ACT · Math · Preparing for Higher Math
Números complejos, exponentes y raíces, vectores y matrices.
Números reales y complejos, exponentes, vectores y matrices.
$(2+3i)(4-i)=8+10i-3i2=11+10i$.
Intenta responder antes de abrir la solución. Las preguntas están en el idioma del examen.
Which of the following is equal to $\frac{27\cdot2-3}{22}$ ?
Respuesta correcta: B. $4$
Add exponents when multiplying and subtract when dividing: $27+(-3)-2=22=4$.
$16=24$ forgets to divide by $22$. $256=28$ adds the 3 instead of subtracting it. $\frac14=2-2$ gets the sign of the result wrong.
Which of the following is equal to $\sqrt{50}+\sqrt{18}$ ?
Respuesta correcta: B. $8\sqrt{2}$
Simplify each radical: $\sqrt{50}=\sqrt{25\cdot2}=5\sqrt{2}$ and $\sqrt{18}=\sqrt{9\cdot2}=3\sqrt{2}$. Like radicals add like terms: $5\sqrt{2}+3\sqrt{2}=8\sqrt{2}$.
$\sqrt{68}$ adds the numbers under the roots, which is not allowed. 16 comes from writing $8\sqrt{4}$ instead of $8\sqrt{2}$. $15\sqrt{2}$ multiplies the coefficients 5 and 3 instead of adding them.
What is the value of $i+i2+i3+\cdots+i50$ ? (Note: $i=\sqrt{-1}$)
Respuesta correcta: C. $-1+i$
The powers of $i$ repeat every 4 terms: $i,\,-1,\,-i,\,1$, and each block of 4 adds to $0$. The first 48 terms are 12 complete blocks, so they add to $0$. What is left is $i49+i50=i+i2=i-1=-1+i$.
$-1$ is only $i50$ (the trap: using $50\div4$ leaves remainder 2 and stopping at $i2$, forgetting $i49$). $0$ assumes all 50 terms form complete blocks. $i$ is only $i49$.