FM Revision 4 !!! - Mixed | | DJS

FM Revision 4 !!! - Mixed

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 that          MISSING !!!!!

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 ℓ
2 goes through S parallel to OP and 
       line ℓ
3 goes through R, perpendicular to both ℓ1 and ℓ2.
(i)    Find an equation for ℓ3  
(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)=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 ⅓ Ma².

Find the inertia when it is rotated 
(ii)  around an end so that the rod is in the plane of rotation 
(iii)  around a point along the line of the rod, but 2a from the centre
(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

y   18.9  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.        Check index !!!!

Hence show

It is given that .                           

Hence show that, while n ≠ 0.5,  

and deduce that     MISSING EXPRESSION !!!!!

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 extrremely unpleasant - and these usually shorten one’s life.

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