FPM Rates of Change 9

This sheet looks at the Pure we haven’t done much work on.


1     The plane A has vector equation  r= i+7j-3k + (i-2j+2k +  µ(2i-3j+2k

The points P & Q, not in the plane have position vectors 4i+5j+7k and 10i+8j +k respectively, and the line l1, which passes through P and Q, meets plane A at the point R. Find the position vector of R.                                                                               [4]

The line l is perpendicular to l, passes through R and lies in the plane. Find the direction of l.                                                                                                                      [5]


2    The curve C has equation y = a/x + b/x² where a and b are constants such that a>0 and b≠0. Show that C has exactly one stationary point and find its co-ordinates in terms of a and b                                                                                                             [4]

On separate diagrams, draw a sketch of C for b>0 and for b<0. In each case mark the intercepts (where the line meets an axis)                                                             [6]

Use your diagrams to show that there are positive values for m for which the equation mx3 – ax – b = 0 has three real roots                                                                   [4]


3    The set S consists of all numbers of the form a + b√5, where a and b are integers. Show that

a)  S is closed under addition and multiplication                                                 [2]

b)  there is an identity in S for addition and one for multiplication                       [2]

c)   not every element of S has an inverse with respect to multiplication            [2]

The conjugate of x = a + b√5 is xc = a - b√5. The relation я is defined on S by:

[(a + b√5) я (a – b√5)]  [ab= ab (mod 5)]

d) Show that я is an equivalence relation, that is, that the relation is symmetric, reflexive and transitive.                                                                                                      [3]

e) Determine the subset T of S such that, for each element x in T, xяq and xc²+q²=0 (mod 5), where    q = 1 + 2√5                                                                                       [5]              


4    Find the general solution of the differential equation  y” + 2y’ +2y = sin3x where y’ means the 1st differential with respect to x.                                                      [5]

Hence show that y = psin3x+ qcos3x, for large positive x, and expressing p and q as rational numbers, irrespective of the initial conditions.                                      [2]


5    Write a definition of sinh x.[1]

Write the equivalent equation to these circular function identities and equations using the hyperbolic functions, developing them from first principles.

sin² x + cos² x = 1                                                                                               [2]

sin2x = 2 sinx cosx                                                                                              [2]

cos 3x = 4 cos³ - 3 cos x                                                                                  [2]


6    The curve C is defined parametrically by x = sinh t, y = tanh t for real t. The section S of C that joins the origin to the point is such that S=

where t=ln2 is rotated about the x-axis.          

Show that the area of the surface generated is given by S as above [don’t attempt the integral]                                (yes, the bracket is raised to the power 1/2)           [4]  

The mean value of z-3(1+z)1/2 over the interval 1≤z≤5/4 is denoted by M. By finding the connection between z and t, show that S= M π/2                                               [4]


7    The system of equations 2x+4y+z = 1; 3x+5y=1; 5x+13y+7z=4 has a solution of the form a + λb where a and b are vectors in three dimensions. Use matrix method to find the solution. 

8    The integral In is defined for n≥1. 

By considering the differential of the expression above,   show that  
   2n 
In+1 = (2n-1)In+2  - n                                                                                   [4]                                                                            

Evaluate I5 exactly.                                                                                             [3]


9    Sketch the curve r – θ = 2π for  |θ| ≤ 6π                                                        [4]

Sketch separately, and for the same range of values as above, the similar curves    
3r – θ = 2π and r – 4θ = 2π.                                                                                 [4]


10    The pair of simultaneous differential equations, dx/dt = 2x+4y and dy/dt = x-y are not readily solvable. However, you can plot the tangent field for |x|<5 and |y|<5. By investigating the value of dy/dx for various values of x and y, sketch the solution sets for the pair of equations. Try to find the (two) lines for which dy/dx = mx. Try to put arrows on your lines indicating increasing values of t.                                                         [6]


















1 w93_p1_q11

(-2, 2, 13); (20, -24, 8) or(); (5, -6, 2)

2  w93_p1_q1  (-2b/a, -a2/4b)

3  w94_p1_q7

4   w93_p1_5(a)

-7/85, -6/85

5   DJS

6  w93_p1_4(a)      Is not as hard as it looks.

z = cosh t          (I thought it was sech t in 2017)

7  Cohn Linear Equations Ex IV Q9

½(-1, 1, 0) + λ(5, -3, 2)

8       Specimen 1992 P1Q7(a)

9    developed in class, Dec’07

10    MEI Diff Eqns (M4, once)  example Qn

© David Scoins 2017