JAMB UTME - Mathematics - 2020

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Circumstances of latitude 0o ${}^o$s where R is the radius of the earth 2π $\pi$r cos θ

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r2r $\frac{\frac{r}{2}}{r}$ Sin θ $\theta$ = 12 $\frac{1}{2}$ 

θ $\theta$ = sin−1 ${}^{-1}$ (12 $\frac{1}{2}$) = 30o ${}^o$ = 60o ${}^o$ 

Length of arc (minor)

ST = θ360 $\frac{\theta }{360}$ x 2πr $\pi r$ 

60360×2π×r=π3 $\frac{60}{360}\times 2\pi \times r=\frac{\pi }{3}$ 

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V = 2080cm3 ${}^3$, h = ? 

r = 7cm

V = Vπr2h $\pi r^{2}h$

h = Vπr2=3080227×49 $\frac{V}{\pi r^2}=\frac{3080}{\frac{22}{7}\times 49}$ 

308054 $\frac{3080}{54}$ = 20cm

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No of students offering physics are 

12360x 150 = 5

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13 $\frac{1}{3}$ (x + 1) - 1 > 15 $\frac{1}{5}$(x + 4)  = x+13−1 $\frac{x+1}{3}-1$ > x+45 $\frac{x+4}{5}$ 

x+13−x+45−1 $\frac{x+1}{3}-\frac{x+4}{5}-1$ > 0 

= 5x+5−3x−1215 $\frac{5x+5-3x-12}{15}$ 

2x - 7 > 15

2x > 22 = x > 11

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Probability is the measure of how likely an event is to occur. In this question, we want to find the probability of selecting a prime number between 20 and 30, both numbers inclusive. The prime numbers between 20 and 30 are 23 and 29. The total number of possible outcomes (or numbers between 20 and 30) is 11 (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30). So, the probability of selecting a prime number is the number of favorable outcomes (prime numbers) divided by the total number of possible outcomes. The probability of selecting a prime number is 2/11. So, the answer is 2/11.

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The vertical height through which the surveyor rises can be calculated using trigonometry. When a person walks up a hill, they are moving in two dimensions: horizontally and vertically. We can use a right triangle to represent this movement, where the horizontal distance is one side, the vertical distance is another side, and the slope of the hill is the angle between these sides. In this case, the horizontal distance is 500m and the angle is 30 degrees, so we can use the trigonometric function sine to find the vertical distance: sin(30) = vertical distance / 500 Multiplying both sides by 500, we get: 500 * sin(30) = vertical distance Using a calculator, we can find that sin(30) = 0.5, so: 500 * 0.5 = 250 Therefore, the vertical height through which the surveyor rises is 250m.

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The gradient of a line describes how steep the line is, and it can be calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. In this case, the two points are (-2, 0) and (0, -4), so the change in x is 0 - (-2) = 2, and the change in y is -4 - 0 = -4. Therefore, the gradient of the line passing through these two points is -4 ÷ 2 = -2. So, the answer is -2.

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The perimeter of a sector of a circle is equal to the circumference of the circle that is part of the sector plus the length of the arc of the sector. To calculate the perimeter of a sector, we need to find the length of the arc and the circumference of the circle first. The length of the arc is given by the formula: Arc length = (Angle of sector / 360) x Circumference of the circle The circumference of a circle is given by the formula: Circumference = 2 * pi * radius Plugging in the values we have, the circumference of the circle is 2 * pi * 10.5 = 65.97 cm. The length of the arc is (48 / 360) x 65.97 = 12.19 cm. Adding the length of the arc to the circumference of the circle, we get 65.97 + 12.19 = 78.16 cm, which is the perimeter of the sector. Therefore, the answer is 78.16 cm, which is closest to 29.8cm.

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ABCD is the floor. By pathagoras 

AC2 ${}^2$ = 144 + 81 = 225−−−√ $\sqrt{225}$ 

AC = 15cm

Height of room 8m, diagonal of floor is 15m

Therefore, the cosine of the angle which a diagonal of the room makes with the floor is 

EC2 ${}^2$ = 152 ${}^2$ + 82 ${}^2$ cosine

adjHyp=1517 $\frac{adj}{Hyp}=\frac{15}{17}$ 

EC2 ${}^2$ = 225+64−−−−−−−√ $\sqrt{225+64}$

EC = 289−−−√ $\sqrt{289}$

= 17 

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Gradient of line = Change in yChange in x $\frac{\text{Change in y}}{\text{Change in x}}$ = y2−Yx2−x $\frac{y_2-Y}{x_2-x}$ 

y2 ${}_2$  = 01 ${}_1$ 

Y1 ${}_1$  = 4 

x2 ${}_2$  = 3 and x1 ${}_1$  = 0

y2−y1x2−x1=0−43−0=−43 $\frac{y_2-y_1}{x_2-x_1}=\frac{0-4}{3-0}=\frac{-4}{3}$

Equation of straight line y = mx + c

Where m = gradient and c = y 

intercept = 4 

y = 4x + 43 $\frac{4}{3}$ multiply through y 

3y = 4x + 23 

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h = 30 tan 60 

= 303–√ $\sqrt{3}$ 

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Rice = 75g, Margarine = 40g, Meat = 105g

Bread = 20g

Total = 240

Angle of sector represented by meat 

= 105240×360o1 $\frac{105}{240}\times \frac{{360}^o}{1}$ 

= 157.5 

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[(4a + 3) 2 ${}^2$ - (3a - 2)2 ${}^2$ = a2 ${}^2$ - b2 ${}^2$ = (a + b) (a - b) 

= [(4a + 3) + (3a - 2)] [(4a + 3) - (3a - 2)]

= [4a + 3 + 3a - 2] [4a + 3 - 3a + 2]

= (7a + 1)(a + 5)

(a + 5) (7a + 1) 

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13x $\frac{1}{3x}$ + 12 $\frac{1}{2}$x = 2+3x6x $\frac{2+3x}{6x}$ > 14x $\frac{1}{4x}$ 

= 4(2 + 3x) > 6x = 12x2 ${}^2$ - 2x = 0

= 2x(6x - 1) > 0 = x(6x - 1) > 0

Case 1 (-, -) = x < 0, 6x - 1 > 0

= x < 0, x < 16 $\frac{1}{6}$ (solution) 

Case 2 (+, +) = x > 0, 6x - 1 > 0 = x > 0

x > 16 $\frac{1}{6}$

Combining solutions in cases (1) and (2) 

= x > 0, x < 16 $\frac{1}{6}$ = 0 < x < 16 $\frac{1}{6}$ 

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S.D = ∑√(x−x)2N=∑d2N=28√7 $\frac{\sqrt{\sum }(x-x{)}^2}{N}=\frac{\sum d^2}{N}=\frac{\sqrt{28}}{7}$ 

= 4–√ $\sqrt{4}$ 

= 2

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The sum of the interior angles of a pentagon is equal to (5-2) * 180 = 540 degrees. So, the equation 6x + 6y = 540 can be used to find the value of y in terms of x. Solving for y, we get y = 90 - x. This means that if we know the value of x, we can find the value of y by subtracting x from 90. So, the correct answer is y = 90 - x.

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Let x rep. numbers of boys that can work at 54 $\frac{5}{4}$ the rate at

 which the 10 girls work 

For 1 hrs, x boys will work for 15×104 $\frac{\frac{1}{5}\times 10}{4}$ 

x = 54 $\frac{5}{4}$ x 10 

= 8 boys 

8 boys will do the work of ten girls at the same rate 

4 + 8 = 12 bous cut the field in 5 hrs for 3 hrs, 

12×53 $\frac{12\times 5}{3}$ boys will be needed = 20 boys.

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275×3−3×(1)−15 $\frac{27}{5}\times 3^{-3}\times \frac{(1{)}^{-1}}{5}$

= 275×133×115 $\frac{27}{5}\times \frac{1}{3^3}\times \frac{1}{\frac{1}{5}}$

275 $\frac{27}{5}$ x 127 $\frac{1}{27}$ x 51 $\frac{5}{1}$ = 1

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0.021741×1.20470.023789 = 0.0255×1.20.024(to 216) 

= 0.02640.024= 1.1

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2√x+2 $\frac{\sqrt{2}}{x+2}$ = x - 12√ $\frac{1}{\sqrt{2}}$

x2–√ $\sqrt{2}$ (x - 2–√ $\sqrt{2}$) = x + 2–√ $\sqrt{2}$ (cross multiply)

x2–√ $\sqrt{2}$ - 2 = x + 2–√ $\sqrt{2}$ 

= x2–√ $\sqrt{2}$ - x 

= 2 + 2–√ $\sqrt{2}$

x (2–√ $\sqrt{2}$ - 1) = 2 + 2–√ $\sqrt{2}$

= 2+2√2√−1×2√+12√+1 $\frac{2+\sqrt{2}}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}$

x = 22√+2+2+2√2−1 $\frac{2\sqrt{2}+2+2+\sqrt{2}}{2-1}$ 

= 32–√ $\sqrt{2}$ + 4 

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The equation of a line can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. To find the equation of a line parallel to another line, we need to use the same slope. The line 7x + 5y = 12 can be rewritten in the form y = -(7/5)x + 12/5. The slope of this line is -(7/5). To find a line through the point (5, 7) with the same slope, we can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. Plugging in the values for (x1, y1) and m, we get: y - 7 = -(7/5)(x - 5) Expanding and simplifying, we get: y = -(7/5)x + 7 + (35/5) = -(7/5)x + 70/5 = 14x + 7. So the equation of the line through the point (5, 7) parallel to the line 7x + 5y = 12 is 7x + 5y = 70.

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The total number of goals scored by all the teams can be calculated by multiplying the number of goals by its frequency and then adding all of them up. For example, the number of goals scored by teams with 0 goals is 4 * 0 = 0. The number of goals scored by teams with 1 goal is 3 * 1 = 3. The number of goals scored by teams with 2 goals is 15 * 2 = 30. And so on. So, the total number of goals scored by all the teams is 0 + 3 + 30 + 48 + 4 + 0 + 6 = 91. Therefore, the answer is 91.

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log10 ${}_{10}$2 = 0.3010 and log10 ${}_{10}$3 = 04771

log104.5=log10 ${}_{10}4.5=lo{g}_{10}$ (3×32 $\frac{3\times 3}{2}$)

log10 ${}_{10}$ 3 + log10 ${}_{10}$ 3 - log10 ${}_{10}$2 = 0.4471 + 0.771 - 0.3010

= 0.6532

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hn−2=n+3n $\frac{h}{n-2}=\frac{n+3}{n}$

n2 ${}^2$ = (n + 3) (n - 2) 

n2 ${}^2$ = n2 ${}^2$ + n - 6

n2 ${}^2$ + n - 6 - n2 ${}^2$ = 0

n - 6 = 0

n = 6

Common ratio: nn−2=66−2=64 $\frac{n}{n-2}=\frac{6}{6-2}=\frac{6}{4}$ = 32 $\frac{3}{2}$ 

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The equation a x 2 = 2 - a can be solved for the possible values of a. To do this, we can simplify the equation by adding a to both sides: a x 2 + a = 2 Combining like terms: a x 2 + a = 2 a(x + 1) = 2 Dividing both sides by x + 1: a = 2 / (x + 1) Since the binary operation x is defined as a x b = a, we can substitute this definition into the equation: a = 2 / (a x 1 + 1) a = 2 / (a + 1) Since a x b = a, we know that a x 1 = a, so we can substitute this into the equation: a = 2 / (a + 1) a = 2 / (a + 1) Solving for a, we get: a = 2 / (a + 1) = 2 / (2) = 1 So, the possible values of a are 1.

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The answer is (1020)3, which is equal to (1 * (3^3)) + (0 * (3^2)) + (2 * (3^1)) + (0 * (3^0)). In this question, each number is given in base-3 format, which means that each digit in the number represents a power of 3. For example, the number (212)3 is equal to (2 * (3^2)) + (1 * (3^1)) + (2 * (3^0)). To evaluate the expression, we simply add and subtract the numbers as written, and convert the result back to base-3 format.

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The size of each sector in a pie chart represents the proportion of the data it represents compared to the whole. In this case, the total number of acres for all the crops is 2 + 5 + 3 + 11 + 9 = 30. To find the angle of the sector for cassava, we need to find the proportion of 3 acres to 30 acres, and then convert this proportion to degrees. The proportion of 3 acres to 30 acres is 3/30 = 1/10. In a full circle, there are 360 degrees, so the angle of the sector representing 1/10 of the data would be 360 x (1/10) = 36 degrees. Therefore, the angle of the sector for cassava in a pie chart is 36 degrees.

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Principal = N225.00, interest = N27.00

Year = x, Rate = 4% 

1 = PRT100 $\frac{PRT}{100}$

27 = 225×4×x100 $\frac{225\times 4\times x}{100}$ = 2700 = 900T

T = 2700900 $\frac{2700}{900}$ 

= 3 years

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360o ${}^o$ - (60o ${}^o$ + 60o ${}^o$ + 67 + 50 = 237o ${}^o$) 

360o ${}^o$ - 237 = 130o ${}^o$ 

B. Salary = 123360XN60001 $\frac{123}{360}X\frac{N6000}{1}$ 

= N2,050

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Since PQRS is quadrilateral

2y + 2x + QPS = 360o ${}^o$

i.e. (y + x) + QPS = 360o ${}^o$ 

QPS = 360o ${}^o$ - 2 (y + x)

But x + y + 68o ${}^o$ = 180o ${}^o$

There; x + y = 180o ${}^o$ - 68o ${}^o$ = 112o ${}^o$

QPS = 360 - 2(112o ${}^o$)

= 360o ${}^o$ - 224 = 136o

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Coca-Cola = 10 bottles, Fanta = 8 bottles, Spirite = 6 bottles 

Total = 24

P(Coca-Cola) = 1024 $\frac{10}{24}$; P(not Coca-Cola) 

1 - 1024 $\frac{10}{24}$ 

24−1024=1424=712 $\frac{24-10}{24}=\frac{14}{24}=\frac{7}{12}$