Download the King`s Factor Year 12 further questions 7 1. [1998 STEP I

the King’s Factor Year 12 further questions 7
1. [1998 STEP I question 4]
Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a
square.
The result changes if, instead of maximising the sum of lengths of sides of the rectangle, we
seek to maximise the sum of nth powers of the lengths of those sides for n ≥ 2. What happens
if n = 2? What happens if n = 3? Justify your answers.
2. [2008 AEA question 7]
Relative to a fixed origin O, the position vectors of the points A, B and C are
−→
OA = −3i + j − 9k ,
−−→
OB = i − k ,
−→
OC = 5i + 2j − 5k
respectively.
(a) Find the cosine of angle ABC.
The line L is the angle bisector of angle ABC.
(b) Show that an equation of L is r = i − k + t(i + 2j − 7k) .
(c) Show that AB = AC .
The circle S lies inside triangle ABC and each side of the triangle is a tangent to S.
(d) Find the position vector of the centre of S.
(e) Find the radius of S
3. [2004 STEP I question 4]
Differentiate sec t with respect to t.
Z
(i) Use the substitution x = sec t to show that
Z
1
p
dx .
(x + 2) (x + 1)(x + 3)
Z
1
dx .
(x + 2) x2 + 4x − 5
(ii) Determine
(iii) Determine
√
1
2
√
2
1
√
dx =
x3 x2 − 1
√
3−2
π
+
.
8
24
the King’s Factor Year 12 further questions 7
4. [2004 STEP I question 9] (Mechanics)
A particle is projected over level ground with speed u at an angle θ above the horizontal.
Derive an expression for the greatest height of the particle in terms of u, θ and g.
9
A particle is projected from the floor of a horizontal tunnel of height 10
d. Point P is 21 d
vertically and d horizontally along the tunnel from the point of projection. The particle
passes through point P and lands inside the tunnel without hitting the roof. Show that
arctan
3
< θ < arctan 3 .
5
2