BTech Computing Programme Mathematics Test 2023
National University of Singapore
BTech Computing Programme - Mathematics Test
6 May 2023
Time Allowed: 2 Hours
NRIC/FIN/Passport No.
Instructions to Candidates
1. Please write your NRIC/FIN/PASSPORT NUMBER only. Do not write your name.
2. This assessment paper contains ELEVEN (11) questions and comprises TWELVE
(12) printed pages.
3. You are required to answer ALL questions and show FULL DETAILS of your
solution.
4. You are to use only BLACK or BLUE pen.
5. This is a CLOSED BOOK assessment.
6. The use of an electronic non-programmable calculator is expected, where
appropriate.
7. A Formula Sheet is provided on page 2 for your reference.
Question
Maximum Marks
1
10
2
5
3
5
4
6
5
7
6
8
7
6
8
7
9
9
10
7
11
10
Total
80
Marks Obtained
1
BTech Computing Programme Mathematics Test 2023
2
BTech Computing Programme Mathematics Test 2023
Question 1
(a) Solve the following simultaneous equations.
x2 - y2 = 7
2y = 2 + x
_________________ [5 marks]
(b) The straight line y - 1 = 2 m does not intersect the curve
.
Find the largest integer value of m.
_________________ [5 marks]
3
BTech Computing Programme Mathematics Test 2023
Question 2
Without using a calculator, solve, for x and y, the simultaneous equations:
64x x 4y = 1
3y x 811/x = 81
_________________ [5 marks]
Question 3
Solve the equation log 4 p - 8 log p 4 = 2.
_________________ [5 marks]
4
BTech Computing Programme Mathematics Test 2023
Question 4
The function f(x) = 3x3 + ax2 - 10x + b, where a and b are constants, is exactly divisible by
x - 4 and leaves a remainder of 5 when divided by x +1.
(a) Find the value of a and b.
_________________ [4 marks]
(b) Hence, solve f(x) = 0.
_________________ [2 marks]
5
BTech Computing Programme Mathematics Test 2023
Questions 5
(i) Given that the term independent of x is the binomial expansion of
is -672, find the value of the positive constant k.
_________________ [4 marks]
(ii) Using the value of k found in part (i), show that the constant term in the expansion of
is -1344.
_________________ [3 marks]
6
BTech Computing Programme Mathematics Test 2023
Question 6
The diagram below shows the trapezium ABCD where AD is parallel to BC. It is given that A
is the point (0,10), B is (-10,8), D is (6,6) and angle ADC is a right angle.
(i) Given that C lies on the x-axis, find the equation of the perpendicular bisector of BC.
_________________ [6 marks]
(ii) Find the area of triangle ABD.
_________________ [2 marks]
7
BTech Computing Programme Mathematics Test 2023
Question 7
Solve each of the following equations, for x is between 0o and 360o.
(i) sin 0.5 x = 0.8
_________________ [3 marks]
(ii) sin x = 3 cos x
_________________ [3 marks]
8
BTech Computing Programme Mathematics Test 2023
Question 8
The equation of a curve is y = (x + 3)(ax2 + b). The curve passes through the point (0,9)
and when x = -1, dy/dx = d2y/dx2 = 0. Find
(i) the value of a and b.
_________________ [3 marks]
(ii) the stationary point of the curve and determine with working, the nature of the stationary
point.
_________________ [4 marks]
9
BTech Computing Programme Mathematics Test 2023
Question 9
Sand is being delivered from a conveyor belt onto a conical container at a rate of 2.5 m3/min.
The height of the cone is twice the length of the radius.
(i) Show that the volume of sand in the container is represented by
_________________ [2 marks]
(ii) Find the rate of increase in the height of the sand when the height is 3 m.
_________________ [3 marks]
(iii) Find the rate of change of area of sand at the top when the height is 3 m.
_________________ [4 marks]
10
BTech Computing Programme Mathematics Test 2023
Question 10
In the diagram, the curve x = (y - 1)2 + 4 and the line y + x = 7 intersect at A and B.
(i) Find the coordinates of A and B.
_________________ [3 marks]
(ii) Calculate the area of the shaded region.
_________________ [4 marks]
11
BTech Computing Programme Mathematics Test 2023
Question 11
A particle moves in a straight line so that, t seconds after leaving a fixed point O, its
displacement, s meters, is given by
(i) Find the minimum velocity.
_________________ [4 marks]
(ii) Show that the particle will never return to its starting point O.
_________________ [2 marks]
(iii) Find the distance travelled by the particle and its average speed in the first 4 seconds.
_________________ [4 marks]
END OF PAPER
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