JAMB UTME - Mathematics - 2022

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

Bayanin Amsa

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.

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

Bayanin Amsa

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

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%.

Bayanin Amsa

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

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

Bayanin Amsa

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.

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

Bayanin Amsa

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

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

Bayanin Amsa

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 $\frac{\mathrm{<br>}}{}$)

If 8, p, q and 26 is an A.P. Find the values of p and q

Bayanin Amsa

The common difference is 6.

Just add 6 to get the next term

8, 14, 20 and 26

A poultry farmer realized 200 eggs from his birds and sold 5 \(\frac{1}{2}\) crates of eggs. What percentages of eggs are left?

Bayanin Amsa

To determine the percentage of eggs left, we need to first calculate the total number of eggs produced by the poultry farmer. We know that the farmer produced 200 eggs, but we need to convert the 5 1/2 crates of eggs sold into a number of eggs. Let's assume that one crate contains 30 eggs (this may vary depending on the size of the crate, but for the purposes of this example, we'll use 30). 5 1/2 crates can be written as 5 + 1/2 crates, which is equivalent to 5 crates + 0.5 x 30 eggs = 5 crates + 15 eggs. So the farmer sold a total of 5 x 30 + 15 = 165 eggs. To find the percentage of eggs left, we can subtract the number of eggs sold (165) from the total number of eggs produced (200) and then divide the result by the total number of eggs and multiply by 100 to get the percentage: Percentage of eggs left = ((200 - 165) / 200) x 100 = 35 / 2 = 17.5% Therefore, the answer is option (C) 17.5%.

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

Bayanin Amsa

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

Let (*) be a binary operation on a natural number defined by a * b = a - b + (ab)2 ${}^2$, then find 3 * 5

Bayanin Amsa

3 * 5 = 3 - 5 + (3 x 5)2 ${}^{\mathrm{<br>}}$

-2 + (15)2 ${}^{\mathrm{<br>}}$

-2 + 225 

= 223

Solve for k in the equation 18k+2 = 1

Bayanin Amsa

(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

Evaluate Log2 ${}_2$ 8√2

Bayanin Amsa

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

Find the determinant of the matrix A = (2133)

Bayanin Amsa

|A| = (2*3) - (1*3)

→ 6 - 3

= 3

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

Bayanin Amsa

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.

The equivalent of (10110.011)\(_2\) in base 10 is?

Bayanin Amsa

To convert a binary number to a decimal number, we need to understand the positional value of each digit in the binary number. In a binary number, each digit represents a power of 2, with the rightmost digit representing 2^0, the next digit to the left representing 2^1, and so on. So, for the binary number (10110.011)_2, we can break it down into its decimal equivalent as follows: 1 x 2^4 + 0 x 2^3 + 1 x 2^2 + 1 x 2^1 + 0 x 2^0 + 0 x 2^-1 + 1 x 2^-2 + 1 x 2^-3 = 16 + 0 + 4 + 2 + 0 + 0 + 0.25 + 0.125 = 22.375 Therefore, the equivalent of (10110.011)_2 in base 10 is 22.375.

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

Bayanin Amsa

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

Find x if the mean of 2x, 4x, 2x - 13 and 6x is 4.

Bayanin Amsa

To find the value of x, we need to use the formula for calculating the mean (also known as the average). The mean of a set of numbers is equal to the sum of the numbers divided by the total number of numbers in the set. In this problem, we have four numbers: 2x, 4x, 2x - 13, and 6x. To find their mean, we need to add them up and divide by 4 (since there are four numbers in the set). Therefore, we have: (2x + 4x + 2x - 13 + 6x) / 4 = 4 Simplifying the left-hand side of the equation, we get: 14x - 13 / 4 = 4 Multiplying both sides by 4, we get: 14x - 13 = 16 Adding 13 to both sides, we get: 14x = 29 Dividing both sides by 14, we get: x = 29/14 Therefore, x is approximately equal to 2.07 (rounded to two decimal places). So the answer is not one of the given options, but rather x = 29/14.

If n(P) = 20 and n(Q) = 30 and n(PuQ) = 40, find the value n(PnQ)

Bayanin Amsa

The values n(P), n(Q), n(PuQ), and n(PnQ) represent the number of elements in the sets P, Q, the union of sets P and Q, and the intersection of sets P and Q, respectively. To find n(PnQ), we need to figure out how many elements are in both sets P and Q. This is because the intersection of two sets only includes the elements that are present in both sets. We can use the formula n(PuQ) = n(P) + n(Q) - n(PnQ) to find n(PnQ). Plugging in the given values, we get: 40 = 20 + 30 - n(PnQ) Solving for n(PnQ), we get: n(PnQ) = 20 + 30 - 40 = 10 So the answer is 10.

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

Bayanin Amsa

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⎞⎠⎟

If a dress is sold for #3800.00 at 20% discount. what is its original price?

Bayanin Amsa

The original price of the dress can be found by reversing the discount. If the dress was sold for #3800.00 at a 20% discount, then 20% of the original price was taken off to get #3800.00. To find the original price, we need to add back the 20% that was taken off. So if we take 100% and add 20% to it, we'll have the original price of the dress. Therefore, the original price of the dress was #3800.00 / (100% - 20%) = #3800.00 / 80% = #4750.00.

Factorize 4a\(^2\) - 9b\(^2\)

Bayanin Amsa

To factorize 4a\(^2\) - 9b\(^2\), we can use the difference of squares formula, which states that a\(^2\) - b\(^2\) = (a+b) (a-b). We just need to recognize that 4a\(^2\) is a\(^2\) multiplied by 4, and 9b\(^2\) is b\(^2\) multiplied by 9. Then we have: 4a\(^2\) - 9b\(^2\) = (2a)\(^2\) - (3b)\(^2\) = (2a + 3b) (2a - 3b) Therefore, the correct answer is (2a+3b) (2a-3b).

The 10th term of an AP is 32. If the first term is 3/2, what is the 4th term?

Bayanin Amsa

The 10th term of an arithmetic progression (AP) is given as 32, and the first term is 3/2. We need to find the value of the 4th term. We can use the formula to find the nth term of an AP: nth term = a + (n-1)d where a is the first term, d is the common difference, and n is the term we want to find. We can first find the common difference, d, using the formula for the 10th term: 10th term = a + (10-1)d 32 = 3/2 + 9d 29.5 = 9d d = 29.5/9 Now, we can find the 4th term using the same formula: 4th term = 3/2 + (4-1)(29.5/9) 4th term = 3/2 + 3.2778 4th term = 35/6 Therefore, the answer is: 35/3.

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?

Bayanin Amsa

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.

The shaded portion in the venn diagram above represents?

Bayanin Amsa

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)

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

Bayanin Amsa

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.

In the diagram above angle LNM and angle YNZ are represented by g and h respectively. Find ∠MNY

Bayanin Amsa

In a triangle, the sum of all the interior angles is equal to 180 degrees. So, if we know two angles in a triangle, we can find the third angle by subtracting the sum of those two angles from 180 degrees. In this case, we know the measures of two angles in triangle MNY: angle LNM (represented by g) and angle YNZ (represented by h). To find the measure of angle MNY, we can subtract the sum of these two angles from 180 degrees: ∠MNY = 180º - gº - hº So, the answer to the question is (180º - gº - hº).

Integrate (2x+1)3

Bayanin Amsa

Recall chain rule: 

u = 2x +1; du = 2dx → dx = du2
u3 ${}^3$ = ∫ u3 ${}^3$ du2 $\frac{\mathrm{<br>}}{}$ → 12 $\frac{1}{2}$ ∫ u3 ${}^3$

=  1∗u42∗4 

=  u48 $\frac{{\mathrm{<br>}}^{}}{}$ →  2x+148 $\frac{{\mathrm{<br>}}^{}}{}$ + C

The number line represented by the inequality

Bayanin Amsa

x ≥ 2 is represented by an arrow with an filled-in dot above the 2.

The arrow points in the direction of all the numbers that are greater than 2.

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

Bayanin Amsa

square both sides to remove the big square root

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

cross multiply

3v = r2 ${}^2$ * πh

v = πr2h3

Evaluate (101two)3

Bayanin Amsa

5\(^3\) = 125

And 125 =  \(1111101_{two}\)

Simplify 13−√2 $\frac{1}{3-\surd 2}$ in the form of p + q√2

Bayanin Amsa

Rationalization with conjugate 3+√2 $3+\surd 2$

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

→ 1∗[3+√2][3−√2][3+√2] $\frac{\mathrm{<br>}}{}$

=  3+√29−3√2+3√2+√4

=  3+√29−2 $\frac{3+\surd 2}{9-2}$ →  

= $\frac{3}{7}$ +  √27

If sin θ = -  35 and θ lies in the third quadrant, find cos θ

Bayanin Amsa

Where sin θ = opphyp → 
$\frac{-3}{5}$

opp = -3, hyp = 5

using pythagoras formula 

hyp2 ${}^2$ = adj2 ${}^2$ + opp2 ${}^2$

adj2 ${}^2$ = hyp2 ${}^2$ - opp2 ${}^2$

adj2 ${}^2$ = 52 ${}^2$ - 32 ${}^2$ → 25 - 9

adj2 ${}^2$ = 16

adj = 4

cos θ = adjhyp → 
$\frac{4}{5}$

In third quadrant: cos θ is negative → - 45

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

Bayanin Amsa

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

Find the length of a chord 3cm from the centre of a circle of radius 5cm

Bayanin Amsa

Using pythagoras formula:

Hyp2 ${}^{\mathrm{<br>}}$ = adj2 ${}^{\mathrm{<br>}}$ + opp2 ${}^{\mathrm{<br>}}$

52 ${}^{\mathrm{<br>}}$ = opp2 ${}^{\mathrm{<br>}}$ + 32 ${}^{\mathrm{<br>}}$

52 ${}^{\mathrm{<br>}}$ - 32 ${}^{\mathrm{<br>}}$ = adj2 ${}^{\mathrm{<br>}}$

4 = adj

length of the chord = 2 * 4 = 8cm

If y varies inversely as x and x = 3 when y =4. Find the value of x when y = 12

Bayanin Amsa

This problem involves an inverse variation, which means that as one variable increases, the other variable decreases, and vice versa. The relationship between the variables can be expressed as: y = k/x where k is the constant of variation. To solve for the value of k, we can use the given information that x = 3 when y = 4: 4 = k/3 Multiplying both sides by 3 gives: k = 12 Now that we know the constant of variation, we can use it to find the value of x when y = 12: 12 = 12/x Multiplying both sides by x gives: 12x = 12 Dividing both sides by 12 gives: x = 1 Therefore, when y = 12, x = 1. Option (B) is the correct answer.

The locus of points equidistant from a fixed point.

Bayanin Amsa

The locus of points equidistant from a fixed point is a circle. A circle is a two-dimensional shape that is defined as a set of points that are all the same distance away from a fixed point, called the center. The distance between the center of the circle and any point on the circle is called the radius. So, when we talk about the locus of points equidistant from a fixed point, we are talking about all the points that are the same distance away from the fixed point. This creates a circular shape, with the fixed point as the center of the circle and the distance between the center and any point on the circle as the radius. Therefore, the correct answer is: circle.

find the limit of y = x3+6x−7x−1 $\frac{{\mathrm{<br>}}^{}}{}$ as x tends to 1

Bayanin Amsa

x3+6x−7x−1 $\frac{{\mathrm{<br>}}^{}}{}$

When numerator is differentiated → 3x2 ${}^2$ + 6  

When denominator is differentiated → 1

: 3x2+61 $\frac{\mathrm{<br>}}{}$

substitute x for 1

 3∗12+61 $\frac{3\ast 1^2+6}{1}$ =  3+61 $\frac{3+6}{1}$ 

=  91 $\frac{9}{1}$

= 9

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?

Bayanin Amsa

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.

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

Bayanin Amsa

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.

Bayanin Amsa

When a fair coin is tossed, there are two possible outcomes: heads or tails, each with a probability of 1/2. If the coin is tossed twice, there are four possible outcomes: - heads on the first toss, followed by heads on the second toss (HH) - heads on the first toss, followed by tails on the second toss (HT) - tails on the first toss, followed by heads on the second toss (TH) - tails on the first toss, followed by tails on the second toss (TT) Out of these four possible outcomes, three of them have at least one head (HH, HT, TH). Therefore, the probability of obtaining at least one head is 3/4 or 0.75. So the answer is: 0.75.