IB Maths AI · Topic 1 · 32 cards

Number and algebra flashcards

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How do you write a number in standard form, and what is 0.000472 in standard form?
Write it as \(a\times10^{k}\) with \(1\le a<10\) and \(k\) an integer. Here \(0.000472=4.72\times10^{-4}\).
How do you decide how many significant figures a number has?
Count from the first non-zero digit. Leading zeros do not count; zeros after the decimal point at the end do. So 0.004050 has 4 significant figures.
When should you round during a multi-step calculation?
Only at the end. Store full-precision values in the GDC memory and use them in later steps; rounding early can change the third significant figure of the final answer.
What is the formula for percentage error, and what is it for an approximate value 7.8 when the exact value is 8.0?
\(\varepsilon=\left|\dfrac{v_A-v_E}{v_E}\right|\times100\%\). Here \(\left|\dfrac{7.8-8.0}{8.0}\right|\times100=2.5\%\).
How are the nth term and the sum of an arithmetic sequence found? Find \(S_{10}\) when \(u_1=5\), \(d=3\).
\(u_n=u_1+(n-1)d\) and \(S_n=\dfrac n2\big(2u_1+(n-1)d\big)\). So \(S_{10}=5(10+27)=185\).
How are the nth term and the sum of a geometric sequence found?
\(u_n=u_1r^{\,n-1}\) and \(S_n=\dfrac{u_1(r^{n}-1)}{r-1}\), where \(r\) is the constant ratio between consecutive terms.
How can you tell whether a sequence is arithmetic or geometric?
Arithmetic: consecutive differences are constant (a linear pattern). Geometric: consecutive ratios are constant (an exponential pattern). For 3, 6, 12, 24 the ratio is 2, so it is geometric.
What is the compound interest formula, and what does it give for £2000 at 3% a year for 5 years?
\(FV=PV\left(1+\dfrac{r}{100k}\right)^{kn}\), with \(k\) compounding periods a year and \(n\) years. Here \(2000(1.03)^5=£2318.55\).
How do you model depreciation, and what is a €18 000 car worth after 4 years at 12% loss a year?
Use a negative rate: \(V=V_0(1-0.12)^n\). So \(V=18000(0.88)^4\approx10\,795\) euros, so about €10 795.
What is the difference between simple and compound interest?
Simple interest is paid only on the original sum: \(I=\dfrac{Prt}{100}\), giving linear growth. Compound interest is paid on the growing balance, giving exponential growth.
Which values does the GDC finance solver ask for, and how do you use it?
Enter N (periods), I% (annual rate), PV, PMT, FV, P/Y and C/Y. Fill in every known value, leave the unknown blank and solve for it. Mind the sign: money paid out is negative.
State the laws of exponents for \(a^m a^n\), \(a^m\div a^n\), \((a^m)^n\) and \(a^{-n}\).
\(a^ma^n=a^{m+n}\); \(a^m\div a^n=a^{m-n}\); \((a^m)^n=a^{mn}\); \(a^{-n}=\dfrac1{a^n}\).
(HL) Evaluate \(8^{2/3}\) without a calculator.
\(8^{2/3}=(\sqrt[3]{8})^2=2^2=4\). Take the root first, then the power.
How do you solve \(2^x=50\) using the GDC?
Graph \(y=2^x\) and \(y=50\) and find the intersection, or use the numerical equation solver. The result is \(x\approx5.64\).
How do you solve \(x+y=7\) and \(2x-y=5\) using the GDC, and how do you check it?
Use the simultaneous-equation solver to get \(x=4,\ y=3\). Check by substituting into both original equations: \(4+3=7\) and \(8-3=5\).
How do you solve \(x^2-3x-10=0\) with a GDC?
Use the polynomial solver, or graph \(y=x^2-3x-10\) and read the x-intercepts. The solutions are \(x=5\) and \(x=-2\).
How do you convert £250 into euros when £1 = €1.17?
Multiply by the rate when converting £ into €: \(250\times1.17=292.50\), so €292.50. Divide by the rate to convert back.
How are rounding and truncating different?
Rounding gives the nearest value at the chosen accuracy (3.678 to 1 d.p. is 3.7). Truncating just cuts digits off (3.678 to 1 d.p. is 3.6).
How do you increase or decrease an amount by a percentage using a multiplier? Increase £60 by 15%.
Multiply by \(1+\tfrac{p}{100}\) for an increase or \(1-\tfrac{p}{100}\) for a decrease. \(60\times1.15=£69\).
How can you estimate an answer to check a calculation? Estimate \(4.9\times19.8\).
Round each number to 1 significant figure: \(5\times20=100\). The exact value is 97.02, so the result is sensible.
How do you find the amount after monthly compounding? Use £1000 at 6% a year for 2 years.
Use \(k=12\): \(1000\left(1+\tfrac{6}{1200}\right)^{24}=1000(1.005)^{24}\approx£1127.16\).
(HL) What are the laws of logarithms?
\(\log_a(xy)=\log_ax+\log_ay\); \(\log_a\dfrac xy=\log_ax-\log_ay\); \(\log_a(x^n)=n\log_ax\). Also \(\log_ab=c\iff a^c=b\).
What is an amortised loan, and how do you find its repayment using technology?
A loan repaid by equal regular payments, each covering that period's interest and reducing the balance. Use the finance solver: enter N, I%, PV (loan), FV = 0 and payments per year, then solve for PMT.
(HL) What does the sum to infinity of a geometric series equal, and when does it exist? Find it for \(u_1=12,\ r=\tfrac13\).
\(S_\infty=\dfrac{u_1}{1-r}\), valid only when \(|r|<1\). Here \(S_\infty=\dfrac{12}{1-\frac13}=18\).
(HL) How do you add, multiply and divide complex numbers? Find \((2+3i)(1-i)\).
Expand using \(i^2=-1\); to divide, multiply top and bottom by the conjugate. \((2+3i)(1-i)=2-2i+3i-3i^2=5+i\).
(HL) How do you write a complex number in modulus-argument (polar) form? Do it for \(1+i\).
\(z=r(\cos\theta+i\sin\theta)\) with \(r=|z|=\sqrt{a^2+b^2}\) and \(\theta=\arg z\). For \(1+i\): \(r=\sqrt2\), \(\theta=\tfrac\pi4\), so \(z=\sqrt2\left(\cos\tfrac\pi4+i\sin\tfrac\pi4\right)\).
(HL) Solve \(x^2+2x+5=0\) over the complex numbers.
The discriminant is \(4-20=-16<0\), so the roots are complex conjugates: \(x=\dfrac{-2\pm4i}{2}=-1\pm2i\).
(HL) When can two matrices be multiplied, and is \(AB=BA\)? Compute \(\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}\).
Columns of the first must equal rows of the second. In general \(AB\ne BA\). The product is \(\begin{pmatrix}2&1\\4&3\end{pmatrix}\).
(HL) How do you find the determinant and inverse of a 2×2 matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\)?
\(\det=ad-bc\). If \(\det\ne0\), the inverse is \(\dfrac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\). If \(\det=0\) there is no inverse. For \(\begin{pmatrix}3&1\\5&2\end{pmatrix}\), \(\det=1\).
(HL) What are eigenvalues and eigenvectors, and how are eigenvalues found?
\(A\mathbf v=\lambda\mathbf v\) with \(\mathbf v\ne\mathbf0\). Solve \(\det(A-\lambda I)=0\). For \(\begin{pmatrix}4&1\\2&3\end{pmatrix}\): \(\lambda^2-7\lambda+10=0\), so \(\lambda=2\) or \(5\).
(HL) How do you use diagonalisation to find a high power of a matrix?
Write \(A=PDP^{-1}\), where the columns of \(P\) are eigenvectors and \(D\) holds the eigenvalues. Then \(A^n=PD^nP^{-1}\), and \(D^n\) just raises each diagonal entry to the power \(n\).
(HL) How do you solve a system of three linear equations in three unknowns?
Write it as \(M\mathbf x=\mathbf b\) and use the GDC system solver, or compute \(\mathbf x=M^{-1}\mathbf b\). Always check the answer in the original equations.

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