2023 Mathematics II MCB MCE MEG MPC MPG PCM

Mathematics II
029
NATIONAL EXAMINATION AND
25/07/2023 08.30 AM -11.30 AM SCHOOL INSPECTION
AUTHORITY
ADVANCED LEVEL NATIONAL EXAMINATIONS, 2022-2023
SUBJECT: MATHEMATICS II
COMBINATIONS:
- MATHEMATICS-CHEMISTRY-BIOLOGY (MCB)
- MATHEMATICS -COMPUTER SCIENCE-ECONOMICS (MCE)
- MATHEMATICS-ECONOMICS-GEOGRAPHY (MEG)
- MATHEMATICS -PHYSICS-COMPUTER SCIENCE (MPC)
- MATHEMATICS-PHYSICS-GEOGRAPHY (MPG)
- PHYSICS-CHEMISTRY-MATHEMATICS (PCM)
DURATION: 3 HOURS
INSTRUCTIONS:
1) Write your names and index number on the answer booklet as written on your
registration form, and DO NOT write your names and index number on
additional answer sheets if provided.
2) Do not open this question paper until you are told to do so.
3) This paper consists of two sections: A and B.
Section A: Attempt all questions. (55 marks)
Section B: Attempt only three questions. (45 marks)
4) Geometrical instruments and silent non-programmable calculators
may be used.
5) Use only a blue or black pen.
2022-2023-NESA (National Examination and School Inspection Authority) 029 - Page 1 of 4

SECTION A: ATTEMPT ALL QUESTIONS (55 marks)
1) State the cosine and sine laws which are used to solve practical
problems involving triangles and angles. (4 marks)
2) In a geometric progression, insert 4 geometric terms that are

between 2 and 6250. (4 marks)
x −1 1− x
3) Solve e − 18e − 3 = 0 (4 marks)
4) Find the differential equation of all straight lines passing through
the origin. (2 marks)
5) Find the vector, parametric and symmetric equations of the line (l )
passing through the point A ( 3, −2, 4 ) with direction vector u = ( 2,3,5) . (4 marks)
6) Find square roots of the complex number 3+4i (3 marks)
7) From the top of a cliff, 100 m above sea level, the angle of
depression to a ship sailing past is 17 degrees. How far is the ship
from the base of the cliff to the nearest meter? (3 marks)
8) Use Gauss – Jordan method of elimination to solve:
− 3 x − 2 y + 4 z = 9

 3y − 2z = 5
 4x − 3y + 2z = 7
 (4 marks)
9) Write the 3 terms of the Maclaurin expansion of f (x) = ln(1 + e x ) (3 marks)

10) Let A be a 2 × 2 matrix with 𝑡𝑟(𝐴) = 6 and 𝑑𝑒𝑡(𝐴) = 5.
Find the eigenvalues of 𝑨. (3 marks)
11) Let 𝑡: ℝ3 → ℝ3 be a linear transformation such that

𝑡(1,0,0) = (2,4, −1), 𝑡(0,1,0) = (1,3, −2) and 𝑡(0,0,1) = (1, −2,2).
Find 𝑡(0,3, −1). (3 marks)
12) Verify if T = E 2 → E 2 defined by T ( ( x1 , x2 ) ) = ( x1 + x2 , x1 − x2 + 1)

is linear or not linear. (3 marks)
13) Suppose that you are observing the behavior of cell duplication in
a laboratory. If, in one of the experiments, you started with
1,000,000 cells and the cell population decreased by ten percent
every minute.
a) Write an equation with base (0.9) to determine the number
of cells after t minutes. (1 mark)
b) Determine how long it would take the population to reach
a size of 10 cells. (3 marks)

y = 2x +1
14) Given that: y = 3 x + 1
x=4
a) Find the coordinate points of intersection of the lines. (3 marks)
b) Sketch a graph of the given lines in the same
two dimensions. (2 marks)
c) Find the area of the region found in (b). (2 marks)
15) a) Find the constant c such that the function :

cx 2 0 x3
f ( x) = 
0 otherwise
is a density function (2 marks)

b) Compute P (1  x  2 ) (2 marks)
SECTION B: ATTEMPT THREE QUESTIONS ONLY (45 marks)

1
16) Give that 𝑓(𝑥) = 𝑥 2 𝑒 𝑥+1
2
a) Find the domain of 𝑓(𝑥), (1 marks)
b) Find relative asymptotes (if any), (4 marks)
c) Study the first and second derivative with variation table, (8 marks)
d) Sketch the curve of 𝑓(𝑥). (2 marks)
17) Given the differential equation
d2y dy
2
− 2k + k 2 y = 12 xekx , k  0
dx dx
a) Find a general solution of differential equation given that
y = Px3ekx where 𝜬 is a constant and part of the solution. (11 marks)
dy
b) Given further that y = 1, = 0 at x = 0 show that
dx
y = ekx (2 x3 − kx + 1) (4 marks)

18) The table below shows the marks scored by 10 students in
Biology and Chemistry test.
Biology (x) 8 7 6 9 8 9 7 8 5 6
Chemistry (y) 7 8 7 9 8 8 7 9 7 5
a) Find Mean, (5 marks)

b) Find Variance, (1 mark)
c) Find Standard deviation of x and y (1 mark)
d) Find covariance of x and y (1 mark)
e) Find correlation coefficient r and interpret it (2 marks)
f) Find an equation of a line that best fits
in the form of 𝑦 = 𝑎 + 𝑏𝑥 (3 marks)
g) If a student scored 7.5 in Biology, predict his/her
score in Chemistry. (2 marks)
19) For the conic defined by 9 x 2 − 16 y 2 − 18 x − 64 y − 199 = 0
a) Write the given conic in a standard form. (2 marks)

b) Name the conic represented in (a). (1 mark)
c) Find the:
i) Coordinates of the centre. (2 marks)

ii) Vertices. (2 marks)
iii) Foci. (4 marks)
iv) Equations of the asymptotes then sketch the graph. (4 marks)
1 + 8i
20) It is given that Z=
1 − 2i
a) Express Z in the form x + iy , where x and y are real numbers. (4 marks)
b) Find the modulus and argument of Z . (4 marks)
2
c) Show clearly that: 𝑎𝑟𝑐𝑡𝑎𝑛 8 + 𝑎𝑟𝑐𝑡𝑎𝑛 2 + 𝑎𝑟𝑐𝑡𝑎𝑛 3 = 𝜋 (7 marks)
-END-

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Mathematics II

ADVANCED LEVEL NATIONAL EXAMINATIONS, 2022-2023

2022-2023-NESA (National Examination and School Inspection Authority) 029 - Page 1 of 4

2) In a geometric progression, insert 4 geometric terms that are

9) Write the 3 terms of the Maclaurin expansion of f (x) = ln(1 + e x ) (3 marks)

11) Let 𝑡: ℝ3 → ℝ3 be a linear transformation such that

12) Verify if T = E 2 → E 2 defined by T ( ( x1 , x2 ) ) = ( x1 + x2 , x1 − x2 + 1)

2022-2023-NESA (National Examination and School Inspection Authority) 029 - Page 2 of 4

15) a) Find the constant c such that the function :

is a density function (2 marks)

SECTION B: ATTEMPT THREE QUESTIONS ONLY (45 marks)

17) Given the differential equation

2022-2023-NESA (National Examination and School Inspection Authority) 029 - Page 3 of 4

a) Find Mean, (5 marks)

19) For the conic defined by 9 x 2 − 16 y 2 − 18 x − 64 y − 199 = 0

a) Write the given conic in a standard form. (2 marks)

i) Coordinates of the centre. (2 marks)

2022-2023-NESA (National Examination and School Inspection Authority) 029 - Page 4 of 4

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