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UNIVERSITY ENTRANCE EXAMINATION 2016

MATHEMATICS (‘AO’ LEVEL EQUIVALENT)

Time allowed : 2 hours

INSTRUCTIONS TO CANDIDATES

1. This examination paper has TWO (2) sections − A and B, and comprises TEN

(10) printed pages.

2. Attempt all sections.

3. Answer all questions in section A. Indicate your answers on the answer paper pro-

vided. Each question carries 2 marks. Marks will not be deducted for wrong

answers.

4. Answer not more than FIVE (5) questions from Section B. Write your answers on

the answer paper provided. Begin each question on a fresh sheet of paper. Write

the question number clearly. Each question carries 12 marks.

5. A non-programmable scientific calculator may be used. However, candidates should

lay out systematically the various steps in the calculation.

6. At the end of the examination, attach the cover paper on top of your answer script.

Complete the information required on the cover page and tie the papers together

with the string provided. The colour of the cover paper for this examination is

GREEN.

7. Do not take any paper, including the question paper and unused answer paper, out

of the examination hall.

1UEE 2016 - Mathematics AO / 2

SECTION A (40 Marks)

Answer all questions in this section. Each question carries 2 marks.

1. Suppose p > 0 and logp 8 + logp n = 0. Then the value of n is

(A) −8

(B) −18

(C) 18

(D) 8

(E) none of the above

2. The maximum value of the function f(x) = (3− x)(x− 4) is

(A) 14

(B) 12

(C) −12

(D) 12


3. The derivative ofln 5x

e5xwith respect to x is

(A) e−5x(1

x− 5 ln 5x

)(B) e−5x

(5

x− 5 ln 5x

)(C) e−5x

(5

x− ln 5x

)(D) e−5x

( 1

5x− 5 ln 5x

)(E) none of the above

2

4. The line y + 3x = k is a tangent to the curve y = x2 + 5. The value of k is

(A) −94

(B) 94

(C) −114

(D) 114


5. Which of the following is the result of completing the square of the expression

−5x2 + 3x− 1?

(A) −5(x− 3

10

)2− 11

20

(B) −5(x− 3

5

)2− 14

5

(C) −5(x− 3

5

)2+

4

5

(D) −5(x− 3

10

)2− 29

20


6. The integral ∫cos 2π dx

equals

(A) sin 2π + C

(B) 12

sin 2π + C

(C) − sin 2π + C

(D) −12

sin 2π + C


3

7. If f(x) = 2mx+12x+2

and f(12) = 1, then the value of m is equal to

(A) 2

(B) 3

(C) 4

(D) 5


8. The derivative of tan(tanx) with respect to x is

(A) sec2(sec2 x)

(B) sec2(tanx) tanx

(C) sec2(tanx) sec2 x

(D) tan(sec2 x) sec2 x


9. Two bags each contain ten balls which are identical apart from colour. Bag A

contains 4 red balls and 6 black balls. Bag B contains 7 red balls and 3 black balls.

A ball is chosen at random from Bag A and placed in Bag B. Then, after thoroughly

mixing, a ball is taken from Bag B and placed in Bag A. The probability that Bag

A still contains exactly 4 red balls is

(A) 24110

(B) 32110

(C) 54110

(D) 56110


4

10. Which option corresponds to the partial fraction decomposition of the rational

function13

−6x2 + 5x+ 6?

(A)2

2x− 3+

3

3x+ 2

(B)2

2x− 3− 3

3x+ 2

(C) − 2

2x− 3+

3

3x+ 2

(D) − 2

2x− 3− 3

3x+ 2


11. Two parallel lines are drawn. Six distinct points are marked on each line. The

number of distinct triangles which may be formed using 3 of the 12 points as

vertices is

(A) 42

(B) 90

(C) 180

(D) 220


12. The integral ∫e2√ex+ 5

dx

equals

(A) e√ex+ 5 + C

(B) 1e

√ex+ 5 + C

(C) 2e√ex+ 5 + C

(D) 2e

√ex+ 5 + C


5

13. The sum of 30 consecutive numbers is 1215. The smallest value of these numbers is

(A) 24

(B) 25

(C) 26

(D) 27


14. The maximum value of the function f(x) = (6 cosx− 5)2 + 6 is

(A) 6

(B) 31

(C) 42

(D) 127


15. The line y = 2x+ 1 is reflected about the line y = 5. The equation of the reflected

line is

(A) y = −2x+ 5

(B) y = −2x+ 9

(C) y = 2x+ 5

(D) y = 2x+ 9


6

16. The position vectors of the points A, B and C, relative to the origin O, are 6i +

10j, −2i + 2j and pj respectively. If−→AB is perpendicular to

−→AC, the value of p is

equal to

(A) −16

(B) −8

(C) 8

(D) 16


17. A geometric progression has first term a and common ratio 1√3. The sum to infinity

of the progression is

(A) a(√

3 + 3)

(B) a2(√

3 + 3)

(C) a4(√

3 + 3)

(D) a8(√

3 + 3)


18. The derivative ofcos(x2)

x2with respect to x is

(A)x sin(x2)− 2 cos(x2)

x3

(B)2x2 sin(x2)− 2 cos(x2)

x3

(C)−x sin(x2)− 2 cos(x2)

x3

(D)−2x2 sin(x2)− 2 cos(x2)

x3


7

19. Suppose lg x = a and lg y = b. Then lg

√100x2

yin terms of a and b is

(A) a− 1 + 12b

(B) a− 1− 12b

(C) a+ 1 + 12b

(D) a+ 1− 12b


20. Suppose −7 ≤ x ≤ −3 and 2 ≤ y ≤ 9. Then the largest value of (y − 5)2 + x2 is

(A) 18

(B) 25

(C) 58

(D) 65


8

SECTION B (60 Marks)

Answer FIVE (5) questions in this section. Each question carries 12 marks.

21(a). Show that the polynomial f(x) = x4−3x3−6x2 +8x is exactly divisible by (x−1).

Find all the factors of f(x) and solve the equation f(x) = 0. [6 Marks]

21(b). Find the range of values of k for which the line 4x = (k − 2)y intersects the curve

xy = 2− x at two distinct points. [6 Marks]

22(a). Differentiate x2 lnx with respect to x. Hence, evaluate

∫ 3

1

x lnx dx. [6 Marks]

22(b). Find the area enclosed between the curves y = x2− 8x+ 13 and y = −7 + 6x− x2.

[6 Marks]

23(a). The eleventh term of an arithmetic progression is 24. The sum of the first eleven

terms is 99. Find the first term, the common difference and an expression for the

nth term in its simplest form. [6 Marks]

23(b). Solve the following equations

(i) ex = 4e−x + 3. [3 Marks]

(ii) log5(x− 1) + log5(2x− 5) = 1. [3 Marks]

24. Find all the angles between 0◦ and 360◦ inclusive which satisfy the equation

(a) tan(2x− 60◦) = cot 60◦. [4 Marks]

(b) 2 cos 2y + 3 sin y = 1. [4 Marks]

(c) 3 cot z − 4 cos z = 0. [4 Marks]

9

25(a). Find the coefficient of x5 in the expansion of (1− 2x)(1 + x3)7. [6 Marks]

25(b). Find the coefficient of x−12 in the expansion of (x3 − 1x)24. [6 Marks]

26(a). A curve is given parametrically by the equations

x = (2t− 3)2 and y = 3t− 7.

At the point P on the curve, the value of the parameter t is 2. The tangent at P

meets the x and y axes at points Q and R respectively. Find the coordinates of Q

and R. [6 Marks]

26(b). The equations of a curve, in terms of a parameter θ, are given by

x = sec θ + tan θ and y = sec θ − 2 tan θ.

Express sec θ and tan θ in terms of x and y. Hence, obtain in simplified form, the

cartesian equation of the curve. [6 Marks]

END OF PAPER

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