senior cycle

SECTION A: OBJECTIVES

1.               Express the product of 0.06 and 0.09 in standard form

[A] 5.4 × 10^-3                               [B ] 5.4 × 10^-1      [C ]5.4 × 10²        [D]  5.4 × 10³

          2             Simplify   36 ^½   ×   64 ^-1/2 ×  5 ⁰

                                   [A] 2               [B] ¾      [C] 2 ¾                  D 4

3                 Find the number whose logarithm to base 10 is 2.6025

[A] 0.404                      [B] 40                    [C] 400.4                              [D] 0.04

       4.            The angles of a pentagon are x⁰,  2x⁰,  (x+60)⁰,  (x+10)⁰,  (x-10)⁰ find x

                                 [A] 30                            [B] 800                  [C] 0.8                   [D] 80

      5.            The angles marked in this diagram are measured in degrees, find x.   

                                         

                       [A] 24⁰                   [B] 48⁰                   [C] 30⁰                   [D] 36⁰

    6.        Convert 101101_two to a number in base ten

                [A] 46                    [B] 45                    [C] 44                    [D] 61

     7.       In the diagram, PQRS is a parallelogram and <QRT=30⁰. Find x.

                [A] 95⁰                    [B] 100⁰                                [C] 150⁰                                         [D] 120⁰

     8.       Correct 0.04945 to two significant figures.

                [A] 0.49                [B] 0.050                              [C]0.040                               [D]0.049

     9.       Evaluate 7¹/₂ –( 2¹/₂ + 3) ÷ 33/2

                [A] 7                      [B] 8                       [C] 32                    [D]14

10.          In the diagram below, IQRI= 5cm, PQR =60 and PSR =45⁰. Find IPSI.

                [A] 24                    [B] 12.25              [C] 13                    [D] 43   

        

11.   Each interior angle of a regular polygon is 162⁰. How many sides has the polygon?

         [A]    8       [B]    12      [c]      20       [d]      16

12.  From the diagram above, find the value of x.

                               

  [A]   25    [B]     40       [C]     20       [D]      30

13.   Which of the following is not true

  [A]   A parallelogram is a quadrilateral

  [B]   A rhombus is a parallelogram

  [C]   A square is a rhombus

  [D}   A square is a polygon

14.  Simplify ( 1⁵/₈ + 1 ³/₅₎ ÷ 5³/₅

       [A]    ⁴/₅     [B]      ²/₃     [C]     ³/₅   [D]     ¹/₂

15.   186 is the result of increasing a number by 20%. Find the number.

     [A]    40     [B]     130    [C]   155    [D]   85

16.  Find the value of   4(3d - e) -2f when d=2, e=4 and f=3

    [A]   -3    [B]    6    [C]    2    [D]    4

17.  If  N = {  odd numbers greater than 11 }, which one of the following is an element of N?

    [A]    9      [B]     10        [C]       13       [D]    14

18.    In the diagram below, ABP = 110⁰ and DCP = 163⁰, calculate BPC

   [A]    47⁰      [B]    63⁰    [C]    93⁰     [D]    180⁰                                                                      

19.  Which of the following statements is false?                                                

   [A]    12 є {2, 4, 6, 8…}

   [B]     {a, b } C { f, a, c, e }

   [C]     {1,2} U { 2, 3} == {2}

  [d]   If Z = {10}, n(Z) = 0

20.  Express 40cm as a percentage of 8m

    [A]   32%    [B]   20%     [C]   5%    [D]    8%

21.  Simplify  (¹/₄)^-1.5

   [A]   21     [B]    8    [C]    2      [D]     4

22.  If a=1, b=2, and n=5,

    Find the value of 

  [A]   7    [B]    6     [C]     3      [D]     2

The table below shows the frequency distribution of a number of choirs in each of 40 rooms of various houses.

Use the table to answer Question 23- 25

23. Find the mean of the distribution.

  [A]     5.7      [B]      4.4       [C]      5.0      [D]   3.5

24.  Find the mode of the distribution.

   [A]    40      [B]     5       [C]      9    [D]    7

25. Find the median of the distribution

   [A]    3     [B]     5       [C]     4      [D]     4.5

26.  In the diagram, EF//QR, PE = 2cm, EQ= 4cm and FR = 6cm. Find x

         

[A]    4cm       [B]      3cm     [C]    5cm      [D]    6cm

27.  In the diagram, QRS is a Straight line, QP//RT, < PQR = 56⁰ , <QPR = 48⁰

        And <TRS = x⁰ Find x

   [A]   84⁰    [B]     76⁰   [C]     21⁰     [D]   33⁰

28. In isosceles ∆ ABC, ІABІ= ІACІ. If  B = 55⁰  calculate A

   [A]    125⁰       [B]     70⁰      [C]     81⁰      [D]     35⁰

29.  ABC is an equilateral triangle. P is a point on IACI such that PBC=46⁰

        Calculate <APB 

 [A]   100⁰      [B]   106⁰        [C]    41⁰      [D]     28⁰

30.  Find the interior angles of a regular polygon which has 6 sides.

  [A]    70⁰    [B]     120⁰       [C]      60⁰      [D]     140⁰

31.  One of the following is not a polygon.

   [A]   Triangle   [B]   Rectangle    [C]     Square     [D]    cube

32.  How many pieces of string each 8¹/₂ cm long can be cut from a string 42¹/₂ cm long

     [A]   50      [B]    35      [C]     ¹/₅    [D]    500

33.  A man gets a 10% pay rise. If his present wage is N 6000 per week, calculate his new weekly wage.

     [A]   N 250 [B]    N 700    [C]    N 300    [D]   N 6, 600

34.  Calculate the principal which earns N 75 075

       Simple interest in 11years at 7% p.a

     [A]   N 500     [B]   N 4 000   [C]     N 23 000     [D]    N 97, 500

35.  Remove the bracket and simplify 2a-(4a – 5a-7)

      [A]   2+a    [B]    -3 +4a    [C]    9a-2    [D]   3a-7

36.  Which of the following is incorrect?

     [A]   U = union   [B]  є = element   [C]  Ф = empty set  [D]  С = infinite set

37. Solve for the value of x in 22= 7 + 2⅟₂x

       [A]  18    [B]   7     [C]  4     [D]   6

38.  t= ^3p/_r  +  S    make r the subject of formular.

      [A]   r= ^t-p/_s       [B]  r= ^t/_p+t        [C]   r= t²/p+s        [D]  r= ^3p/_t-s

39. The square of  111_two  is

      [A]   10001   [B]    11101    [C]    10111    [D]  110001

40. ABCDEF is a regular hexagon with O at its centre. What kind of quadrilateral is ABCO

       [A] Rectangle   [B] Square   [C]   parallelogram   [D]   Rhombus

41. In the diagram below, IPRI = IPQI, IOSI = IQRI and RPQ = 40⁰ , calculate PQS.

     

       [A]   30⁰    [B]    70⁰     [C]    40⁰     [D]    85⁰  

Use the diagrams below to answer question 42 to 44

42. Which of the following triangles are congruent?

 [A]  ii   and iii     [B]   I and iv    [C]    iii and iv      [D]   I and iii

43.  Which of the following is/ are embedded angle?

  [A]  I   and ii    [B]   ii   and iii   [C]    I and iv   [D]    iii and iv

44. Which of the following is (AAS)

   [A]   i    [B]   ii      [C]    iv    [D]    iii

45.  Simplify    5x – 8x + x + 3y

   [A]   3y-2x     [B]    2-y       [C]   y-3x       [D]    3y+3x

46. Find the sum of angles in a regular heptagon.

   [A] 320⁰    [B]   900⁰    [C]   700⁰    [D]    540⁰

47. In the diagram below IXYI = IYZI, IXZI = IZHI and XYZ = 52⁰ calculate ZHX

   

48. The perimeter of a rectangle is 20m and the length is Xm. Find the area of     

   the rectangle in terms of x.

   [A]  2x+10   [B]  x-5    [C]    2x-20     [D]     10x-x²

49. If 8x-4 = 6x-10, find the value of 5x

   [A]   -15       [B]    6      [C]     7      [d]     3

50. Which of the following is not necessarily true of a rectangle?

    [A]   The diagonals are equal.

    [B]   Each diagonal divides the rectangle into

    [C]   The diagonals are perpendicular

    [D] Each diagonal divides the area of the rectangle equally.

SECTION B: THEORY

PART1: ANSWER ALL THE QUESTION IN THIS PART

1a. Simplify 125^-⅟₃ × 49^-½ × 10⁰

  b. list  (i)  four basic laws of indices.

              (ii) Two fractional indices.

1                     Use mathematical tables to evaluate

(a)         

(b)       

 

3a.    The sum of the interior angles of a regular polygon is 900⁰.      

         Calculate (i) The   number of sides   (ii)   The size of one   

         Exterior angle of the polygon.

  b.   Prove that the exterior angle of a triangle is equal to sum 

         of the opposite interior angles.                                                                         

4a.    Simplify 7¹/₂ – ( 2¹/₂ + 3 ) ÷ ³³/₂

  b.    Solve for the value of x in  3x-[3 (1+x) – 2x] =3

PART II:  ANSWER FOUR QUESTIONS ONLY FROM THIS PART

5a. The result of taking 3 from x and multiplying the answer by   

        4 is the same as

       Taking 3 from the x

b. 4m/₃ – ¹⁷/₂₁ = ^6m-1/₇  solve for m.

6a. Convert (i)    57_ten to binary

                       (ii)  1101_two to base Ten.

  b.  (i)   find ( 101₂ )² expressing the answer in base 2

        (ii)   Convert 1.101 from bicimals to decimals.

7a. Isosceles triangle ABC and ABD are drawn on opposite sides   

      of a common base AB. If ABC = 50⁰ and ADB = 110⁰,  

      Calculate ACB and CBD.

                            

    b.   List 2 properties each of the following:

        (i)  Parallelogram      (ii)    A rhombus

        (iii) A rectangle          (iv)   A square.

8a.     S = { 1,2,3,4,5,6 } ,  T = { 2,4,5,7 } and R = { 1,4,5 } , find

(i)                                 { S n T } U R

(ii)                               { S U T } n R

    b. Define the following:

           (i)    Ф     (ii)   є   (iii)    n    (iv)    С

9a.  Remove the brackets and simplify.

(i)                               y( 2y-3) -  3(2y-3)

(ii)                            3( 3a-b) – 5(a-4b)

  b.   factorise

       (i)    -18pq – 12                                           (ii) 100 – 49x²

10a.  The following give the summary for congruency of two

         triangles. Draw the diagrams to represent each of the

         following.

(i)                                 SAS   (ii)   ASA     (iii)   SSS     (iv)   RHS

   b.   In a given parallelogram ABCD, simply show that the

          opposite sides are equal.

 

      

What on Earth is a Logarithm?

What on Earth is a Logarithm?

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Interestingly, after I had this guide up for a while, this turned out to be the question I was asked most frequently, usually in terms that included phrases like "Greek to me", "beats me", or, as above, "what on earth"...

To understand what a logarithm is you first have to understand what a power is. Follow that link first if you don't!

OK, you do know what a power is. So it makes sense to you to write something like

 b^x = y.      (*)

In the preceding equation, the x should look like a superscript of b. If it does not you have an underpowered browser.

After these preliminaries, we can now get into the meat of the matter. The equation (*) is the key to everything. The number b is the base, the number x the exponent, and the expression that equals y is apower. If we think of x as the independent variable and y as the dependent variable then (*) defines an exponential function.

In the equation (*) we can now pretend that two of the variables are given, and solve for the third. If the base and the exponent are given we compute a power, if the the exponent and the power are given we compute a root (or radical ), and, if the power and the base are given, we compute a logarithm.

In other words, The logarithm of a number y with respect to a base b is the exponent to which we have to raise b to obtain y.

We can write this definition as

x = log_by   <--->  b^x = y

and we say that x is the logarithm of y with base b if and only if b to the power x equals y.

Let's illustrate this definition with a few examples. If you have difficulties with any of these powers go back to my page on  10² = 100      log₁₀100 = 2

·          10^-2 = 0.01      log₁₀0.01 = -2

·          10⁰ = 1      log₁₀1 = 0

·          2³ = 8      log₂8 = 3

·          3² = 9      log₃9 = 2

·          25^1/2 = 5      log₂₅5 = 1/2

·          8^-2/3 = 1/4      log₈1/4 = -2/3

·          2^1/2 =  1.4142135623...      log₂1.414.. = 1/2

Special Bases

Logarithms with respect to the base b=10 are called common logarithms, and logarithms with respect to the base e=2.71828... are called natural logarithms.

More Information

You should find extensive information on logarithms in any textbook on College Algebra. To check your understanding and guide your further study figure out answers to the following questions:

• Why are logarithms important?
• Why are exponential functions important?
• How do you convert a logarithm with respect to one base to a logarithm with respect to another base?
• Why does the base have to be positive?
• Why is the power always positive?
• What is it that makes natural logarithms natural?

A Logarithm Calculator

helping their students develop strategies and build concepts needed for a robust understanding of numbers and place value.

In today's math classroom, we want children to do more than just memorize math facts. We want them to understand the math facts they are being asked to 

MULTIPLE CHOICE

INSTRUCTION: Answer all.

[1]Write in figures: two million, two hundred and twenty –two thousand, and two hundred and twenty two. [A] 202 222 [B] 2 222 222 [C] 2 222 [D] 22 222 [E] 222 222  

[2] The H C F OF 16, 12 and 20 is_________ [A] 8 [B] 4 [C] 240 [D] 48 [E] 12

[3] What does 8 stand for in 17.682? __________

 [A] 8 Thousands [B] 8 tenths [C] 8 hundredths [D] 8 tens [E] hundreds

[4] 360 expressed as the product of   its prime factors is _________ 

[A] 2² x 3 [B] 2² X 3² [C] 2² X 5 [D] 2² X 3 X 7 [E] 2³ X 3² X5

[5] Which of the following numbers will not divide 3960 exactly?

[A] 9   [B] 3 [C] 7   [D] 11 [E] 8

[6] How many lines of symmetry has an isosceles triangle?

[A] 0 [B] 1 [C] 2 [D] 3 [E] 6

[7]   Which of the following statement is/are true?

(i) 1089 is divisible by 6 (ii)   1089 is divisible by 9 (iii) 1089 is a perfect square

[A] (i) only [B] (ii) only [C] (iii) only [D] (i) and (ii) only [E]  (ii) and (iii) only

 [8] Find the value of a if:    9a – 7 = 7a + 5

 [A] -6 [B] -2 [C] -1 [D] 1 [E] 6

[9] If x is a set of counting numbers such that x-3 ≤ 1, find x:

[A] x = 1 [B] x = 2 [C] x = 3 [D] x = (1,2,3) [E] x = (1,2,3,4)

[10] Three years ago Bolu was t years old. How old will he be in three years’ time?

[A] 3t years [B] (t + 3) years [C] 6t years [D] (t + 6) years [E] (2t + 6) years

[11] Express 540 as a product of prime factors

[A] 2² x 3² x 5    [B] 2³ x 3² x 5   [C] 2² x 3² x 5   [D] 2 x 3³ x 5   [E] 2² x 3 x 5².

[12] Express 0.27 as percentage

[A] 27% [B] 2.7% [C] 0.27% [D] O.O27% [E] 0.0027%

[13] What percentage of 4 is 5?

[A] 125% [B] 120% [C] 80% [D] 25% [E] 20%

[14] A trader gains #4000 on a sale which was equivalent to 8% profit. What was the cost price? [A] #54 000 [B] 50 000 [C] 32 000 [D] 500 [E] 320

[15] A boy spent half of his money on sweet and one quarter on Kola nut. If he has N6.00 left how much did he have initially?

[A]#8.00 [B] #2.00 [C] #24.00 [D] #36.00 [E] #48.00

[16] In an examination 154 out of 175 candidates passed. What percentage failed?

[A] 6⅔% [B] 12% [C] 13⅘% [D] 15% [E] 88%

[17] Express 250g as a percentage of 2kg:

[A] 125% [B] 0.125% [C] 15% [D] 20.5% [E] 12.5%

[18] Express product of 0.005 and 0.045 in standard form:

[A] 0.00225 [B] 2.2 x 10³ [C] 2.25 x 10³ [D] 2.5 x 10^-1 [E] 2.25 x 10^-4

[19] Which of the following is not a Pythagorean triples?

[A] (5, 12, 13) [B] (3, 4, 5) [C] (10, 40, 41) [D] (6, 8, 10)

[20]The difference between the square of 25.6 and the square of 12.5 is:

[A] 49.911 [B] 499.11 [C] 0.499 [D] 500.11 [E] 49900

[21] Express 83kg in tones.

[A] 8500At [B] 0.85t [C] 0.085t [D] 0.0085t [E] 850t

[22] Find the value of -10 + (-35) + (-7)

[A] 5 [B] 15 [C] 20 [D] -52 [E] 4⁵/₇   

[23] -40 – (-90) will simplify to:

 [A] 50 [B] -50 [C] -40 [D] -130 [E] 130

[24] The product of the square root of 16 and the square of 9 is:

[A] 3240 [B] 1296 [C] 36 [D] 144 [E] 324

[25] Express 250g as a percentage of 2kg.

[A] 125% [B] 0.125% [C] 15% [D] 20.5% [E] 12.5%

[26] 30% of #900 is:

[A] #290 [B]# 27 [C]# 90 [D] #270 [E] #670

[27] Express the ratio of ¹/₅:  ²/₃ in its simplest form.

[A] 2 : 5 [B] 3 : 5 [C] 10 : 2 [D] 3 : 2 [E] 3 : 10

[28]Which of the following is 20%   of 1 hour?

  [A] 5min          [B]      6min      [C] 12min     [D] 20min     [E] 30min

[29] Which of the following plane shapes does not have a line of symmetry?

[A] Rectangle [B] Parallelogram [C] Kite [D] Pentagon [E] Trapezium.

  [30] A triangle with two sides or two angles equal is called:

[A] Scalene [B] Isosceles   [C] Right-angled triangle [D] Equilateral

[E] Obtuse-angled triangle  

[31] When two angles add up to 360° they are said to be:

[A] Supplementary angles [B] Adjacent angles [C] Complementary angles

 [D] Vertically opposite angles [E] Conjugate angles.

[32] A triangle with each angle equals to 60° is called:      

[A] Isosceles [B] Scalene [C] Equilateral [D] Right-angled [E] Acute-angled

[33] Find the distance round a semi-circle of radius 1.4cm. (Take π = 3¹/₇)

[A] 7.2cm [B] 8.8cm [C] 11.6cm [D] 4.4cm [E] 3.08cm

[34] Find the surface area of a cube of side 5cm.

[A] 25cm2 [B] 150cm2 [C]  15cm2 [D] 125cm2 [E] 200cm2

[35] When recording data, the tally marks IIII IIII IIII   III represent the number:

[A] 13 [B] 15 [C] 18 [D] 20 [E] 33

[36] If 6x + 7 = 55, then x =

[A] 7     [B] 8   [C] 10⅓      [D] 42      [E] 48

Use the following set of numbers in questions 38, 39 and 40:  2, 2, 2, 5, 5, 8, 9, 10,11.

[37] The mode of the above set of numbers is:

[A] 2 [B] 3 [C] 5 [D] 6 [E] 9

[38] The median of the above set of numbers is:

[A] 2 [B] 3 [C] 5 [D] 6 [E] 9

[39] The mean of the above set of numbers is:

[A] 2 [B] 3 [C] 5 [D] 6 [E] 9

[40] A number is trebled and then 17 is subtracted. If the result is 40, the original number is: [A] 7⅔ [B] 11 [C] 19 [D] 57 [E] 69

[41] The mean of three numbers is 6. The mode of the three numbers is 7. The lowest of the three numbers is:  [A] 2 [B] 3 [C] 4 [D] 6 [E] 7

[42] If ^x/₁₂ = 3, then x = _________        [A] ⅟_⁴ [B] 3 [C] 4 [D] 9 [E] 36

[43] Which one of the following is most likely to be the correct value of 3.90 × 7.8?

[A] 0.50 [B] 20 [C] 21.50 [D] 30.42 [E] 33.00

[44] Add 10110₂ plus 11011₂, leaving your answer in base two.

[A] 110001₂ [B] 101101₂ [C] 110011₂ [D] 101010₂ [E] 11O1O1₂

[45] Find the square of 0.021

[A] 0.441 [B] 0.00441 [C] 0.0441 [D] 0.000441 [E] 0.0000441

[46] Express the decimal number 9 in binary form:

[A] 1O11₂ [B] 1100₂ [C] 1001₂ [D] 1101₂ [E] 1010₂

[47] The two prime numbers between 20 and 30 are:

[A] 21 and 29 [B] 23 and 27 [C] 23 and 29 [D] 21 and 27 [E] 27 and 29

[48] A round stool has diameter 42m. Find its perimeter. [Take π = ²²/₇]

[A] 66cm [B] 132cm [C]  264cm [D] 44cm [E] 396cm

[49] A man walks round a rectangular field of length 1.22km and breadth 0.78km twice. Calculate the total distance covered. 

 [A] 4km [B] 16km [C] 8km [D] 18km [E] 12km

[50] How much interest does #400 make in 5 years at 7% per annum?

[A] #7 [B] #20 [C] #28 [D] #35 [E] #140

ESSAY

Instruction: Answer three Questions only.

[1](a) Find the total of 1 week 5 days, 6days and 3weeks 4days.

Give the answer :(i) in weeks and days, (ii)in days.

 (b) What number does MCMXXXIX represent?

(c) Translate I   LOVE MATHS into simple code using the numbers that stand for letters of the alphabet.

(d) Write 9 999 999 999 in words.                                                                   

[2](a) Find the HCF and LCM of 63, 42 and 21.

(b) During a week, a student spends 1/3 of her time in bed, 5/24 of her time in lessons and 1/8 of her time doing homework. What fraction of her time is left for doing other things?

(c) In an examination a student scored 60 marks out of a possible 80. What percentage is this?

[3](a) A skeleton square-based pyramid is made from wire. All its edges are the same length. If 40cm of wire is used, find the length of each edge of the pyramid. (Make a sketch).

(b) Name two solids which have (i) five faces (ii) no vertices.

(c)Copy and complete the table below:

S/N	Shapes	Number of vertices	Number of faces	Number of edges
1	Triangular-based pyramid
2	square-based pyramid
3	hexagonal-based pyramid
4	Triangular prism
5	Cube
6	Cuboid
7	cylinder

[4] Simplify the following by grouping:

(a) 3m-8m-2m+16m-4m.

(b) p-15p-5p-2p+25p.

(c) 4y-2x+5x-3y.

(d) 3x+2=7x-4+5x+6.

THEORY OF GRADE 8

INSTRUCTION:  Answer THREE Questions ONLY. Question TWO is compulsory!

[1] (a) Sketch the graphs of the following in  equalities : (i) x ≤ -2 (ii) x > 1 (iii) x ≥ 4 (iv) x < -2.                  (b) What is Pythagoras’ theorem? (c) Which of the following are Pythagorean triples?                     (i) (10, 24, 26) (ii) (14, 49, and 50). (d) A ladder 9.6m long leans against a wall. It touches the wall at a point 9m above the ground. Find the distance of the foot of the ladder from the wall.

[2] The table below gives corresponding values of x and y for the equation y = x – 4.

X	-3	0	+3
y	-7	-4	-1

(a) Using a scale of 2cm to 1 unit on both axes, draw the graph of y = x-4. Using the graph to:              (b) find the value of y when x = 1; (c) find the value of x when y = -5; (d) write down the coordinates of the points where the graph cuts the axes.

[3] (a) What is the instrument used in measuring the angles of elevation and depression?                      (b) The angle of elevation of the top of a tower from a point 42m away from its base on level ground is 36°. Find the height of the tower. (c) From the top of a building 50m high, the angle of depression of a car is 55°. Find the distance of the car from the foot of the building.

[4] (a) A bag contains 6 blue balls, 4 red balls, and 10 black balls. What is the probability of selecting a ball from the bag at random to obtain: (i) a blue ball; (ii) a red ball; (iii) a black or a red ball            (iv) neither a blue nor a red ball; (v) not a red ball; (vi) a yellow ball?  (b) A card is selected from a well shuffled standard pack of 52 cards. What is the probability of getting:  (i) a diamond (ii) an ace (iii) a red card (iv)the ace of spades (v) any card other than an ace?