Questions (1 Viewer)

181jsmith · Oct 30, 2011

1. Solve the equation $\sqrt{2}cos(x+60)-1=0 for 0\leq x\leq 360$

2. P (2ap, ap^2) and Q (2aq, aq^2) are 2 points on the parabola x^2= 4ay with parameter values p=4 and q=-6. Show that the lines OP and OQ are inclined at 45 degrees to each other.

3. Solve the inequality $\frac{1}{x} - \frac{1}{x-2}> 0$

4. Show that 2cos(A - B)sin(A+B)= sin2A + sin 2B

5. The polynomial eq $x^{3} + bx^{2} + cx +d =0$ has roots
a, a^2 and a^3

Find in terms of b,c and d $\frac{1}{a}+\frac{1}{a^{2}}+\frac{1}{a^{3}}$

Show that $b^{3}d-c^{3}=0$

thanks.

SpiralFlex · Oct 30, 2011

1. $\cos (x+60)=\frac{1}{\sqrt{2}}$

$x+60 = 315, 390$

$x=255, 330$

2. $P(8a, 16a), Q(-12,36a), O(0,0)$

Find the gradients, you should get $2, -3$ respectively.

$PO: y=2x, QO: y=-3x$

$\tan \theta = |\frac{2-(-3)}{1+3*-2}|$

$\therefore \theta = 45$

3. $\frac{x-2-x}{x(x-2)}>0$

$\frac{-2}{x(x-2)}>0$

$-2x(x-2)>0$

$2<x<0$

4. $=(\cos A \cos B + \sin A \sin B)(\sin A \cos B + \cos A \sin B)$

$=\cos A \cos^2 B \sin A +\cos^2 A \cos B \sin B +\sin^2 A \sin B \cos B + \sin A \sin^2 B \cos A$

$=(\sin^2 A + \cos^2 A)(2 \sin B \cos B)+(\sin^2 B + \cos^2 B)(2\sin A \cos A)$

$=\sin 2A + \sin 2B$

5 i. Sum of roots: $a+a^2+a^3=-b...(1)$

Two at a time: $a^3+a^4+a^5=c...(2)$

Product: $a^6=-d...(3)$

$\frac{(2)}{(1)}$

$a^2=-\frac{c}{b}$

$\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}=\frac{a^2(a+a^2+a^3)}{a^6}$

$=\frac{\sqrt[3]{-d}*-b}{(-\frac{c}{b})^3}$

$=\frac{\sqrt[3]{-d}*b^4}{c^3}$

ii. $(a^2)^3=a^6$

$(-\frac{c}{b})^3=-d$

$b^3d-c^3=0$

taeyang · Oct 30, 2011

SpiralFlex said:
1. $\cos (x+60)=\frac{1}{\sqrt{2}}$

$x+60 = 315, 390$

$x=255, 330$

2. $P(8a, 16a), Q(-12,36a), O(0,0)$

Find the gradients, you should get $2, -3$ respectively.

$PO: y=2x, QO: y=-3x$

$\tan \theta = |\frac{2-(-3)}{1+3*-2}|$

$\therefore \theta = 45$

3. $\frac{x-2-x}{x(x-2)}>0$

$\frac{-2}{x(x-2)}>0$

$-2x(x-2)>0$

$2<x<0$

4. $=(\cos A \cos B + \sin A \sin B)(\sin A \cos B + \cos A \sin B)$

$=\cos A \cos^2 B \sin A +\cos^2 A \cos B \sin B +\sin^2 A \sin B \cos B + \sin A \sin^2 B \cos A$

$=(\sin^2 A + \cos^2 A)(2 \sin B \cos B)+(\sin^2 B + \cos^2 B)(2\sin A \cos A)$

$=\sin 2A + \sin 2B$

5 i. Sum of roots: $a+a^2+a^3=-b...(1)$

Two at a time: $a^3+a^4+a^5=c...(2)$

Product: $a^6=-d...(3)$

$\frac{(2)}{(1)}$

$a^2=-\frac{c}{b}$

$\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}=\frac{a^2(a+a^2+a^3)}{a^6}$

$=\frac{\sqrt[3]{-d}*-b}{(-\frac{c}{b})^3}$

$=\frac{\sqrt[3]{-d}*b^4}{c^3}$

ii. $(a^2)^3=a^6$

$(-\frac{c}{b})^3=-d$

$b^3d-c^3=0$

Juices are flowing.

bleakarcher · Oct 30, 2011

for q5 i) can u just write -c/d as an equivalent expression or does it have to be in terms of b also

SpiralFlex · Oct 30, 2011

I would assume by the question, in terms of "b, c, d".

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