Math II, B & C Formulas

Substituting into the distance formula |ax₀+by₀+c|/√(a²+b²) gives |3+8−1|/5=10/5=2. If you forget the absolute value in the numerator, or forget to divide by √(a²+b²), you end up with 10. Learn the two steps as a set: substitute and take the absolute value, then divide by the root.

Completing the square gives (x−2)²+(y+3)²=9. The trap is the sign: (x−2)² means the x-coordinate of the centre is +2, the opposite of the sign you see in the bracket. Note also that the radius is not 9 but √9=3.

The point dividing in the ratio m:n is ((n·x₁+m·x₂)/(m+n), (n·y₁+m·y₂)/(m+n)). For 2:1 this gives ((1·1+2·7)/3, (1·2+2·5)/3)=(5, 4). Reversing the ratio gives (3, 3), and dividing externally gives (13, 8). Remember it as 'multiply by the part of the ratio on the opposite side' (cross-multiplying).

The distance from the centre (0, 0) to the line 2x−y+5=0 is 5/√5=√5, exactly equal to the radius √5, so the line is tangent to the circle. If d=r they touch, if d<r they meet at two points, and if d>r there is no common point. Comparing the distance d from the centre with the radius r is quicker than using the discriminant.

From AP:BP=1:2 we get 4AP²=BP², and rearranging gives (x+2)²+y²=16. When the ratio of the distances from two fixed points is not 1:1, the locus is a 'circle of Apollonius'. The trap is that only the 1:1 case gives the perpendicular bisector. Check as well that the circle lies on the side of A, the point with the smaller part of the ratio.

The condition for parallel lines is a₁b₂−a₂b₁=0, so 1·3−2·a=0 and a=3/2. The answer a=−6 comes from using the perpendicularity condition a₁a₂+b₁b₂=0, a classic trap. Keep them apart: parallel means the cross-multiplied form equals 0, perpendicular means the dot-product form equals 0.

The sine addition formula runs in the order 'sine-cosine, cosine-sine', and for (α+β) the sign stays as +. The − sign belongs to sin(α−β). Say the rhythm 'sin cos, cos sin' out loud to fix the order.

The cosine addition formula runs as 'cos cos, sin sin', and the point to watch most is that the sign flips to − for (α+β). It becomes + for cos(α−β). Remember: cosine reverses the sign inside the bracket, sine keeps it.

Substituting cos²θ=1−sin²θ into cos2θ=cos²θ−sin²θ gives 1−2sin²θ. It is easy to confuse with the sign-reversed 2sin²θ−1 or with the cosine version 2cos²θ−1. Sort it out like this: written with sin the 1 comes first, written with cos the −1 comes last.

Split it as 75°=45°+30° and use the addition formula: sin45°cos30°+cos45°sin30°=(√6+√2)/4. The value (√6−√2)/4 is sin15°, the typical sign slip. You can check with 'the larger the angle here, the larger the value, so 75° takes the + version'.

The amplitude is √(1²+(√3)²)=2, and cosα=1/2, sinα=√3/2 give α=π/3. Reading the coefficients the other way round and getting π/6 is a very common mistake. Confirm with the argument of the point (1, √3): the coefficient of sin matches cosα, and the coefficient of cos matches sinα.

Squaring both sides gives sin²θ+cos²θ+2sinθcosθ=1/4, so 1+2sinθcosθ=1/4 and sinθcosθ=−3/8. Forgetting sin²θ+cos²θ=1 and writing 2sinθcosθ=1/4 leads to 1/8. The golden rule: when a sum is given, square it first and a 1 will appear.

Since 8=2³, we have 8^(2/3)=(2³)^(2/3)=2²=4, using the index law (aᵐ)ⁿ=aᵐⁿ. You can also work in two steps: the power 2/3 means 'take the cube root, then square', so ³√8=2 and 2²=4. Forgetting to multiply the exponents produces answers such as 16.

A sum of logarithms is the logarithm of the product: log₁₀2+log₁₀5=log₁₀(2×5)=log₁₀10=1. Adding the arguments to get log₁₀7 is the classic error. The basic law of logarithms is 'addition turns into multiplication'.

Since 1/27=3⁻³, log₃3⁻³=−3. The knack is to reword it as 'to what power must 3 be raised to give 1/27?'. Do not drop the minus sign and answer 3, and do not confuse it with −1/3; check with 'if the argument is smaller than 1, the logarithm is negative'.

By the change of base formula, log₄8=log₂8/log₂4=3/2. Swapping numerator and denominator to get 2/3 is the trap. Remember the layout: the new base is yours to choose, the argument goes on top and the original base goes underneath. Check with 4^(3/2)=8.

Writing everything with base 3 gives 3²ˣ=3³·3^(1/2)=3^(7/2), so 2x=7/2 and x=7/4. If you forget the conversion √3=3^(1/2) and compare with 27 alone, you get 3/2. The first move for any exponential equation is to make the bases the same.

log₁₀2⁵⁰=50×0.3010=15.05, so 10¹⁵ < 2⁵⁰ < 10¹⁶ and the number has 16 digits. Answering 15, the integer part of the logarithm, is the most frequent mistake. Remember: number of digits = integer part of the logarithm + 1, so always add the 1.

Applying (xⁿ)′=nxⁿ⁻¹ to each term gives f′(x)=3x²−6x. Forgetting to multiply by the coefficient 3, or differentiating −3x² by bringing down only the coefficient to get −3x, are the standard slips. Apply 'bring the exponent down to the front, then reduce the exponent by 1' carefully to every term.

An antiderivative is x³+x², so [x³+x²]₀¹=1+1=2. Substituting x=1 into the integrand without integrating gives 5. Never break the order: integrate first, then substitute the endpoints.

The slope of the tangent is the derivative value f′(3). From y′=2x, substituting x=3 gives 6. Be careful not to answer the y-coordinate 9 or the x-coordinate 3 as they stand. The catchphrase: if you want a slope, differentiate first.

From f′(x)=3x²−3=0 we get x=±1. A table of signs shows that f′ changes from + to − around x=−1, so there is a local maximum there, with f(−1)=−1+3=2. At x=1 there is a local minimum (value −2). Remember the shape: if the coefficient of x³ is positive, the hill comes first, so the local maximum is on the left.

The curve meets the x-axis at x=0 and x=3. By the 1/6 formula, |1|·(3−0)³/6=27/6=9/2. Since y≤0 on this interval, forgetting to attach the minus sign when integrating causes a sign error. The enclosed area is (β−α)³/6 in one stroke.

From x²=2x+3 we get x²−2x−3=0, so the intersections are x=−1 and x=3. By the 1/6 formula, (3−(−1))³/6=64/6=32/3. Forgetting to simplify and hunting for 64/6 among the options, or squaring (β−α) instead of cubing it and landing on 16/3, are the usual errors. Cube the difference and divide by 6, precisely.

'(first term + last term) × number of terms ÷ 2' is the formula for the sum of an arithmetic sequence. Picture it as the area of a trapezium ((top + bottom) × height ÷ 2) and you will not forget it. Forgetting the ÷2, or forgetting to multiply by the number of terms n, are the typical mistakes.

The sum of the natural numbers from 1 to n is n(n+1)/2. The expression n(n−1)/2 is the sum from 1 to n−1, an error caused by shifting the range. Check with n=2: 1+2=3=2·3/2. Remember it as 'half the product of two neighbouring numbers'.

The formula for the sum of squares is n(n+1)(2n+1)/6. It is easy to confuse with n²(n+1)²/4, which is the formula for the sum of cubes (Σk³). Checking with n=2 makes it certain: 1+4=5=2·3·5/6. Keywords: divide by 6, and 2n+1 in the middle.

Substituting into a(rⁿ−1)/(r−1) gives 3(2⁵−1)/(2−1)=3×31=93. You can also confirm by adding directly: 3+6+12+24+48=93. Watch out for forgetting to multiply by the first term 3, which gives 31, and for mistaking the fifth term 48 for the answer.

The difference is 2n, so for n≥2 we get aₙ=1+Σ(k=1→n−1)2k=1+(n−1)n=n²−n+1 (which also holds for n=1). Taking the upper limit of the sum to be n instead gives n²+n+1, and that is the biggest trap. Chant it: 'the sum of the differences runs only up to n−1'. Check with a₂=3.

From the characteristic equation α=3α+2 we get α=−1. The recurrence can be rewritten as aₙ₊₁+1=3(aₙ+1), so {aₙ+1} is a geometric sequence with first term 2 and common ratio 3, giving aₙ+1=2·3ⁿ⁻¹ and therefore aₙ=2·3ⁿ⁻¹−1. The easy slip is the sign in the last step: you must undo the shift you added, not add it again. Check with a₂=5 and you can relax.

The dot product is defined as 'product of the magnitudes × cosθ'. The version with sinθ is used for things such as the area of a parallelogram and is something quite different. Link it to a picture: the dot product uses cos, the idea of a shadow (orthogonal projection). This also fits the fact that the dot product is 0 when θ=90°.

In components the dot product is the sum of the product of the x-parts and the product of the y-parts: 1×3+2×(−1)=3−2=1. Dropping the sign of the y-component (−1) and writing 3+2=5 is the typical mistake. Note as well that the answer to a dot product is a single number, not a vector.

If they are perpendicular then θ=90° and cosθ=0, so the dot product is 0. The condition a₁b₂−a₂b₁=0 is the condition for the vectors to be parallel, and it is the one most easily confused with perpendicularity. Memorise them as a pair: perpendicular means dot product zero, parallel means cross-multiplication zero.

From the parallel condition a₁b₂−a₂b₁=0 we get 2×6−3x=0, so x=4. Solving with dot product = 0, the perpendicularity condition, gives 2x+18=0 and hence x=−9. Thinking in terms of the ratio of components, 2:3=x:6, also gives x=4. Confirm with 'parallel means equal ratios'.

Dividing internally in the ratio m:n gives (n·a→+m·b→)/(m+n). For 2:1 this is (1·a→+2·b→)/3. The most frequent error is attaching each number of the ratio to the nearer point, so remember 'each part of the ratio multiplies the point on the far side'. The denominator is always m+n, the sum of the ratio.

|a→+b→|²=|a→|²+2a→·b→+|b→|²=9−6+4=7, so |a→+b→|=√7. Turning the sign of the dot product into + and writing 9+6+4=19 is the typical mistake. Always square the magnitude before expanding, and do not forget to take the square root again at the end.

The modulus is √(1²+1²)=√2, and the point (1, 1) has argument π/4. The standard errors are using 1²+1²=2 as the modulus without the root, and getting the sign of the argument wrong. Make it a habit: r always carries a square root, and check the argument by plotting the point.

|z|=√(3²+4²)=√25=5. Common errors are adding the real and imaginary parts as they stand to get 7, or answering the sum of squares 25 without taking the root. Recall the 3:4:5 right-angled triangle and it takes an instant. The modulus is the distance from the origin.

Raising to the n-th power multiplies the argument by n: that is de Moivre's theorem. Raising the angle θ itself to the n-th power, or raising cos to the n-th power, are all wrong. Remember it as rotation: multiplying adds the arguments, so the n-th power makes the argument n times as large.

(1+i)²=1+2i+i²=2i, so (1+i)⁴=(2i)²=−4. Forgetting to use i²=−1 and answering +4 is the typical mistake. In polar form the modulus is (√2)⁴=4 and the argument is 4×45°=180°, so you can answer −4 straight away. A question that lets you feel the power of de Moivre's theorem.

A rotation through an angle θ means multiplying by (cosθ+i sinθ). For θ=90° we have cos90°+i sin90°=i, so the answer is iz. Here −iz is a rotation through −90°, −z is a rotation through 180°, and z̄ is reflection in the real axis, not a rotation. 'Multiplying by i means turning 90° anticlockwise' is the watchword of the complex plane.

For a hyperbola c²=a²+b², so c²=4+5=9 and the foci are (±3, 0). The biggest trap is confusing this with the ellipse relation c²=a²−b² and computing 4−5 or |4−5|=1. Keep them firmly apart: subtract for an ellipse, add for a hyperbola. Since the x² term is positive, the foci lie on the x-axis.

Expectation is linear, so E(aX+b)=aE(X)+b. The point is that the constant b remains as it is. Confusing this with the variance formula, where b disappears, and dropping the b is the typical mistake. Picture it: shift all the data and the mean shifts by exactly the same amount.

Variance measures spread, so a shift by b leaves it unchanged and the b disappears, while the scale factor a acts squared, giving a²V(X). The two frequent errors are keeping the b and forgetting to square the a. Learn the rhythm: for variance, a is squared and b vanishes.

The variance of a binomial distribution is np(1−p)=100×0.2×0.8=16. The value np=20 is the expectation and is easily mistaken for the variance. The value 4 is the standard deviation √16. Learn them as a set: expectation np, and for the variance multiply by (1−p) as well.

Standardizing means z=(X−μ)/σ=(70−50)/10=2. Forgetting to divide by σ gives 20, and dividing by the variance 100 gives 0.2. The point to watch most is that the '10²' in N(50, 10²) is the variance, while what you divide by is the standard deviation 10. Two steps: subtract, then divide by σ.

The standard deviation of the sample mean is σ/√n, so for 95% the interval is x̄±1.96·σ/√n. The two big mistakes are using σ as it is, forgetting to divide by √n, and dividing by n instead. The value 2.58 belongs to the 99% confidence interval, another standard trap. Make it a mantra: 95% goes with 1.96, and the width uses σ/√n.

The standard deviation of the sample mean is σ/√n=12/√36=12/6=2. The most frequent mistake is dropping the root and writing 12/36=1/3, and answering the population standard deviation 12 is also wrong. Understand it this way: the larger n is, the less the sample mean varies, and it shrinks by a factor of 1/√n.