JAMB UTME - Mathematics - 2019

Question 1 Report

Find the polynomial if given $q(x) = x^2 - x - 5$, $d(x) = 3x - 1$ and $r(x) = 7$.

3x

^{3}

- 4x

^{2}

- 14x + 12

3x

^{2}

+ 3x - 7

3x

^{3}

+ 4x

^{2}

+ 14x - 12

3x

^{2}

- 3x + 4

Answer Details

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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Polynomials

Progression

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Question 2 Report

Age in years	7	8	9	10	11
No of pupils	4	13	30	44	9

The table above shows the number of pupils in a class with respect to their ages. If a pie chart is constructed to represent the age, the angle corresponding to 8 years old is

48.6°

56.3°

46.8°

13°

Answer Details

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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Percentages

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Question 3 Report

The histogram above represents the number of candidates who did Further Mathematics examination in a school. How many candidates scored more than 40?

100

79

150

90

Answer Details

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

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Sets

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Question 4 Report

Given $ \sin 58^\circ = \cos p^\circ $, find p.

48°

58°

32°

52°

Answer Details

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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Read lesson note on Angles Of Elevation And Depression (WAEC)

Sine, Cosine And Tangent Of An Angle

Angles Of Elevation And Depression

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Question 5 Report

If $\left|\begin{matrix}2 & -4 \\ x & 9\end{matrix}\right| = 58$, find the value of x.

10

30

14

28

Answer Details

| \begin{matrix} 2 & - 4 \\ x & 9 \end{matrix} | = 58

⟹ (2 \times 9) - (- 4 \times x) = 58

18 + 4 x = 58 ⟹ 4 x = 58 - 18 = 40

x = 10

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Number Bases

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Question 6 Report

The weight of a day-old chick was measured to be 0.21g. If the actual weight of the chick is 0.18g, what was the percentage error in the measurement?

15.5%

18.2%

14.8%

16.7%

Answer Details

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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Read lesson note on Fractions, Decimals, Approximations And Percentage (JAMB)

Modular Arithmetic

Fractions, Decimals, Approximations And Percentage

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Question 7 Report

In a committee of 5, which must be selected from 4 males and 3 females. In how many ways can the members be chosen if it were to include 2 females?

144 ways

15 ways

185 ways

12 ways

Answer Details

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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Question 8 Report

Evaluate $ \dfrac{2\log_{3} 9 \times \log_{3} 81^{-2}}{\log_{5} 625} $

4

-32

-8

36

Answer Details

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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Read lesson note on Logarithms (WAEC)

Number Bases

Logarithms

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Question 9 Report

Calculate the volume of the regular three-dimensional figure drawn above, where

394 cm

^{3}

425 cm

^{3}

268 cm

^{3}

540 cm

^{3}

Answer Details

|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 b a s e \times l e n g t h \times h e i g h t

=

\frac{1}{2} \times 5 \times 12 \times 18

= 540 cm

^{3}

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Volumes

Mensuration

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Question 10 Report

In the figure above, |CD| is the base of the triangle CDE. Find the area of the figure to the nearest whole number.

56 cm

^{2}

24 cm

^{2}

42 cm

^{2}

34 cm

^{2}

Answer Details

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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Volumes

Coordinate Geometry

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Question 11 Report

Simplify $81^{-\frac{3}{4}} \times 25^{\frac{1}{2}} \times 243^{\frac{2}{5}}$

\frac{2}{5}

\frac{3}{5}

\frac{5}{2}

\frac{5}{3}

Answer Details

81

^{\frac{- 3}{4}}

x 25

^{\frac{1}{2}}

x 243

^{\frac{2}{5}}

=

(\sqrt[4]{81})^{- 3} \times \sqrt{25} \times (\sqrt[5]{243})^{2}

=

\frac{5 \times 3^{2}}{3^{- 3}}

=

\frac{5}{3}

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Circles

Number Bases

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Question 12 Report

Find the value of $ \frac{(0.5436)^3}{0.017 \times 0.219} $ to 3 significant figures.

46.2

43.1

534

431

Answer Details

\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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Modular Arithmetic

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Question 13 Report

Given matrix $ M = \begin{vmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{vmatrix} $, find $ M^T + 2M $

| \begin{matrix} - 4 & 2 & 1 \\ 6 & 0 & 5 \\ 0 & 6 & 2 \end{matrix} |

| \begin{matrix} - 6 & 0 & 13 \\ 0 & - 3 & 18 \\ 14 & 18 & 9 \end{matrix} |

| \begin{matrix} 5 & 2 & 6 \\ 0 & 1 & 1 \\ 3 & 4 & - 7 \end{matrix} |

| \begin{matrix} - 4 & 0 & 8 \\ 0 & - 2 & - 16 \\ 10 & 12 & 6 \end{matrix} |

Answer Details

M =

| \begin{matrix} - 2 & 0 & 4 \\ 0 & - 1 & 6 \\ 5 & 6 & 3 \end{matrix} |

M

^{T}

=

| \begin{matrix} - 2 & 0 & 5 \\ 0 & - 1 & 6 \\ 4 & 6 & 3 \end{matrix} |

2M =

| \begin{matrix} - 4 & 0 & 8 \\ 0 & - 2 & 12 \\ 10 & 12 & 6 \end{matrix} |

M

^{T}

+ 2M =

| \begin{matrix} - 6 & 0 & 13 \\ 0 & - 3 & 18 \\ 14 & 18 & 9 \end{matrix} |

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Matrices And Determinants

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Question 14 Report

If P(2, m) is the midpoint of the line joining Q(m, n) and R(n, -4), find the values of m and n.

m = 0, n = 4

m = 4, n = 0

m = 2, n = 2

m = -2, n = 4

Answer Details

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 . . . (i)

n - 4 = 2 \times m ⟹ n - 4 = 2 m . . . (i i)

Solving (i) and (ii) simultaneously,
m = 0 and n = 4.

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Polynomials

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Question 15 Report

Tade bought 200 mangoes at 4 for ₦2.50. 30 out of the mangoes got spoilt and the remaining were sold at 2 for ₦2.40. Find the percentage profit or loss.

43.6% loss

35% profit

63.2% profit

28% loss

Answer Details

200 mangoes at 4 for N2.50

⟹

Total cost price =

\frac{200}{4} \times N 2.50

= 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 N 2.40

= 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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Percentages

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Question 16 Report

If the 3rd and 7th terms of a G.P are 9 and 1/9 respectively. Find the common ratio.

\frac{1}{3}

\frac{1}{9}

\frac{2}{3}

\frac{2}{9}

Answer Details

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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Sequence And Series

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Question 17 Report

Determine the values for which $x^2 - 7x + 10 \le 0$

2

\leq

x

\geq

5

-2

\leq

x

\leq

3

-2

\leq

x

\geq

3

2

\leq

x

\leq

5

Answer Details

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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Inequalities

Quadratic Equations

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Question 18 Report

If given two points A(3, 12) and B(5, 22) on a x-y plane. Find the equation of the straight line with intercept at 2.

y = 5x + 2

y = 5x + 3

y = 12x + 2

y = 22x + 3

Answer Details

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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Question 19 Report

Solve for x in $ \frac{4x-6}{3} \leq \frac{3+2x}{2} $

x \leq 1 \frac{1}{2}

x \leq \frac{21}{2}

x \geq \frac{21}{2}

x \geq 1 \frac{1}{2}

$x \geq 1 \frac{1}{2}$

Answer Details

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

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Inequalities

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Question 20 Report

Find the probability that a number selected at random from 21 to 34 is a multiple of 3

\frac{3}{11}

\frac{2}{9}

\frac{5}{14}

\frac{5}{13}

Answer Details

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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Probability

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Question 21 Report

The angle of elevation of the top of a tree from a point on the ground 60m away from the foot of the tree is 78°. Find the height of the tree correct to the nearest whole number.

148m

382m

282m

248m

Answer Details

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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Bearings

Angles Of Elevation And Depression

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Question 22 Report

The locus of a point which moves so that it is equidistant from two intersecting straight lines is the

bisector of the two lines

line parallel to the two lines

angle bisector of the two lines

perpendicular bisector of the two lines

Loci

Triangles And Polygons

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Question 23 Report

Each of the interior angles of a regular polygon is 140°. Calculate the sum of all the interior angles of the polygon.

1080°

1260°

2160°

1800°

Answer Details

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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Question 24 Report

Find the distance between the points C(2, 2) and D(5, 6).

13 units

7 units

12 units

5 units

Answer Details

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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Question 25 Report

This table below gives the scores of a group of students in a Further Mathematics Test.

Score	1	2	3	4	5	6	7
Frequency	4	6	8	4	10	6	2

Find the mode of the distribution.

7

10

5

4

Answer Details

Mode = Score with the highest frequency
= 5

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Question 26 Report

A binary operation $ \otimes $ is defined by $ m \otimes n = mn + m - n $ on the set of real numbers, for all m, n $ \in $ R. Find the value of $ 3 \otimes (2 \otimes 4) $.

6

25

15

18

Answer Details

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.

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Binary Operations

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Question 27 Report

This table below gives the scores of a group of students in a Further Mathematics Test.

Score	1	2	3	4	5	6	7
Frequency	4	6	8	4	10	6	2

Calculate the mean deviation for the distribution

4.32

2.81

1.51

3.90

Answer Details

Score(x)	1	2	3	4	5	6	7	Total
Frequency (f)	4	6	8	4	10	6	2	40
fx	4	12	24	16	50	36	14	156
x – $\bar{x}$	-2.9	-1.9	-0.9	0.1	1.1	2.1	3.1
\|x – $\bar{x}$ \|	2.9	1.9	0.9	0.1	1.1	2.1	3.1
f\|x – $\bar{x}$ \|	11.6	11.4	7.2	0.4	11	12.6	6.2	60.4

Mean = $\frac{\sum f x}{\sum f}$

= $\frac{156}{40}$

= 3.9

M.D = $\frac{\sum f | x - \bar{x} |}{\sum f}$

= $\frac{60.4}{40}$

= 1.51

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Statistics

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Question 28 Report

Express $ (0.0439 \div 3.62) $ as a fraction.

\frac{21}{100}

\frac{21}{1000}

\frac{12}{1000}

\frac{12}{100}

Answer Details

(0.0439 \div 3.62)

= 0.01213

≊

0.012
=

\frac{12}{1000}

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Fractions, Decimals And Approximations

Solution Of Linear Equations

Areas

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Question 29 Report

Points X and Y are 20km North and 9km East of point O, respectively. What is the bearing of Y from X? Correct to the nearest degree.

24°

56°

127°

156°

Answer Details

\tan θ = \frac{9}{20} = 0.45

θ = \tan^{- 1} (0.45)

= 24.23°

∴

The bearing of Y from X = 180° - 24.23°
= 155.77°
= 156° (to the nearest degree)

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Bearings

Angles And Intercepts On Parallel Lines

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Question 30 Report

Evaluate $ \left(\frac{6}{0.32} \div \frac{2}{0.084}\right)^{-1} $ correct to 1 decimal place.

1.3

2.5

4.6

3.2

Answer Details

(\frac{6}{0.32} \div \frac{2}{0.084})^{- 1}

=

(\frac{600}{32} \div \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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Fractions, Decimals And Approximations

Number Bases

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Question 31 Report

From the cyclic quadrilateral MNOP above, find the value of x.

16°

25°

42°

39°

Answer Details

The sum of two opposite angles of a cyclic quadrilateral = 180°

∴

(2x + 18)° + 84° = 180°
2x + 102° = 180°

⟹

2x = 78°
x = 39°

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Question 32 Report

Simplify $ \frac{0.0839 \times 6.381}{5.44} $ to 2 significant figures.

0.2809

2.51

3.5

0.098

Fractions, Decimals And Approximations

Positive And Negative Integers, Rational Numbers

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Question 33 Report

A factory worker earns ₦50,000 per month out of which he spends 15% on his children's education, ₦13,600 on Food, 3% on electricity and uses the rest for his personal purpose. How much does he have left?

N21,850

N18,780

N27,400

N32,500

Answer Details

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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Financial Arithmetic

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Question 34 Report

If $\left|\begin{matrix}2 & -5 & 3 \\ x & 1 & 4 \\ 0 & 3 & 2\end{matrix}\right| = 132$, find the value of x.

5

8

6

3

Answer Details

Explanation:

| \begin{array}{lll} 2 & - 5 & 3 \\ x & 1 & 4 \\ 0 & 3 & 2 \end{array} | = 132

⟹ 2 | \begin{matrix} 1 & 4 \\ 3 & 2 \end{matrix} | - (- 5) | \begin{matrix} x & 4 \\ 0 & 2 \end{matrix} | + 3 | \begin{matrix} x & 1 \\ 0 & 3 \end{matrix} | = 132

2 (2 - 12) + 5 (2 x) + 3 (3 x) = 132

- 20 + 10 x + 9 x = 132

19 x = 152

x = 8

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Logarithms

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Question 35 Report

The angles of a polygon are given by 2x, 5x, x and 4x respectively. The value of x is

31°

30°

26°

48°

Answer Details

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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Indices

Triangles And Polygons

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Question 36 Report

Find the value of x and y in the simultaneous equation: 3x + y = 21; xy = 30

x = 3 or 7, y = 12 or 8

x = 6 or 1, y = 11 or 5

x = 2 or 5, y = 15 or 6

x = 1 or 5, y = 10 or 7

Answer Details

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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Solution Of Linear Equations

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Question 37 Report

$ \dfrac{d}{dx}[\log(4x^3 - 2x)] $ is equal to

\frac{12 x - 2}{4 x^{2}}

\frac{43 x^{2} - 2 x}{7 x}

\frac{4 x^{2} - 2}{7 x + 6}

\frac{12 x^{2} - 2}{4 x^{3} - 2 x}

Answer Details

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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Logarithms

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Question 38 Report

If the volume of a frustrum is given as $V=\frac{\pi h}{3}(R^2+Rr+r^2)$, find $\frac{\mathrm{d}V}{\mathrm{d}R}$.

\frac{π h}{3} (2 R + r)

2R+r+πh3

\frac{π h}{3} (2 R^{2} + r + 2 r)

\frac{2 R^{2}}{3} π h

Answer Details

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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Question 39 Report

If $y = 8x^3 - 3x^2 + 7x - 1$, find $\frac{d^2y}{dx^2}$.

48x - 6

11x

^{2}

+ 6x - 7

32x + 7

24x

^{2}

- 6x + 7

Answer Details

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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Differentiation

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Question 40 Report

If $2x^2 + x - 3$ divides $x - 2$, find the remainder.

7

3

5

6

Answer Details

When you divide a polynomial p(x) by (x - a), the remainder = p(a)
i.e. In the case of 2x