x FM Revision 4 (Mixed) copy

1    Establish the rank of the matrix A

and reduce it to echelon form, E.

Given that Ax = ( 6  30  11  15 )

show that x = x0e is a valid form for vectors x0 and e and a variable, λ. Show that there are no points in the positive quadrant.

[Helpful check:   e = (…, …, …, 27) ]

2  Use induction to show this formula holds         

and find three prime factors of S100.

3    Point       P is 3i – 4j + 5k,   

                      Q is 2i +3j –k,   

                      R is –1i + 3j +k   and 

                      S is 10i + 9j -8k.

The line ℓ1 includes P and Q,

       line ℓ2 goes through S parallel to OP and

       line ℓ3 goes through R, perpendicular to both ℓ1 and ℓ2.

(i)    Find an equation for ℓ3 

  1. 1.(ii)      Find the shortest distance from S to ℓ3. Find also the foot of this                    perpendicular in the form 1/43 (p, q, r)

(iii)Find the equation of the plane π3, that includes Q and ℓ3.

(iv)Find the distance from S to π3.

4    Find the eigenvectors of the matrix M

Find P and D such that (M-kI)n=PDP-1 where I is the identity matrix, k is a constant scalar and n is a positive integer.

5    (i) Prove that the inertia of a rod length 2a rotated around its centre of mass along an axis perpendicular to the rod is 1/3 Ma2.

Find the inertia when it is rotated

  1. 1.(ii)around an end so that the rod is in the plane of rotation

  2. 2.(iii)around a point along the line of the rod, but 2a from the centre

  3. 3.(iv)by suspension from the middle of a light string of length 2λ, connected to the ends of the rod so that the rod remains in the plane of rotation.

6     The continuous random variable X has a probability function

           f(x) = k e q-px for x≥3.              f(x) is otherwise zero.

(i) Find an expression connecting p,q & k.

While p = k, (working entirely in terms of p)

(ii)  Calculate the probability distribution

(iii) Write an expression for the median

(iv) Find expressions for the expectation and variance.

7    There are five points found for the curve y=f(x)

(i)   Sketch possible curves that will fit the following cases:

a) the linear pmcc is close to 1  (product moment correlation coefficient )

b)the pmcc is close to zero

The table shows the age, x, and %body fat, y, for a random sample of ten adults

x 23    27    39    41    45    49     53    54    57    61

y18.9  28.7  40.9  33.9  38.6  35.8  43.9  39.7  40.8  46.5

(ii) Calculate the linear pmcc for the sample. They are Americans. Comment.

(iii) To test whether body fat increases with age, calculate a rank correlation coefficient and test it at the 5% level. State your conclusions. Clearly.

8    Differentiate x(1+x2)-n and show that this is also

2n(1+x2)-n-1 – (2n-1) (1+x2)-n.

Hence show

It is given that .

Hence show that, while n ≠ 0.5, 

and deduce that

2    I developed this formula myself. I think the even powers have curious formulae and the odd powers are predictable. N(N+1) (2N+1) seem to be common factors.

3(i) I think the direction vector of ℓ3 could be written as   (a b c) where b2 + ac +18 = 0

(ii-iv) I found I wanted roots of 579, 95106 and 3038. The 1/43 didn’t help and should have been something else; these questions are hard to write with ‘nice’ numbers, which may explain why at least one question has been re-used.

6    The CIE examiner uses probability distribution, F(x) when he means cumulative probability distribution. Last reminder ! The target questions go on to define related functions….

7    The US population suffers from obesity. It might well be a bimodal distribution, very fat and very thin. Contributing factors are too much money. too much ‘fast’ food, not enough exercise and a social acceptability of extreme size. Being slim becomes an attribute of being well off, unlike other cultures where success is echoed by a matching weight. Or overweight. Obesity leads to type II diabetes and other disorders, some of which are extremely unpleasant - and these usually shorten one’s life.

© David Scoins 2017