JAMB UTME - Mathematics - 1993

In the diagram, O is the centre of the circle and POQ a diameter. If POR = 96${}^{?}$, find the value of ORQ.

Maelezo ya Majibu

OQ = OR = radii

< ROQ = 180 - 86 = 84${}^{?}$

$?$OQR = Isosceles

R = Q

R + Q + 84 = 180(angle in a $?$)

2R = 96 since R = Q

R = 48${}^{?}$

ORQ = 48${}^{?}$

If two angles of a triangle are 30o each and the longest side is 10cm. Calculate the length of each of the other sides

Maelezo ya Majibu

102 = 2x2 - 2x2 cos 120o (Cosine rule)
100 = 242 - 2x2 - 2x2 x -$\frac{1}{2}$
100 = 3x2 + x2
= 3x2
x = $\sqrt{\frac{100}{3}}$
= $\frac{10}{\sqrt{3}}$ x $\frac{3}{\sqrt{3}}$
x = 10$\frac{3}{3}$cm

PQRST is a regular pentagon and PQVU is a rectangle with U and V lying on TS and SR respectively as shown in the diagram. Calculate TUP

Maelezo ya Majibu

Simplify $\frac{1}{p}$ - $\frac{1}{q}$ + $\frac{p}{q}$ - $\frac{q}{p}$

Maelezo ya Majibu

$\frac{1}{p}$ - $\frac{1}{q}$ + $\frac{p}{q}$ - $\frac{q}{p}$ = $\frac{q?p}{pq}$ + $\frac{p^2?q^2}{pq}$
$\frac{q?p}{pq}$ x $\frac{pq}{p^2{q}^2}$ = $\frac{q?p}{p^2{q}^2}$
$\frac{?(p?q)}{(p+q)(p?q)}$
= $\frac{?1}{p+q}$

The mean of the ages of ten secondary school pupils is 16 but when the age of their teacher is added to it the men becomes 19. Find the age of the teacher

Maelezo ya Majibu

Let the age of the teacher be T. According to the problem, the mean age of the 10 pupils is 16. Therefore, the sum of their ages is 10 x 16 = 160. When the teacher's age is added to it, the sum becomes 160 + T. Also, it is given that the new mean age after adding the teacher's age is 19. So, we have: (160 + T) / 11 = 19 Multiplying both sides by 11, we get: 160 + T = 209 Subtracting 160 from both sides, we get: T = 49 Therefore, the age of the teacher is 49.

In the diagram, QPS = SPR, PR = 9cm. PQ = 4cm and QS = 3cm, find SR.

Maelezo ya Majibu

Using angle bisector theorem: line PS bisects angle QPR

QS/QP = SR/PR

3/4 = SR/g

4SR = 27

SR = $\frac{27}{4}$

= 6$\frac{3}{4}$cm

Make x the subject of the relation $\frac{1+ax}{1-ax}$ = $\frac{p}{q}$

Maelezo ya Majibu

To make x the subject of the given relation, we need to isolate x on one side of the equation. First, we can simplify the expression by finding the common denominator of the two fractions in the numerator: (1 + ax)/(1 - ax) = (1 - ax + ax + a^2x)/(1 - ax) = (1 + a^2x)/(1 - ax) Now we can cross-multiply to get rid of the denominator: (1 + a^2x) = pq - axpq Next, we can move the term with x to the left-hand side of the equation: a^2x + axpq = pq - 1 Finally, we can factor out x from the two terms on the left-hand side: x(a^2 + apq) = pq - 1 Therefore, we can isolate x by dividing both sides by (a^2 + apq): x = (pq - 1)/(a^2 + apq) So the correct option is: p-q / a(p+q) which is not among the given options.

In the diagram, PQRs is a circle with 0 as centre and PQ/RT. If RTS = ${}^{\circ }$. Find PSQ

Maelezo ya Majibu

< PSO = $\frac{1}{2}$ < SOQ = $\frac{1}{2}$(180) = 90${}^{\circ }$

< RTS = < PQS = 32${}^{\circ }$ (Alternative angle)

< PSQ = 90 - < PSQ = 90${}^{\circ }$ - 32${}^{\circ }$

= 58${}^{\circ }$

Quantities in the proportions 1, 4, 6, 7 are to be represented in a pie chart. Calculate the angle of the sector with proportion 7

Maelezo ya Majibu

To calculate the angle of the sector with proportion 7, we first need to find the total of all the quantities in the proportion. Total quantity = 1 + 4 + 6 + 7 = 18 Next, we need to find the proportion of quantity 7 in relation to the total quantity. Proportion of quantity 7 = 7/18 To find the angle of the sector that represents this proportion, we need to use the formula: Angle of sector = Proportion of quantity * 360 degrees Angle of sector = (7/18) * 360 degrees Angle of sector = 140 degrees Therefore, the answer is option D, which is 140 degrees.

The three sides of an isosceles triangle are length of lengths (x + 3), (2x + 3), (2x - 3) respectively. Calculate x.

Maelezo ya Majibu

2x + 3 $\ne$ 2x - 3 for any value of x
∴ for the $△$ to be isosceles, either
2x - 3 = x + 3 or 2x + 3 = x + 3
solve the two equations we arrive at
x = 6 or x = 0
When x = 6, the sides are 9, 15, 9
When x = 0, the sides are 3, 4, -3 since lengths of a $△$can never be negative then the value of x = 6

Evaluate (x + $\frac{1}{x}$ + 1)2 - (x + $\frac{1}{x}$ + 1)2

Maelezo ya Majibu

(x + $\frac{1}{x}$ + 1)2 - (x + $\frac{1}{x}$ + 1)2
= (x + $\frac{1}{x}$ + 1 + x + $\frac{-1}{x}$ - 1)(x - $\frac{1}{x}$ + 1 - x + $\frac{1}{x}$ + 1)
= (2x) (2 + $\frac{2}{x}$) = 2x x 2(1 + $\frac{1}{x}$)
4x (1 + $\frac{1}{x}$) = 4x + 4
= 4(1 + x)

simplify $\frac{1}{\surd 3-2}$ - $\frac{1}{\surd 3+2}$

Maelezo ya Majibu

$\frac{1}{\surd 3-2}$ - $\frac{1}{\surd 3+2}$
L.C.M = (3- 2) (3 + 2)
∴ $\frac{1}{\sqrt{3-2}}$ - $\frac{1}{\sqrt{3-2}}$ = $\frac{\sqrt{3+2}-\sqrt{3-2}}{\sqrt{3-2}+\sqrt{3-2}}$

$\frac{\surd 3+2-\surd 3+2}{3-2\surd 3+2\surd 3-4}$ = $\frac{4}{3-2}$
= $\frac{4}{-1}$
= -4

$\begin{array}{cccccc}Weight\left(s\right)& 0-10& 10-20& 20-30& 40-50& \\ \text{Number of coconuts}& 10& 27& 19& 6& 2\end{array}$

Estimate the mode of the frequency distribution above.

Maelezo ya Majibu

Mode = a + $\frac{(b-a)(F_m-F_b)}{2F_m-F_a-F_b}$
= $L_1+\frac{{\Delta }_1{x}^{\text{c}}}{{\Delta }_1+{\Delta }_2}$
= $10+\frac{(20-10)(27-10)}{2\left(27\right)-10-19}$
= 10 + $\frac{170}{25}$
= 10 + 6.8
= 16.8

Solve the following equation (3x - 2)(5x - 4) = (3x - 2)2

Maelezo ya Majibu

(3x - 2)(5x - 4) = (3x - 2)2 = 5x2 - 22x + 6
= 9x2 = 12x + 4
6x2 - 10x + 4 = 0
6x2 - 6x - 4x + 4 = 0
6x(x - 1) -4(x - 1) = (6x - 4)(x -1) = 0
x = 1 or $\frac{2}{3}$

The bar chart shows the distribution of marks in a class test. How many students took the test?

Maelezo ya Majibu

Number of students that took the test = $\sum f$. Where $f$ is the frequencies
= 2 + 5 + 0 + 3 + 4 + 0 + 0 + 0 + 1 + 0 + 2 = 17

A man's initial salary is ₦540.00 a month and increases after each period of six months by ₦36.oo a month. Find his salary in the eighth month of the third year

Maelezo ya Majibu

Initial salary = ₦540
increment = ₦36 (every 6 months)
Period of increment = 2 yrs and 6 months
amount(increment) = ₦36 x 5 = ₦180
The man's new salary = ₦540 = ₦180
= ₦720.00

Calculate the length in cm. of the area of a circle of diameter 8cm which subtends an angle of 22$\frac{1}{2}$o at the centre of the circle

Maelezo ya Majibu

Diameter = 8cm
∴ Radius = 4cm
Length of arc = $\frac{\theta }{360}$ x 2 $\pi$r but Q = 22$\frac{1}{2}$
∴ Length $\frac{22\frac{1}{2}}{360}$ x 2 x $\pi$ x 4
= $\frac{22\frac{1}{2}\times 8\pi }{360}$
= $\frac{180}{360}$
= $\frac{\pi }{2}$

In the diagram above, |PQ| = |QR|, |PS| = |RS|, ∠PSR = 30o and ∠PQR = 80o. Find ∠SPQ.

Maelezo ya Majibu

Join PR
QRP = QPR
= 180 - 80 = 100/20 = 50o
SRP = SPR
= 180 - 30 = 150/2 = 75o
∴ SPQ = SPR - QPR
= 75 - 50 = 25o

Which of the following is a factor of 15 + 7x - 2x2

Maelezo ya Majibu

Factorize 15 + 7x - 2x2
(5 - x)(3 + 2x); suppose 15 + 7x = 2x2 = 0
∴ (5 - x)(3 + 2x) = 0
x = 5 or x = -$\frac{3}{2}$
Since 5 is a root, then (x - 5) is a factor

If $\sqrt{x^2+9}$ = x + 1, solve for x

Maelezo ya Majibu

$\sqrt{x^2+9}$ = x + 1
x2 + 9 = (x + 1)2 + 1
0 = x2 + 2x + 1 - x2 - 9
= 2x - 8 = 0
2(x - 4) = 0
x = 4

If x is negative, what is the range of values of x within which $\frac{x+1}{3}$ > $\frac{1}{X+3}$

Maelezo ya Majibu

$\frac{x+1}{3}$ > $\frac{1}{X+3}$ = $\frac{x+1}{3}$ > $\frac{x+3}{X+3}$
= (x + 1)(x + 3)2 > 3(x + 3) = (x + 1)[x2 + 6x + 9] > 3(x + 3)
x3 + 7x2 + 15x + 9 > 3x + 9 = x3 + 7x2 + 12x > 0
= x(x + 3)9x + 4) > 0
Case 1 (+, +, +) = x > 0 , x + 3 > 0, x + 4 > 0
= x > -4 (solution only)
Case 2 (+, -, -) = x > 0, x + 4 < 0

= x > 0, x < -3, x < -4 = x < -3(solution only)

Case 3 (-, +, -) = x < 0, x > -3, x < -4 = x < -0, -4 < x < 3(solutions)

Case 4 (-, -, +) = x < 0, x + 3 < 0, x + 4 > 0
= x < 0, x < -5, x > -4 = x < -0, -4 < x < -3(solution)

combining the solutions -4 < x < -3

The following marks were obtained by twenty students in an examination: 53, 30, 70, 84, 59, 43, 90, 20, 78, 48, 44, 60, 81, 73, 50, 37, 67, 68, 64, 52. Find the numbers of students who scored at least 50 marks

Maelezo ya Majibu

To find the number of students who scored at least 50 marks, we need to count the number of marks that are greater than or equal to 50. Looking at the marks obtained by the students, we can see that there are 14 marks that are greater than or equal to 50. Therefore, the number of students who scored at least 50 marks is 14. We can count these marks by going through the list one by one, or by using a table or chart to organize the data. It's important to pay close attention to the wording of the question and to make sure we're answering the question being asked. In this case, the question is asking for the number of students, not the number of marks.

In the figure, the line segment ST is tangent to two circles at S and T. O and Q are the centres of the circles with OS = 5cm. QT = 2cm and OR = 14cm. Find ST

Maelezo ya Majibu

SQ2 + OS2 = OQ2 + 52 = 142

SQ2 = 142 - 52

196 - 25 = 171

ST2 + TQ2 = SQ2

ST2 + 22 = 171

ST2 = 171 - 4

= 167

ST = $\sqrt{167}$

= 12.92 = 12.9cm

Solve without using tables log5(62.5) - log5($\frac{1}{2}$)

Maelezo ya Majibu

log5(62.5) - log5($\frac{1}{2}$)
= log5$\frac{\left(62.5\right)}{\frac{1}{2}}$ - log5(2 x 62.5)
= log5(125)
= log553 - 3log55
= 3

From the figure, calculate TH in centimeters

Maelezo ya Majibu

TH = tan 45${}^{\circ }$, TH = QH

$\frac{TH}{5+QH}$ = tan 30${}^{\circ }$

TH = (b + QH) tan 30${}^{\circ }$

QH = 56 (5 + QH) $\frac{1}{\sqrt{3}}$

QH(1 - $\frac{1}{\sqrt{3}}$) = $\frac{5}{\sqrt{3}}$

QH = $\frac{5\sqrt{3}}{\sqrt{3}}-\frac{1}{\sqrt{3}}$

= $\frac{5}{\sqrt{3}-1}$

If k + 1; 2k - 1, 3k + 1 are three consecutive terms of a geometric progression, find the possible values of the common ratio

Maelezo ya Majibu

$\begin{array}{cc}Class& Frequency\\ 1-5& 2\\ 6-10& 4\\ 11-15& 5\\ 16-20& 2\\ 21-25& 3\\ 26-30& 2\\ 31-35& 1\\ 36-40& 1\end{array}$

Find the median of the observation in the table given.

Maelezo ya Majibu

The shaded portion in the Venn diagram is

Maelezo ya Majibu

The shaded part exists on x $\cap$ z but not in y

Find the area of the sector of a circle with radius 3m, if the angle of the sector is 60o

Maelezo ya Majibu

To find the area of a sector, we use the formula: Area of sector = (θ/360) x πr² Where θ is the angle of the sector in degrees, r is the radius of the circle, and π (pi) is a mathematical constant approximately equal to 3.14. In this problem, the radius is given as 3m and the angle of the sector is 60°. Substituting these values into the formula, we get: Area of sector = (60/360) x π(3)² = (1/6) x π(9) = (1/6) x 28.27 = 4.71m² (rounded to two decimal places) Therefore, the area of the sector of the circle with radius 3m and angle 60° is approximately 4.71m². The answer is option C.

find the radius of a sphere whose surface area is 154cm2 ($\pi =\frac{22}{7}$)

Maelezo ya Majibu

Surface area = 154cm2 (area of sphere)
4$\pi$r2 = 154
r$\sqrt{\frac{154}{4\pi }}$
= 3.50cm

If the function f is defined by f(x + 2) = 2x2 + 7x = 5, find f(-1)

Maelezo ya Majibu

A binary operation $\ast$ is defined on a set of real numbers by x $\ast$ y = xy for all real values of x and y. If x $\ast$ 2 = x. Find the possible values of x

Maelezo ya Majibu

x $\ast$ y = xy
x $\ast$ 2 = x2
x $\ast$ 2 = x
∴ x2 - x = 0
x(x - 1) = 0
x = 0 or 1

The angle between latitudes 30oS and 13oN is

Maelezo ya Majibu

The angle between 2 latitudes one in northern hemisphere and the other in southern hemisphere and the other in southern hemisphere is the sum of their latitudes.
∴ Total angle difference = (30 + 13) = 43o

Solve for y in the equation 101 x 5(2x - 2) x 4(y - 1) = 1

Maelezo ya Majibu

10y x 5(2y - 2) x 4(y - 1) = 1
but 10y - (5 x 2)y = 5y x 2y
= (Law of indices)
5y x 2y x 5(2y - 2) x 4(y - 1) = 1
but 4(y - 1) = 22(y - 1)
= 2y - 2 (Law of indices)
5y x 5(2y -2) x 2(- 2) = 1
5(3y -2) x 2y x 2(2y -2) = 1
= 5(3y -2) x 2(3y -2) = 1
But ao = 1
10(3y -2) = 10o
3y - 2 = 0
? y = $\frac{2}{3}$

If sin x = cos 50o, then x equals

Maelezo ya Majibu

We know that sin x = cos 50°. To solve for x, we need to find the angle whose sine is equal to cos 50°. We can use the trigonometric identity that sin (90° - θ) = cos θ. So, sin x = cos 50° can be rewritten as sin x = sin (90° - 50°). Using the identity, we have: sin x = sin 40° Now, we need to find the angle x whose sine is equal to sin 40°. Since sine is a periodic function, there are multiple angles whose sine is equal to a given value. One such angle is x = 40°. However, sine is also negative in the third and fourth quadrants. In the third quadrant, x = 180° - 40° = 140° and in the fourth quadrant, x = 360° - 40° = 320° also satisfy the equation sin x = sin 40°. However, since x has to be between 0° and 360°, we can eliminate the30°. Therefore, the possible values of x are 40°, 140°, and 320°. However, since we know that sin x = cos 50° and cos is positive in the first quadrant, x cannot be in the third or fourth quadrants. Therefore, the only possible value of x is x = 40°. Hence, the answer is x = 40°.

The figure represents the graphs of y = x(2 - x) and y = (x - 1)(x - 3). What are the x-coordinates of P, Q and F respectively?

Maelezo ya Majibu

To find the x-coordinates of P, Q, and F, we need to solve the system of equations formed by setting the two equations equal to each other: x(2 - x) = (x - 1)(x - 3) Expanding the right-hand side and simplifying, we get: 2x - x^2 = x^2 - 4x + 3 Rearranging and simplifying, we get a quadratic equation in standard form: x^2 - 6x + 3 = 0 Using the quadratic formula, we get: x = (6 ± sqrt(6^2 - 4*1*3)) / (2*1) = 3 ± sqrt(3) Therefore, the x-coordinates of P, Q, and F are 1, 3, and 3 - sqrt(3) or approximately 1.268, respectively. To see why, we can plot the two functions and visually determine the x-coordinates of the points where they intersect. Point P is where the blue line intersects the x-axis, which occurs at x = 1. Point Q is where the red line intersects the x-axis, which occurs at x = 3. Point F is where the two lines intersect above the x-axis, which occurs at x = 3 - sqrt(3) or approximately 1.268. Therefore, the answer is 1, 2, 3.

Solve the inequality y2 - 3y > 18

Maelezo ya Majibu

y2 - 3y > 18 = 3y - 18 > 0
y2 - 6y + 3y - 18 > 0 = y(y - 6) + 3 (y - 6) > 0
= (y + 3) (y - 6) > 0
Case 1 (+, +) $\to$ (y + 3) > 0, (y - 6) > 0
= y > -3 y > 6
Case 2 (-, -) $\to$ (y + 3) < 0, (y - 6) < 0

= y < -3, y < 6

Combining solution in case 1 and Case 2
= x < -3y < 6

= -3 < y < 6

A number is selected at random between 20 and 30, both numbers inclusive. Find the probability that the number is a prime

Maelezo ya Majibu

To solve this problem, we need to first determine the set of possible numbers that could be selected at random between 20 and 30. Since the problem states that both 20 and 30 are inclusive, the set of possible numbers is: {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} To find the probability that the selected number is prime, we need to determine the number of prime numbers in the set of possible numbers, and then divide that by the total number of possible numbers. Prime numbers are numbers that are only divisible by 1 and themselves. The prime numbers in the set of possible numbers are: { 23, 29 } Therefore, the probability that the selected number is prime is 2/11 or approximately 0.18.

Divide the expression x3 + 7x2 - x - 7 by -1 + x2

Maelezo ya Majibu

If ₦225.00 yields ₦27.00 in x years simple interest at the rate of 4% per annum, find x

Maelezo ya Majibu

To solve this problem, we use the formula for simple interest: I = P*r*t Where: I = Interest earned P = Principal amount r = Rate of interest per year (as a decimal) t = Time period in years In this case, we are given: P = ₦225.00 (the principal amount) I = ₦27.00 (the interest earned) r = 4% = 0.04 (the rate of interest per year) Substituting these values into the formula, we get: 27 = 225 * 0.04 * t Simplifying the equation, we have: 27 = 9t Dividing both sides by 9, we get: t = 3 Therefore, the answer is 3 years. Explanation: The interest earned on the principal amount of ₦225.00 is ₦27.00. This means that for every year, the interest earned is 27/225 = 0.12 or 12% of the principal amount. We know that the rate of interest per year is 4%, which is much lower than the interest earned per year of 12%. This means that it will take more than 1 year for the interest earned to reach the rate of interest. In fact, it takes 3 years for the interest earned to reach the rate of interest of 4%, which is why the answer is 3.

In the figure, the area of the square of the square PQRS is 100cm2. If the ratio of the area of the square TUYS to the area of the area of the square XQVU is 1 : 16, Find YR

Maelezo ya Majibu

Since area of square PQRS = 100cm2

each lenght = 10cm

Also TUYS : XQVU = 1 : 16

lengths are in ratio 1 : 4, hence TU : UV = 1: 4

Let TU = x

UV = 1: 4

hence TV = x + 4x = 5x = 10cm

x = 2cm

TU = 2cm

UV = 8cm

But TU = SY and UV = YR

YR = 8cm

Two chords PQ and RS of a circle when produced meet at K. If ∠KPS = 31o and ∠PKR = 42o, find ∠KQR

Maelezo ya Majibu

QPS - QRK = 31o
QRK + RKQ + KQR = 180
31 + 42 + KQR = 180o
KQR = 180 - 73 = 107o

Solve the following simultaneous equation for x. x2 + y - 5 = 0, y - 7x + 3 = 0

Maelezo ya Majibu

x2 + y - 5 = 0.....(i)
y - 7x + 3 = 0.........(ii)
y = 7x - 3, substituting the value of y in equation (i)
x2 + (7x - 3) - 5 = 0
x2 + 7x + 3 = 0
(x + 8)(x - 1) = 0
x = -8 or 1

A rectangular polygon has 150o as the size of each interior angle. How many sides has the polygon?

Maelezo ya Majibu

A rectangular polygon has each interior angle to be 150o
let the polygon has n-sides
therefore, Total interior angle 150 x n = 150n
hence 150n = (2n - 4)90
150n = 180n - 360
360 = (180 - 150)n
30n = 360
n = 12

In the diagram, QP//ST:PQR = 34${}^{\circ }$ QRS = 73${}^{\circ }$ and RS = RT. Find SRT

Maelezo ya Majibu

Construction joins R to P such that SRP = straight line

R = 180${}^{\circ }$ - 107${}^{\circ }$

< p = 180${}^{\circ }$ - (107${}^{\circ }$ - 34${}^{\circ }$)

108 - 141${}^{\circ }$ = 39${}^{\circ }$

Angle < S = 39${}^{\circ }$ (corr. Ang.) But in $△$SRT

< S = < T = 39${}^{\circ }$

SRT = 180 - (39${}^{\circ }$ + 39${}^{\circ }$)

= 180${}^{\circ }$ - 78${}^{\circ }$

= 102${}^{\circ }$

Calculate the standard deviation of the following data: 7, 8, 9, 10, 11, 12, 13.

Maelezo ya Majibu

To calculate the standard deviation, we need to first find the mean (average) of the data. Mean = (7 + 8 + 9 + 10 + 11 + 12 + 13) / 7 = 10 Next, we need to find the difference between each data point and the mean, and square the differences. (7-10)^2 = 9 (8-10)^2 = 4 (9-10)^2 = 1 (10-10)^2 = 0 (11-10)^2 = 1 (12-10)^2 = 4 (13-10)^2 = 9 Then, we add up all of these squared differences and divide by the total number of data points. (9 + 4 + 1 + 0 + 1 + 4 + 9) / 7 = 28 / 7 = 4 Finally, we take the square root of this result to find the standard deviation. Standard deviation = √4 = 2 Therefore, the standard deviation of the given data is 2.

Two chords QR and NP of a circle intersect inside the circle at x. If RQP = 37o, RQN = 49o and QPN = 35o, find PRQ

Maelezo ya Majibu

In PNO, ONP
= 180 - (35 + 86)
= 180 - 121
= 59
PRQ = QNP = 59(angles in the same segment of a circle are equal)

If sin $\theta$ = cos $\theta$, find $\theta$ between 0o and 360o

Maelezo ya Majibu

sin $\theta$ = cos $\theta$ 0 $\le$ $\theta$ $\le$ 360o
The acute angle where sin $\theta$ = cos $\theta$ = 45o
But at the fourth Quadrant Cos $\theta$ = +ve
at the 4th quadrant, value with respect to Q is
(360 - Q) where Q = acute angle
(360 - 45) = 315o
The two solution are 45o, 315o

The chances of three independent events X, Y, Z occurring are $\frac{1}{2}$, $\frac{2}{3}$, $\frac{1}{4}$ respectively. What are the chances of Y and Z only occurring?

Maelezo ya Majibu

Chance of x = $\frac{1}{2}$
Change of Y = $\frac{2}{3}$
Chance of Z = $\frac{1}{4}$
Chance of Y and Z only occurring
= Pr (Y ? Z ? Xc)
where Xc = 1 - Pr(X)
1 = $\frac{1}{2}$ = 1$\frac{1}{2}$
= Pr(Y) x Pr(Z) x Pr(Xc)
= $\frac{2}{3}$ x $\frac{1}{4}$ x $\frac{1}{2}$
= $\frac{1}{12}$

If 2log3 y + log3 x2 = 4, then y is

Maelezo ya Majibu

The given equation is 2log3 y + log3 x2 = 4. We can simplify this equation using the laws of logarithms. First, we can use the power rule of logarithms to rewrite x2 as (x)2. Then, we can use the product rule of logarithms to combine the two logarithms on the left-hand side of the equation: 2log3 y + log3 (x)2 = log3 y2 + log3 (x)2 Now, we can use the sum rule of logarithms to combine the two logarithms on the right-hand side of the equation: log3 y2(x)2 = log3 (x2y2) We can now rewrite the original equation as: log3 (x2y2) = 4 Using the definition of logarithms, we know that log3 (x2y2) = 4 is equivalent to 3^4 = x2y2. Therefore, we have: x2y2 = 81 Taking the square root of both sides, we get: xy = ±9 Since y is a positive real number, we can discard the negative solution. Therefore, we have: xy = 9 Finally, we can solve for y: y = 9/x So, the answer is "±9/x".