Study questions platform-wide or filter by specific tests with correct answers revealed.
Let $O$ be the vertex and $Q$ any point on the parabola $x^2=8y$. If point $P$ divides segment $OQ$ internally in the ratio $1:3$, find the locus of $P$.
Let $Q=(4t, 2t^2)$ be a point on $x^2=8y$ (using parametrization $x=4t, y=2t^2$). $O=(0,0)$.
$P$ divides $OQ$ in ratio $1:3$: $P=\left(\frac{1\cdot4t+3\cdot0}{4},\frac{1\cdot2t^2+3\cdot0}{4}\right)=\left(t,\frac{t^2}{2}\right)$
Let $P=(h,k)$: $h=t$ and $k=\frac{t^2}{2}=\frac{h^2}{2}$
So $h^2=2k$, i.e., the locus is $x^2=2y$.
Speed of sound in ideal gas does NOT depend on:
For ideal gas, v = โ(ฮณRT/M). Pressure has no effect at constant temperature; moisture slightly affects density but the question says ideal gas: strictly, pressure independent.
Cronbach's Alpha (\(\alpha\)) measures the internal consistency of a multi-item scale โ the degree to which items within the scale measure the same construct.
Interpretation benchmarks:
| Cronbach's \(\alpha\) | Internal Consistency |
|---|---|
| \(\alpha \geq 0.90\) | Excellent |
| \(0.80 \leq \alpha < 0.90\) | Good |
| \(0.70 \leq \alpha < 0.80\) | Acceptable |
| \(0.60 \leq \alpha < 0.70\) | Questionable |
| \(\alpha < 0.60\) | Poor โ tool needs revision |
Let $I_n=\displaystyle\int\tan^n x\,dx$ for $n>1$. If $I_4+I_6 = a\tan^5x + bx^5 + C$, find the ordered pair $(a,b)$.
Use the reduction: $I_n+I_{n-2}=\displaystyle\int\tan^{n-2}x\cdot\tan^2x\,dx+\int\tan^{n-2}x\,dx=\int\tan^{n-2}x\sec^2x\,dx=\dfrac{\tan^{n-1}x}{n-1}+C$
$I_4+I_6$: set $n=5$: $I_4+I_6=\dfrac{\tan^5x}{5}+C$
Comparing with $a\tan^5x+bx^5+C$: $a=\dfrac{1}{5}$, $b=0$.
$(a,b) = \left(\dfrac{1}{5},0\right)$
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