JAMB UTME - Mathematics - 2019

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To solve this system of equations, we can use substitution or elimination. Here's how to use substitution: From the first equation, we can rearrange it to solve for y in terms of x: 3x + y = 21 y = 21 - 3x Then we can substitute this expression for y into the second equation: xy = 30 x(21 - 3x) = 30 Expanding the left side: 21x - 3x^2 = 30 Rearranging and factoring: 3x^2 - 21x + 30 = 0 Dividing by 3: x^2 - 7x + 10 = 0 This quadratic equation can be factored as: (x - 2)(x - 5) = 0 So the possible values of x are 2 and 5. To find the corresponding values of y, we can substitute each value of x into the expression we found for y: When x = 2: y = 21 - 3(2) = 15 When x = 5: y = 21 - 3(5) = 6 Therefore, the solution to the system of equations is: x = 2 or 5, y = 15 or 6 So the correct option is: - x = 2 or 5, y = 15 or 6

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The sum of two opposite angles of a cyclic quadrilateral = 180°
$\therefore$ (2x + 18)° + 84° = 180°
2x + 102° = 180° $⟹$ 2x = 78°
x = 39°

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|AC| = |DF| = 13 cm
Using Pythagoras theorem,
|AC|${}^2$ = |AB|${}^2$ + |BC|${}^2$
13${}^2$ = 12${}^2$ + |BC|${}^2$
|BC|${}^2$ = 169 - 144 = 25
|BC| = $\sqrt{25}$
= 5 cm
Volume of triangular prism = $\frac{1}{2}\times base\times length\times height$
= $\frac{1}{2}\times 5\times 12\times 18$
= 540 cm${}^3$

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Q(m, n) and R(n, -4)
Midpoint : P(2, m)
$⟹(\frac{m+n}{2},\frac{n-4}{2})=(2,m)$
$m+n=2\times 2⟹m+n=4...\left(i\right)$
$n-4=2\times m⟹n-4=2m...\left(ii\right)$
Solving (i) and (ii) simultaneously,
m = 0 and n = 4.

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The sum of the interior angles of a polygon with n sides is (n-2) times 180 degrees. Therefore, for a polygon with 4 sides, the sum of the interior angles is (4-2) times 180 degrees, which is 360 degrees. Using the given angles in terms of x, we can write the equation: 2x + 5x + x + 4x = 360 Simplifying this equation, we get: 12x = 360 Dividing both sides by 12, we get: x = 30 Therefore, the value of x is 30.

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Given q(x) [quotient], d(x) [divisor] and r(x) [remainder], the polynomial is gotten by multiplying the quotient and the divisor and adding the remainder.
i.e In this case, the polynomial = (x${}^2$ - x - 5)(3x - 1) + 7.
= (3x${}^3$ - x${}^2$ - 3x${}^2$ + x - 15x + 5) + 7
= (3x${}^3$ - 4x${}^2$ - 14x + 5) + 7
= 3x${}^3$ - 4x${}^2$ - 14x + 12

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We can simplify the expression using the properties of logarithms. First, we can rewrite the expression as: 2log3(3^2) * log3(3^4) - 2log5(5^4) Using the power rule of logarithms, we can simplify the first term: 2log3(3^2) * log3(3^4) = 2(2) * 4 = 16 Using the power rule of logarithms again, we can simplify the second term: 2log5(5^4) = 8log5(5) = 8 Substituting these simplified terms back into the original expression, we get: 16 - 8 = 8 Therefore, the value of the expression is 8, which is option (C) in the given choices.

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$(\frac{6}{0.32}÷\frac{2}{0.084}{)}^{-1}$
= $(\frac{600}{32}÷\frac{2000}{84}{)}^{-1}$
= $(\frac{600}{32}\times \frac{84}{2000}{)}^{-1}$
= $(\frac{63}{80}{)}^{-1}$
= $\frac{80}{63}$
= 1.3 (to 1 decimal place)

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$\tan \theta =\frac{9}{20}=0.45$
$\theta ={\tan }^{-1}\left(0.45\right)$
= 24.23°
$\therefore$ The bearing of Y from X = 180° - 24.23°
= 155.77°
= 156° (to the nearest degree)

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Number of students that scored above 40 = 55 + 45 + 30 + 15 + 5
= 150 students.

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To find dV/dR, we need to take the derivative of V with respect to R, while treating all other variables as constants. Using the product rule of differentiation, we get: dV/dR = πh/3 [2R + r] Therefore, the answer is πh/3 [2R + r]. To understand this, we need to remember that the derivative of a function gives us the rate of change of the function. In this case, the volume of a frustrum is a function of its height (h) and radii (R and r). By taking the derivative of the volume with respect to one of the radii (R), we get the rate of change of the volume with respect to that radius. In other words, dV/dR tells us how much the volume changes for a small change in R, while holding h and r constant.

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To find the equation of a straight line, we need to determine its slope and y-intercept. We can use the two given points to find the slope of the line using the formula: slope (m) = (y2 - y1) / (x2 - x1) where (x1, y1) and (x2, y2) are the coordinates of the two points. Plugging in the values of the two points A(3, 12) and B(5, 22), we get: m = (22 - 12) / (5 - 3) = 10/2 = 5 Now we have the slope of the line. To find the y-intercept, we can use the fact that the line intersects the y-axis at 2. The y-intercept is the value of y when x is 0, so we can substitute x=0 and y=2 into the equation y = mx + b, where m is the slope and b is the y-intercept, to get: 2 = 5(0) + b b = 2 Now we have the slope and y-intercept of the line. We can plug them into the equation y = mx + b to get: y = 5x + 2 Therefore, the correct answer is: - y = 5x + 2

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Let's suppose that the first term of the GP is 'a' and the common ratio is 'r'. Then, using the given information, we can create the following equations: Third term: ar^2 = 9 ---(1) Seventh term: ar^6 = 1/9 ---(2) To find 'r', we can divide equation (2) by equation (1) as follows: (ar^6) / (ar^2) = (1/9) / 9 r^4 = 1/81 r = (1/81)^(1/4) r = 1/3 Therefore, the common ratio is 1/3.

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Actual weight = 0.18g
Error = 0.21g - 0.18g
= 0.03g
% error = $\frac{0.03}{0.18}\times 100$
= 16.7

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M = $\left|\begin{array}{ccc}-2& 0& 4\\ 0& -1& 6\\ 5& 6& 3\end{array}\right|$
M${}^T$ = $\left|\begin{array}{ccc}-2& 0& 5\\ 0& -1& 6\\ 4& 6& 3\end{array}\right|$
2M = $\left|\begin{array}{ccc}-4& 0& 8\\ 0& -2& 12\\ 10& 12& 6\end{array}\right|$
M${}^T$ + 2M = $\left|\begin{array}{ccc}-6& 0& 13\\ 0& -3& 18\\ 14& 18& 9\end{array}\right|$

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$\left|\begin{array}{cc}2& -4\\ x& 9\end{array}\right|=58$
$⟹(2\times 9)-(-4\times x)=58$
$18+4x=58⟹4x=58-18=40$
$x=10$

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$\frac{4x-6}{3}\le \frac{3+2x}{2}$
2(4x - 6) $\le$ 3(3 + 2x)
8x - 12 $\le$ 9 + 6x
8x - 6x $\le$ 9 + 12
2x $\le$ 21
$x\le \frac{21}{2}$

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Mode = Score with the highest frequency
= 5

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Explanation:

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S = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}
n(S) = 14
multiples of 3 = {21, 24, 27, 30, 33}
n(multiples of 3) = 5
Prob( picking a multiple of 3) = 5/14

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The factory worker earns ₦50,000 per month. He spends 15% of his income on his children's education. 15% of ₦50,000 = (15/100) x ₦50,000 = ₦7,500 He spends ₦13,600 on food. He spends 3% of his income on electricity. 3% of ₦50,000 = (3/100) x ₦50,000 = ₦1,500 Therefore, the total amount he spends on education, food and electricity is: ₦7,500 + ₦13,600 + ₦1,500 = ₦22,600 To find out how much he has left for personal purposes, we subtract the total amount he spends from his income. ₦50,000 - ₦22,600 = ₦27,400 Therefore, he has ₦27,400 left for his personal purpose. So the answer is N27,400.

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$(0.0439÷3.62)$
= 0.01213
$\approxeq$ 0.012
= $\frac{12}{1000}$

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Since each interior angle = 140°;
Each exterior angle = 180° - 140° = 40°
Number of sides of the polygon = $\frac{360°}{40°}$
= 9 
Sum of angles in the polygon = 140° x 9
= 1260°

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81${}^{\frac{-3}{4}}$ x 25${}^{\frac{1}{2}}$ x 243${}^{\frac{2}{5}}$
= $(\sqrt[4]{481}{)}^{-3}\times \sqrt{25}\times (\sqrt[5]{5243}{)}^2$
= $\frac{5\times 3^2}{3^{-3}}$
= $\frac{5}{3}$

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Mean = $\frac{\sum fx}{\sum f}$

= $\frac{156}{40}$

= 3.9

M.D = $\frac{\sum f|x–\overline{x}|}{\sum f}$

= $\frac{60.4}{40}$

= 1.51

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$\frac{(0.5436{)}^3}{0.017\times 0.219}$
= $\frac{0.16063}{0.017\times 0.219}$
= 43.1 (to 3 s.f)

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To find d2y/dx2, we need to take the second derivative of y with respect to x. dy/dx = 24x^2 - 6x + 7 Taking the derivative of dy/dx, we get: d2y/dx2 = 48x - 6 Therefore, the answer is 48x - 6. To understand this, we need to remember that the second derivative of a function gives us the rate of change of the slope of the function. In this case, the first derivative of y gives us the slope of the function, and the second derivative gives us the rate of change of that slope.

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To solve this trigonometric problem, we need to use the fact that the sine of an angle and the cosine of its complement are equal. That is, if x is an acute angle, then:

sin(x) = cos(90° - x)

Using this identity, we can rewrite the given equation:

sin(58°) = cos(p°)

cos(90° - 58°) = cos(p°) (using the identity)

cos(32°) = cos(p°) (simplifying)

Now, since the cosine function is periodic with a period of 360°, any two angles whose cosine values are equal must differ by a multiple of 360°. That is:

p° = 32° + 360°n (where n is an integer)

So, there are infinitely many possible values of p° that satisfy the equation. Some examples are:

• p° = 32° (n = 0)
• p° = 392° (n = 1)
• p° = -328° (n = -1)

Note that we can find these values by adding or subtracting multiples of 360° to the initial value of 32°, since the cosine function has the same value for an angle and its coterminal angles.

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200 mangoes at 4 for N2.50
$⟹$ Total cost price = $\frac{200}{4}\times N2.50$
= N 125.00
Since 30 mangoes got spoilt $⟹$ Left over = 200 - 30
= 170 mangoes 
170 mangoes at 2 for N 2.40
$⟹$ Total selling point = $\frac{170}{2}\times N2.40$
= N 204.00
Profit : N (204.00 - 125.00) = N 79.00
% profit = $\frac{79}{125}\times 100$
= 63.2% profit.

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To solve this problem, we can use basic trigonometry.

Let's draw a diagram to visualize the problem:

           * 
         / | \
        /  |  \
     h /   |   \ 60m
      /    |    \
     /     |78° \
    *------x------*
         60m

In the diagram, the tree is represented by a point at the top, the point on the ground where the angle of elevation is measured is represented by "x", and the height of the tree is represented by "h".

We know that the angle of elevation from point "x" to the top of the tree is 78°. Therefore, the angle between the horizontal and the line from point "x" to the top of the tree is also 78°.

Using trigonometry, we can find the height of the tree "h" by using the tangent function:

tan(78°) = h/60m

To solve for "h", we can multiply both sides by 60m:

h = 60m * tan(78°)

Using a calculator, we can find that:

h ≈ 282.79m

Therefore, the height of the tree is approximately 282m (rounded to the nearest whole number).

So, the correct answer is:

282m.

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When you divide a polynomial p(x) by (x - a), the remainder = p(a)
i.e. In the case of 2x${}^2$ + x - 3 $÷$ (x - 2), the remainder = p(2).
= 2(2)${}^2$ + 2 - 3
= 8 + 2 - 3
= 7.

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To determine the values of x that satisfy x² - 7x + 10 ≤ 0, we need to find the roots of the quadratic equation x² - 7x + 10 = 0 and then analyze the behavior of the quadratic function. To find the roots, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a In this case, a = 1, b = -7, and c = 10, so: x = (-(-7) ± √((-7)² - 4(1)(10))) / 2(1) x = (7 ± √9) / 2 x1 = 5 and x2 = 2 The roots of the equation are x = 5 and x = 2. Now, we need to analyze the behavior of the quadratic function in the intervals between the roots and to the left and right of the roots. We can do this by creating a sign chart: Interval | x² - 7x + 10 --------------------------------- (-∞, 2) | + (2, 5) | - (5, +∞) | + In the interval (-∞, 2), the quadratic function is positive because all the factors are positive. In the interval (2, 5), the function is negative because x - 2 is negative and x - 5 is positive. In the interval (5, +∞), the function is positive again because both factors are positive. Therefore, the solution to the inequality x² - 7x + 10 ≤ 0 is the interval [2, 5], because the function is non-positive in that interval and positive elsewhere. In interval notation, we can write: 2 ≤ x ≤ 5 So the correct answer is: 2 ≤ x ≤ 5.

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To select a committee of 5 members, including 2 females, we can break down the problem into the following steps: Step 1: Select 2 females from the 3 available females. This can be done in $\binom{3}{2} = 3$ ways. Here, $\binom{n}{r}$ denotes the number of ways to choose r items from a set of n items, also known as "n choose r". Step 2: Select 3 members from the remaining 4 males and 1 female. This can be done in $\binom{4}{3} \cdot \binom{1}{0} = 4$ ways. Here, we choose 3 males from the 4 available males, and 0 females from the 1 remaining female. Step 3: Multiply the results of Steps 1 and 2 to obtain the total number of ways to choose the committee. Therefore, the total number of ways to select a committee of 5 members, including 2 females, is: $\binom{3}{2} \cdot \binom{4}{3} \cdot \binom{1}{0} = 3 \cdot 4 \cdot 1 = 12$ Hence, there are 12 ways to choose the members of the committee. Therefore, the answer is 12 ways.

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To find d/dx[log(4x^3−2x)], we can apply the chain rule of differentiation. Let u = 4x^3−2x, then y = log(u). Applying the chain rule, we have: dy/dx = dy/du * du/dx To find du/dx, we need to differentiate u with respect to x, giving: du/dx = 12x^2 - 2 To find dy/du, we can use the formula for differentiating the natural logarithm, which gives: dy/du = 1/u Putting these together, we have: dy/dx = dy/du * du/dx = 1/u * (12x^2 - 2) = (12x^2 - 2)/(4x^3 - 2x) = 2(6x^2 - 1)/(2x(2x^2 - 1)) Therefore, the answer is (12x^2 - 2)/(4x^3 - 2x), which is equivalent to.

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To find the angle corresponding to 8 years old in the pie chart, we need to first calculate the total number of pupils in the class. The total number of pupils = 4 + 13 + 30 + 44 + 9 = 100 Next, we need to find the fraction of pupils that are 8 years old. From the table, we can see that there are 13 pupils that are 8 years old. Therefore, the fraction of pupils that are 8 years old is: 13/100 To find the angle corresponding to this fraction, we can use the formula: Angle = Fraction x 360° So, the angle corresponding to 8 years old is: (13/100) x 360° = 46.8° Therefore, the answer is option C) 46.8°.

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Area of rectangle ABCD = length x breadth
= 7 x 4 
= 28 cm${}^2$
Area of triangle CDE = $\frac{1}{2}$ base x height
= $\frac{1}{2}\times 3\times 4$
= 6 cm${}^2$
Area of the figure = 28 cm${}^2$ + 6 cm${}^2$
= 34 cm${}^2$

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To find the distance between two points, we can use the distance formula which is: distance = √((x2 - x1)^2 + (y2 - y1)^2) Where (x1, y1) and (x2, y2) are the coordinates of the two points. Using this formula, let's find the distance between points C(2, 2) and D(5, 6). distance = √((5 - 2)^2 + (6 - 2)^2) distance = √(3^2 + 4^2) distance = √(9 + 16) distance = √25 distance = 5 units Therefore, the distance between points C and D is 5 units.

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We are given the binary operation ⊗ defined by m⊗n = mn + m - n for all m, n in the set of real numbers R. So, to find the value of 3⊗(2⊗4), we need to evaluate each operation in the parentheses first: 2⊗4 = 2 * 4 + 2 - 4 = 6 Next, we evaluate 3⊗6: 3⊗6 = 3 * 6 + 3 - 6 = 15 Therefore, 3⊗(2⊗4) = 15.