# Sulvme!

### Unified Tertiary Matriculation Examination Mathematics 2004

1:

Find x and y respectively in the subtraction below carried out in base 5

A:

2,4

B:

3,2

C:

4,2

D:

4,3

E:

No option

2:

Find p, if 4516 - P7 = 3056

A:

6117

B:

1427

C:

1167

D:

627

E:

No option

3:

Evaluate $\frac{\frac{1}{10} \times \frac{2}{3} + \frac{1}{4}}{\frac{1}{2} + \frac{3}{5} - \frac{1}{4}}$

A:

$\frac{2}{25}$

B:

$\frac{19}{60}$

C:

$\frac{7}{12}$

D:

$\frac{19}{35}$

E:

No option

4:

A farmer planted 5,000 grains of maize and harvested 5,000 cobs each bearing 500 grains What is the ratio of the number of grains sowed to the number harvested ?

A:

1:500

B:

1:5,000

C:

1:25,000

D:

1:250,000

E:

No option

5:

Three teachers shared a packet of chalk. The first teacher got $\frac{2}{5}$ of the chalk and the second teacher received $\frac{2}{15}$ of the remainder. What fraction did the third teacher receive?

A:

$\frac{11}{25}$

B:

$\frac{12}{25}$

C:

$\frac{13}{25}$

D:

$\frac{8}{15}$

E:

No option

6:

Given that $\sqrt[3]{4^{2x}} = 16$, find the value of x.

A:

2

B:

3

C:

4

D:

6

E:

No option

7:

Simplify $\frac{1}{\sqrt{3} + 2}$ in the form a + b $\sqrt{3}$

A:

-2 - $\sqrt{3}$

B:

-2 + $\sqrt{3}$

C:

2 - $\sqrt{3}$

D:

2 + $\sqrt{3}$

E:

No option

8:

If 6logx2 - 3logx3 = 3log50.2, find x

A:

$\frac{3}{8}$

B:

$\frac{3}{4}$

C:

$\frac{4}{3}$

D:

$\frac{8}{3}$

E:

No option

9:

The shaded region in the venn diagram below is

A:

$P^{c} \cap (Q \cap R)$

B:

$P \cap Q$

C:

$P^{c} \cup (Q \cap R)$

D:

$P^{c} \cap (Q \cup R)$

E:

No option

10:

In a class of 40 students, each student offers at least one of Physics and Chemistry. If the number of students that offer Physics is three times the number that offer both subjects and the number that offer Chemistry is twice the number that offer Physics, find the number of students that offer Physics only

A:

25

B:

15

C:

10

D:

5

E:

No option

11:

Find the values of x where the curve y = x3 + 2x2 - 5x - 6 crosses the x-axis

A:

-2, -1 and 3

B:

-2, 1 and -3

C:

2, -1 and -3

D:

2, 1 and 3

E:

No option

12:

Find the remainder when 3x3 + 5x2 - 11x + 4 is divided by x + 3

A:

4

B:

1

C:

-1

D:

-4

E:

No option

13:

Factorize completely ac - 2bc - a2 + 4b2

A:

(a - 2b)(c + a - 2b)

B:

(a - 2b)(c - a - 2b)

C:

(a - 2b)(c + a + 2b)

D:

(a - 2b)(c - a + 2b)

E:

No option

14:

y is inversely proportional to x and y and 4 when x = 1/2. Find x when y = 10

A:

$\frac{1}{10}$

B:

$\frac{1}{5}$

C:

2

D:

10

E:

No option

15:

The length L of a simple pendulum varies directly as the square of its period T. If a pendulum with period 4 sec. is 64cm long, find the length of a pendulum whose period is 9 sec.

A:

36cm

B:

96cm

C:

144cm

D:

324cm

E:

No option

16:

The shaded area in the diagram below is represented by

A:

{(x,y): y + 3x < 6}

B:

{(x, y) : y + 3x < -6}

C:

{(x, y) : y - 3x < 6}

D:

{(x, y) : y - 3x < -6}

E:

No option

17:

What are the integral values of x which satisfy the inequality $-1 < 3 - 2x \leq 5$ ?

A:

-2, 1, 0, -1

B:

-1, 0, 1, 2

C:

-1, 0, 1

D:

0, 1, 2

E:

No option

18:

The nth term of two sequences are Qn = 3.2n-2 and Um = 3.22m-3. Find the product of Q2 and U2

A:

3

B:

6

C:

12

D:

18

E:

No option

19:

Given that the first and fourth terms of a G.P are 6 and 162 respectively, find the sum of the first three terms of the progression.

A:

8

B:

27

C:

48

D:

78

E:

No option

20:

Find the sum to infinity of the series $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, ...$

A:

1

B:

$\frac{3}{4}$

C:

$\frac{2}{3}$

D:

$\frac{1}{3}$

E:

No option

21:

If the operation * on the set of integers is defined by $p * q = \sqrt{pq}$, find the value of 4*(8*32)

A:

16

B:

8

C:

4

D:

3

E:

No option

22:

The inverse of the matrix $\begin{pmatrix} 2 & 1\\ 1 & 1 \end{pmatrix}$ is

A:

$\begin{pmatrix} 1 & 1\\ -1 & 2 \end{pmatrix}$

B:

$\begin{pmatrix} 1 & -1\\ 1 & 2 \end{pmatrix}$

C:

$\begin{pmatrix} 1 & 1\\ 1 & 2 \end{pmatrix}$

D:

$\begin{pmatrix} 1 & -1\\ -1 & 2 \end{pmatrix}$

E:

No option

23:

If P $\begin{pmatrix} 1 & 0 & -1\\ 3 & 4 & 5\\ -1 & 0 & 1 \end{pmatrix}$, then $|P|$ is

A:

-8

B:

0

C:

4

D:

1

E:

No option

24:

The sum of the interior angles of a pentagon is 6x + 6y. Find in terms of x.

A:

y = 60 - x

B:

y = 90 - x

C:

y = 120 - x

D:

y = 150 - x

E:

No option

25:

PQRSTV  is a regular polygon of side 7 cm inscribed in a circle. Find the circumference of the circle PQRSTV

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

A:

22cm

B:

42cm

C:

44cm

D:

56cm

E:

No option

26:

P, R, and S lie on a circle centre O as shown above, while Q lies outside the circle. Find <PS

A:

350

B:

400

C:

450

D:

550

E:

No option

27:

In the diagram above, PQ = 4cm and TS = 6cm. If the area of parallelogram PQTU is 32cm2, find the area of the trapezium PQRU

A:

24cm2

B:

48cm2

C:

60cm2

D:

72cm2

E:

No option

28:

An arc of a circle of length 22cm subtends an angle of 3x0 at the centre of the circle. Find the value of x if the diameter of the circle is 14cm.

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

A:

300

B:

600

C:

1200

D:

1800

E:

No option

29:

Determine the locus of a point inside a square PQRS which is equidistant from PQ and QR.

A:

The diagonal PR

B:

The diagonal QS

C:

Side SR

D:

The perpendicular bisector of PQ

E:

No option

30:

The locus of a point which is 5cm from the line LM is

A:

A pair of lines on opposite sides of LM and parallel to it, each distance 5cm from LM

B:

Line parallel to LM and 5cm from LM

C:

pair of parallel lines on one side of LM and parallel to LM

D:

line distance 10cm from LM and parallel to LM

E:

No option

31:

Find the value of $\alpha^{2} + \beta^{2}$ if $\alpha + \beta = 2$ and the distance between the points $(1, \alpha)$ and $(\beta, 1)$ is 3 units

A:

3

B:

5

C:

11

D:

14

E:

No option

32:

Find the midpoint of the line joining P(-3, 5) and Q(5, -3)

A:

(4, -4)

B:

(4, 4)

C:

(2,2)

D:

(1,1)

E:

No option

33:

Find the value of x in the figure below

A:

$20 \sqrt{6}$

B:

$15 \sqrt{6}$

C:

$5 \sqrt{6}$

D:

$3 \sqrt{6}$

E:

No option

34:

The shadow of a pole $5 \sqrt{3}$ high is 5m. Find the angle of elevation of the sun.

A:

300

B:

450

C:

600

D:

750

E:

No option

35:

Find the derivative of (2+3x)(1-x) with respect to x

A:

6x - 1

B:

1 - 6x

C:

6

D:

-3

E:

No option

36:

Find the derivative of the function y = 2x2 (2x - 1) at the point x = -1

A:

-6

B:

-4

C:

16

D:

18

E:

No option

37:

If $y = 3cos\frac{x}{3} = -1$, Find  $\frac{dy}{dx}$ when $x = \frac{3 \pi}{2}$

A:

2

B:

1

C:

-1

D:

-3

E:

No option

38:

What is the rate of change of the volume v of a hemisphere with respect to its radius r when r = 2 ?

A:

$2 \pi$

B:

$4 \pi$

C:

$8 \pi$

D:

$16 \pi$

E:

No option

39:

Evaluate $\int_{1}^{3}(x^{2} - 1)dx$

A:

$6\frac{2}{3}$

B:

$\frac{2}{3}$

C:

$-\frac{2}{3}$

D:

$-6\frac{2}{3}$

E:

No option

40:

The pie chart below shows the distribution of the crops harvested from a farmland in a year. If 3000 tonnes of millet is harvested, what amount of beans is harvested?

A:

9000 tonnes

B:

6000 tonnes

C:

1500 tonnes

D:

1200 tonnes

E:

No option

41:

I. Rectangular bars of equal width

II. The height of each rectangular bar is proportional to the frequency of the corresponding class interval

III. Rectangular bars have common sides with no gaps in-between

A histogram is described completely by

A:

I and II

B:

I and III

C:

I, II and III

D:

II and III

E:

No option

42:

The graph above shows the cumulative frequency curve of the distribution of marks in a class test. What percentage of the students scored more than 20 marks?

A:

68%

B:

28%

C:

17%

D:

8%

E:

No option

43:

The mean age of a group of students is 15 years. When the age of a teacher, 45 years old, is added to the ages of the students, the mean of their ages becomes 18 years. Find the number of students in the group

A:

7

B:

9

C:

15

D:

42

E:

No option

44:

The weights of 10 pupils in class are 15kg, 16kg, 17kg, 18kg, 16kg, 17kg, 17kg, 17kg, 18kg and 16kg. What is the range of this distribution

A:

1

B:

2

C:

3

D:

4

E:

No option

45:

Find the mean deviation of 1, 2, 3 and 4

A:

1.0

B:

1.5

C:

2.0

D:

2.5

E:

No option

46:

In how many ways can 2 students be selected from a group of 5 students in a debating competition?

A:

10 ways

B:

15 ways

C:

20 ways

D:

25 ways

E:

No option

47:

A committee of six is to be formed by a state governor from nine state commissioners and three members of the state house of assembly. In how many ways can the members of the committee be chosen so as to include one member of the house of assembly?

A:

924 ways

B:

840 ways

C:

462 ways

D:

378 ways

E:

No option

48:

Some white balls were put in a basket containing twelve red balls and sixteen black balls. If the probability of picking a white ball from the basket is $\frac{3}{7}$, how many white balls were introduced ?

A:

32

B:

28

C:

21

D:

12

E:

No option

49:

An unbiased die is rolled 100 times and the outcome is tabulated as follows:

A:

$\frac{1}{6}$

B:

$\frac{1}{5}$

C:

$\frac{1}{4}$

D:

$\frac{1}{2}$

E:

No option

50:

A container has 30 gold medals, 22 silver medals and 18 bronze medals. If one medal is selected at random from the container, what is the probability that it is not a gold medal?

A:

$\frac{4}{7}$

B:

$\frac{3}{7}$

C:

$\frac{11}{35}$

D:

$\frac{9}{35}$

E:

No option