Specialist U3&4 Quiz

If a = i + j – 4k and b = –3i + 3j – 4k, find the angle between a and b to the nearest degree.

• 40º

• 50º

• 57º

• 130º

• 140º

Find a vector normal to the plane containing the points

(–3, 2, 3), (–2, 4, 5) and (0, 11, 12).

• (0, –3, 3)

• (–1, –2, 0)

• (2, –9, 12)

• (2, 5, 0)

Find the quadratic equation which has roots 8 ± 5i.

• x² + 8x + 25

• x² – 8x + 25

• x² + 16x + 25

• x² – 16x + 89

A complex number in polar form has modulus 6 and argument $\frac{5\pi}{6}$ . Write this number in polar form.

• $z=6\left[\cos\left(\frac{5\pi}{6}\right)-i\sin\left(\frac{5\pi}{6}\right)\right]$

• $z=6\left[\cos\left(\frac{5\pi}{6}\right)+i\sin\left(\frac{5\pi}{6}\right)\right]$

• $z=6\left[\sin\left(\frac{5\pi}{6}\right)+i\cos\left(\frac{5\pi}{6}\right)\right]$

• $z=6\left[\sin\left(\frac{5\pi}{6}\right)-i\cos\left(\frac{5\pi}{6}\right)\right]$

Use De Moivre’s theorem to find z⁵ if z = 2[cos (q) + i sin (q)].

• 32[cos (5q) + i sin (5q)]

• 32[cos⁵ (q) + i sin⁵ (q)]

• 10[cos (5q) + i sin (5q)]

• 10[cos⁵ (q) + i sin⁵ (q)]

The complex number c = 5 – 2i. Describe the locus of z if

|z – c| = 3.

• A circle centre (5, –2), radius 3.

• A circle centre (–5, 2), radius 9.

• A circle centre (–5, 2), radius 3.

• A circle centre (5, –2), radius 9.

 $Sketch\ the\ locus\ of\ z\ if\ \frac{\pi}{3}<\arg\left(z\right)\le\frac{3\pi}{4}$   

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If r = 2t⁶i – 3t⁴j, find r´.

• r'= 6t⁵i – 4t³j

• r'= 10t⁵i – 9t³j

• r'= 10t⁶i – 9t⁴j

• r'= 12t⁵i – 12t³j

A particle’s position at any time t is given by

r = t cos (t)i + t sin (t)j. Find its speed when t = 9.

• $\sqrt{71}$

• $\sqrt{76}$

• $\sqrt{80}$

• $\sqrt{82}$

A projectile is fired from a point O, defined as the origin, at a speed of 42 m/s at an angle of 49º to the horizontal. What is the maximum height it will reach, correct to one decimal place? Use g = 9.8 m/s².

• 54.2 m

• 51.3 m

• 63.7 m

• 66.2 m