IB Mathematics Applications & Interpretation Topic…

In a signal model, the complex amplitude is $w=3\sqrt{3}-3\text{i}$ . The voltage at time $t$ is .

(a) Express $w$ in the form $r\text{e}^{\text{i}\theta}$ .

(b) $V(t)$ in the form $A\cos(3t+\alpha)$ .

(e) the revised diagonal lengths, and the change in courtyard area.

The courtyard must have area at least $60$ square metres and neither diagonal may exceed $14$ m.

(f) whether that revision meets both limits.

(g) Find the complete interval of positive scales $r$ that meets both limits when $w=re^{-0.6i}$ .

(h) the area if the lower-bound scale is stored to two decimal places, and whether that stored setting qualifies.

The final controller offers positive scales in steps of $0.05$ , while retaining the common angle $-0.6$ radians.

(i) Choose an available controller scale strictly inside the admissible interval and verify both limits. why rounding either endpoint to the nearest $0.05$ step is unsuitable.

A simulation compares complex gains $z_1=8\mathrm e^{\frac{3\pi}{4}\mathrm i}$ and $z_2=2\mathrm e^{\frac\pi{12}\mathrm i}$ using $w=\frac{z_1}{z_2}$ . The gain after $n$ identical comparison stages is $w^n$ . All phase angles are in radians. The principal argument lies in $(-\pi,\pi]$ , and $\operatorname{cis}\theta=\cos\theta+\mathrm i\sin\theta$ .

(c) A programmer adds the arguments and obtains $w_{\mathrm{bad}}=4\mathrm e^{\frac{5\pi}{6}\mathrm i}$ . For each of $w$ and $w_{\mathrm{bad}}$ , find the smallest positive integer $n$ for which the $n$ -stage gain is real, and that gain exactly. the consequence of the error.

Let $z=-\sqrt{3}+i$ , where the point representing $z$ lies in the second quadrant of the Argand diagram and angles are measured in radians.

(a) Find the modulus and argument of $z$ , and write $z$ in the form $r(\cos\theta+i\sin\theta)$ .

(b) Hence find $z^3$ in Cartesian form.

A survey firm designs a courtyard as the parallelogram with corners $O=0$ , $A=a$ , $B=b$ and $C=a+b$ , where $a=5e^{0.45i}$ and $b=7e^{1.95i}$ , in metres. A revision multiplies every position by the same nonzero pair $w$ . The planner must preserve the parallelogram while fitting limits on its diagonal length and area. Use radians throughout. AHL_1.13-q7-fig1

(b) Find the position of $C$ and the displacement from $A$ to $B$ .

A graphics program represents positions in centimetres by complex coordinates. It transforms a point $z$ to $T(z)=wz+c$ , where $w$ is a dimensionless complex multiplier and $c$ is a displacement in centimetres. The program gives $T(0)=1-\mathrm i$ and $T(1+2\mathrm i)=-3+\mathrm i$ . All phase angles are in radians. The principal argument lies in $(-\pi,\pi]$ , and $\operatorname{cis}\theta=\cos\theta+\mathrm i\sin\theta$ .

(a) $w$ in exponential form and in Cartesian form, with Cartesian components to three significant figures.

(b) all positive integers $n\le12$ for which $w^n$ is real, including the sign of each value.

(a) $a$ and $b$ in Cartesian form, and plot $A$ and $B$ on Figure 1.

(d) the area of triangle $OAB$ and the courtyard area.

The first proposed revision uses $w=1.4e^{-0.6i}$ .

(b) the geometric effect of $T$ as a sequence of a stretch, a rotation and a translation.

IB Mathematics Applications & Interpretation AHL 1.13 Complex Numbers Continued Questionbank

IB Mathematics Applications & Interpretation AHL 1.13 Complex Numbers Continued Questionbank

IB Mathematics Applications & Interpretation AHL 1.13 Complex Numbers Continued Questionbank

IB Mathematics Applications & Interpretation AHL 1.13 Complex Numbers Continued Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 1.13 Complex Numbers Continued Questionbank