WAEC SSCE - General Mathematics - 2021
Frage 1 Bericht
correct 0.007985 to three significant figures.
Frage 2 Bericht
From a point T, a man moves 12km due west and then moves 12km due south to another point Q. Calculate the bearing of T from Q.
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Frage 3 Bericht
For what value of x is \(\frac{4 - 2x}{x + 1}\) undefined.
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A fraction is undefined when its denominator is equal to zero. Therefore, we need to find the value of x that makes the denominator x + 1 equal to zero.
x + 1 = 0
x = -1
Therefore, the value of x that makes the fraction undefined is x = -1
Frage 5 Bericht
A fair die is tossed twice what is the probability of get a sum of at least 10.
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Frage 6 Bericht
A man will be (x+10)years old in 8years time. If 2years ago he was 63 years., find the value of x
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A man will be (x+10) years old in 8years time.
As at today, he is x + 2 years of age.
The man was 63 years old 2 years ago, so he is 63+2=65 now.
8 years from now, he will be 65+8=73.
He will be (x+10) years old when he is 73. So
x+10=73
x=73-10=63
Frage 7 Bericht
If log\(_{10}\) 2 = m and log\(_{10}\) 3 = n, find log\(_{10}\) 24 in terms of m and n.
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Frage 8 Bericht
500 tickets were sold for a concert tickets for adults and children were sold at $4.50 and $3.00 respectively if the total receipts for the concerts was $1987.50 how many tickets for adults were sold?
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Let's assume that x is the number of tickets sold for adults, and y is the number of tickets sold for children. We know that the total number of tickets sold is 500, so we can write an equation: x + y = 500 We also know that the price of an adult ticket is $4.50 and the price of a child ticket is $3.00. So we can write another equation based on the total receipts: 4.5x + 3y = 1987.5 Now we have two equations with two unknowns, and we can solve for x, the number of adult tickets sold. One way to do this is to use the first equation to solve for y: y = 500 - x Then we can substitute this expression for y into the second equation: 4.5x + 3(500 - x) = 1987.5 Simplifying and solving for x: 4.5x + 1500 - 3x = 1987.5 1.5x = 487.5 x = 325 So the number of adult tickets sold is 325. Therefore, the correct answer is option (A): 325.
Frage 10 Bericht
In the diagram, line \(\overline{EC}\) is a diameter of the circle ABCDE.
If angle ABC equals 158°, find ?ADE
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Frage 11 Bericht
In the diagram O is the centre of the circle PQRS, ?PQR = 72° and OR is parallel to PS. Find ∠ OPS
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Frage 14 Bericht
The circumference of a circular track is 9km. A cyclist rides round it a number of times and stops after covering a distance of 302km. How far is the cyclist from the starting point?
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Frage 15 Bericht
if p = {-3<x<1} and Q = {-1<x<3}, where x is a real number, find P n Q.
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Frage 16 Bericht
In the diagram, \(\overline{MP}\) is a tangent to the circle NQR, ∠NQR, ∠PNQ = 64 and | \(\overline{RQ}\) | = | \(\overline{RN}\) |. Find the angle market t.
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Frage 17 Bericht
A box contains 40 identical balls of which 10 are red and 12 are blue. if a ball is selected at random from the box what is the probability that it is neither red nor blue?
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Frage 18 Bericht
A solid brass cube is melted and recast as a solid cone of height h and base radius r. If the height of the cube is h, find r in terms of h.
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Frage 19 Bericht
Find, correct to two decimal, the mean of 1\(\frac{1}{2}\), 2\(\frac{2}{3}\), 3\(\frac{3}{4}\), 4\(\frac{4}{5}\), and 5\(\frac{5}{6}\).
Frage 21 Bericht
In the diagram, ?XYZ is produced to T. if |XY| = |ZY| and ?XYT = 40°, find ?XZT
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Frage 22 Bericht
If 16 * 2\(^{(x + 1)}\) = 4\(^x\) * 8\(^{(1 - x)}\), find the value of x.
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Frage 23 Bericht
The distance d between two villages east more than 18 KM but not more than 23KM.
which of these inequalities represents the statements?
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Frage 25 Bericht
The pie chart represents the distribution of fruits on display in the shop if there are 60 apples on display how many oranges are there?
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Frage 28 Bericht
The sum of the interior angles of a regular polygon with k sides is (3k-10) right angles. Find the size of the exterior angle?
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Frage 29 Bericht
If \(\frac{2}{x-3}\) - \(\frac{3}{x-2}\) = \(\frac{p}{(x-3)(x -2)}\), find p.
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Frage 30 Bericht
What number should be subtracted from the sum of 2 \(\frac{1}{6}\) and 2\(\frac{7}{12}\) to give 3\(\frac{1}{4}\)?
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Frage 32 Bericht
A cone has a base radius of 8cm and height 11cm. calculate , correct to 2d.p, the curved surface area
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Frage 33 Bericht
The height of an equilateral triangle of side is 10 3√ cm. calculate its perimeter.
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Frage 34 Bericht
The equation of a line is given as 3 x - 5y = 7. Find its gradient (slope)
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The equation of a line in the slope-intercept form is y = mx + c, where m is the gradient (slope) of the line and c is the y-intercept. However, the given equation of the line is not in the slope-intercept form, but in the standard form. To find the slope of the line, we need to rewrite the equation in the slope-intercept form. We can do this by solving the equation for y: 3x - 5y = 7 -5y = -3x + 7 y = (3/5)x - 7/5 Comparing this with the slope-intercept form, we can see that the gradient (slope) of the line is 3/5. Therefore, the answer is 3/5.
Frage 35 Bericht
Mensah is 5 years old and joyce is thrice as old as mensah. In how many years will joyce be twice as old as Mensah?
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Frage 36 Bericht
consider the statements:
P = All students offering Literature(L) also offer History(H);
Q = Students offering History(H) do not offer Geography(G).
Which of the Venn diagram correctly illustrate the two statements?
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Frage 37 Bericht
A trader paid import duty of 38 kobo in the naira on the cost of an engine. If a total of #22,800.00 was paid as import duty, calculate the cost of the engine.
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Frage 38 Bericht
| Height(cm) | 160 | 161 | 162 | 163 | 164 | 165 |
| No. of players | 4 | 6 | 3 | 7 | 8 | 9 |
the table shows the height of 37 players of a basketball team calculates correct to one decimal place the mean height of the players.
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Frage 39 Bericht
Which of the following is not an exterior angle of a regular polygon?
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Frage 41 Bericht
\(\overline{XY}\) is a line segments with the coordinates X (- 8,- 12) and Y(p,q). if the midpoint of \(\overline{XY}\) is (-4,-2) find the coordinates of Y.
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The midpoint of a line segment is the point that is exactly halfway between the two endpoints of the segment. To find the midpoint, we take the average of the x-coordinates and the y-coordinates of the endpoints. So, for line segment XY with endpoints X (-8,-12) and Y (p,q), the midpoint is (-4,-2). Therefore, the average of the x-coordinates of X and Y is -4: (-8 + p)/2 = -4 Solving for p, we find: p = -8 + 2 * -4 = 0 Similarly, the average of the y-coordinates of X and Y is -2: (-12 + q)/2 = -2 Solving for q, we find: q = -12 + 2 * -2 = 8 Therefore, the coordinates of Y are (0,8).
Frage 42 Bericht
A trapezium of parellel sides 10cm and 21cm and height 8cm is inscribed in a circle of radius 7cm. calculate the area of the region not covered by the trapezium.
π =\(\frac{22}{7}\)
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Frage 43 Bericht
A cyclist moved at a speed of Xkm/h for 2 hours. He then increased his speed by 2 km/h for the next 3 hours.
If the total distance covered is 36 km, calculate his initials speed.
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Frage 46 Bericht
In △LMN, |LM| = 6cm, ∠LNM = x and sin x = sin x = \(\frac{3}{5}\).
Find the area of △LMN
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Frage 49 Bericht
In the diagram, ∠ABC and ∠BCD are right angles, ∠BAD = t and ∠EDF = 70°. Find the value of t.
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Frage 50 Bericht
.(a) In a class of 80 students,\(\frac{3}{4}\) study Biology and \(\frac{3}{5}\) study Physics.
If each student studies at least one of the subjects: (i) draw a Venn diagram to represent this information
(ii) how many students study both subjects
(iii) find the fraction of the class that study Biology but not Physics.
(b) Johnson and Jocatol Ltd. owned a business office with floor measuring 15m by 8 m which was to be carpeted.
The cost of carpeting was Gh¢ 890.00 per square metre. If a total of GH 216,120.00 was spent on painting and carpeting, how much was the cost of painting?
(a) Number studying Biology \(=\tfrac{3}{4}\times80=60\); Physics \(=\tfrac{3}{5}\times80=48\). Every student takes at least one subject, so none is outside.
(ii) Both subjects: let \(x\) study both.
\[60+48-x=80\Rightarrow 108-x=80\Rightarrow x=28.\]
(i) Venn diagram: Biology-only \(=60-28=32\); Physics-only \(=48-28=20\); overlap \(=28\). Check: \(32+28+20=80.\)
(iii) Fraction studying Biology but not Physics:
\[\frac{60-28}{80}=\frac{32}{80}=\frac{2}{5}.\]
(b) Floor area \(=15\times8=120\text{ m}^{2}\). Carpeting cost \(=120\times890=\text{GH}\!\cent\,106{,}800\).
Cost of painting \(=216{,}120-106{,}800=\text{GH}\!\cent\,109{,}320.00.\)
Antwortdetails
(a) Number studying Biology \(=\tfrac{3}{4}\times80=60\); Physics \(=\tfrac{3}{5}\times80=48\). Every student takes at least one subject, so none is outside.
(ii) Both subjects: let \(x\) study both.
\[60+48-x=80\Rightarrow 108-x=80\Rightarrow x=28.\]
(i) Venn diagram: Biology-only \(=60-28=32\); Physics-only \(=48-28=20\); overlap \(=28\). Check: \(32+28+20=80.\)
(iii) Fraction studying Biology but not Physics:
\[\frac{60-28}{80}=\frac{32}{80}=\frac{2}{5}.\]
(b) Floor area \(=15\times8=120\text{ m}^{2}\). Carpeting cost \(=120\times890=\text{GH}\!\cent\,106{,}800\).
Cost of painting \(=216{,}120-106{,}800=\text{GH}\!\cent\,109{,}320.00.\)
Frage 51 Bericht
(a) Mr Sarfo borrowed $25,000 from Afiak financial services at 21% simple interest per annum for 3 years if he was able to pay back the loan in two years at equal yearly installments how much did he pay each year?
(b) Two consecutive numbers are such that the sum of thrice the smaller and twice the larger is 17.
find correct through three significant figures the smaller number as a percentage of the sum of the two numbers
(a) If Mr. Sarfo borrowed $25,000 at a simple interest rate of 21% per annum for 3 years, the total amount of interest he would have to pay is:
$25,000 x 21% x 3 = $15,750
So, the total amount he would have to pay back is:
$25,000 + $15,750 = $40,750
If he pays back the loan in two years at equal yearly installments, he would have to pay:
$40,750 / 2 = $20,375 per year
Therefore, Mr. Sarfo would have to pay $20,375 each year to pay off the loan in two years.
(b) Let's assume that the two consecutive numbers are x and x+1 (where x is the smaller number).
According to the problem, the sum of thrice the smaller and twice the larger is 17. So, we can write an equation as:
3x + 2(x+1) = 17
Simplifying this equation, we get:
5x + 2 = 17
5x = 15
x = 3
So, the smaller number is 3 and the larger number is 4.
The sum of the two numbers is 7, so the percentage that the smaller number represents of the sum is:
(3/7) x 100% = 42.86% (rounded to three significant figures)
Therefore, the smaller number represents 42.86% of the sum of the two numbers.
Antwortdetails
(a) If Mr. Sarfo borrowed $25,000 at a simple interest rate of 21% per annum for 3 years, the total amount of interest he would have to pay is:
$25,000 x 21% x 3 = $15,750
So, the total amount he would have to pay back is:
$25,000 + $15,750 = $40,750
If he pays back the loan in two years at equal yearly installments, he would have to pay:
$40,750 / 2 = $20,375 per year
Therefore, Mr. Sarfo would have to pay $20,375 each year to pay off the loan in two years.
(b) Let's assume that the two consecutive numbers are x and x+1 (where x is the smaller number).
According to the problem, the sum of thrice the smaller and twice the larger is 17. So, we can write an equation as:
3x + 2(x+1) = 17
Simplifying this equation, we get:
5x + 2 = 17
5x = 15
x = 3
So, the smaller number is 3 and the larger number is 4.
The sum of the two numbers is 7, so the percentage that the smaller number represents of the sum is:
(3/7) x 100% = 42.86% (rounded to three significant figures)
Therefore, the smaller number represents 42.86% of the sum of the two numbers.
Frage 52 Bericht
The points X, Y and Z are located such that Y is 15 km south of X, Z is 20 km from X on a bearing of 270".
Calculate, correct: (a) two significant figures, |YZ|
(b) The nearest degree, the bearing of Y from Z
Take \(X\) as origin. \(Y\) is 15 km due south of \(X\); \(Z\) is 20 km on bearing \(270^{\circ}\), i.e. due west of \(X\). So \(\angle YXZ=90^{\circ}\) (south and west are perpendicular).
(a) \(|YZ|\) by Pythagoras:
\[|YZ|=\sqrt{15^{2}+20^{2}}=\sqrt{225+400}=\sqrt{625}=25\text{ km (2 s.f.)}.\]
(b) Bearing of \(Y\) from \(Z\)
Place \(Z=(-20,0)\), \(Y=(0,-15)\). The vector \(Z\!\to\!Y=(20,-15)\): from \(Z\), \(Y\) lies east and south. The angle east of south:
\[\tan\theta=\frac{20}{15}=1.333\Rightarrow\theta=53.13^{\circ}\ \text{(east of due south)}.\]
Bearing \(=180^{\circ}-53.13^{\circ}=126.87^{\circ}\approx \mathbf{127^{\circ}}\) (nearest degree).
Antwortdetails
Take \(X\) as origin. \(Y\) is 15 km due south of \(X\); \(Z\) is 20 km on bearing \(270^{\circ}\), i.e. due west of \(X\). So \(\angle YXZ=90^{\circ}\) (south and west are perpendicular).
(a) \(|YZ|\) by Pythagoras:
\[|YZ|=\sqrt{15^{2}+20^{2}}=\sqrt{225+400}=\sqrt{625}=25\text{ km (2 s.f.)}.\]
(b) Bearing of \(Y\) from \(Z\)
Place \(Z=(-20,0)\), \(Y=(0,-15)\). The vector \(Z\!\to\!Y=(20,-15)\): from \(Z\), \(Y\) lies east and south. The angle east of south:
\[\tan\theta=\frac{20}{15}=1.333\Rightarrow\theta=53.13^{\circ}\ \text{(east of due south)}.\]
Bearing \(=180^{\circ}-53.13^{\circ}=126.87^{\circ}\approx \mathbf{127^{\circ}}\) (nearest degree).
Frage 53 Bericht
(a) A man shared his property among his children as follows:
| Child's name |
Ann | Afia | Kojo | Nuno | Akom |
| Percentage share |
5 | 15 | 10 | 45 | 25 |
Represent the information on a pie chart
(b) A box contains 5 red, 3 green and 4 blue identical beads. Calculate the probability th a girl takes away two red beads, one after the other, from the box.
(a) Pie chart of the property share.
| Child | Percentage share |
|---|---|
| Ann | 5 |
| Afia | 15 |
| Kojo | 10 |
| Nuno | 45 |
| Akom | 25 |
The shares sum to \(5+15+10+45+25 = 100\). Each sector angle is the percentage times \(\frac{360}{100}=3.6^\circ\):
| Child | Percentage | Sector angle |
|---|---|---|
| Ann | 5 | \(18^\circ\) |
| Afia | 15 | \(54^\circ\) |
| Kojo | 10 | \(36^\circ\) |
| Nuno | 45 | \(162^\circ\) |
| Akom | 25 | \(90^\circ\) |
Check: \(18+54+36+162+90 = 360^\circ\). Draw a circle and mark each labelled sector with a protractor.
(b) Probability of drawing two red beads. The box holds \(5 + 3 + 4 = 12\) beads, of which 5 are red. The two beads are taken one after the other without replacement.
\[P(\text{1st red}) = \frac{5}{12}, \qquad P(\text{2nd red}\mid\text{1st red}) = \frac{4}{11}.\]
\[P(\text{both red}) = \frac{5}{12}\times\frac{4}{11} = \frac{20}{132} = \frac{5}{33}.\]
Antwortdetails
(a) Pie chart of the property share.
| Child | Percentage share |
|---|---|
| Ann | 5 |
| Afia | 15 |
| Kojo | 10 |
| Nuno | 45 |
| Akom | 25 |
The shares sum to \(5+15+10+45+25 = 100\). Each sector angle is the percentage times \(\frac{360}{100}=3.6^\circ\):
| Child | Percentage | Sector angle |
|---|---|---|
| Ann | 5 | \(18^\circ\) |
| Afia | 15 | \(54^\circ\) |
| Kojo | 10 | \(36^\circ\) |
| Nuno | 45 | \(162^\circ\) |
| Akom | 25 | \(90^\circ\) |
Check: \(18+54+36+162+90 = 360^\circ\). Draw a circle and mark each labelled sector with a protractor.
(b) Probability of drawing two red beads. The box holds \(5 + 3 + 4 = 12\) beads, of which 5 are red. The two beads are taken one after the other without replacement.
\[P(\text{1st red}) = \frac{5}{12}, \qquad P(\text{2nd red}\mid\text{1st red}) = \frac{4}{11}.\]
\[P(\text{both red}) = \frac{5}{12}\times\frac{4}{11} = \frac{20}{132} = \frac{5}{33}.\]
Frage 54 Bericht
.(a) In APQR, ∠PQR= 90°. If its area is 216cm\(^2\) and |PQ|:|QR| is 3:4, find |PR|.
(b) The present ages of a man and his son are 47 years and 17 years respectively. In how many years would the man's age be twice that of his son?
(a) In \(\triangle PQR\), \(\angle PQR=90^{\circ}\), so \(PQ\) and \(QR\) are the perpendicular sides. Let \(PQ=3k,\ QR=4k\).
\[\text{Area}=\tfrac12\,PQ\cdot QR=216\Rightarrow\tfrac12(3k)(4k)=216\Rightarrow 6k^{2}=216\Rightarrow k^{2}=36\Rightarrow k=6.\]
So \(PQ=18\text{ cm},\ QR=24\text{ cm}\). By Pythagoras:
\[|PR|=\sqrt{18^{2}+24^{2}}=\sqrt{324+576}=\sqrt{900}=30\text{ cm}.\]
(b) Let the man's age be twice his son's after \(x\) years:
\[47+x=2(17+x)\Rightarrow 47+x=34+2x\Rightarrow x=13.\]
In 13 years' time the man will be twice as old as his son (60 and 30).
Antwortdetails
(a) In \(\triangle PQR\), \(\angle PQR=90^{\circ}\), so \(PQ\) and \(QR\) are the perpendicular sides. Let \(PQ=3k,\ QR=4k\).
\[\text{Area}=\tfrac12\,PQ\cdot QR=216\Rightarrow\tfrac12(3k)(4k)=216\Rightarrow 6k^{2}=216\Rightarrow k^{2}=36\Rightarrow k=6.\]
So \(PQ=18\text{ cm},\ QR=24\text{ cm}\). By Pythagoras:
\[|PR|=\sqrt{18^{2}+24^{2}}=\sqrt{324+576}=\sqrt{900}=30\text{ cm}.\]
(b) Let the man's age be twice his son's after \(x\) years:
\[47+x=2(17+x)\Rightarrow 47+x=34+2x\Rightarrow x=13.\]
In 13 years' time the man will be twice as old as his son (60 and 30).
Frage 55 Bericht
The table shows the distribution of the number of hours per day spent in studying by 50 students.
| Number of hours per day |
4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Number of students |
5 | 7 | 5 | 9 | 12 | 4 | 3 | 5 |
Calculate, correct to two decimal places,
the: (a) mean; (b) standard deviation.
Given distribution.
| Hours per day (x) | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|
| Students (f) | 5 | 7 | 5 | 9 | 12 | 4 | 3 | 5 |
Total number of students: \(\sum f = 50\).
(a) Mean. Compute \(\sum fx\):
| x | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|
| f | 5 | 7 | 5 | 9 | 12 | 4 | 3 | 5 |
| fx | 20 | 35 | 30 | 63 | 96 | 36 | 30 | 55 |
\[\sum fx = 20+35+30+63+96+36+30+55 = 365.\]
\[\bar{x} = \frac{\sum fx}{\sum f} = \frac{365}{50} = 7.30.\]
(b) Standard deviation. Compute \(\sum fx^{2}\):
| x | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|
| fx\(^2\) | 80 | 175 | 180 | 441 | 768 | 324 | 300 | 605 |
\[\sum fx^{2} = 80+175+180+441+768+324+300+605 = 2873.\]
\[\text{Variance} = \frac{\sum fx^{2}}{\sum f} - \bar{x}^{2} = \frac{2873}{50} - (7.3)^{2} = 57.46 - 53.29 = 4.17.\]
\[\text{S.D.} = \sqrt{4.17} = 2.04 \text{ (2 d.p.).}\]
The mean study time is 7.30 hours and the standard deviation is 2.04 hours.
Antwortdetails
Given distribution.
| Hours per day (x) | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|
| Students (f) | 5 | 7 | 5 | 9 | 12 | 4 | 3 | 5 |
Total number of students: \(\sum f = 50\).
(a) Mean. Compute \(\sum fx\):
| x | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|
| f | 5 | 7 | 5 | 9 | 12 | 4 | 3 | 5 |
| fx | 20 | 35 | 30 | 63 | 96 | 36 | 30 | 55 |
\[\sum fx = 20+35+30+63+96+36+30+55 = 365.\]
\[\bar{x} = \frac{\sum fx}{\sum f} = \frac{365}{50} = 7.30.\]
(b) Standard deviation. Compute \(\sum fx^{2}\):
| x | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|
| fx\(^2\) | 80 | 175 | 180 | 441 | 768 | 324 | 300 | 605 |
\[\sum fx^{2} = 80+175+180+441+768+324+300+605 = 2873.\]
\[\text{Variance} = \frac{\sum fx^{2}}{\sum f} - \bar{x}^{2} = \frac{2873}{50} - (7.3)^{2} = 57.46 - 53.29 = 4.17.\]
\[\text{S.D.} = \sqrt{4.17} = 2.04 \text{ (2 d.p.).}\]
The mean study time is 7.30 hours and the standard deviation is 2.04 hours.
Frage 56 Bericht
In the diagram, \(\overline{AD}\) is a diameter of a circle with Centre O. If ABD is a triangle in a semi-circle ∠OAB=34",
find: (a) ∠OAB (b) ∠OCB
(a) ∠OAB = 34°: Given in the problem.
(b) ∠OCB: We know that in a circle, opposite angles are equal. So, ∠OCB = ∠OAB = 34°.
Explanation: When a straight line cuts a circle at two points, it is called a chord of the circle. And when the chord is a diameter of the circle, it is called the diameter. The angle formed between the chord and the line drawn from the center of the circle to the midpoint of the chord is called the angle subtended by the chord at the center of the circle. And it is equal to half of the angle formed by the chord at the circumference of the circle.
So, in this case, ABD is a triangle inscribed in a semicircle. And ∠OAB is half of the central angle subtended by the chord AB. Hence, ∠OCB, which is opposite to ∠OAB is equal to ∠OAB.
Antwortdetails
(a) ∠OAB = 34°: Given in the problem.
(b) ∠OCB: We know that in a circle, opposite angles are equal. So, ∠OCB = ∠OAB = 34°.
Explanation: When a straight line cuts a circle at two points, it is called a chord of the circle. And when the chord is a diameter of the circle, it is called the diameter. The angle formed between the chord and the line drawn from the center of the circle to the midpoint of the chord is called the angle subtended by the chord at the center of the circle. And it is equal to half of the angle formed by the chord at the circumference of the circle.
So, in this case, ABD is a triangle inscribed in a semicircle. And ∠OAB is half of the central angle subtended by the chord AB. Hence, ∠OCB, which is opposite to ∠OAB is equal to ∠OAB.
Frage 57 Bericht
In the diagram, \(\overline{PQ//RS}\) is a trapezium with QR//PS. U and T are points on \(\overline{PS}\) such that \(\overline{|PU|}\) = 5 cm,
\(\overline{|QU|}\) = 12 cm and ?PUQ= ?STR =90°. If the area of PQR = 20 cm\(^2\),
calculate, correct to the nearest whole number, the:
(a) perimeter; (b) area; of the trapezium
Reading the diagram. The trapezium has \(QR \parallel PS\). \(QU\) and \(RT\) are drawn perpendicular to \(PS\), so \(QU = RT = 12\text{ cm}\) is the vertical height of the trapezium. From the figure: \(|PU| = 5\text{ cm}\), \(|QU| = 12\text{ cm}\), \(\angle PUQ = \angle STR = 90^\circ\), the angle at \(S\) is \(50^\circ\), and the area of \(\triangle PQR = 20\text{ cm}^2\).
Step 1: The slant side \(PQ\). Triangle \(PUQ\) is right-angled at \(U\):
\[|PQ| = \sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13\text{ cm}.\]Step 2: The top side \(QR\). Since \(QU \perp PS\), \(RT \perp PS\) and \(QR \parallel PS\), the figure \(QURT\) is a rectangle, so \(QR = UT\). Vertices \(Q\) and \(R\) both lie \(12\text{ cm}\) above line \(PS\), while \(P\) lies on \(PS\); hence the height of \(\triangle PQR\) on base \(QR\) is \(12\text{ cm}\):
\[\text{Area }\triangle PQR = \tfrac{1}{2}\times QR \times 12 = 6\,QR.\]\[6\,QR = 20 \;\Rightarrow\; QR = \tfrac{20}{6} = 3.33\text{ cm}.\]Step 3: The right-hand side using \(\angle S = 50^\circ\). Triangle \(STR\) is right-angled at \(T\) with \(RT = 12\):
\[|TS| = \frac{RT}{\tan 50^\circ} = \frac{12}{1.1918} = 10.07\text{ cm},\qquad |RS| = \frac{RT}{\sin 50^\circ} = \frac{12}{0.7660} = 15.67\text{ cm}.\]Step 4: The base \(PS\).
\[|PS| = |PU| + |UT| + |TS| = 5 + 3.33 + 10.07 = 18.40\text{ cm}.\](a) Perimeter of the trapezium.
\[P = |PQ| + |QR| + |RS| + |SP| = 13 + 3.33 + 15.67 + 18.40 = 50.40\text{ cm}.\]Perimeter \(\approx \mathbf{50\text{ cm}}\) (nearest whole number).
(b) Area of the trapezium. Parallel sides \(QR = 3.33\) and \(PS = 18.40\), height \(12\):
\[A = \tfrac{1}{2}(QR + PS)\times h = \tfrac{1}{2}(3.33 + 18.40)\times 12 = \tfrac{1}{2}(21.73)(12) = 130.4\text{ cm}^2.\]Area \(\approx \mathbf{130\text{ cm}^2}\) (nearest whole number).
Antwortdetails
Reading the diagram. The trapezium has \(QR \parallel PS\). \(QU\) and \(RT\) are drawn perpendicular to \(PS\), so \(QU = RT = 12\text{ cm}\) is the vertical height of the trapezium. From the figure: \(|PU| = 5\text{ cm}\), \(|QU| = 12\text{ cm}\), \(\angle PUQ = \angle STR = 90^\circ\), the angle at \(S\) is \(50^\circ\), and the area of \(\triangle PQR = 20\text{ cm}^2\).
Step 1: The slant side \(PQ\). Triangle \(PUQ\) is right-angled at \(U\):
\[|PQ| = \sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13\text{ cm}.\]Step 2: The top side \(QR\). Since \(QU \perp PS\), \(RT \perp PS\) and \(QR \parallel PS\), the figure \(QURT\) is a rectangle, so \(QR = UT\). Vertices \(Q\) and \(R\) both lie \(12\text{ cm}\) above line \(PS\), while \(P\) lies on \(PS\); hence the height of \(\triangle PQR\) on base \(QR\) is \(12\text{ cm}\):
\[\text{Area }\triangle PQR = \tfrac{1}{2}\times QR \times 12 = 6\,QR.\]\[6\,QR = 20 \;\Rightarrow\; QR = \tfrac{20}{6} = 3.33\text{ cm}.\]Step 3: The right-hand side using \(\angle S = 50^\circ\). Triangle \(STR\) is right-angled at \(T\) with \(RT = 12\):
\[|TS| = \frac{RT}{\tan 50^\circ} = \frac{12}{1.1918} = 10.07\text{ cm},\qquad |RS| = \frac{RT}{\sin 50^\circ} = \frac{12}{0.7660} = 15.67\text{ cm}.\]Step 4: The base \(PS\).
\[|PS| = |PU| + |UT| + |TS| = 5 + 3.33 + 10.07 = 18.40\text{ cm}.\](a) Perimeter of the trapezium.
\[P = |PQ| + |QR| + |RS| + |SP| = 13 + 3.33 + 15.67 + 18.40 = 50.40\text{ cm}.\]Perimeter \(\approx \mathbf{50\text{ cm}}\) (nearest whole number).
(b) Area of the trapezium. Parallel sides \(QR = 3.33\) and \(PS = 18.40\), height \(12\):
\[A = \tfrac{1}{2}(QR + PS)\times h = \tfrac{1}{2}(3.33 + 18.40)\times 12 = \tfrac{1}{2}(21.73)(12) = 130.4\text{ cm}^2.\]Area \(\approx \mathbf{130\text{ cm}^2}\) (nearest whole number).
Frage 58 Bericht
(a) On Sam's first birthday celebration, his grandfather deposited an amount of S 1,000.00 in a bank compound at 4 % interest annually.
Find how much is in the account if Sam is 4 years old.
In the diagram, ABCD are points on the circle centre O. If |AB| = |BC| and ∠ADC= 50°, find ∠BAD.
(a) Amount at end of the 2nd year = \(\frac{104}{100}\) x $1000
= $1,040.00
Amount at end of 3rd year = \(\frac{104}{100}\) x $1040
= $1,081.60
Amount at end of 4th year = \(\frac{104}{100}\) x $1081.6
= $1,124.86
(b) ∠ADC + ∠DCA+ ∠CAD = 180°
50° + 90°+ ∠CAD = 180°
∠CAD = (180 - 140)°
= 40
∠ADC + ∠ABC = 180°
50° + ∠ABC = 180°
∠ABC =130°
∠BAC+ ∠BCA + ∠ABC = 180°
But ∠BAC = ∠BCA
2∠BAC+ ∠130° = 180°
2 ∠BAC=50°
∠BAC = 25°
∠BAD = ∠CAD + ∠BAC
= 40°+25° = 65°
(a) If Sam's grandfather deposited $1,000.00 in a bank at 4% interest annually, after 4 years, the total amount in the account would be $1,124.86.
(b) To find ∠BAD, we need to find ∠CAD and ∠BAC first. We know that ∠ADC = 50° and ∠ADC + ∠DCA + ∠CAD = 180°, so ∠CAD = 180° - 50° - 90° = 40°. Next, we know that ∠ADC + ∠ABC = 180°, so ∠ABC = 180° - 50° = 130°. Also, ∠BAC + ∠BCA + ∠ABC = 180° and ∠BAC = ∠BCA, so 2∠BAC + 130° = 180° and ∠BAC = (180° - 130°) / 2 = 25°. Finally, ∠BAD = ∠CAD + ∠BAC = 40° + 25° = 65°.
Antwortdetails
(a) If Sam's grandfather deposited $1,000.00 in a bank at 4% interest annually, after 4 years, the total amount in the account would be $1,124.86.
(b) To find ∠BAD, we need to find ∠CAD and ∠BAC first. We know that ∠ADC = 50° and ∠ADC + ∠DCA + ∠CAD = 180°, so ∠CAD = 180° - 50° - 90° = 40°. Next, we know that ∠ADC + ∠ABC = 180°, so ∠ABC = 180° - 50° = 130°. Also, ∠BAC + ∠BCA + ∠ABC = 180° and ∠BAC = ∠BCA, so 2∠BAC + 130° = 180° and ∠BAC = (180° - 130°) / 2 = 25°. Finally, ∠BAD = ∠CAD + ∠BAC = 40° + 25° = 65°.
Frage 59 Bericht
(a) Copy and complete the table of values for the relation y=2x\(^2\) - x - 2 for 4 ≤ x ≤ 4.
| x | -4 | -3 | -2 | -2 | 0 | 1 | 2 | 3 | 4 |
| y | 19 | -2 | 26 |
(b) Using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of y = 2x\(^2\) - x - 2 for 4 ≤ x ≤ 4.
(c) On the same axes, draw the graph of y = 2x + 3.
(d) Use the graph to find the: (i) roots of the equation 2x-3r-5 0; (i) range of values of x for which 2x\(^2\) -x - 2<0.
(a)
To complete the table of values for the relation y=2x\(^2\) - x - 2 for 4 ≤ x ≤ 4, we need to substitute each value of x into the equation and calculate the corresponding value of y.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 30 | 19 | 8 | -1 | -2 | -1 | 4 | 13 | 26 |
(b)
Using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, we can plot the points from the completed table of values to draw the graph of y = 2x\(^2\) - x - 2 for 4 ≤ x ≤ 4. The graph should be a smooth curve passing through the plotted points.
(c)
On the same axes, we can draw the graph of y = 2x + 3 by plotting some points and drawing a straight line passing through them. For example, when x = -2, y = -1, and when x = 2, y = 7. This gives us two points (-2,-1) and (2,7), which we can use to draw the line.
(d)
(i)
To find the roots of the equation 2x - 3r - 5 = 0 using the graph, we need to look for the points where the graph intersects the line y = 2x - 5. These points correspond to the values of x that satisfy the equation. From the graph, it appears that the intersection point is around x = 3. Therefore, we can estimate that the root of the equation is x ≈ 3.
(ii)
To find the range of values of x for which 2x\(^2\) -x - 2 < 0 using the graph, we need to look for the points where the graph is below the x-axis. These points correspond to the values of x that satisfy the inequality. From the graph, it appears that the graph is below the x-axis for values of x between approximately -1.5 and 1.5. Therefore, the range of values of x for which 2x\(^2\) -x - 2 < 0 is -1.5 < x < 1.5.
Antwortdetails
(a)
To complete the table of values for the relation y=2x\(^2\) - x - 2 for 4 ≤ x ≤ 4, we need to substitute each value of x into the equation and calculate the corresponding value of y.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 30 | 19 | 8 | -1 | -2 | -1 | 4 | 13 | 26 |
(b)
Using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, we can plot the points from the completed table of values to draw the graph of y = 2x\(^2\) - x - 2 for 4 ≤ x ≤ 4. The graph should be a smooth curve passing through the plotted points.
(c)
On the same axes, we can draw the graph of y = 2x + 3 by plotting some points and drawing a straight line passing through them. For example, when x = -2, y = -1, and when x = 2, y = 7. This gives us two points (-2,-1) and (2,7), which we can use to draw the line.
(d)
(i)
To find the roots of the equation 2x - 3r - 5 = 0 using the graph, we need to look for the points where the graph intersects the line y = 2x - 5. These points correspond to the values of x that satisfy the equation. From the graph, it appears that the intersection point is around x = 3. Therefore, we can estimate that the root of the equation is x ≈ 3.
(ii)
To find the range of values of x for which 2x\(^2\) -x - 2 < 0 using the graph, we need to look for the points where the graph is below the x-axis. These points correspond to the values of x that satisfy the inequality. From the graph, it appears that the graph is below the x-axis for values of x between approximately -1.5 and 1.5. Therefore, the range of values of x for which 2x\(^2\) -x - 2 < 0 is -1.5 < x < 1.5.
Frage 60 Bericht
(a) A cottage is on a bearing of 200° and 110° from Dogbe's and Manu's farms respectively. If Dogbe walked 5 km and Manu 3 km from the cottage to their farms, find, correct to: (i) two significant figures, the distance between the two farms, (ii) the nearest degree, the bearing of Manu's farm from Dogbe's.
(b) A ladder 10 m long leaned against a vertical wall xm high. The distance between the wall and the foot of the ladder is 2 m longer than the height of the wall.
Calculate the value of x
(a) The cottage \(C\) is at bearing \(200^{\circ}\) from Dogbe's farm \(D\) and \(110^{\circ}\) from Manu's farm \(M\). Reversing bearings, from the cottage:
- Bearing of \(D\) from \(C=200^{\circ}-180^{\circ}=020^{\circ}\), with \(|CD|=5\) km.
- Bearing of \(M\) from \(C=110^{\circ}+180^{\circ}=290^{\circ}\), with \(|CM|=3\) km.
Angle at the cottage \(=290^{\circ}-020^{\circ}=270^{\circ}\Rightarrow\) reflex; the angle between the two directions is \(360^{\circ}-270^{\circ}=90^{\circ}\).
(i) With \(\angle DCM=90^{\circ}\):
\[|DM|=\sqrt{5^{2}+3^{2}}=\sqrt{34}=5.83\approx\mathbf{5.8\text{ km}}\ (2\text{ s.f.}).\]
(ii) Using coordinates \(D=(1.71,4.70),\ M=(-2.82,1.03)\): \(D\!\to\!M=(-4.53,-3.67)\) points south-west.
\[\tan\alpha=\frac{4.53}{3.67}=1.234\Rightarrow\alpha=51^{\circ};\quad\text{bearing}=180^{\circ}+51^{\circ}=\mathbf{231^{\circ}}.\]
(b) Ladder 10 m, wall \(x\) m, foot distance \((x+2)\) m:
\[x^{2}+(x+2)^{2}=10^{2}\Rightarrow 2x^{2}+4x+4=100\Rightarrow x^{2}+2x-48=0\Rightarrow (x+8)(x-6)=0.\]
Since \(x>0,\ \mathbf{x=6\text{ m}}.\)
Antwortdetails
(a) The cottage \(C\) is at bearing \(200^{\circ}\) from Dogbe's farm \(D\) and \(110^{\circ}\) from Manu's farm \(M\). Reversing bearings, from the cottage:
- Bearing of \(D\) from \(C=200^{\circ}-180^{\circ}=020^{\circ}\), with \(|CD|=5\) km.
- Bearing of \(M\) from \(C=110^{\circ}+180^{\circ}=290^{\circ}\), with \(|CM|=3\) km.
Angle at the cottage \(=290^{\circ}-020^{\circ}=270^{\circ}\Rightarrow\) reflex; the angle between the two directions is \(360^{\circ}-270^{\circ}=90^{\circ}\).
(i) With \(\angle DCM=90^{\circ}\):
\[|DM|=\sqrt{5^{2}+3^{2}}=\sqrt{34}=5.83\approx\mathbf{5.8\text{ km}}\ (2\text{ s.f.}).\]
(ii) Using coordinates \(D=(1.71,4.70),\ M=(-2.82,1.03)\): \(D\!\to\!M=(-4.53,-3.67)\) points south-west.
\[\tan\alpha=\frac{4.53}{3.67}=1.234\Rightarrow\alpha=51^{\circ};\quad\text{bearing}=180^{\circ}+51^{\circ}=\mathbf{231^{\circ}}.\]
(b) Ladder 10 m, wall \(x\) m, foot distance \((x+2)\) m:
\[x^{2}+(x+2)^{2}=10^{2}\Rightarrow 2x^{2}+4x+4=100\Rightarrow x^{2}+2x-48=0\Rightarrow (x+8)(x-6)=0.\]
Since \(x>0,\ \mathbf{x=6\text{ m}}.\)
Frage 61 Bericht
A man left town am at 10:00 AM and traveled by car to town N at an average speed of 72 km/h.
He spent 2hours for a meeting and returned through town M by bus at an average speed of 40KM/H.
If the distance covered by the bus was 2km longer than that of the car and he arrived at town M at 1 :55PM.
calculate distance from M to N.
Setting up the time equation
Total time from 10:00 a.m. to 1:55 p.m. \(=3\text{ h }55\text{ min}=3\tfrac{55}{60}\text{ h}=3\tfrac{11}{12}\text{ h}\).
Removing the 2-hour meeting, the driving+bus time is
\[3\tfrac{11}{12}-2=1\tfrac{11}{12}\text{ h}=\tfrac{23}{12}\text{ h}.\]
Let the car distance \(M\!\to\!N\) be \(d\) km (at 72 km/h). The bus distance is \((d+2)\) km (at 40 km/h). Then
\[\frac{d}{72}+\frac{d+2}{40}=\frac{23}{12}.\]
Multiply through by 360:
\[5d+9(d+2)=690\Rightarrow 14d+18=690\Rightarrow 14d=672\Rightarrow d=48.\]
Check: car \(=48/72=40\) min, bus \(=50/40=1\text{ h }15\text{ min}\); driving \(=1\text{ h }55\text{ min}\); with the 2 h meeting the total is 3 h 55 min, arriving 1:55 p.m. \(\checkmark\)
Distance from \(M\) to \(N\) \(= 48\) km.
Antwortdetails
Setting up the time equation
Total time from 10:00 a.m. to 1:55 p.m. \(=3\text{ h }55\text{ min}=3\tfrac{55}{60}\text{ h}=3\tfrac{11}{12}\text{ h}\).
Removing the 2-hour meeting, the driving+bus time is
\[3\tfrac{11}{12}-2=1\tfrac{11}{12}\text{ h}=\tfrac{23}{12}\text{ h}.\]
Let the car distance \(M\!\to\!N\) be \(d\) km (at 72 km/h). The bus distance is \((d+2)\) km (at 40 km/h). Then
\[\frac{d}{72}+\frac{d+2}{40}=\frac{23}{12}.\]
Multiply through by 360:
\[5d+9(d+2)=690\Rightarrow 14d+18=690\Rightarrow 14d=672\Rightarrow d=48.\]
Check: car \(=48/72=40\) min, bus \(=50/40=1\text{ h }15\text{ min}\); driving \(=1\text{ h }55\text{ min}\); with the 2 h meeting the total is 3 h 55 min, arriving 1:55 p.m. \(\checkmark\)
Distance from \(M\) to \(N\) \(= 48\) km.
Frage 62 Bericht
In the diagram, PQRS is a circle. = . ?SPR = 26° and the interior angles of PQS are in the ratio 2:3 :3.
Calculate: (i) PQR; (ii) RPQ; (iii) PRQ
(b) The coordinates of two points P and Q in a plane are (7, 3) and (5, x) respectively, where X is a real number.
If |PQ| = 29 units, find the value of x.
(a)
(i) Since PQRS is a circle and |PQ| = |QS|, we have ?PQR = ?QRS (angles subtended by equal chords are equal).
Let ?PQS = 2x. Then, ?PQR = 3x and ?QRS = 3x.
We know that the sum of interior angles of a triangle is 180°. So, in ?PQS, we have:
2x + 3x + 3x = 180°
8x = 180°
x = 22.5°
Now, in ?PQR, we have:
?PQR + ?QPR + ?RPQ = 180° (sum of interior angles of a triangle)
3x + 90° + 26° = 180° (since ?QPR is a right angle)
3x = 64°
x = 21.33°
Therefore, PQR = 3x = 64°.
(ii) RPQ = 180° - ?PQR - ?QPR = 180° - 64° - 90° = 26°.
(iii) PRQ = 180° - ?PQR - ?RPQ = 180° - 64° - 26° = 90°.
(b) Using the distance formula, we can find the distance between P and Q:
|PQ|² = (5-7)² + (x-3)²
|PQ|² = 4 + (x-3)²
Since |PQ| = 29 units, we have:
29² = 4 + (x-3)²
841 = (x-3)² + 4
837 = (x-3)²
Taking the square root of both sides, we get:
x-3 = ±?837
x = 3 ± ?837
Since x is a real number, we take the positive square root:
x = 3 + ?837
Therefore, the value of x is 3 + ?837 units.
Antwortdetails
(a)
(i) Since PQRS is a circle and |PQ| = |QS|, we have ?PQR = ?QRS (angles subtended by equal chords are equal).
Let ?PQS = 2x. Then, ?PQR = 3x and ?QRS = 3x.
We know that the sum of interior angles of a triangle is 180°. So, in ?PQS, we have:
2x + 3x + 3x = 180°
8x = 180°
x = 22.5°
Now, in ?PQR, we have:
?PQR + ?QPR + ?RPQ = 180° (sum of interior angles of a triangle)
3x + 90° + 26° = 180° (since ?QPR is a right angle)
3x = 64°
x = 21.33°
Therefore, PQR = 3x = 64°.
(ii) RPQ = 180° - ?PQR - ?QPR = 180° - 64° - 90° = 26°.
(iii) PRQ = 180° - ?PQR - ?RPQ = 180° - 64° - 26° = 90°.
(b) Using the distance formula, we can find the distance between P and Q:
|PQ|² = (5-7)² + (x-3)²
|PQ|² = 4 + (x-3)²
Since |PQ| = 29 units, we have:
29² = 4 + (x-3)²
841 = (x-3)² + 4
837 = (x-3)²
Taking the square root of both sides, we get:
x-3 = ±?837
x = 3 ± ?837
Since x is a real number, we take the positive square root:
x = 3 + ?837
Therefore, the value of x is 3 + ?837 units.
