A differential equation links a quantity with its rate of change. Solving one recovers the quantity itself, as in population growth, cooling and decay.
✂️ Separating the variables
If dy/dx = f(x) · g(y), rearrange so all the y terms are on one side and all the x terms on the other, then integrate both sides. Remember the constant of integration, then use any given condition to find it.
dy/dx = 2xy, y = 1 when x = 0
∫ 1/y dy = ∫ 2x dx
ln y = x² + c
y = A e^(x²)
y = 1 at x = 0: A = 1, so y = e^(x²)
🖼️ See it
Separate the variables, integrate both sides, then use the initial condition to find the constant: the rate of growth is proportional to y.
📉 Growth and decay
If the rate of change is proportional to the amount, dN/dt = kN, then N = N₀ e^(kt). If k > 0 this is exponential growth; if k < 0 it is decay. Half-life T satisfies e^(kT) = ½, so T = ln 2 ÷ |k|.
dN/dt = −0.1 N, N₀ = 200
N = 200 e^(−0.1t)
after t = 0: N = 200
half-life = ln 2 ÷ 0.1 ≈ 6.93
Always check by differentiating your answer: it must reproduce the original equation.
🧪 Worked examples: from a sentence to a solution
Forming the equation. Translate each phrase into symbols before you integrate. “Rate of change” is a derivative, and “proportional to” introduces a constant k.
“The rate of increase of the population P is proportional to P”
dP/dt = kP
“The rate of cooling is proportional to the excess temperature over 20 °C”
dT/dt = −k(T − 20) (k > 0, so T falls towards 20)
“The volume V grows at a rate proportional to √V”
dV/dt = k√V
A full solution with a condition.
dy/dx = x ÷ y, y = 5 when x = 0
y dy = x dx
∫ y dy = ∫ x dx
y²/2 = x²/2 + c
y² = x² + A
at (0, 5): 25 = 0 + A, so A = 25
y² = x² + 25 (take y > 0 because y = 5 at the start)
when x = 12: y² = 169, y = 13
Doubling time. If a quantity doubles every T units of time, then N = N₀ × 2^(t/T), which is the same as N₀ e^(kt) with k = ln 2 ÷ T.
bacteria double every 3 hours, starting from 100
after 9 hours = 3 doublings: 100 × 2³ = 800
Check. Differentiate your answer. For y² = x² + 25, 2y dy/dx = 2x, so dy/dx = x ÷ y, which is the original equation.
🎯 Exam focus: setting out and modelling
A typical question has four steps: form the equation, separate and integrate, use the condition to find the constant, and interpret the answer.
general solution contains an arbitrary constant
particular solution the constant has been found from a condition
ln|y| = kt + c → y = A e^(kt) where A = ± e^c
Common mistakes. Forgetting the constant of integration before using the condition. Integrating the x side and y side with the wrong variable, for example writing ∫ y dx. Dropping the modulus: ∫ 1/y dy = ln|y|, and it is only legitimate to drop the bars after you know the sign of y. Writing e^(c) + e^(kt) instead of A e^(kt).
Limits of a model. Exponential growth predicts unlimited growth, which is unrealistic for real populations that run out of food or space. In an exam, say that the model is only reasonable while resources are not limiting.
✏️ Quick check · 1 of 12
First step
To solve dy/dx = 3y by separation, you first write:
Divide by y.
✏️ Quick check · 2 of 12
Constant
After integrating both sides of a separated equation, you must include:
+ c.
✏️ Quick check · 3 of 12
Sign of k
In dN/dt = kN, a decay corresponds to:
Decay means decreasing.
✍️ Type your answer · 4 of 12
dN/dt = 0.2N and N = 50 when t = 0.
What is the rate of growth dN/dt at t = 0?
Enter an exact answer, without rounding.
Use the equation with N = 50.
✍️ Type your answer · 5 of 12
y = e^(x²) solves dy/dx = 2xy with y = 1 at x = 0.
What is ln y when x = 3?
Enter an exact answer, without rounding.
Take ln of both sides.
✍️ Type your answer · 6 of 12
dy/dx = 4x with y = 3 when x = 0.
What is y when x = 2?
Enter an exact answer, without rounding.
Integrate then use the condition.
✍️ Type your answer · 7 of 12
A quantity decays as N = 200 e^(−0.1t), so dN/dt = −0.1N.
What is dN/dt at t = 0?
Enter an exact answer, without rounding.
N = 200 at t = 0.
✍️ Type your answer · 8 of 12
dy/dx = 6x² and y = 5 when x = 1.
What is y when x = 0?
Enter an exact answer, without rounding.
Find c first.
✍️ Type your answer · 9 of 12
dy/dx = x ÷ y with y = 5 when x = 0. The solution is y² = x² + 25, and y is positive.
What is y when x = 12?
Enter an exact answer, without rounding.
Substitute x = 12 and take the positive square root.
✍️ Type your answer · 10 of 12
A culture of bacteria doubles every 3 hours. At the start there are 100.
How many bacteria are there after 9 hours?
Enter an exact answer, without rounding.
Count how many doublings fit in 9 hours.
✍️ Type your answer · 11 of 12
A radioactive sample follows dN/dt = −kN. It has a half-life of 5 days and starts at 80 g.
What mass is left after 15 days, in grams?
Enter an exact answer, without rounding.
g
How many half-lives is that?
✏️ Quick check · 12 of 12
Forming an equation
The rate at which a hot drink cools is proportional to the difference between its temperature T and the room temperature of 20 °C. Which equation models this?
Which expression is the temperature difference?
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The solution of dy/dx = ky is:
Question 2 of 5
To separate variables you:
Question 3 of 5
∫ 1/y dy =
Question 4 of 5
A general solution contains:
Question 5 of 5
The half-life of N = N₀ e^(−kt) is:
Lesson 2 · Pure Maths
🧭 Vectors in Three Dimensions
Vectors extend neatly into 3D, where they describe lines, planes and angles in space.
📦 3D vectors
A 3D vector is written (x, y, z) or xi + yj + zk. Add and scale component by component. The magnitude is √(x² + y² + z²). The position vector of a point is the vector from the origin, and AB = OB − OA.
Work component by component. A negative dot product means the angle between the vectors is obtuse.
📐 Scalar product and angle
The scalar (dot) product is a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. If a·b = 0 the vectors are perpendicular. The angle between vectors: cos θ = a·b ÷ (|a||b|). A line through a with direction d is r = a + λd.
Do two lines meet? Write each line in components, equate the x, y and z parts, solve two equations for the two parameters, and then check the third.
l₁: r = λ(1, 2, 3) l₂: r = (1, 1, 1) + μ(1, 3, 5)
x: λ = 1 + μ
y: 2λ = 1 + 3μ → 2 + 2μ = 1 + 3μ → μ = 1, λ = 2
z: 3λ = 1 + 5μ → 6 = 6 ✓ the third equation holds
the lines meet at λ = 2: the point (2, 4, 6)
Change l₂ to start at (1, 1, 2) and the z equation becomes 3λ = 2 + 5μ. With λ = 2 and μ = 1, 6 ≠ 7, so the lines do not meet. They are not parallel (the directions are different), so they are skew.
Perpendicular test. Find an unknown with a·b = 0. For example (2, p, 1)·(1, 3, −5) = 2 + 3p − 5 = 0, so p = 1.
🎯 Exam focus: three kinds of line pair
parallel direction vectors are multiples of each other
intersect one value of λ and μ satisfies ALL THREE component equations
skew not parallel and no common point (only possible in 3D)
To find the angle between two lines use their direction vectors, not their position vectors. If cos θ comes out negative, the angle between the lines is the acute angle 180° − θ.
Common mistakes. Using AB = OA − OB (it is OB − OA: end minus start). Stopping after two equations and not checking the third. Calling lines “parallel” when the direction vectors are not multiples of each other. Using the position vectors in the angle formula. Writing a line as a column of numbers without the “r =”.
Show that. When a question says “show that the lines intersect”, you must show all three equations are satisfied and quote the point of intersection.
✏️ Quick check · 1 of 12
Perpendicular
Two non-zero vectors are perpendicular when their scalar product is:
cos 90°.
✏️ Quick check · 2 of 12
Line
The vector equation of a line through a with direction d is:
Start plus a multiple.
✏️ Quick check · 3 of 12
Journey
The vector from A to B is:
End minus start.
✍️ Type your answer · 4 of 12
The vector (1, 2, 2).
What is its magnitude?
Enter an exact answer, without rounding.
Use the 3D Pythagoras.
✍️ Type your answer · 5 of 12
A is (1, 0, 2) and B is (4, 4, 2).
What is the length of AB?
Enter an exact answer, without rounding.
Subtract, then find the magnitude.
✍️ Type your answer · 6 of 12
a = (1, 2, 3) and b = (2, 1, −2).
What is a·b?
Enter an exact answer, without rounding.
Multiply matching components and add.
✍️ Type your answer · 7 of 12
a = (2, k, 1) and b = (1, 3, −2) are perpendicular.
What is k?
Enter an exact answer, without rounding.
Set a·b = 0.
✍️ Type your answer · 8 of 12
The vector (2, 3, 6).
What is its magnitude?
Enter an exact answer, without rounding.
Use the 3D Pythagoras.
✍️ Type your answer · 9 of 12
The vectors a = (1, 0, 1) and b = (0, 1, 1) have a·b = 1 and |a| = |b| = √2.
What is the angle between them, in degrees?
Enter an exact answer, without rounding.
°
cos θ = a·b ÷ (|a||b|).
✍️ Type your answer · 10 of 12
The vectors (2, p, 1) and (1, 3, −5) are perpendicular.
What is p?
Enter an exact answer, without rounding.
Set the scalar product to zero.
✍️ Type your answer · 11 of 12
l₁: r = λ(1, 2, 3) and l₂: r = (1, 1, 1) + μ(1, 3, 5) intersect.
What is the value of λ at the point of intersection?
Enter an exact answer, without rounding.
Solve the x and y equations together.
✏️ Quick check · 12 of 12
Skew lines
Two lines in three dimensions are skew when:
Skew needs neither meeting nor parallel.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The magnitude of (x, y, z) is:
Question 2 of 5
The scalar product of two vectors gives:
Question 3 of 5
The unit vector in the direction of (0, 0, 5) is:
Question 4 of 5
(3, 0, 0)·(0, 4, 0) =
Question 5 of 5
Two lines are parallel if their direction vectors are:
Lesson 3 · Statistics
🌧️ The Poisson Distribution
The Poisson distribution models how many times a rare event occurs at random in a fixed interval: calls per hour, flaws per metre, emails per day.
🎲 What it models
X ~ Po(λ) counts random events that occur independently, at a constant average rate λ per interval, and one at a time. Both the mean and the variance equal λ.
P(X = r) = e^(−λ) λ^r ÷ r!. For a Poisson distribution the mean and the variance are both equal to λ.
➕ Adding and approximating
If X ~ Po(λ) and Y ~ Po(μ) are independent, then X + Y ~ Po(λ + μ). Rates scale with the interval: 3 calls per hour is 6 per two hours. For λ > 10 the Poisson is approximated by a normal distribution N(λ, λ).
Poisson as an approximation to the binomial. When n is large and p is small, B(n, p) ≈ Po(np).
200 items, each faulty with probability 0.01
X ~ B(200, 0.01) ≈ Po(200 × 0.01) = Po(2)
P(no faulty items) ≈ e⁻² = 0.135
Normal approximation. For a large mean (λ greater than about 15), X ~ Po(λ) is approximately N(λ, λ). Use a continuity correction, because Poisson is discrete and the normal is continuous.
Scaling the rate. If calls arrive at 3 per hour, then in 20 minutes the mean is 1, so use Po(1).
🎯 Exam focus: conditions and wording
To say a situation can be modelled by a Poisson distribution, state all three conditions and apply each to the context:
1 events occur singly (not in groups)
2 events occur independently of one another
3 events occur at a constant average rate in the interval
(plus: the interval is fixed, such as an hour or a metre)
Checking suitability from data. For a Poisson distribution the mean and variance are equal. If a sample has a variance much larger than its mean, events are clustering and Poisson is not a good model.
Common mistakes. Using the rate for a different interval from the one in the question (always rescale first). Forgetting the e^(−λ) factor. Writing P(X ≥ 3) as 1 − P(X ≤ 3): it should be 1 − P(X ≤ 2). Omitting the continuity correction in a normal approximation. Using Poisson when there is a fixed number of trials with two outcomes: that is binomial.
Calculator tip. Check that your probabilities for all values add up to 1; this catches slips in e^(−λ) and factorials quickly.
✏️ Quick check · 1 of 12
When to use it
Which situation is best modelled by a Poisson distribution?
Events in an interval.
✏️ Quick check · 2 of 12
Mean and variance
For a Poisson distribution, the variance equals:
Same as the mean.
✏️ Quick check · 3 of 12
Adding
If X ~ Po(2) and Y ~ Po(3) are independent, then X + Y follows:
Add the rates.
✍️ Type your answer · 4 of 12
Calls arrive at a rate of 3 per hour.
What is the mean number of calls in 2 hours?
Enter an exact answer, without rounding.
Scale the rate.
✍️ Type your answer · 5 of 12
X ~ Po(4).
What is the variance of X?
Enter an exact answer, without rounding.
Variance equals the mean.
✍️ Type your answer · 6 of 12
X ~ Po(2) and Y ~ Po(5) are independent.
What is the mean of X + Y?
Enter an exact answer, without rounding.
Add the means.
✍️ Type your answer · 7 of 12
X ~ Po(1) (use e⁻¹ = 0.368).
What is P(X = 0) to 3 decimal places?
Round to 3 decimal places.
P(0) = e^(−λ).
✍️ Type your answer · 8 of 12
X ~ Po(3) (use e⁻³ = 0.0498).
What is P(X = 1) to 3 decimal places?
Round to 3 decimal places.
P(1) = e^(−λ) × λ.
✍️ Type your answer · 9 of 12
X ~ Po(2). Use e⁻² = 0.1353.
What is P(X = 3), to 3 decimal places?
Round your answer as the question asks.
Use e^(−λ) λ^r ÷ r!.
✍️ Type your answer · 10 of 12
A batch of 200 items is checked. Each is faulty with probability 0.01, and X ~ B(200, 0.01) is approximated by a Poisson distribution.
What is the mean of the approximating Poisson distribution?
Enter an exact answer, without rounding.
Mean of a binomial is np.
✍️ Type your answer · 11 of 12
X ~ Po(36) is approximated by a normal distribution.
What is the standard deviation of the normal distribution?
Enter an exact answer, without rounding.
Mean and variance are both 36.
✏️ Quick check · 12 of 12
Poisson conditions
Which of these is NOT a requirement for a Poisson model?
The word is random.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The Poisson probability formula is:
Question 2 of 5
Which is NOT an assumption of the Poisson model?
Question 3 of 5
If X ~ Po(λ), then E(X) =
Question 4 of 5
P(X ≥ 1) equals:
Question 5 of 5
For large λ the Poisson can be approximated by:
Lesson 4 · Pure Mathematics
🧮 Polynomials, Modulus & Circles
Can you find a factor without dividing? And can a single equation describe a whole circle? Both come down to one neat substitution.
✂️ Factor and remainder theorems
When p(x) is divided by (x − a), the remainder is p(a). If p(a) = 0, then (x − a) is a factor. Use this to factorise cubics.
A circle with centre (a, b) and radius r has the equation (x − a)² + (y − b)² = r². Complete the square to find the centre. The modulus |x| is the distance from zero, so it is never negative.
x² + y² − 4x + 6y − 3 = 0
(x − 2)² + (y + 3)² = 16
centre (2, −3), radius 4
|x − 3| = 5 means x = 8 or x = −2
✏️ Quick check · 1 of 6
Theorem
If p(3) = 0, then:
Zero remainder.
✏️ Quick check · 2 of 6
Centre
The circle (x + 1)² + (y − 4)² = 9 has centre:
Watch the signs.
✍️ Type your answer · 3 of 6
p(x) = x³ + 2x − 5
What is the remainder when p(x) is divided by (x − 2)?
Enter an exact answer, without rounding.
Find p(2).
✍️ Type your answer · 4 of 6
(x − 1) is a factor of x³ + kx − 4.
What is k?
Enter an exact answer, without rounding.
Set p(1) = 0.
✍️ Type your answer · 5 of 6
A circle has equation (x − 2)² + (y + 3)² = 49.
What is its radius?
Enter an exact answer, without rounding.
Square root of the right side.
✍️ Type your answer · 6 of 6
|x − 3| = 5 has two solutions.
What is the larger solution?
Enter an exact answer, without rounding.
Try + and −.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
Division of p(x) by (x + 2) gives a remainder of:
Question 2 of 5
|−7| equals:
Question 3 of 5
A tangent to a circle meets the radius at:
Question 4 of 5
x² + y² = 25 has its centre at:
Question 5 of 5
y = |x| has a graph shaped like a:
Lesson 5 · Pure Mathematics
➰ Implicit & Parametric Differentiation
Not every curve can be written as y = something. These two methods let you find gradients for curves such as circles and paths in motion.
🔗 Implicit differentiation
Differentiate each term with respect to x. Every y term picks up a factor of dy/dx by the chain rule.
Evaluate ∫ from 0 to 1 of x eˣ dx, using [x eˣ − eˣ].
What is the value?
Enter an exact answer, without rounding.
Upper minus lower.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The integration by parts formula is:
Question 2 of 5
∫ 1/x dx equals:
Question 3 of 5
∫ ln x dx is found by taking:
Question 4 of 5
To integrate 1/((x − 1)(x + 1)), you should first:
Question 5 of 5
∫ 2/(x + 1) dx equals:
Lesson 7 · Mechanics
⚖️ Mechanics: Forces & Equilibrium
Why does a heavy box stay still on a slope, until it suddenly slides? Mechanics models the world with arrows and a few reliable laws.
📐 Resolving forces
Split a force F at angle θ into F cos θ along and F sin θ across a chosen direction. An object is in equilibrium when the resultant force is zero: the components balance in each direction. Take g = 10 m/s².
A force of 20 N at 30° to the horizontal
horizontal = 20 cos 30° ≈ 17.3 N
vertical = 20 sin 30° = 10 N
🛷 Friction
Friction satisfies F ≤ μR. At limiting equilibrium, F = μR, where R is the normal contact force and μ is the coefficient of friction.
A 5 kg box on a rough horizontal floor, μ = 0.4
R = 5 × 10 = 50 N
limiting friction = 0.4 × 50 = 20 N
A 12 N push does not move it: friction = 12 N
A 30 N push moves it: net force = 10 N
✏️ Quick check · 1 of 6
Component
The component of F along a line at θ to F is:
Adjacent to θ.
✏️ Quick check · 2 of 6
Equilibrium
A particle is in equilibrium when:
No overall force.
✍️ Type your answer · 3 of 6
A 10 N force acts at 60° to the horizontal, with cos 60° = 0.5.
What is its horizontal component, in N?
Enter an exact answer, without rounding.
N
F cos θ.
✍️ Type your answer · 4 of 6
A 5 kg box rests on a rough horizontal floor with μ = 0.4 (g = 10 m/s²).
What is the limiting friction, in N?
Enter an exact answer, without rounding.
N
μR.
✍️ Type your answer · 5 of 6
A box with limiting friction 20 N is pushed with 12 N and stays still.
What is the friction force, in N?
Enter an exact answer, without rounding.
N
It balances the push.
✍️ Type your answer · 6 of 6
A box with limiting friction 20 N is pushed with 30 N.
What is the net force on it, in N?
Enter an exact answer, without rounding.
N
Push minus limiting friction.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Weight of a particle of mass m is:
Question 2 of 5
On a smooth surface, friction is:
Question 3 of 5
For a particle on a slope at angle θ, the weight component down the slope is:
Question 4 of 5
Normal contact force on a slope at angle θ is:
Question 5 of 5
If F < μR, the object is:
Lesson 8 · Mechanics
🛗 Mechanics: Newton’s Laws & Connected Particles
A lift, a train with carriages, two masses over a pulley: all are solved by writing F = ma for each part.
🚂 F = ma in action
Newton’s second law: resultant force = mass × acceleration. A lift changes the normal reaction. Work done, kinetic and potential energy link to motion: W = Fd, and P = Fv.
A 60 kg person in a lift accelerating up at 2 m/s²
R − 600 = 60 × 2
R = 720 N
A car’s engine: P = Fv
20 kW at 25 m/s gives F = 800 N
🪢 Connected particles
For particles joined by a light string over a smooth pulley, both have the same acceleration and the tension is the same along the string. Write F = ma for each and solve.
Masses 3 kg and 2 kg over a pulley
3g − T = 3a
T − 2g = 2a
adding: g = 5a, so a = 2 m/s²
T = 2 × 10 + 2 × 2 = 24 N
✏️ Quick check · 1 of 6
Law
Newton’s second law states that:
F = ma.
✏️ Quick check · 2 of 6
String
For a light string over a smooth pulley, the tension is:
Light and smooth.
✍️ Type your answer · 3 of 6
A 4 kg block is pulled by a 20 N force on a smooth floor.
What is its acceleration, in m/s²?
Enter an exact answer, without rounding.
m/s²
a = F ÷ m.
✍️ Type your answer · 4 of 6
A 60 kg person is in a lift accelerating upward at 2 m/s² (g = 10 m/s²).
What is the reaction from the floor, in N?
Enter an exact answer, without rounding.
N
R − mg = ma.
✍️ Type your answer · 5 of 6
Masses of 3 kg and 2 kg hang over a smooth pulley (g = 10 m/s²).
What is the acceleration, in m/s²?
Enter an exact answer, without rounding.
m/s²
Net force ÷ total mass.
✍️ Type your answer · 6 of 6
An engine works at 20 000 W at a speed of 25 m/s.
What is its driving force, in N?
Enter an exact answer, without rounding.
N
P = Fv.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
In a lift moving down at a steady speed, the reaction on a person is:
Question 2 of 5
In a lift accelerating downward, the reaction is:
Question 3 of 5
The work done by a force F moving through d in its direction is:
Question 4 of 5
In the 3 kg and 2 kg pulley system, the tension is:
Question 5 of 5
Newton’s third law says that forces come in:
Lesson 9 · Statistics
📈 Continuous Random Variables
You can’t list every possible height or waiting time. So, in place of a table of probabilities, we use a curve and find areas.
📊 Probability density
A continuous random variable has a probability density function (PDF) f(x) with f(x) ≥ 0 and total area 1. Probabilities are areas: P(a < X < b) = ∫ f(x) dx from a to b. P(X = a) is always 0.
f(x) = kx for 0 ≤ x ≤ 2
∫ kx dx from 0 to 2 = 2k = 1
k = 1/2
P(X < 1) = ∫ x/2 dx from 0 to 1 = 1/4
🧮 Mean and combinations
E(X) = ∫ x f(x) dx, and Var(X) = ∫ x² f(x) dx − [E(X)]². The median m satisfies P(X < m) = 1/2. For independent variables, E(aX + bY) = aE(X) + bE(Y) and Var(aX + bY) = a²Var(X) + b²Var(Y).
For f(x) = x/2 on 0 to 2:
E(X) = ∫ x²/2 dx = 4/3
median: m²/4 = 1/2, so m = √2
Var(X) = 2, Var(Y) = 3, independent
Var(2X + Y) = 4 × 2 + 3 = 11
✏️ Quick check · 1 of 6
Area
The total area under a PDF is:
Certain outcome.
✏️ Quick check · 2 of 6
Exact value
For a continuous X, P(X = 2) is:
A point has no width.
✍️ Type your answer · 3 of 6
f(x) = kx on 0 ≤ x ≤ 2.
What is k, as a decimal?
Enter an exact answer, without rounding.
Set the area equal to 1.
✍️ Type your answer · 4 of 6
f(x) = x/2 on 0 ≤ x ≤ 2.
What is P(X < 1), as a decimal?
Enter an exact answer, without rounding.
Integrate from 0 to 1.
✍️ Type your answer · 5 of 6
E(X) = 3 and E(Y) = 4.
What is E(2X + Y)?
Enter an exact answer, without rounding.
Linear combination.
✍️ Type your answer · 6 of 6
Var(X) = 2 and Var(Y) = 3, with X and Y independent.
What is Var(2X + Y)?
Enter an exact answer, without rounding.
Square the coefficients.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The formula for E(X) is:
Question 2 of 5
The median m satisfies:
Question 3 of 5
Var(X − Y) for independent X and Y is:
Question 4 of 5
A PDF can never be:
Question 5 of 5
Var(3X) equals:
Final Challenge
🏆 21-Question Challenge Quiz
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
0 / 21
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · Differential Equations
The solution of dy/dx = 2 with y = 1 when x = 0 is:
Q2 · Differential Equations
The general solution of dy/dx = 2x is:
Q3 · Differential Equations
dy/dx = ky with k > 0 models:
Q4 · Vectors in Three Dimensions
The angle between a and b satisfies:
Q5 · Vectors in Three Dimensions
The magnitude of the vector (2, 3, 6) is:
Q6 · Vectors in Three Dimensions
Two non-zero vectors are perpendicular if their scalar (dot) product is:
Q7 · The Poisson Distribution
Emails arrive at 5 per hour. The mean number in 30 minutes is: