Sulvme!

Unified Tertiary Matriculation Examination Mathematics 2005

1:

Find the value of m if 13m + 24m = 41m

A:

8

B:

6

C:

5

D:

2

E:

No option

2:

If 3214 is divided by 234 and leaves a remainder r, what is the value of r?

A:

0

B:

1

C:

2

D:

3

E:

No option

3:

Simplify $3\frac{1}{2} - \left ( 2\frac{1}{3} \times 1\frac{1}{4} \right ) + \frac{3}{5}$

A:

$2\frac{11}{60}$

B:

$2\frac{1}{60}$

C:

$1\frac{11}{60}$

D:

$1\frac{1}{60}$

E:

No option

4:

A father decided to give 20% of his monthly income to his three children as their monthly allowance. The eldest child got 45% of the allowance and the youngest got 25%. How much was the father's monthly income if the second child got #3,000 ?

A:

#33,000

B:

#45,000

C:

#50,000

D:

#60,000

E:

No option

5:

If the interest on #150.00 for $2\frac{1}{2}$ years is #4.50, find the interest on #250.00 for 6 months at the same rate.

A:

#1.50

B:

#7.50

C:

#15.00

D:

#18.00

E:

No option

6:

Three boys shared some oranges, the first received $\frac{1}{3}$ of the oranges and the second received $\frac{2}{3}$ of the remainder. If the third boy received the remaining 12 oranges, how many oranges did they share?

A:

60

B:

54

C:

48

D:

42

E:

No option

7:

Evaluate $\frac{81^{\frac{3}{4}} - 27^{\frac{1}{3}}}{3 \times 2^{3}}$

A:

3

B:

1

C:

$\frac{1}{3}$

D:

$\frac{1}{8}$

E:

No option

8:

If log102 = 0.3010 and log103 = 0.4771, evaluate log104.5

A:

0.9542

B:

0.6532

C:

0.4771

D:

0.3010

E:

No option

9:

Simplify $\frac{\sqrt{12} - \sqrt{3}}{\sqrt{12} + \sqrt{3}}$

A:

0

B:

$\frac{1}{3}$

C:

$\frac{3}{5}$

D:

1

E:

No option

10:

If U = {x/x is a positive integer less than 10} and P = {x/ x is a prime factor of 30}, find the complement of P

A:

{1, 2, 4. 7. 8. 9}

B:

{1, 2, 4. 6, 7. 8. 9}

C:

{1, 4. 6, 7. 8. 9}

D:

{1, 4. 7. 8. 9}

E:

No option

11:

The venn diagram below shows a class of 40 students with the games they play. How many of the students play two games only?

A:

19

B:

16

C:

15

D:

4

E:

No option

12:

If m = 3, p = -3, q = 7 and $r = \frac{5}{2}$, evaluate m(p + q + r)

A:

19.50

B:

19.15

C:

18.95

D:

18.05

E:

No option

13:

Divide 6x2 - 13x + 5 by 2x - 1

A:

3x + 5

B:

3x - 5

C:

5x - 3

D:

5x + 3

E:

No option

14:

A polynomial in x whose zeros are -2, -1 and 3 is

A:

x3 - 7x + 6

B:

x3 + 7x - 6

C:

x3 + 7x + 6

D:

x3 - 7x - 6

E:

No option

15:

The time taken to do a piece of work is inversely proportional to the number of men employed. If it takes 30 men to do a piece of work in 6 days, how many men are required to do the work in 4 days?

A:

20

B:

35

C:

45

D:

60

E:

No option

16:

The weight W kg of a metal bar varies jointly as its length L metres and the square of its diameter d metres. If W = 140 and d = $4\frac{2}{3}$ and L = 54, find d in terms of W and L

A:

$\sqrt{\frac{42W}{5L}}$

B:

$\sqrt{\frac{5L}{42W}}$

C:

$\frac{42W}{5L}$

D:

$\frac{5L}{42W}$

E:

No option

17:

Find the range of values of x for which 7x - 3 > 25 + 3x

A:

x > 7

B:

x < 7

C:

x > -7

D:

x < -7

E:

No option

18:

The diagram below is the graph of the function f(x). Determine the range of values of x for which $f(x) \leq 0$

A:

$x \leq 2$

B:

$0 \leq x \leq 2$

C:

$-2 \leq x \leq 0$

D:

$x \leq -2, 0 \leq x \leq 2$

E:

No option

19:

If the 7th term of an AP is twice the third term and the sum of the first four terms is 42, find the common difference

A:

6

B:

3

C:

2

D:

1

E:

No option

20:

Find the sum of the first 20 terms of the series 8, 12, 16, ........., 96

A:

1400

B:

1040

C:

960

D:

920

E:

No option

21:

An operation * is defined on the set of real numbers by a * b = ab + 2 (a + b + 1). Find the identity element.

A:

2

B:

1

C:

-1

D:

-2

E:

No option

22:

If M and N are two matrices defined by

$M = \begin{pmatrix} 1 & 3 & 2\\ 4 & 5 & -1\\ -3 & 2 & 0 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & -2 & 3\\ 4 & -1 & 5\\ 2 & -3 & -1 \end{pmatrix}$

Evaluate 2M - 3N

A:

$\begin{pmatrix} -1 & 12 & 5\\ 4 & 7 & 13\\ 0 & -5 & -3 \end{pmatrix}$

B:

$\begin{pmatrix} -1 & 0 & -5\\ -4 & 7 & -17\\ 0 & -5 & 3 \end{pmatrix}$

C:

$\begin{pmatrix} -1 & 12 & -5\\ -4 & 13 & -17\\ -12 & 13 & 3 \end{pmatrix}$

D:

$\begin{pmatrix} -1 & 12 & -5\\ 4 & 13 & 13\\ -12 & 13 & 3 \end{pmatrix}$

E:

No option

23:

If $P = \begin{pmatrix} 1 & 3 & 2\\ 4 & 5 & -1\\ -3 & 2 & 0 \end{pmatrix}$, find the determinant of matrix P

A:

75

B:

57

C:

-57

D:

-75

E:

No option

24:

In the diagram below, calculate the value of x.

A:

600

B:

1000

C:

1200

D:

1400

E:

No option

25:

Three straight lines EF, GH and LK intersect at O as shown below. If $\angle KOF = 52^{0}$ and $\angle LOH = 85^{0}$, calculate the size of $\angle EOG$

A:

260

B:

430

C:

520

D:

850

E:

No option

26:

The sum of the interior angles of a regular polygon is 18000. Calculate the size of one exterior angle of the polygon

A:

300

B:

240

C:

180

D:

120

E:

No option

27:

In the diagram below, O is the centre of the circle, $\angle UOT = 70^{0}$ and $\angle RST = 100^{0}$, Calculate $\angle RUO$

A:

200

B:

250

C:

500

D:

800

E:

No option

28:

A chord of a circle subtends an angle of 600 at the centre of a circle of radius 14cm. Find the length of the chord

A:

7cm

B:

14cm

C:

21cm

D:

28cm

E:

No option

29:

A sector of a circle has an area of 55cm2. If the radius of the circle is 10cm, calculate the angle of the sector.

($\pi = \frac{22}{7}$)

A:

450

B:

630

C:

750

D:

900

E:

No option

30:

Find the curved surface area of a cone with circular base diameter 10cm and height 12cm

A:

$25 \pi cm^{2}$

B:

$65 \pi cm^{2}$

C:

$120 \pi cm^{2}$

D:

$156 \pi cm^{2}$

E:

No option

31:

Two lines PQ and ST intersect at 750. The locus of points equidistant from PQ and ST lies on the

A:

perpendicular bisector of PQ

B:

perpendicular bisector of ST

C:

bisector of the angles between lines PQ and ST

D:

bisector of the angles between lines PT and QS

E:

No option

32:

Find the equation of the perpendicular at point (4, 3) to the line y + 2x = 5

A:

2y - x = 4

B:

y + 2x = 3

C:

y + 2x = 5

D:

2y - x = 2

E:

No option

33:

Find the coordinates of the midpoint of the line joining (3, -4) and (-1, 10)

A:

(1, 3)

B:

(1, 2)

C:

(2, 3)

D:

(3, 2)

E:

No option

34:

If sin $\theta$ = $-\frac{1}{2}$ for 0 < $\theta$ < 3600, the value of $\theta$ is

A:

300 and 1500

B:

1500 and 2100

C:

2100 and 3300

D:

1500 and 3300

E:

No option

35:

From the diagram below, find the bearing of R from S

A:

2260

B:

2240

C:

1360

D:

1340

E:

No option

36:

If y = (1 - 2x)3, find the value of $\frac{dy}{dx}$ at x = -1

A:

57

B:

27

C:

-6

D:

-54

E:

No option

37:

Find the derivative of y = sin (2x3 + 3x - 4)

A:

cos (2x3 + 3x - 4)

B:

-cos (2x3 + 3x - 4)

C:

(6x2 + 3) cos (2x3 + 3x - 4)

D:

- (6x2 + 3) cos (2x3 + 3x - 4)

E:

No option

38:

The radius r of a circular disc is increasing at the rate of 0.5cm/sec. At what rate is the area of the disc increasing when its radius is 6cm?

A:

36 $\pi$ cm2 /sec

B:

18 $\pi$ cm2 /sec

C:

$\pi$ cm2 /sec

D:

$\pi$ cm2 /sec

E:

No option

39:

The maximum value of the function

f(x) = 2 + x - x2 is

A:

$\frac{9}{4}$

B:

$\frac{7}{4}$

C:

$\frac{3}{2}$

D:

$\frac{1}{2}$

E:

No option

40:

Find the area of the figure bounded by the given pair of curves y = x2 - x + 3 and y = 3

A:

$\frac{17}{6}$ units (sq)

B:

$\frac{7}{6}$ units (sq)

C:

$\frac{5}{6}$ units (sq)

D:

$\frac{1}{6}$ units (sq)

E:

No option

41:

Evaluate $\int_{0}^{\frac{\pi}{2}}sin 2xdx$

A:

1

B:

0

C:

$-\frac{1}{2}$

D:

-1

E:

No option

42:

The histogram below shows the distribution of the monthly incomes of the workers in a company. How many workers earn more than #700.00?

A:

16

B:

12

C:

8

D:

6

E:

No option

43:

The grades of 36 students in a test are shown in the pie chart above. How many students had excellent?

A:

7

B:

8

C:

9

D:

12

E:

No option

44:

The table below shows the scores of a group of students in a test. If the average score is 3.5, find the value of x

A:

1

B:

2

C:

3

D:

4

E:

No option

45:

The modal height and range of heights 1.35, 1.25, 1.35, 1.40, 1.35, 1.45, 1.50, 1.35, 1.50 and 1.20 are m and r respectively. Find m + 2r

A:

1.35

B:

1.65

C:

1.95

D:

3.00

E:

No option

46:

Find the value of t if the standard deviation of 2t, 3t, 4t, 5t and 6t is $\sqrt{2}$

A:

1

B:

2

C:

3

D:

4

E:

No option

47:

In how many ways can 6 coloured chalks be arranged if 2 are of the same colour?

A:

60

B:

120

C:

240

D:

360

E:

No option

48:

How many possible ways are there of seating seven people P, Q, R, S, T, U and V at a circular table?

A:

360

B:

720

C:

2520

D:

5040

E:

No option

49:

A box contains 5 blue balls, 3 red balls and 2 white balls. Two balls are selected from the box with replacement. Find the probability of obtaining two blue or two red balls

A:

$\frac{17}{50}$

B:

$\frac{3}{25}$

C:

$\frac{1}{50}$

D:

$\frac{3}{250}$

E:

No option

50:

What is the probability that an integer x, $(1 \leq x \leq 20)$ chosen at random is divisible by both 2 and 3

A:

$\frac{1}{20}$

B:

$\frac{1}{3}$

C:

$\frac{3}{20}$

D:

$\frac{7}{10}$

E:

No option