# Sulvme!

### Unified Tertiary Matriculation Examination Mathematics 2009

1:

Subtract 164189 from 186309.

A:

11219

B:

21129

C:

21139

D:

22119

E:

No option

2:

If 55x + 52x = 7710 , find x

A:

5

B:

6

C:

7

D:

10

E:

No option

3:

Simplify  $7\frac{1}{12} - 4\frac{3}{4} + 2\frac{1}{2}$

A:

4

B:

$4\frac{1}{6}$

C:

$4\frac{5}{6}$

D:

$5\frac{1}{6}$

E:

No option

4:

Evaluate  $\frac{81.81 + 99.44}{20.09 + 36.16}$ correct to 3 significant figures

A:

6.21

B:

3.22

C:

2.78

D:

2.13

E:

No option

5:

A man bought a second-hand photocopying machine for #34,000. He serviced it at a cost of #2,000 and then sold it at a profit of 15%. What was the selling price?

A:

#37,550

B:

#40,400

C:

#41,400

D:

#42,400

E:

No option

6:

A student spent 1/5 of his allowances on books, 1/3 of the remainder on food and kept the rest for contingencies. What fraction was kept?

A:

$\frac{7}{15}$

B:

$\frac{8}{15}$

C:

$\frac{2}{3}$

D:

$\frac{4}{5}$

E:

No option

7:

Solve  $5^{2(x-1)} \times 5^{x+1} = 0.04$

A:

$\frac{1}{3}$

B:

$\frac{1}{4}$

C:

$-\frac{1}{5}$

D:

$-\frac{1}{3}$

E:

No option

8:

If log102 = 0.3010 and log107 = 0.8451, evaluate log10280.

A:

3.4471

B:

2.4471

C:

1.4471

D:

1.4071

E:

No option

9:

Simplify  $\frac{5 + \sqrt{7}}{3 + \sqrt{7}}$

A:

$17 - \sqrt{7}$

B:

$4 - \sqrt{7}$

C:

$15 + \sqrt{7}$

D:

$7 - \sqrt{7}$

E:

No option

10:

If x = {n2 + 1: n is a positive integer and $1\leq n \leq5$ },

y = {5n: n is a positive integer and $1\leq n \leq5$ },

find x $\cap$ y

A:

{5, 10}

B:

{5, 10, 15}

C:

{2, 5, 10}

D:

{5, 10, 15, 20}

E:

No option

11:

I. $S \cap T \cap W = S$

II. $S \cup T \cup W = W$

III. $T \cap W = S$

If  $S \subset T \subset W$, which of the above statements are true?

A:

I and II

B:

I and III

C:

II and III

D:

I, II and III

E:

No option

12:

If  $P = \sqrt{\frac{rs^{3}}{t}}$, express r in terms of p, s and t

A:

$\frac{p^{2}t}{s^{3}}$

B:

$\frac{p^{3}t}{s^{3}}$

C:

$\frac{p^{3}t}{s^{2}}$

D:

$\frac{pt}{s^{3}}$

E:

No option

13:

A polynomial in x whose roots are 4/3 and -3/5 is

A:

15x2 - 11x - 12

B:

15x2 + 11x - 12

C:

12x2 - x - 12

D:

12x2 + 11x - 15

E:

No option

14:

Which of the following equations represents the graph below?

Image coming soon

A:

y = 2 + 7x + 4x2

B:

y = 2 - 7x + 4x2

C:

y = 2 + 7x - 4x2

D:

y = 2 - 7x - 4x2

E:

No option

15:

W is directly proportional to U. If W = 5 when  U = 3, find U when W = $\frac{2}{7}$

A:

$\frac{6}{35}$

B:

$\frac{10}{21}$

C:

$\frac{21}{10}$

D:

$\frac{35}{6}$

E:

No option

16:

Determine the value of x for which (x2 - 1) > 0

A:

x < -1 or x > 1

B:

-1 < x < 1

C:

x > 0

D:

x < -1

E:

No option

17:

Find the range of values of x for which $3x - 7 \leq 0$ and $x + 5 > 0$

A:

$-5< x< \frac{7}{3}$

B:

$-5\leq x\leq \frac{7}{3}$

C:

$-5< x\leq \frac{7}{3}$

D:

$-5\leq x<\frac{7}{3}$

E:

No option

18:

The sum of the first n terms of the arithmetic progression 5, 11, 17, 23, 29, 35,........ is

A:

n(3n - 0.5)

B:

n(3n + 2)

C:

n(3n + 2.5)

D:

n(3n + 5)

E:

No option

19:

Find to infinity the sum of the sequence 1, $\frac{9}{10}$, $\left ( \frac{9}{10} \right )^{2}$, $\left ( \frac{9}{10} \right )^{3}$,......

A:

10

B:

9

C:

$\frac{10}{9}$

D:

$\frac{9}{10}$

E:

No option

20:

If m * n = n - (m + 2) for any real numbers m and n, find the value of 3 * (-5)

A:

-6

B:

-8

C:

-10

D:

-12

E:

No option

21:

A binary operation $\otimes$ defined on the set of integers is such that m $\otimes$ n = m + n + mn for all integers m and n. Find the inverse of -5 under this operation, if the identity element is 0

A:

$-\frac{5}{4}$

B:

$-\frac{5}{6}$

C:

0

D:

5

E:

No option

22:

If  $Q = \begin{pmatrix} 9 & -2\\ -7 & 4 \end{pmatrix}$, then $\left | Q \right |$ is

A:

-50

B:

-22

C:

22

D:

50

E:

No option

23:

If  $P = \begin{pmatrix} x+3 & x+2\\ x+1 & x-1 \end{pmatrix}$, evaluate x if $\left | P \right | = -10$

A:

-5

B:

-2

C:

2

D:

5

E:

No option

24:

Find the acute angle between the straight lines y = x and $y = \sqrt{3x}$

A:

15o

B:

30o

C:

45o

D:

60o

E:

No option

25:

A regular polygon has 150o as the size of each interior angle. How many sides does it have?

A:

12

B:

10

C:

9

D:

8

E:

No option

26:

In the figure below, TS//XY and XY//TY, <STZ = 34o, <TXY = 47o, find the angle marked n.

A:

47o

B:

52o

C:

56o

D:

99o

E:

No option

27:

If the hypothenus of a right-angled isosceles triangle is 2cm, what is the area of the triangle?

A:

$\frac{1}{\sqrt{2}}cm^{2}$

B:

$1 cm^{2}$

C:

$\sqrt{2}cm^{2}$

D:

$2\sqrt{2}cm^{2}$

E:

No option

28:

A chord is drawn 5cm away from the center of a circle of radius 13cm. Calculate the length of a chord

A:

7cm

B:

9cm

C:

12cm

D:

24cm

E:

No option

29:

Find the radius of a sphere whose surface area is 154cm2 ($\pi$ = 22/7)

A:

7.00cm

B:

3.50cm

C:

3.00cm

D:

1.75cm

E:

No option

30:

Find the locus of a particle which moves in the first quadrant so that it is equidistant from the lines x = 0 and y = 0

A:

x + y = 0

B:

x - y = 0

C:

x + y + k = 0

D:

x - y - k = 0

E:

No option

31:

What is the locus of the mid-point of all chords of length 6cm with a circle of radius 5cm and with center O?

A:

A circle of radius 4cm and with Center O

B:

The perpendicular bisector of the chords

C:

A straight line passing through center O

D:

A circle of radius 6cm and with center O

E:

No option

32:

What is the value of p, if the gradient of the line joining (-1, p) and (p, 4) is $\frac{2}{3}$

A:

-2

B:

-1

C:

1

D:

2

E:

No option

33:

What is the value of r if the distance between the points (4,2) and l, r is 3 units?

A:

1

B:

2

C:

3

D:

4

E:

No option

34:

Find the value of sin 45o - cos 30o

A:

$\frac{2 + \sqrt{6}}{4}$

B:

$\frac{\sqrt{2} + \sqrt{3}}{4}$

C:

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

D:

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

E:

No option

35:

A cliff on the bank of a river is 300 meters high. If the angle of depression of a point on the opposite side of the river is 60o, find the width of the river.

A:

$100m$

B:

$75\sqrt{3}m$

C:

$100 \sqrt{3}m$

D:

$200 \sqrt{3}m$

E:

No option

36:

If y = 3 cos 4x, $\frac{dy}{dx}$ equals

A:

6 sin 8x

B:

-24 sin 4x

C:

12 sin 4x

D:

-12 sin 4x

E:

No option

37:

If s = (2 + 3t)(5t - 4), find $\frac{ds}{dt}$ when t = 4/5 secs

A:

0 units per sec

B:

15 units per sec

C:

22 units per sec

D:

26 units per sec

E:

No option

38:

What value of x will make the function x(4 - x) a maximum?

A:

4

B:

3

C:

2

D:

1

E:

No option

39:

The distance travelled by a particle from a fixed point is given as s = (t3 - t2 - t + 5) cm. Find the minimum distance that the particle can cover from the fixed point

A:

2.3 cm

B:

4.0 cm

C:

5.2 cm

D:

6.0 cm

E:

No option

40:

Evaluate  $\int sec^{2}\theta d \theta$

A:

$sec \theta tan \theta + k$

B:

$tan \theta + k$

C:

$2 sec \theta + k$

D:

$sec \theta + k$

E:

No option

41:

The distribution above shows the number of days a group of 260 students were absent from school in a particular term. How many students were absent for at least four days in the term?

A:

180

B:

120

C:

110

D:

40

E:

No option

42:

The histogram below represents the number of candidates that sat for Mathematics examination in a school. How many candidates scored more than 50 marks?

A:

80

B:

90

C:

100

D:

115

E:

No option

43:

The pie chart below represents 400 fruits on display in a grocery store. How many apples are in the store?

A:

45

B:

50

C:

60

D:

75

E:

No option

44:

The cummulative frequency curve above shows the distribution of the scores of 50 students in an examination. Find the 36th percentile score.

A:

18%

B:

28%

C:

36%

D:

50%

E:

No option

45:

5,8,6 and k occur with frequencies 3,2,4 and 1 respectively and have a mean of 5.7. Find the value of k

A:

4

B:

3

C:

2

D:

1

E:

No option

46:

What is the mean deviation of x, 2x, x+1 and 3x, if their mean is 2?

A:

0.5

B:

1.0

C:

1.5

D:

2.0

E:

No option

47:

In how many ways can a delegation of 3 be chosen from 5 men and 3 women, if at least 1 man and 1 woman must be included?

A:

15

B:

28

C:

30

D:

45

E:

No option

48:

In how many ways can 9 people be seated if 3 chairs are available?

A:

720

B:

504

C:

336

D:

210

E:

No option

49:

The probability of a student passing any examination is 2/3. If the student takes three examinations, what is the probability that he will not pass any of them?

A:

2/3

B:

4/9

C:

8/27

D:

1/27

E:

No option

50:

The table above shows the distribution of marks of students in a test. Find the probability of passing the test if the pass mark is 5

A:

3/5

B:

4/9

C:

7/20

D:

1/5

E:

No option