## KNEC KCSE Mathematics Paper 2 – 2014 KCSE COMA Joint Exam

### 2014 KCSE COMA Joint Exam

#### Mathematics Paper 2

### SECTION I (50 Marks)

**Answer all questions in this section.**

A pyramid block has a square base whose side is exactly 7.5cm. Its height measured to the nearest millimeter is 3.5cm. Find the percentage error in calculating its volume correct to 3 decimal places.

3 marks

A blend of juice is made from pineapple and passion. The cost of two limes of pineapple is

120/= and three limes of passion is 270/=. In what ratio should the juice be mixed such that

by selling the mixture at 84/= per lime a profit of 20% is realized?

3 marks

Solve for* x *in (log_{2} *x*)^{2} + log_{2} 8 = log_{2} *x*^{4}

3 marks

In the figure shown below, angle PTS = 54° and PT and ST are tangents to the circle and that PR is parallel to TS.

Giving reasons; find the values of angles:

(i) PRQ. (2 marks)

(ii) RQS. (2 marks)

4 marks

Given that tan15°=2+3 , find without using tables or a calculator, in the form a+2 c, the value of tan 75°.

3 marks

Make P the subject of the formula:

3 marks

Expand up to the 5^{th} team. Hence use the expansion to evaluate (3.25)^{5} correct to 4 decimal places.

3 marks

A commercial plot is valued at shs.500,000. The plot depreciates at a rate of 10% per six months for a period of 2 years. It then appreciates at a rate of 4% per quarter yearly for three years.

Find the value of the plot after 5 years to nearest shillings.

3 marks

The equation of a curve is y =*x ^{3}* – 3

*x*+ K

^{2}*x*+ 2 and a normal is 9y +

*x*= 18. If they intersect at

*x*= 0; Find the value of K.

3 marks

The figure below drawn toscalerepresentsa field in the shape of an equilateral triangle of sides 120m.

Mr. Mutai wants to plant some tea seedlings in the field. The seedling must be atmost 90m from A and nearer to B than to C. If no seedling is to be more than 60m from BC, show by shading, the exact region where theseedling may be planted within the triangle.

4 marks

The product ofthe digits ina two digit number is 24. Four times the ten digit exceeds the unit digit by

10. Calculate the number.

3 marks

Solve for *x* in the equation sin^{2}(3*x* + 30°) =34 for 0^{0}≤x≤180^{0}

3 marks

AKenya airways plane flies from point P(40°N, 45°W) to a point Q(35°N, 45°W), thento point T(35°N, 135°E). Find the shortest distance between Q and T in nautical miles.

2 marks

The position vectors of points A and B are 2i~-j~+4K~ and 4i~+3j~ respectively. If point R is the midpoint of AB→ . Find the magnitude of AR→

3 marks

Water flows through a pipe whose cross sectional radius is 3.5cm at a rate of 3m/min. Calculate how long it will take the pipe to fill a 22000 line Ken tank.

2 marks

The figure below shows an arc of a circle throughthree points A, B and C.

Calculate the co-ordinates of the centre of the circle.

4 marks

The diagram above shows triangle ABC, such that CA~→=2a~ and CB~→=3b~. M is the midpoint of CA→. N is a point on AB such that 2 AN = NB and R is a point on AB produced such that 2 AR = 5RB. If K is the point of intersection of MR and CN,

(a) Express in terms of a~ and b~.

(i) AB. (1 mark)

(ii) CN. (2 marks)

(iii) BR. (1 mark)

(iv) MR. (2 marks)

(v) CK. (2 marks)

(b) Find the ratio CK : KN. (2 marks)

10 marks

### SECTION II (0 Marks)

**Answer any five questions.**

Fill the table below using the following function y=3+4*x*+2*x*^{2} for -3≤×≤5 (2 marks)

x |
-3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |

-2x^{2} |
-18 | _ | _ | 0 | _ | -8 | _ | -50 | |

4x |
_ | _ | -4 | _ | 4 | _ | 12 | _ | _ |

3 | _ | _ | _ | _ | _ | _ | _ | _ | _ |

y | _ | _ | _ | _ | _ | _ | _ | _ | _ |

(b) On the grid provided, draw the graph of the function y=3+4*x*+2*x*^{2} for -3≤x≤5. (3 marks)

(c) Using your graph; estimate the roots of the equations:-

(i) 3 + 4*x* = 2*x ^{2}*. (2 marks)

(ii) 2

*x*

^{2}– 3

*x*– 6 = 0. (2 marks)

(d) State the y – co-ordinate of the maximum turning point. (1 mark)

10 marks

(a) P, Q and R are three quantities such that P varies directly as the square of Q and inversely as

the square root of R.

(i) Given that P = 12 when Q = 24 and R = 36, find P when Q =27 and R = 121. (3 marks)

(ii) If Q increases by 10% and R decreases by 25%, find the percentage increase in P.(4 marks)

(b) If Q is inversely proportional to the square root of P and P = 4 when Q = 3. Calculate the

value of P when Q = 8. (3 marks)

10 marks

Every morning during class time, Brenda either reads a novel or solves Mathematics questions. The

probability that she reads a novel is 45. If she reads a novel, there is a probability of 37 that she will fall asleep. If she solves Math’s questions there is a probability of 12 that she will fall asleep. Sometimes the teacher on duty enters Brenda’s classroom. When Brenda is asked whether she had been a sleep, there is a probability of 15 that she will admit that she had been asleep and a probability of 35 that she will claim to have been asleep.

Using a tree diagram, find the probability that

(i) She sleeps and admits it. (2 marks)

(ii) She sleeps and does not admit. (2 marks)

(iii) She does not sheep but claims to have been asleep. (2 marks)

(iv) She does not sleep and says that she has not been a slept. (2 marks)

(v) She sleeps and admits and changes her mind. (2 marks)

10 marks

The table below shows the distribution of marks scored by 50 students of Afraha high.

Marks | 11-20 | 21-20 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 | 81-90 | 91-100 |

No. of students | 2 | 3 | 5 | 6 | 12 | 10 | 6 | 4 | 2 |

Calculate:-

(a) interquartile range. (3 marks)

(b) Mean mark. (3 marks)

(c) Standard deviation (4 marks)

10 marks

Two quantities P and r are connected by the equation P = Krⁿ. Where k and n are constants.The table

of values of P and r is given below.

P | 1.2 | 1.5 | 2.0 | 2.5 | 3.5 | 4.5 |

r | 1.58 | 2.25 | 3.39 | 4.774 | 7.86 | 11.5 |

(a) State the linear equation connecting P and r. (1mark)

(b) (i) Using a suitable scale, draw a suitable line graph from the above data on the grid provided. (5 marks)

(ii) Using your graph, estimate the values of K and n. (3 marks)

(c) Find the relation connecting P and r. (1 mark)

10 marks

The product of the first three terms of geometric progression is 729. If the first term is a and

the common ratio is r.

(a) Express r in terms of a. (2 marks)

(b) Given the sum of the three terms is 39.

(i) Find the values of a and r and hence write down two possible sequences each

up to the 4^{th} term. (6 marks)

(ii) Find the product of the 10^{th} term of the two sequences. (2 marks)

10 marks

The velocity of a particle, Vm/s, moving in a straight line after t seconds is given by:- V = 3t² – 3t – 6 Find:-

(i) the acceleration of the particle after 2 seconds. (2 marks)

(ii) the distance covered by the particle between t = 1 and t = 4 seconds. (3 marks)

(iii) the time when the particle is momentarily at rest. (2 marks)

(iv) The maximum velocity attained by the particle. (3 marks)

10 marks