JUPEB 2019 – MATHEMATICS: Essay Questions

(a) In a class of 100 students, 35 offer History, 43 offer Geography, and 50 offer Economics. 14 students offer History and Geography, 13 offer Geography and Economics, and 11 offer History and Economics. The number of students that offer none of the subjects is four times the number of those that offer three subjects.

i. How many students offer three subjects? (4 marks)

ii. How many students offer none (1 mark)

iii. How many students offer only one of the subjects? (2 marks)

iv. How many students offer History and Geography only? (1 mark)

(b) Consider the Venn diagram below.

Find the following sets.

$$\begin{gathered} \left(\mathrm{i}\right) \mathrm{A}\cap {\mathrm{B}}^{\mathrm{c}}. 1\mathrm{mark} \\ \left(\mathrm{ii}\right) {\mathrm{A}}^{\mathrm{c}}\cup \mathrm{B}. 1 \mathrm{mark} \\ \left(\mathrm{c}\right) \mathrm{Solve} \left(2\sqrt{\mathrm{x}+5} –\sqrt{2\mathrm{x}+8}\right)=\left(2\right). 5\mathrm{marks} \end{gathered}$$

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(a) Prove by mathematical induction or otherwise that

$\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{k}}^2=\frac{\mathrm{n}\left(\mathrm{n}+1\right)\left(2\mathrm{n}+1\right)}{6}. 5\mathrm{marks}$

(b) Let R be the set of real numbers if binary operations ⊗is defined on R by α⊗b=α+1b+1 where a and b belong to R.2+4n ⊗4-n=5n⊗n. 5marks

(c) An A.P has a common difference of 2 and G.P has a common ratio of 3. The corresponding terms are added together to produce a new sequence. If the 1^st term of the sequence is 8 and its third term is 52, find.

(i) the A.P. (2 Marks)

(ii) the G.P. (2 Marks)

(iii) The new sequence (1 Mark)

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(a) A vehicle starts from rests and its velocity is measured every second for 8 seconds as follows:

The distance travelled in 8 seconds is given by

${\int }_0^8\mathrm{v} \mathrm{dt}.$

Estimate this distance using Simpson’s rule. 6 marks

$$\begin{gathered} \left(\mathrm{b}\right) \mathrm{Evaluate} \int \frac{\mathrm{x}+1}{{\mathrm{x}}^2–3\mathrm{x}+2}\mathrm{dx} \left[6 \mathrm{marks}\right] \\ \left(\mathrm{c}\right) \mathrm{Find} \mathrm{the} \mathrm{differential} \mathrm{coeffient} \mathrm{of} \\ \mathrm{Y}=3{\mathrm{x}}^2 \sin 2\mathrm{x} \left[3 \mathrm{marks}\right] \end{gathered}$$

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$$\begin{gathered} \left(\mathrm{a}\right) \mathrm{Obtain} \mathrm{the} \mathrm{expansion} \mathrm{of} {\mathrm{e}}^{\mathrm{x}} \sin \mathrm{x} \mathrm{in} \mathrm{ascending} \\ \mathrm{power} \mathrm{of} \mathrm{x} \mathrm{as} \mathrm{far} \mathrm{as} \mathrm{the} \mathrm{term} \mathrm{in} {\mathrm{x}}^5. \left[8 \mathrm{marks}\right] \\ \left(\mathrm{b}\right) \mathrm{Evaluate} \\ \left(\mathrm{i}\right) {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \cos \mathrm{x}d\mathrm{x} \left[3 \mathrm{marks}\right] \\ \left(\mathrm{ii}\right) {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} \left({\sec }^2 \mathrm{x} + {\tan }^2\mathrm{x}\right)\mathrm{dx} \left[4 \mathrm{marks}\right] \end{gathered}$$

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(a) A block of mass 3.5kg rests on a rough plane which is inclined at an angle of 42⁰ to the horizontal. If the coefficient of friction between the block and the plane is 0.75, what is the least force P parallel to the plane required to:

i. Start moving the block up the plane? (5 Marks)

ii. Prevent the block from sliding down the plane? (Take g = 9.8ms^-2). (5 Marks)

(b) A body of mass 100kg is placed in a lift. Find the reaction between the floor of the lift and the body when the lift moves upward with an acceleration of 3.5m^s-2 (Take g = 9.8ms^-2). (5 Marks)

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(a) Three points A, B and C are uniquely orientated in space with coordinates (a, -1, 2), (5, 2, 4), and (14, b, 10) respectively. Find the values of a and b such that the three points are collinear.

(b) A car of mass 160kg is placed on a rough horizontal road with coefficient of friction 0.5. Finding the driving force when the car moves with

i. an acceleration of 5m^s-2  ( 4 Marks)

ii. a uniform speed of 3 ms^-1  (3 Marks)

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(a) Suppose the probability function of a discrete random variable X is given by:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)= \left\{\frac{4!}{\left(4–\mathrm{x}\right)!\mathrm{x}!}.\frac{1}{16},\right. \\ \mathrm{x}=0, 1, 2, 3, 4 \\ \mathrm{elsewhere} \\ \left(\mathrm{i}\right) \mathrm{Find} \mathrm{P}\left(\mathrm{X}\le 3\right). \left[3\mathrm{marks}\right] \\ \left(\mathrm{ii}\right) \mathrm{Obtain} \mathrm{the} \mathrm{distribution} \mathrm{function} \mathrm{of} \mathrm{x}. \left[1\mathrm{mark}\right] \\ \left(\mathrm{iii}\right) \mathrm{Find} \mathrm{P}\left(\mathrm{X}>0\right) \left[2\mathrm{marks}\right] \\ \left(\mathrm{iv}\right) \mathrm{Find} \mathrm{the} \mathrm{mean} \mathrm{and} \mathrm{variance} \mathrm{of} \mathrm{X}. \left[4\mathrm{marks}\right] \end{gathered}$$

(b) The following tales lists the frequency distribution for 60 rolls of a die.

Test at the 5% significance level whether the null hypothesis that the given die is fair true (5 Marks)

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(a)(i) Distinguish between type I and type II errors.

(ii) Past record shows that the means score of year II Accountancy students Business Statistics is 56% with a standard deviation of 7%. A random sample of 45 years  II students of this department was tested and the mean score was 52%. Verify at a = 0.5 if the average score of the students in this course has changed. (9 Marks)

(b) An Accountancy student claims he is 90% sure of the posting he made in all transactions.  Would you agree with this claim if, in a given transaction, he makes 20 out of 25 postings correctly? Use a = 0.005.

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