e−0.1(10)) - (200e−0.1(0))
=−200e−1+200
≈−200(0.3679)+200
≈−73.58+200
≈126.42 liters.
An annuity pays out $500 at the end of each year for 10 years. If the interest rate is 6% per annum, compounded annually, what is the present value of this annuity?
Approximately $3680.04
The cost of producing x items is given by C(x)=0.1x2+5x+100. Find the average rate of change of the cost as the number of items produced increases from x=10 to x=20
$8 per item
A surveyor needs to find the distance across a lake. They measure the distance from point A to point C as 80 meters and the distance from point B to point C as 120 meters. The angle ACB is 60∘. Find the distance across the lake (distance AB)
AB2=802+1202−2(80)(120)cos60∘
=6400+14400−19200(0.5)
=20800−9600
=11200.
AB= sqrt(11200)≈105.8 meters.
A random variable X follows a binomial distribution with n=10 trials and a probability of success p=0.4. What is the expected value (mean) of X?
The expected value of a binomial distribution is E(X)=np=10×0.4=4.
Find the area enclosed by the curve y=6x−x2 and the x-axis.
The curve intersects the x-axis when y=0, so 6x−x2=0⟹x (6−x)=0, giving x=0 and x=6.
The area is ∫ 0 to 6 (6x−x2)dx=108−72=36.