JAMB UTME - Mathematics - 2022

Evaluate n2+1 ${}^{{\mathrm{<br>}}^{}+1}$Cn+5
 if n = 3 

Answer Details

32+1 ${}^{{\mathrm{<br>}}^{}}$C3+5 ${}_{\mathrm{<br>}}$

9+1 ${}^{\mathrm{<br>}}$C3+5 ${}_{\mathrm{<br>}}$

10 ${}^{}$C
${}_8$ = 10!8!2! $\frac{\mathrm{<br>}}{}$

10∗9∗8!8!2! $\frac{\mathrm{<br>}}{}$ = 10∗92 $\frac{\mathrm{<br>}}{}$

= 45

The shaded portion in the venn diagram above represents?

Answer Details

The shaded portion in the Venn diagram represents the set of elements that belong to set F, but do not belong to the intersection of sets E and F (E ∩ F) or the intersection of sets G and F (G ∩ F). In symbols, we can represent this set as: F - (E ∩ F) - (G ∩ F) This can be read as "the set of elements in F that are not in both E and F, and are not in both G and F". For example, if we think of the sets as categories, where F is the category of "fruits", E is the category of "apples", and G is the category of "oranges", then the shaded region represents the fruits that are not both apples and fruits, and are not both oranges and fruits. So, this would include fruits that are not apples or oranges, such as bananas, grapes, or strawberries, as well as fruits that are both apples and oranges, such as tangelos. Therefore, the answer is: F - (E ∩ F) - (G ∩ F)

Given that S = 3t2 $\mathrm{<br>}{\mathrm{<br>}}^{\mathrm{<br>}}$ + 5t - 10 is displacement of a particle in metres, calculate it initial velocity.

Answer Details

To calculate the initial velocity of the particle, we need to find the first derivative of the displacement equation. The first derivative of the equation 3t^2 + 5t - 10 would give us the velocity equation.

The derivative of 3t^2 is 6t, the derivative of 5t is 5, and the derivative of a constant like -10 is 0.

So, the velocity equation would be 6t + 5. This is the initial velocity of the particle, which is 5 m/s.

Calculate the median of 14, 17, 10, 13, 18 and 10.

Answer Details

The median of a set of numbers is the middle value when the numbers are arranged in order. If there is an odd number of values, the median is the middle value, and if there is an even number of values, the median is the average of the two middle values. In this case, the set of numbers {14, 17, 10, 13, 18, 10} has 6 values, which is an even number. To find the median, we need to first arrange the values in order: {10, 10, 13, 14, 17, 18} The median is the average of the two middle values, which are 13 and 14, so: Median = (13 + 14) / 2 = 13.5 So, the median of the set of numbers {14, 17, 10, 13, 18, 10} is 13.5.

Three times a certain number (x), minus 2 is less than the number minus 6.Find the possible values of x. 

Answer Details

The problem states that "Three times a certain number (x), minus 2 is less than the number minus 6." Mathematically, this can be written as: 3x - 2 < x - 6 To solve for x, we want to isolate x on one side of the inequality sign. First, we can simplify the inequality by subtracting x from both sides: 2x - 2 < -6 Next, we can add 2 to both sides: 2x < -4 Finally, we can divide both sides by 2: x < -2 Therefore, the possible values of x are x < -2. This means that any number less than -2 will satisfy the original inequality. Option (a) is correct: x < -2.

Given that r = 3vπh−−−√ make v the subject of the formula

Answer Details

square both sides to remove the big square root

→ r2 ${}^2$ = 3vπh $\frac{\mathrm{<br>}}{}$

cross multiply

3v = r2 ${}^2$ * πh

v = πr2h3

Mr Adu spends his annual salary on food(f), rent(r), car maintenance, gifts(g), savings(s) and some miscellaneous (m) as indicate in the table below:

If the above information is represented on a pie chart. What angle represents his spending on food?

Answer Details

The percentage of Mr. Adu's annual salary spent on food is 28%. To find the angle that represents his spending on food in a pie chart, we need to know that the total angle of a circle is 360 degrees. To find the angle that represents Mr. Adu's spending on food, we need to calculate 28% of 360 degrees: 28% of 360 degrees = 0.28 * 360 degrees = 100.8 degrees So the angle that represents Mr. Adu's spending on food is 100.8 degrees. Therefore, the answer is: 100.8.

If the mean of 2, 5, (x+1), (x+2), 7 and 9 is 6. Find the median 

Answer Details

Firstly; solving for x

6 = 2+5+x+1+x+2+7+96 $$

cross multiply to have:

6 * 6 = 2 + 5 + x+1 + x+2 + 7 + 9

36 = 2x + 26

36 - 26 = 2x

10 = 2x

x = 5

Median = 7+62 $\frac{7+6}{2}$

→ 6.5

Evaluate (101two)3

Answer Details

5\(^3\) = 125

And 125 =  \(1111101_{two}\)

If sec2 ${}^2$θ + tan2 ${}^2$θ = 3, then the angle θ is equal to?

Answer Details

We're given the equation sec²θ + tan²θ = 3, and we need to solve for θ. One way to approach this is to use the trigonometric identity: tan²θ + 1 = sec²θ Substituting this into the given equation, we get: tan²θ + 1 + tan²θ = 3 Simplifying this equation, we get: 2tan²θ = 2 Dividing both sides by 2, we get: tan²θ = 1 Taking the square root of both sides, we get: tanθ = ±1 This means that θ must be one of the angles whose tangent is ±1. These angles are 45º and 225º (or -135º), since tangent is positive in the first and third quadrants, and negative in the second and fourth quadrants. However, we need to check whether these values of θ satisfy the original equation. Let's start with θ = 45º: sec²45º + tan²45º = 2 + 1 = 3 So this value of θ does satisfy the equation, and therefore it is the solution. Therefore, the angle θ is 45º. So the answer is (c) 45º.

A rectangular pyramid has an area 24cm2 ${}^2$ and height 7.5cm. Find its volume?

Answer Details

Volume of a rectangular pyramid = length∗width∗height3or area∗height3 $\frac{\mathrm{<br>}}{}$

= 24∗7.53 $\frac{\mathrm{<br>}}{}$ → 1803 $\frac{\mathrm{<br>}}{}$

Volume of the rectangular pyramid = 60cm3

Find the equation of a straight line parallel to the line 2x - y = 5 and having intercept of 5

Answer Details

To find the equation of a straight line parallel to 2x - y = 5, we need to determine the slope of the given line. The slope of a line is defined as the change in y divided by the change in x, which can be written as Δy/Δx. We can rewrite 2x - y = 5 as y = 2x - 5, which is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. From this form, we can see that the slope of the line is 2. A line parallel to this line will have the same slope of 2. We also know that the new line has an intercept of 5, which means it passes through the point (0, 5). Using the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute m = 2 and (x1, y1) = (0, 5) to get the equation: y - 5 = 2(x - 0) Simplifying this equation gives: y = 2x + 5 Therefore, the equation of a straight line parallel to 2x - y = 5 and having intercept of 5 is 2x + y = 5. Option (a) is the correct answer.

If A =  $\left(\begin{array}{cc}2& 1\\ 2& 3\\ 1& 2\end{array}\right)$ and B = $\left(\begin{array}{cc}3& 2\\ 4& 2\end{array}\right)$. Find AB

Answer Details

Given A = ⎛⎝⎜221132⎞⎠⎟ $<br><br>$ and B = (3422) $\begin{array}{cc}& \end{array}$

We can multiply these matrices since the number of colums in A = number of rows in B

AB = ⎛⎝⎜(2∗3)+(1∗4)(2∗3)+(3∗4)(1∗3)+(2∗4)(2∗2)+(1∗2)(2∗2)+(3∗2)(1∗2)+(2∗2)⎞⎠⎟

AB = ⎛⎝⎜(6+4)(6+12)(3+8)(4+2)(4+6)(2+4)⎞⎠⎟ $\begin{array}{cc}& \end{array}$

= ⎛⎝⎜1018116106⎞⎠⎟

Solve for k in the equation 18k+2 = 1

Answer Details

(8?1)k+2 $(8^{?1}{)}^{?+2}$ = (80) $(8^0)$

base 8 cancel out on both sides

-1(k+2) = 0

-k -2 = 0

: k = -2

In how many ways can the letter of ZOOLOGY be arranged?

Answer Details

Zoology has 7 letters in total, with O repeated thrice

\(\frac{7!}{3!}\) → \(\frac{7*6*5*4*3*2*1}{3*2*1}\)

= 840ways

The coordinates of the mid-point of the line joining the points (-3,5) and (2,10) is given by?

Answer Details

Given Data: x1 ${}_1$ = -3,  x2 ${}_2$ = 2,  y1 ${}_1$ = 5,  y2 ${}_2$ = 10

coordinates of the mid-point of the line = (x1+x22 $\frac{{\mathrm{<br>}}_{}}{}$ , y1+y22 $\frac{{\mathrm{<br>}}_{}}{}$)

(−3+22 $\frac{<br>}{}$ , 5+102 $\frac{\mathrm{<br>}}{}$)

=  −12 $\frac{<br>}{}$) , 152 $\frac{\mathrm{<br>}}{}$)

In the diagram above, XY = 8cm and OX = 5cm. Find Oz

Answer Details

hyp = 5cm, adj =  8cm2 $\frac{\mathrm{<br>}\mathrm{<span\; style="font-weight:\; var(--bs-body-font-weight);\; word-spacing:\; normal;">4cm,\; opp\; =\; xcm</span>}}{}$

Pythagoras theorem: 

hyp22
 = opp2 ${}^{\mathrm{<br>}}$ + adj2 ${}^{\mathrm{<br>}}$

52 ${}^{\mathrm{<br>}}$ = x2 ${}^{\mathrm{<br>}}$ + 42 ${}^{\mathrm{<br>}}$

x
${}^2$ = 25 - 16

x = √9

x = 3cm

What will be the result obtained when the numerator of 9650 $\frac{96}{50}$ is decreased by 37.5% and its denominator decreased by 20%.

Answer Details

Numerator: 96 → 37.5% of 96 = 36

Decreased by 36 → 96 - 36

New numerator = 60

Denominator: 50 → 20% of 50 = 10

Decreased by 10 → 50 - 10 = 40

New Denominator = 40

New fraction = 6040 $\frac{\mathrm{<br>}}{}$ or 1.5

Evaluate Log2 ${}_2$ 8√2

Answer Details

where Log2 ${}_2$ 8√2 → Log ${}_2$ √128 

→ Log22
 12812 ${}^{\frac{1}{2}}$

=  $\frac{1}{2}$ * (Log2 ${}_2$ 128) → 
$\frac{1}{2}$ * (Log
${}_2$ 2 ${}^{\mathrm{<br>7}}$)

=  7 * 12 * (Log
${}_2$ 2)

where (Log ${}_2$ 2) = 1 

→  7 * 
$\frac{1}{2}$ * 1

= 
$\frac{7}{2}$ or 3.5

The cost C of running a school is directly proportional to the number of students N, if 20 students cost #10,000, How many students can #1,000,000 cover?

Answer Details

To answer this question, we need to use the formula for direct proportion: C = k * N, where C is the cost, N is the number of students, and k is the proportionality constant. Since we know that 20 students cost #10,000, we can find the value of k by substituting the known values into the formula: 10,000 = k * 20 Solving for k: k = 10,000 / 20 = 500 Now that we have the value of k, we can use it to find the number of students that #1,000,000 would cover: C = k * N 1,000,000 = 500 * N N = 1,000,000 / 500 = 2000 So, #1,000,000 would cover 2000 students.