20230929-AM2

Given taht $$\left|\overline{a}\right|=7$$ , $$\left|\overline{b}\right|=8$$ , $$\overline{a}\cdot\overline{b}=-28$$ , $$\left|\overline{a}+\overline{b}\right|=\sqrt[]{c}$$ , find $$c$$ .

(a)  

Given the remainder is $$f\left(a\right)$$ when the polynomial $$f\left(x+2\right)$$  is divided by $$x-2$$ , find $$a$$ .

(a)  

Given that the complex number $$w=\frac{-1+\sqrt[]{3}i}{2}$$ satisfies $$z^3-1=0$$ . Find $$-2w^4+11w^3+7w^2+9w$$ .

(a)  

Find the cooefficient of $$x^3$$ in the expansion of $$\left(1+3x\right)^{\frac{1}{2}}$$ .

In a city, 12% of the residents are virus carriers.
It is known that 94% of the carriers and 15% of the non-carriers would be tested positive in a diagnostic test.
Find the probability that a randomly selected resident is tested positive.

Given that a curve has parametric equations $$x=3t-1$$ and $$y=t^2+t-2$$ ,
find the Cartesian eqaution of this curve.

Given that the foci of the ellipse $$17x^2+8y^2=136$$  and the hyperbola $$\frac{y^2}{104}-\frac{x^2}{k^2}=\frac{1}{25}$$  coincide.
Find the positive number $$k$$ .

(a)  

Find the coordinates of the focus of the parabola $$y=2x^2-8x+1$$ .

Given that the polar equation of a circle is $$r^2-3=4r\cos\theta-6r\sin\theta$$ , find the radius of the circle.

(a)  

After a rotation of the coordinate axes about the origin, the point $$\left(4,13\right)$$ in the new coordinate system are $$\left(11,k\right)$$ .
Find the positive values of $$k$$ .

(a)