# Sulvme!

### Unified Tertiary Matriculation Examination Mathematics 2006

1:

The table above shows the scores of a group of students in a Physics test. If the mode is m and the number of students who scored 4 or more is n, what is (n, m)?

A:

(33, 4)

B:

(22, 4)

C:

(33, 12)

D:

(12, 4)

E:

No option

2:

A final examination requires that a student answer any 4 out of 6 questions. In how many ways can this be done?

A:

15

B:

20

C:

30

D:

45

E:

No option

3:

This response of 160 pupils in a school asked to indicate their favourite subjects is given in the bar chart above. What percentage of the pupils have English and Health Education as their favourite subjects?

A:

36%

B:

52%

C:

55%

D:

22%

E:

No option

4:

If the mean of five consecutive integers is 30, find the largest of the numbers.

A:

28

B:

30

C:

32

D:

34

E:

No option

5:

A bag contains 5 blacks, 4 white and x red marbles. If the probability of picking a red marble is $\frac{2}{5}$, find the value of x

A:

8

B:

10

C:

4

D:

6

E:

No option

6:

Find the variance of 2x, 2x - 1 and 2x + 1.

A:

$\frac{2}{3}$

B:

$2$

C:

$\sqrt{\frac{2}{3}}$

D:

$1$

E:

No option

7:

The table below shows the distribution of recharge cards of four major GSM operators. What is the probability that a recharge card selected at random will be GTN or Qtel?

A:

$\frac{3}{20}$

B:

$\frac{1}{4}$

C:

$\frac{2}{5}$

D:

$\frac{3}{4}$

E:

No option

8:

The pie chart below shows the expenditure of a family whose income is #30,000. If the expenditure on food is twice that on housing and that on school fees is twice that on transport, how much does the family spend on food?

A:

#28,000

B:

#25,500

C:

#15,000

D:

#12,500

E:

No option

9:

For what value of n is $^{n+1}C_{3} = 4(^{n}C_{3})?$

A:

6

B:

5

C:

4

D:

3

E:

No option

10:

The gradient of a curve is 2x + 7 and the curve passes through point (2, 0). Find the equation of the curve

A:

y = x2 + 7x + 9

B:

y = x2 + 7x - 18

C:

y = x2 + 7x + 18

D:

y = x2 + 14x + 11

E:

No option

11:

Differentiate $(x^{2} - \frac{1}{x})^{2}$with respect to x

A:

$4x^{3} - 4x - \frac{2}{x}$

B:

$4x^{3} - 2 + \frac{2}{x^{3}}$

C:

$4x^{3} - 2 - \frac{2}{x^{3}}$

D:

$4x^{3} - 3x + \frac{2}{x}$

E:

No option

12:

Find the value of x for which the function 3x3 - 9x2 is minimum

A:

0

B:

2

C:

3

D:

5

E:

No option

13:

If $\frac{dy}{dx} = x + cos(x)$, find y

A:

x2 - sin x + c

B:

x2 + sin x + c

C:

$\frac{x^{2}}{2} - sin(x)+c$

D:

$\frac{x^{2}}{2} + sin(x)+c$

E:

No option

14:

Differentiate $(cos \theta - sin \theta)^{2}$ with respect to $\theta$

A:

-2 cos 2$\theta$

B:

-2 sin 2$\theta$

C:

1 - 2 cos 2$\theta$

D:

1 - 2 sin 2$\theta$

E:

No option

15:

Evaluate $\int_{-4}^{1}(1 - 2x)dx$

A:

-16

B:

-20

C:

20

D:

10

E:

No option

16:

If $E \subseteq G \subseteq U$, where U is the universal set, then the shaded venn diagram representing U - E or Ec is

A:

coming

B:

coming

C:

coming

D:

coming

E:

No option

17:

Simplify $\left ( \frac{7}{9} - \frac{2}{3} \right ) + \left ( \frac{1}{3} + \frac{2}{5} \div \frac{4}{5} \right )$

A:

$\frac{11}{12}$

B:

$\frac{5}{6}$

C:

$\frac{1}{5}$

D:

$\frac{2}{15}$

E:

No option

18:

If m : n = 13 : 11, find m2 - n2 : (m + n)2.

A:

1 : 11

B:

1 : 13

C:

1 : 10

D:

1 : 12

E:

No option

19:

Calculate the logarithm to base 9 of $3^{-4} \times 9^{2} \times (81)^{-1}$

A:

2

B:

0

C:

-2

D:

-4

E:

No option

20:

If $(k2)_{6} \times 3_{6} = 3_{5}(k4)_{5}$, what is the value of k?

A:

1

B:

4

C:

3

D:

2

E:

No option

21:

In a small village of 500 people, 350 speak the local language while 200 speak pidgin English. What percentage of the population speak both?

A:

30%

B:

10%

C:

50%

D:

14%

E:

No option

22:

Find the tax on an income of #20,000 if no tax is paid on the first #10,000 and tax is paid at #50 in #1,000 on the next #5,000 and at #55 in #1,000 on the remainder.

A:

#500

B:

#552

C:

#255

D:

#525

E:

No option

23:

Rationalize $\frac{2}{6 - 5 \sqrt{3}}$

A:

$-\frac{12}{39} - \frac{10 \sqrt{3}}{39}$

B:

$\frac{12}{39} - \frac{10 \sqrt{3}}{39}$

C:

$-\frac{12}{39} + \frac{10 \sqrt{3}}{39}$

D:

$\frac{12}{39} + \frac{10 \sqrt{3}}{39}$

E:

No option

24:

Compute 1100112 + 111112

A:

10010102

B:

10100102

C:

10001102

D:

10001002

E:

No option

25:

Simplify $\left ( 25 \right )^{-\frac{1}{2}} \times \left ( 27 \right )^{\frac{1}{3}} + \left ( 121 \right )^{-\frac{1}{2}} \times \left ( 625 \right )^{-\frac{1}{4}}$

A:

$\frac{34}{55}$

B:

$\frac{9}{11}$

C:

$\frac{14}{5}$

D:

$\frac{3}{275}$

E:

No option

26:

Convert 22324 to a number in base six

A:

4506

B:

2546

C:

5536

D:

5406

E:

No option

27:

In the diagram above, |QR| is the diameter of the semicircle QR. Find the area of the figure to the nearest whole number

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

A:

89cm2

B:

70cm2

C:

90cm2

D:

80cm2

E:

No option

28:

If tan$\theta$ = $\frac{5}{4}$, find sin2$\theta$ - cos2$\theta$

A:

$\frac{5}{4}$

B:

$\frac{41}{9}$

C:

$\frac{9}{41}$

D:

$1$

E:

No option

29:

PQ and RS are two parallel lines. If the coordinates of P, Q, R, S are (1,q), (3,2), (3,4), (5,2q) respectively, find the value of q

A:

3

B:

4

C:

1

D:

2

E:

No option

30:

In a triangle XYZ, $\angle XYZ$ = 150, $\angle XZY$= 450 and |XY| = 7cm, Find |YZ|

A:

$14 \sqrt{2}cm$

B:

$\frac{7}{2} \sqrt{6}cm$

C:

$7 \sqrt{2} cm$

D:

7cm

E:

No option

31:

In the diagram above, find the value of x.

A:

550

B:

500

C:

450

D:

400

E:

No option

32:

In the diagram below, POQ is a diameter of a the circle PQRS. If $\angle PSR$ = 1450, find x0

A:

550

B:

450

C:

350

D:

250

E:

No option

33:

What is the locus of points equidistant from the line ax + by + c = 0 ?

A:

A line bx - ay + q = 0

B:

A line ax - by + q = 0

C:

A line bx + ay + q = 0

D:

A line ax + by + q = 0

E:

No option

34:

PQRSTW is a rectangular hexagon and QS intersects RT at V. Calculate $\angle TVS$

A:

1200

B:

900

C:

300

D:

600

E:

No option

35:

If the locus of the points which are equidistant from points P and Q meets line PQ at point N, then PN equals

A:

NQ

B:

-NQ

C:

2NQ

D:

-2NQ

E:

No option

36:

In the diagram below, PQ = 10cm, PS = 8cm and $\angle PSR$ is 600 while  $\angle SRQ$ is a right angle. Find SR

A:

14cm

B:

$14 \sqrt{3}$ cm

C:

10 cm

D:

$10 \sqrt{3}$ cm

E:

No option

37:

A binary operation $\Theta$ is defined on the set of real numbers is such that x$\Theta$y = $\frac{xy}{6}$ for all x, y $\epsilon$ R. Find the inverse of 20 under this operation when the identity element is 6

A:

$\frac{1}{12}$

B:

$\frac{10}{3}$

C:

$\frac{1}{20}$

D:

$\frac{9}{5}$

E:

No option

38:

Find the roots of x3 - 2x2 - 5x + 6 = 0.

A:

1, 2, -3

B:

-1, -2, 3

C:

-1, 2, -3

D:

1, -2, 3

E:

No option

39:

The solution set of the shaded area above is

A:

$y \geq 0, y \geq x$ and $y + x \leq 4$

B:

$y \leq x$, $y + x \leq 4$

C:

$y + x \geq 4$, $y \leq x$

D:

$y \leq x$, $y + x \leq 4$ and $y \geq 0$

E:

No option

40:

If x = $\begin{pmatrix} 1 & 0 & 1\\ 2 & -1 & 0 \\ -1 & 0 & 1 \end{pmatrix}$ and y = $\begin{pmatrix} -1 & 1 & 2\\ 0 & -1 & -1 \\ 2 & -1 & 1 \end{pmatrix}$ Find 2x - y

A:

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

B:

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

C:

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

D:

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

E:

No option

41:

If p varies inversely as the cube of q and q varies directly as the square of r, what is the relationship between p and r?

A:

p varies directly as r3

B:

p varies inversely as r6

C:

p varies inversely as $\sqrt[6]{r}$

D:

p varies directly as r6

E:

No option

42:

A binary operation * on the set of rational numbers is defined as x * y = $\frac{x^{2} - y^{2}}{2xy}$, Find -5 * 3

A:

$-\frac{8}{15}$

B:

$\frac{8}{15}$

C:

$\frac{17}{15}$

D:

$-\frac{17}{15}$

E:

No option

43:

Find the value of k if the expression kx3 + x2 - 5x - 2 leaves a remainder 2 when it is divided by 2x + 1

A:

10

B:

8

C:

-10

D:

-8

E:

No option

44:

Solve the inequality for which

$\frac{x + 4}{3} - \frac{(x - 3)}{2} < 4$

A:

x < 7

B:

x > -7

C:

x < -7

D:

x > 7

E:

No option

45:

Find p, q for which $\begin{pmatrix} 2p & 8\\ 3 & -5q\end{pmatrix}$$\begin{pmatrix} 1\\ 2 \end{pmatrix}$ $= \begin{pmatrix} 24\\ -17 \end{pmatrix}$

A:

-4, 2

B:

4, 2

C:

-4, -2

D:

4, -2

E:

No option

46:

The cost of removing a 6m square room is #540. What is the cost of renovating a 9m square room?

A:

#1,215

B:

#720

C:

#1,620

D:

#810

E:

No option

47:

The sum of the first n positive integers is

A:

$\frac{1}{2}n(n - 1)$

B:

$n(n + 1)$

C:

$n(n - 1)$

D:

$\frac{1}{2}n(n + 1)$

E:

No option

48:

If $T = 2 \pi \sqrt{\frac{l}{g}}$, make g the subject of the formula.

A:

$\frac{4 \pi l^{2}}{T}$

B:

$\frac{4 \pi^{2} l}{T^{2}}$

C:

$\frac{4 \pi^{2} l^{2}}{T^{2}}$

D:

$\frac{2 \pi \sqrt{l}}{T}$

E:

No option

49:

If y = x2 - x - 12, find the range of values of x for which $y \geq 0$

A:

x < -3 or x > 4

B:

$x \leq -3$ or $x \geq 4$

C:

$-3 < x \leq 4$

D:

$-3 \leq x \leq 4$

E:

No option

50:

How many terms of the series 3, -6, +12, -24, +... are needed to make a total of 1 - 28 ?

A:

12

B:

10

C:

9

D:

8

E:

No option