# WAEC

## The table shows the distribution of ages of a number of children in a school. If the mean of the distribution is 7, find the; (a) value of x, (b) standard deviation of their ages.

Age 3 4 5 6 7 8 9 10
No. of children 2 6 5 4 6 9 8 5
The table shows the distribution of ages of a number of children in a school. If the mean of the distribution is 7, find the;
(a) value of x, (b) standard deviation of their ages.
Solution & Explanation
Mean = (3Ã—2 +4Ã—6 + 5Ã—5 + 6x + 7Ã—6 + 8Ã—9 + 9Ã—8 + 10Ã—5) Ã· (2 + 6 + 5 + x + 6 + 9 + 8 + 5)
7 = (291 + 6x) Ã· (41+x)
7(41+x) = 291 + 6x
287 + 7x = 291 + 6x
x = 4
(b)
Age(x in yrs) No.of child Fx2
3 2 18
4 6 96
5 5 125
6 4 144
7 6 294
8 9 576
9 8 648
10 5 500

Î£f = 45 Î£fx2 = 2401

Standard deviation = âˆš196/45 = âˆš4.3555
= 2.087

## (a) Copy and complete the table of values for y = 3Sinx + 7Cosx for 0Â°

(a) Copy and complete the table of values for y = 3Sinx + 7Cosx for 0Â°
xÂº 0 20 40 60 80 100 120 140 160 180
y 7.0 4.2 -0.9

(b) Using a scale of 2cm to 20Â° on the x-axis and 2cm to 2 units on the y-axis, draw the graph of y = 3Sinx + 7Cosx for 0Â°
(c) Using the graph, find the;
(i) value of y when x= 150Â°,
(i) range of values of x for which y > 0.

Solution& Explanation
a
xÂº 0 20 40 60 80 100 120 140 160 180
y 7.0 7.6 7.3 6.1 4.2 1.7 -0.9 -3.4 -5.6 7.0

B
(C)(i) From the graph, When x is 150Â° y = -4.6 Â± 0.2
(ii) Range of value of x for which y > is x: 0Âº â‰¤ x â‰¤ 113

## The table shows the monthly expenditure (in percentages) of Mr. Okaforâ€™s salary. (a) Calculate the percentage of Mr. Okaforâ€™s salary that was. put into salary. (b) Illustrate the information on a pie chart. (c) If Mr. Okaforâ€™s annual gross salary is $28,800.00 and he pays tax of 12%. Calculate: (i) his monthly tax; (ii) amount saved each month. Item food & drinks fuel Rent Building project Education Saving Percentage% 35 7.5 1.0 15 17.5 x The table shows the monthly expenditure (in percentages) of Mr. Okaforâ€™s salary. (a) Calculate the percentage of Mr. Okaforâ€™s salary that was. put into salary. (b) Illustrate the information on a pie chart. (c) If Mr. Okaforâ€™s annual gross salary is$28,800.00 and he pays tax of 12%.
Calculate: (i) his monthly tax; (ii) amount saved each month.

Solution& Explanation
(a) 35 + 7.5 + 10 + 15 + 17.5 + x = 100
85 + x = 100.
x = 100 â€“ 85
x = 15
(b)
(c) income tax = 12/100âˆ—28,800
= $3,456.00 Monthly tax = 3456/12 =$288.00

(ii) Monthly net salary = 1/12(288âˆ’3456)
= $2,112.00 Amount saved each month = 15/100âˆ—2112 =$316.80

(a) A man purchased 180 copies of a book at N250.00 each. He sold y copies at N300.00 each and the rest at a discount of 5 kobo in the Naira of the cost price.
If he made a profit of N7,125.00, find the value of y.
(b) A trader bought x bags of rice at a cost C = 24x + 103 and sold them at a price, S = x220âˆ’33x 220âˆ’33 .
Find the expression for the profit (i) If 20 bags of rice were sold,
(ii) calculate the percentage profit.

Solution& Explanation
(a) Total cost 180 x 250
=N45,000.00
Total selling price = 300y + (180-y) x 250 x 95100
300y + 42750 -237.5y

= 62.5y + 42750

Profit = 62.5y + 42750 -4500
7125 = 62.5y â€“ 2250
9375 = 62.5y
y = 150

(b) Profit = x2 /20âˆ’33x
(24x+ 103)
9x â€“ x2 /20 â€“ 103

(ii) Cost = 24(20) +103 = 583.00
Selling Price =33(20) â€“ 202 /20
= 640
Percentage profit = 640âˆ’583/583 x 100%
= 9.78%

## (a) Given that (7 -2x), 9, (5x + 17) are consecutive terms of a Geometric Progression (G. P) with common ratio, r>0, find the values of x. (b) Two positive numbers are in the ratio 3:4. The sum of thrice the first number and twice the second is 68. Find the smaller number.

MATHEMATICS 2022 WAEC THEORY

(a) Given that (7 -2x), 9, (5x + 17) are consecutive terms of a Geometric Progression (G. P) with common ratio, r>0, find the values of x.
(b) Two positive numbers are in the ratio 3:4. The sum of thrice the first number and twice the second is 68. Find the smaller number.

Solution & Explanation

a. Hereâ€™s how to find the values of x:

Set up the equation using the common ratio:
â€“ In a geometric progression, the ratio of consecutive terms is constant.
â€“ Therefore, we have:
(9 / (7 â€“ 2x)) = ((5x + 17) / 9)

Cross-multiply and simplify:
â€“ Multiply both sides by 9 * (7 â€“ 2x) to eliminate fractions:
9 * 9 = (5x + 17) * (7 â€“ 2x)
â€“ Expand both sides:
81 = 35x â€“ 10x^2 + 119 â€“ 34x
0 = -10x^2 + x + 38

â€“ The equation factors as:
-10x^2 + x + 38 = -(5x â€“ 2)(2x â€“ 19) = 0

Solve for x:
â€“ Set each factor equal to 0 and solve for x:
5x â€“ 2 = 0 â†’ x = 2/5
2x â€“ 19 = 0 â†’ x = 19/2

Therefore, the possible values of x are x = 2/5 or x = 19/2.

b. Letâ€™s solve this step-by-step:

1Represent the numbers:

Since the ratio is 3:4, let the first number be 3x and the second number be 4x.
Set up the equation based on the information:

We know that the sum of thrice the first number and twice the second is 68. So, the equation is:
3(3x) + 2(4x) = 68
9x + 8x = 68

3Combine like terms and solve for x:
17x = 68
x = 4

4Find the smaller number:
Remember, the first number was represented by 3x. So, the smaller number is:
Smaller number = 3 * 4 = 12

Therefore, the smaller number in this ratio is 12

## Solve 6Ã—2 = 5x â€“ 1

Solve 6Ã—2 = 5x â€“ 1
A. x = 2,3
B. x = 0,3
C. x = 1/2, 1/3
D. x = 1/2, âˆ’1/3

## Given that log3 27 = 2x + 1, find the value of x.

Given that log3 27 = 2x + 1, find the value of x.
A. 0
B. 1
C. 2
D. 3

## A trader made a loss of 15% when an article was sold. Find the ratio of the selling price : cost price

A trader made a loss of 15% when an article was sold. Find the ratio of the selling price : cost price
A. 3:20
B. 3:17
C. 17:20
D. 20:23

When a trader makes a loss of 15%, it means that the selling price (SP) is 85% of the cost price (CP).
Letâ€™s denote the selling price as SP and the cost price as CP. We can write the equation:
SP = 85% of CP
Mathematically, this can be represented as:
SP = 0.85 * CP
To find the ratio of the selling price to the cost price, we divide the selling price by the cost price:
SP / CP = (0.85 * CP) / CP
Simplifying:
SP / CP = 0.85
Therefore, the ratio of the selling price to the cost price is 0.85 : 1, or 17 : 20

## If the equations x2 â€“ 5x + 6 = 0 and x + px + 6 = 0 have the same roots, find the value of p.

If the equations x2 â€“ 5x + 6 = 0 and x + px + 6 = 0 have the same roots, find the value of p.
A. 5
B. 6
C. -5
D. -6

x2 â€“ 5x + 6 = 0 (Equation 1) x + px + 6 = 0 (Equation 2)
In Equation 1, the coefficients of x2, x, and the constant term are 1, -5, and 6, respectively. In Equation 2, the coefficients of x2, x, and the constant term are 0, p, and 6, respectively.
To have the same roots, the coefficients of corresponding terms in the two equations must be proportional. This means:
1/0 = -5/p = 6/6
Since 1/0 is undefined, we can focus on the other proportion:
-5/p = 6/6
Simplifying:
-5/p = 1
Cross-multiplying:
-5 = p
Therefore, the value of p is -5.

## The straight line y = mx â€“ 4 passes through the point(-4,16). Calculate the gradient of the line

The straight line y = mx â€“ 4 passes through the point(-4,16). Calculate the gradient of the line
A. -5
B. -3
C. 3
D. 5