1. Find the magnitude and direction of the resultant of the forces $\left[\begin{array}{c}-3\\ 1\end{array}\right]$ N , $\left[\begin{array}{c}4\\ 2\end{array}\right]$ N and $\left[\begin{array}{c}1\\ 2\end{array}\right]$ N

2. Use the trapezium rule with five subintervals to estimate ${\int }_2^4\frac{5}{\left(\mathrm{x}+1\right)}\mathrm{dx}$ Give your answer correct to three decimal places.

3. The table below shows the mass of boys in in a certain school.

Calculate the mean.

4. Two cyclists A and B are 36m apart on a straight road. Cyclist B starts from rest with an acceleration of 6ms-2 while Ais in pursuit of B with velocity of 20 m/s and acceleration of 4 ms-2.Find the time when A overtakes B.

5. Events A and Bare independent. P(A) = x,P(B) = x+ 0.2 and P(A∪B) = 0.65. find the value of x.

6. The table below shows the values of a function f(x) for a given value of x.

Use linear interpolation or extrapolation to find:

(a) f(10.4)

(b) the value of xcorresponding to f(x) = 1.46

7. The marks in an examination were found to be normally distributed with mean 53.9 and standard deviation 16.5. 20% of the candidates who sat this examination failed. Find the pass mark for the examination

8. A fixed hollow hemisphere has centre O and is fixed so that the plane of the rim is horizontal. A particle A of weight 30 2N can move on the inside surface of the hemisphere.The particle is acted upon by a horizontal P, whose line of action is in a vertical plane through O and A. OA makes an angle of 45°with the vertical. If the coefficient of friction between the particle and the hemisphere is 0.5 and the particle is just about to clip downwards, find the:

(a) normal reaction.

(b) value of P.

9. The probability density function (p.d.f) of a random variable Y is given by

(a) the value of k

(b) the expectation of Y

(c) P(1≤Y≤1.5)

10. The numbers A= 6.341 and B= 2.6 have been rounded to the given number of decimal places.

(a)Find the maximum possible error in AB.

(b)Determine the interval within which $\frac{{\mathrm{A}}^2}{\mathrm{B}}$ can be expected to lie.

11. The diagram below shows a 12kg mass on a horizontal rough plane. The 6kg and 4kg masses are on rough planes inclined at angles 60°and 30°respectively. The masses are connected to each other by light inextensible strings passing over light smooth fixed pulleys A and B.

The planes are equally rough with coefficient of friction $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. If the system is released from rest, the:

(a)acceleration of the system.

(b)extension in the string.

12. The table below gives the points awarded to eight schools by three judges J1, J2 and J3 during a music competition. J1 was the chief judge.

(a)Determine the rank correlation coefficient between the judges of:

(i) J1 and J2

(ii) J1 and J3

13. A ball is projected from a point A and falls at a point B which is in level with A and at a distance of 160m from A. The greatest height of the ball attained is 40m. Find the:

(a)angle and velocity at which the ball is projected.

(b)time taken for the ball to attain its greatest height.

14. (a) Show that the iterative formula based on Newton Raphson’s method for approximating the root of the equation 2 ln x–1 + 1 = 0 is given by

${\mathrm{X}}_{\mathrm{n}+1}=\left(\frac{2{\mathrm{Inx}}_{\mathrm{n}}-1}{{\mathrm{x}}_{\mathrm{n}}-1}\right)$,n = 0,1,2,......

(b) Draw a flow chart that:

(i) reads the initial approximation xo of the root.

(ii) computes and prints the root correct to 2 decimal places, using the formula in (a).

 

 

15. The diagram below represents a lamina formed by welding together a rectangular metal. Find the position of the centre of gravity of the lamina from the side OA.

16. A box Acontains 4 white and 2 red balls. Another box contains 3 white and 3 red balls. A box is selected at random at two balls are picked one after the other without replacement.

(a)Find the probability that the two balls picked are red.

(b)Given that two white balls are picked, what is the probability that they are from box B?