WAEC SSCE - General Mathematics - 2018

The diagonals of a rhombus WXYZ intersect at M. If |MW| = 5cm and |MX| = 12cm, calculate its perimeter

The table shows the distribution of goals scored by 25 teams in a football competition. Calculate the probability that a team selected at random scored at most 3 goals.

The surface area of a sphere is \(\frac{792}{7} cm^2\). Find, correct to the nearest whole number, its volume. [Take \(\pi = \frac{22}{7}\)]

Find the value of x for which \(32_{four} = 22_x\)

If 3x\(^o\) 4(mod 5), find the least value of x

There are 250 boys and 150 girls in a school, if 60% of the boys and 40% of the girls play football, what percentage of the school play football?

The angle of elevation of the top of a tree from a point 27m away and on the same horizontal ground as the foot of the tree is 30\(^o\). Find the height of the tree.

Factorise completely the expression

\((x + 2)^2\) - \((2x + 1)^2\)

Find the value of x for which \(\frac{x - 5}{x(x - 1)}\) is defined

Donations during the launching of a church project were sent in sealed envolopes. The table shows the distribution of the amount of money in the envelope. How much was the donation?

Given that y varies inversely as the square of x. If x = 3 when y = 100, find the equation connecting x and y.

Evaluate: \((64^{\frac{1}{2}} + 125^{\frac{1}{3}})^2\)

Solved the equation \(2x^2 - x - 6\) = 0

The angles of a polygon are x, 2x, 2x, (x + \(30^o\)), (x + \(20^o\)) and (x - \(10^o\)). Find the value of x

If y + 2x = 4 and y - 3x = -1, find the value of (x + y)

Find the value of t in the diagram

If M and N are the points (-3, 8) and (5, -7) respectively, find |MN|

In the diagram, PS and RS are tangents to the circle centre O,

In the diagram, PR is a tangent to the circle at Q, QT//RS,

Given that Y is 20cm on a bearing of 300\(^o\) from x, how far south of y is x?

If x : y : z = 3 : 3 : 4, evaluate \(\frac{9x + 3y}{6z - 2y}\)

If F = \(\frac{9}{5}\)C + 32, find C when F = 98.6

Find the mean deviation of 20, 30, 25, 40, 35, 50, 45, 40, 20 and 45

The mean of 1, 3, 5, 7 and x is 4. Find the value of x

The equation of the line through the points (4,2) and (-8, -2) is 3y = px + q, where p and q are constants. Find the value of p.

In the diagram, which of the following ratios is equal to \(\frac{|PN|}{|PQ|}\)?

If \(\log_{10}\)(6x - 4) - \(\log_{10}\)2 = 1, solve for x.

The table shows the distribution of goals scored by 25 teams in a football competition. Calculate the probability that a team selected at randon scored either 4 or 7 goals.

Simplify; 2\(\frac{1}{4} \times 3\frac{1}{2} \div  4 \frac{3}{8}\)

Expression 0.612 in the form \(\frac{x}{y}\), where x and y are integers and y \(\neq\) 0

If tan x = \(\frac{4}{3}\), 0\(^o\) < x < 90\(^o\), find the value of sin x - cos x

Find  the \(n^{th}\) term of the sequence 2 x 3, 4 x 6, 8 x 9, 16 x 12...

Find the median of 2, 1, 0, 3, 1, 1, 4, 0, 1 and 2

The total surface area of a hemispher is 75\(\pi cm^2\). Find the radius.

In the diagram, PQ is a straight line, (m + n) = 110\(^o\) and (n + r) = 130\(^o\) and (m + r) = 120\(^o\). Find the ratio of m : n : r

If P and Q are two statements, under what condition would p|q be false?

M and N are two subsets of the universal set (U). If n(U) = 48, n(M) = 20, n(N) = 30 and n(MUN) = 40, find n(M \(\cap\) N)

In the diagram, WXYZ is a rectangle with diamension 8cm by 6cm. P, Q, R and S are the midpoints of the rectangle as shown. Using this information calculate the area of the part of the rectangle that is not shaded

The solution of x + 2 \(\geq\) 2x + 1 is illustrated

A piece of thread of length 21.4cm is used to form a sector of a circle of radius 4.2cm on a piece of cloth. Calculate, correct to the nearest degree, the angle of the sector. [Take \(\pi = \frac{22}{7}\)]

Simplify; \(\frac{2 - 18m^2}{1 + 3m}\)

The diagram shows a trapezium inscribed in a semi-circle. If O is the mid-point of WZ and |WX| = |XY| = |YZ|, calculate the value of m

The volume of a cylindrical tank, 10m high is 385 m\(^2\). Find the diameter of the tank. [Take \(\pi = \frac{22}{7}\)]

In the diagram, PQ//RS. Find x in terms of y and z

In the diagram, WXYZ is a rectangle with dimension 8cm by 6cm. P, Q, R and S are the midpoints of the sides of the rectangle as shown. Using this information, what type of quadrilateral is the shaded region?

Find the inter-quartile range of 1, 3, 4, 5, 8, 9, 10, 11, 12, 14, 16

A curve is such that when y = 0, x = -2 or x = 3. Find the equation of the curve.

Simplify: \(\sqrt{108} + \sqrt{125} - \sqrt{75}\)

If  x : y = \(\frac{1}{4} : \frac{3}{8}\) and y : z = \(\frac{1}{3} : \frac{4}{9}\), find x : z

The graph of y = x\(^2\) and y = x intersect at which of these points?

(a) Lamin bought a book for N300.00 and sold it to Bola at a profit of x%. Bola then sold the same book at a profit of x%. If James paid \(N(6x + \frac{3}{4})\) more for the book than Lamin paid, find the value of x.

(b) Find the range of values of x which satisfies the inequality \(3x - 2 < 10 + x < 2 + 5x\).

(a) If the mean of m, n, s, p and q is 12, calculate the mean of (m + 4), (n - 3), (s + 6), (p - 2) and (q + 8).

(b) In a community of 500 people, the 75th percentile age is 65 years while the 25th percentile age is 15 years. How many of the people are between 15 and 65 years?

(a) Find the equation of the line passing through the points (2, 5) and (-4, -7).

(b) Three ships P, Q and R are at sea. The bearing of Q from P is 030° and the bearing of P and R is 300°.  If |PQ| = 5 km and |PR| = 8 km,

(i) Illustrate the information in a diagram.

(ii) Calculate, correct to three significant figures, the:

(1) distance between Q and R

(2) bearing of R from Q.

The frequency distribution shows the marks distribution of a class of 30 students in an examination.

The mean mark of the distribution is 52.

(a) Find the values of x and y.

(b) Construct a group frequency distribution table starting with a lower class limit of 1 and class interval of 10.

(c) Draw a histogram for the distribution

(d) Use the histogram to estimate the mode.

Using ruler and a pair of compasses only, construct:

(a) (i) quadrilateral PQRS with |PQ| = 6 cm, |PS| = 8 cm, < PSR = 90\(^o\), |SR| = 12 cm and |QR| = 11 cm;

(ii) perpendicular from Q to cut \(\over{SR}\) at K.

(b) Measure:

(i) IQRI;

(ii) < QRS.
 

(a) The diagram, = IWYI = IXZI and < WXY = 80\(^o\). What is the size of < XWZ?

(b) A man was charged 2 kobo per month for every N1.00 he borrowed from a bank. At what rate per annum was the interest charged?

(a) If \(x = \begin{pmatrix} 2 \\ 3 \end{pmatrix}, y = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\) and \(z = \begin{pmatrix} -4 \\ 13 \end{pmatrix}\), find the scalars p and q such that \(px + qy = z\).

(b)(i) Using the scale of 2cm to 2 units on both axis, draw on a graph paper two perpendicular axis x and y for \(-5 \leq x \leq 5, -5 \leq y \leq 5\) respectively.

(ii) Draw, on the graph paper, indicating clearly the vertices and their coordinates,

(1) the quadrilateral WXYZ with W(2, 3), X(4, -1), Y(-3, -4) and Z(-3, 2).

(2) the image \(W_{1}X_{1}Y_{1}Z_{1}\) of the quadrilateral WXYZ under an anti-clockwise rotation of 90° about the origin where \(W \to W_{1}, X \to X_{1}, Y \to Y_{1}\) and \(Z \to Z_{1}\).

(a) The diagonals of a rhombus are 10.2 cm and 9.3 cm long. Calculate, correct to one decimal place, the perimeter of the rhombus.

(b) Given that \(\sin x = \frac{3}{5}, 0° < x < 90°\), find the value of \(5\cos x - 4\tan x\).

In the diagram. PQR is an isosceles triangle. If the perimeter of the triangle is 28 cm, find the:

a. values of x and y;

b. lengths of the sides of the triangle. 

(a) Using a scale of 2cm to 2units on both axes, draw on a sheet of graph paper two perpendicular axes 0x and 0y for - 10 \(\leq\) x \(\leq\) 10  and -10 \(\leq\) y \(\leq\)10

(b) Given the points P(3, 2). Q(-1. 5). R(0. 8) and S(3, 7). draw on the same graph, indicating clearly the vertices and their coordinates, the:

(i) quadrilateral PQRS;

(ii) image P\(_1\)Q\(_1\)R\(_1\)S\(_1\) of PQRS under an anticlockwise rotation of 90\(^o\) about the origin where P \(\to\) P\(_1\), Q \(\to\) Q\(_{1}\), R \(\to\) R\(_{1}\) and S \(\to\) S\(_{1}\)

(iii) image P\(_2\)Q\(_2\)R\(_2\)S\(_2\) of P\(_1\)Q\(_1\)R\(_1\)S\(_1\) under a reflection in the line y - x = 0 where P\(_1\) \(\to\) P\(_2\), Q\(_1\) \(\to\) Q\(_{2}\), R\(_1\) \(\to\) R\(_{2}\) and S\(_1\) \(\to\) S\(_{2}\)

(c) Describe precisely the single transformation T for which T : PQRS \(\to\)  P\(_2\)Q\(_2\)R\(_2\)S\(_2\)

(d) The side P\(_1\)Q\(_1\) of the quadrilateral P\(_1\)Q\(_1\)R\(_1\)S\(_1\) cuts the x-axis at the point W. What type of quadrilateral is P\(_1\)S\(_1\)R\(_1\)W?

In the diagram, |PT| = 4 cm, |TS| = 6 cm, |PQ| = 6 cm and < SPR = 30°. Calculate, correct to the nearest whole number:

(a) |SR| ;

(b) area of TQRS.

A shop had two reduction sales during which prices of all items were reduced by 40% in the first sales and 30% in the second.

1. If a shirt was sold at GH₵35.00 during the second reduction sales, find the price before the first sale.
2. If the price of an article before the first reduction was GH₵180.00, find the total:
3. reduction in the price due to the two sales;
4. percentage reduction in the price of the article.

The table shows the distribution of marks scored by students in a test.

(a) Construct a cumulative frequency table for the distribution.

(b) Draw a cumulative frequency curve for the distribution.

(c) Use the curve to estimate the:

(i) median;

(ii) probability that a student selected at random obtained distinction, if the lowest mark for distinction is 75%.

 

a. Find the range of values of x which satisfy the following inequalities simultaneously: 5 - x > 1 and 9 + x \(\geq\) 8

In the diagram, O is the centre of the circle, IPQI = IQRI and < PSR = 56°. Find < QRS. 

The table shows the distribution of sources obtained when a fair diwe was rolled 50 times.

1. Draw a bar chart for the distribution

2. Calculate the mean score of the distribution

a. A textbook company discovered that the profit made from selling its books is given by y = \(\frac{x^2}{8}\) + 5x, where x is the number of textbooks sold (in thousands) and y is the corresponding profit (in Ghana Cedis). If the company made a profit of GH₵ 20,000.00

i. form a quadratic equation in x;

ii. (using the quadratic formula, find, correct to the nearest whole number, the number of textbooks sold to make the profit.

b. The angle of elevation of the top T of a tree from a point P on the same ground level as the foot Q of a tree is 28\(^o\). A bird perched at a point R, halfway up the tree.

i. Represent the information in a diagram.

ii. Calculate, correct to the nearest degree, the angle of elevation of R from P.

(a) 

In the diagram

(i) The value of x; (ii)

(b)  If \(2N4_{seven} = 15N_{nine}\), find the value of N.

(a) Copy and complete the table of values for y = 2x\(^{2}\) + x - 10 for -5 \(\leq\) x \(\leq\) 4.

(b) Using scales of 2cm to 1 unit on the x- axis and 2cm to 5 units on the y- axis, Draw the graph of y = 2x\(^{2}\) + x - 10 for -5 \(\leq\) x \(\leq\) 4.

(c) Use the graph to find the solution of :

(i) 2x\(^{2}\) + x = 10

(ii) 2x\(^{2}\) + x - 10 = 2x

(a) In a right-angled triangle, sin ⁡X = \(\frac{3}{5}\). Evaluate, leaving the answer as a fraction, 5 (cosX)\(^2\) – 3.

(b) The base of a pyramid, 12 cm high, is a rectangle with dimensions 42 cm by 11 cm. if the pyramid is filled with water and emptied into a conical container of equal height and volume, calculate, leaving the answer in surd form (radicals), the base radius of the container. [Take π=\(\frac{22}{7}\)]

A used car was purchased at N900,000.00. Its value depreciated by 30% in the first year. In each subsequent year, the depreciation was 22% of its value at the beginning of the year. If the car was bought on the 1st of March, 2011, calculate, correct to the nearest hundred naira, the value of the car on the 28th of February, 2015.

(a) The graph of \(y = 2px^{2} - p^{2}x - 14\) passes through the point (3, 10). Find the values of p.

(b) Two lines, \(3y - 2x = 21\) and \(4y + 5x = 5\) intersect at the point Q. Find the coordinates of Q.

In a road worthiness test on 240 cars, 60% passed. The number that failed had faults in Clutch, Brakes and Steering as follows: Clutch only - 28, Clutch and Steering - 14; Clutch, Steering and Brakes - 8; Clutch and Brakes - 20; Brakes and Steering only - 6. The number of cars with faults in Steering only is twice the number of cars with faults in Brakes only.

(a) Draw a Venn Diagram to illustrate this information.

(b) How many cars had : (i) Faulty Brakes?  (ii)  Only one fault?

(a) Evaluate without using calculator, (\(\frac{1}{4} \times 9\frac{1}{7} + \frac{2}{5} (\frac{2}{2} + \frac{3}{4})) \div (\frac{2}{5} - \frac{1}{4})\)

(b) A hunter walked 250 m from point P to Q on a bearing 042\(^o\). Calculate, correct to the nearest meter the vertical distance he has moved.

a) Copy and complete the following table of values for y = 2 cos x – sin x ,\(0^o \leq x \leq 300^o\)

\[\begin{array}{c|c} x & 0^o & 30^o & 60^o & 90^o & 120^o & 150^o & 180^o & 210^o & 240^o & 270^o& 300^o \\ \hline Y & 2.00 & & 0.13 &  & -1.87 & & -2.00 &  & -0.13 & &  \end{array}\]

(b) Using scales of 2 cm to 30\(^o\) on the x-axis and 2cm to 1 unit on the y-axis, draw the graph of y = 2 cos x – sin x for \(0^o \leq x \leq 300^o\)

(c) Use the graph to find the value(s) of x for which:

(i) 2 cos x – sin x = 1;

(ii) tan x = 2.

(a)
(i) Copy and complete the addition \(\oplus\) and multiplication \(\otimes\) tables in modulo 5 on the set {2, 3, 4}.

(ii) Use the tables to:

(a) solve the equation 4 \(\otimes\)e\(\oplus\) 2 \(\equiv\) 1 (mod 5):
(b) find the value of n, if 4 \(\oplus\)n\(\otimes\) 2 \(\equiv\) (mod 5).

(b) Consider the following statements:

p: Landi has cholera,

q: Landi is in the hospital.

If p = q, state whether or not the following statements are valid:

(i) If Landi is in the hospital, then he has cholera.

(ii) If Landi is not in the hospital, then he does not have cholera.

(iii) If Landi does not have cholera, then he is not in the hospital.

(a) Using ruler a pair of compasses only, construct:

(i) a trapezium PQRS such that |PQ| = 6.8 cm, < PQR = 120\(^o\), QR||PS, |PS| =10.6 cm, and IPRI = 93 cm;

(ii) locus \(l_1\) of points equidistant from P and R;

(iii) locus \(l_2\) of points equidistant from Q and R

 (b) Measure: (i) |QR|;

ii. < PSR

ii. < PSR

iii. |QY|, where Y is the point of intersection h and h. 

(a) If tan x = \(\frac{5}{12}\), \(0^o\). < x < 90°, evaluate, without using Mathematical tables or calculator, \(\frac{sin x}{(sin x)^2 + cosx}\) 

(b) The diagram shows a rectangular lawn measuring 14m by 11m. A path of uniform width \(x\)m surrounds it. If the total area of the path is 186 m\(^2\), how wide is the path? 
 

(a) Evaluate: \(\int \limits_1^2 (2x^3 - 4x + 3) dx\)

(b) Given that P\(^{-1}\) = \(\begin{pmatrix} -1 & 1 \\ 4 & -3 \end{pmatrix}\), find the matrix P.

 

Musa is three years older than Manya. Seven years ago, Musa was twice as old as Manya.
(1) How old are they now?
(2) When  will the sum of them be 45?

If log\(_a\)(y + 2) = 1 + log\(_a\) x, find x in terms of y.

2. The table shows the distribution of timber production in five communities in a certain year

1. Draw a pie chart to represent the information.

2. What percentage of timber produced that year was from Amenfi?

3. If a tonne of timber is sold at $560.00, how much more revenue would Oda community receive than Bibiani?

(a) Mr John paid N4,800.00 in N1.00 ordinary shares of a company which sold at N2.50 per share. If dividend was declared at 25k per share, how much dividend did he get?

(b) Using the method of completing the square, solve \(\frac{1 - x}{x} + \frac{x}{1 - x} = \frac{5}{2}\)

 

A man starts from a point X and walk 285 m to Y on a bearing of 078\(^o\). He then walks due South to a point Z which is 307 m from X.

(a) Illustrate the information on a diagram.

(b) Find, correct to the nearest whole number, the:

(i) bearing of X from Z;
(ii) distance between Y and Z.
 

A container, in the form of a cone resting on its vertex, is full when 4.158 litres of water is poured into it.

(a) If the radius of its base is 21 cm,

(i) represent the information in a diagram;

(ii) calculate the height of the container.

(b) A certain amount of water is drawn out of the container such that the surface diameter of the water drops to 28 cm. Calculate the volume of the water drawn out. (Take \(\pi\) = \(\frac{22}{7}\))

The sum of the first ten terms of an Arithmetic Progression (A.P.) is 130. If the fifth term is 3 times the first term, find the:

1. common difference;
2. first term;
3. number of terms of the A.P. if the last term is 28.

(a)       In < PQS, |PQ| = 12 cm, |PS| = 5 cm, < SPQ = < PRQ = 90°, Find, correct to three significant figures, |PR|.

(b) The length of two ladders, L and M are 10m and 12m respectively. They are placed against a wall such that each ladder makes angle with the horizontal ground. If the foot of L is 8m from the foot of the wall.

(i) Draw a diagram to illustrate this information;  (ii) Calculate the height at which M touches the wall.

(a) The frequency distribution shows the range of prices of a brand of a car sold by a dealer and the corresponding quantity demanded. 

(b) Represent the information in a histogram and use the histogram to determine the most preferred selling price for the brand of car.

The diagram shows an athletics track with two parallel sides and two semi-circular ends Each of the parallel sides is 60 metres, long and the diameter of each semi-circular end is 120 metres long.

(a) Calculate the distance covered by an athlete who runs round the tack the two times. [Take \(\pi\) = \(\frac{22}{7}\)]

(b) If the athlete spends 200 seconds for the race, calculate the speed in km/h.

1. A donkey is tied with a rope to a post which is 15 m from a fence. If the length of the rope between the donkey and the post is 17m, calculate the length of the fence within the reach of the donkey.

2. The base of a right pyramid with vertex, V, is a square, PQRS, of side 15 cm. If the slant height is 32 cm long:

3. represent the information in a diagram;

4. calculate its:

5. height, correct to one decimal place;

6. volume, correct to the nearest \(cm^3\)