JAMB UTME - Mathematics - 1990

If x + $\frac{1}{x}$ = 4, find x2 + $\frac{1}{x^2}$

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x + $\frac{1}{x}$ = 4, find x2 + $\frac{1}{x^2}$
= (x + $\frac{1}{x}$)2 = x2 + $\frac{1}{x^2}$ + 2
x2 + $\frac{1}{x^2}$ = ( x + $\frac{1}{x^2}$)2 - 2
= (4)2 - 2
= 16 - 2
= 14

If a = 2, b = -2 and c = -$\frac{1}{2}$, evaluate (ab2 - bc2)(a2c - abc)

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(ab2 - bc2)(a2c - abc)
[2(2)2 - (- 2x$\frac{1}{2}$)] [22(-$\frac{1}{2}$) - 2(-2)(-$\frac{1}{2}$)]
[8 = $\frac{1}{2}$][-2 - 2] = $\frac{17}{2}$ x 42
= -34

Two numbers are removed at random from the numbers 1, 2, 3 and 4. What is the probability that the sum of the numbers removed is even?

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$\begin{array}{cccc}1& 2& 3& 4\\ 1(1,1)& (1,2)& (1,3)& (1,4)\\ 2(2,1)& (2,2)& (2,3)& (2,4)\\ 3(3,1)& (3,2)& (3,3)& (3,4)\\ 4(4,1)& (4,2)& (4,3)& (4,4)\end{array}$
sample space = 16
sum of nos. removed are (2), 3, (4), 5
3, (4), 5, (6)
(4), 5, (6), 7
(5), 6, 7, (8)
Even nos. = 8 of them
Pr(even sum) = $\frac{8}{16}$
= $\frac{1}{2}$

Find the probability that a number selected at random from 40 to 50 is a prime

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From 40 to 50 = 11 & number are prime i.e. 41, 43, 47
prob. of selecting a prime No. is $\frac{3}{11}$

Simplify 4$\frac{3}{4}$ - 6$\frac{1}{4}$

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4$\frac{3}{4}$ - 6$\frac{1}{4}$
$\frac{19}{4}$ - $\frac{25}{4}$............(A)
$\frac{21}{5}$ - $\frac{5}{4}$.............(B)
Now work out the value of A and the value of B and then find the value $\frac{A}{B}$
A = $\frac{19}{4}$ - $\frac{25}{4}$
= $\frac{-6}{4}$
B = $\frac{21}{5}$ x $\frac{5}{20}$
= $\frac{105}{20}$
= $\frac{21}{4}$
But then $\frac{A}{B}$ = $\frac{-6}{4}$
$\frac{21}{4}$ = $\frac{-6}{4}$ $÷$ $\frac{21}{4}$
= $\frac{-6}{4}$ x $\frac{4}{21}$
= $\frac{-24}{84}$
= $\frac{-2}{7}$

Simplify ($\frac{1}{x^{?1}}+\frac{1}{y^{?1}}$)-1

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Simplify $\sqrt{160r^2}$ + $\sqrt{71r^4}$ + $\sqrt{100r^2}$

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$\sqrt{160r^2+71r^4+100r^8}$
Simplifying from the innermost radical and progressing outwards we have the given expression
$\sqrt{160r^2+71r^4+100r^8}$ = $\sqrt{160r^2+81r^4}$
$\sqrt{160r^2+9r^2}$ = $\sqrt{169r^2}$
= 13r

The angle of a sector of s circle, radius 10.5cm, is 48 o, Calculate the perimeter of the sector

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To find the perimeter of the sector, we need to find the arc length and add the radii. The formula for the arc length of a sector is: Arc length = (angle/360) x 2πr where r is the radius of the circle, and angle is the angle of the sector in degrees. In this case, the radius is given as 10.5 cm, and the angle is 48 degrees. So, the arc length is: Arc length = (48/360) x 2π x 10.5 Arc length = 0.133 x 2 x 3.14 x 10.5 Arc length = 8.82 cm (rounded to two decimal places) Now, we add the radii to the arc length to get the perimeter of the sector: Perimeter = 2 x 10.5 + 8.82 Perimeter = 28.82 cm (rounded to two decimal places) Therefore, the answer is 28.8cm.

A car painter charges ₦40.00 per day for himself and ₦10.00 per day for his assistant. if a fleet of cars were painted for ₦2000.00 and the painter worked 10days more than his assistant, how much did the assistant receive?

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To solve this problem, we can use a system of equations. Let's call the number of days the assistant worked "x". Then the number of days the car painter worked would be "x + 10" (since he worked 10 days more than the assistant). We can then set up the following equations: 40(x+10) + 10x = 2000 (this represents the total amount the car painter and his assistant earned for painting the cars) Simplifying this equation: 50x + 400 = 2000 50x = 1600 x = 32 So the assistant worked for 32 days. To find out how much he earned, we can plug this value back into one of the original equations: 10x = 10(32) = 320 Therefore, the assistant earned ₦320.00.

A cylinder pipe, made of metal is 3cm thick.If the internal radius of the pope is 10cm.Find the volume of metal used in making 3m of the pipe.

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To find the volume of metal used to make 3 meters of the pipe, we first need to find the volume of one meter of the pipe, and then multiply it by 3. The pipe is a cylinder with an internal radius of 10cm and a thickness of 3cm, which means that the external radius of the pipe is 13cm (10cm + 3cm). To find the volume of one meter of the pipe, we need to find the volume of the outer cylinder (with radius 13cm) and subtract the volume of the inner cylinder (with radius 10cm): Volume of outer cylinder = π x radius^2 x height = π x 13^2 x 100 = 16,900π cm^3 Volume of inner cylinder = π x radius^2 x height = π x 10^2 x 100 = 10,000π cm^3 Volume of metal in one meter of the pipe = Volume of outer cylinder - Volume of inner cylinder = 16,900π - 10,000π = 6,900π cm^3 Therefore, the volume of metal used to make 3 meters of the pipe is: Volume of metal in 3 meters of the pipe = Volume of metal in 1 meter x 3 = 6,900π x 3 = 20,700π cm^3 So the answer is 20,700π cubic cm.

Correct 241.34(3 x 10-3)2 to 4 significant figures

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To correctly express 241.34 x 10^-3 to 4 significant figures, we need to start by determining the first significant digit, which is the first non-zero digit. In this case, it is 2. Next, we count the next three digits after the first significant digit, which are 4, 1, and 3. Therefore, the number 241.34 x 10^-3 to 4 significant figures is 0.2413. Out of the given options, the number that matches 0.2413 to 4 significant figures is: 0.002172.

If the shadow of a pole 7m high is 1/2 its length what is the angle of elevation of the sun, correct to the nearest degree?

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tan θ - 7/3.5 = 2
tan θ= 2.0000
θ = 63o

What must be added to 4x2 - 4 to make it a perfect square?

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Find the curved surface area of the frustrum in the figure.

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$\frac{x}{4}=\frac{6+x}{6}$

6x = 4(6 + x) = 24 + 4x

x = 12 = c = $\pi RL-\pi L$

= $\pi \left(6\right)\sqrt{{18}^2}+6^2-\pi \times 4\times \sqrt{{12}^2}+4^2$

= 6$\pi \sqrt{360}-4\pi \sqrt{160}$

= 36$\pi \sqrt{10}-16\pi \sqrt{10}$

= 20$\pi \sqrt{10}$cm2

Find the length of a side of a rhombus whose diagonals are 6cm and 8cm

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In a rhombus, the diagonals intersect at a right angle and bisect each other. This means that the diagonals divide the rhombus into four congruent right triangles. Let's call the length of the side we are looking for "x". Using the Pythagorean theorem, we can find the length of each of the two right triangles that are formed by the diagonals: $$(\frac{6}{2})^2 + (\frac{8}{2})^2 = x^2$$ Simplifying this equation gives: $$9+16=x^2$$ Adding 9 and 16 gives us 25, so: $$25=x^2$$ To solve for x, we take the square root of both sides: $$\sqrt{25}=\sqrt{x^2}$$ Which simplifies to: $$5=x$$ Therefore, the length of the side of the rhombus is 5cm.

If Cos $\theta$ = $\frac{12}{13}$. Find $\theta$ + cos2$\theta$

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Cos $\theta$ = $\frac{12}{13}$
x2 + 122 = 132
x2 = 169- 144 = 25
x = 25
= 5
Hence, tan$\theta$ = $\frac{5}{12}$ and cos$\theta$ = $\frac{12}{13}$
If cos2$\theta$ = 1 + $\frac{1}{ta{n}^2\theta }$
= 1 + $\frac{1}{\frac{(5{)}^2}{12}}$
= 1 + $\frac{1}{\frac{25}{144}}$
= 1 + $\frac{144}{25}$
= $\frac{25+144}{25}$
= $\frac{169}{25}$

The perimeter of a rectangular lawn is 24m. If the area of the lawn is 35m2; how wide is the lawn?

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In the figure, PQR is a semicircle. Calculate the area of the shaded region.

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If the heights of two circular cylinder are in the ratio 2 : 3 and their volumes?

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$\frac{h_1}{h_2}$ = $\frac{2}{3}$
h2 = $\frac{2h_1}{3}$
$\frac{r_1}{r_2}$ = $\frac{9}{8}$
r2 = $\frac{9r_1}{8}$
v1 = $\pi$($\frac{9r_1}{8}$)2($\frac{2h_1}{3}$)
= $\pi$r1 2h1 x $\frac{27}{32}$
v = $\frac{\pi r_{1}2h_1\times \frac{27}{32}}{\pi r_{1}2h_1}$ = $\frac{27}{32}$
v2 : v1 = 27 : 32

4, 16, 30, 20, 10, 14 and 26 are represented on a pie chart. Find the sum of the angles of the bisectors representing all numbers equals to or greater than 16.

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Given that 4, 16, 30, 20, 10, 14 and 26
Adding up = 120
nos $\ge$ 16 are 16 + 30 + 20 + 26 = 92
The requires sum of angles = $\frac{92}{120}$ x $\frac{{360}^o}{1}$
= 276o

Find the solution of the equation x - 8$\sqrt{x}$ + 15 = 0

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x - 8$\sqrt{x}$ + 15 = 0
x + 15 = 8$\sqrt{x}$
square both sides = (x + 15)2 = (8 $\sqrt{x}$2
x2 + 225 + 30x = 64x
x2 + 225 + 30x - 64x = 0
x2 - 34x + 225 = 0
(x - 9)(x - 25) = 0
x = 9 or 25

Divide x3 - 2x2 - 5x + 6 by (x - 1)

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

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Given that x2 + y2 + z2 = 194, calculate z if x = 7 and $\sqrt{y}$ = 3

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Given that x2 + y2 + z2 = 194, calculate z if x = 7 and $\sqrt{y}$ = 3
x = 7
∴ x2 = 49
$\sqrt{y}$ = 3
∴ y2 = 81 = x2 + y2 + z2 = 194
49 + 81 + z2 = 194
130 + z2 = 194
z2 = 194 - 130
= 64
z = $\sqrt{64}$
= 8

If a2 + b2 = 16 and 2ab = 7.Find all the possible values of (a - b)

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a2 + b2 = 16 and 2ab = 7
To find all possible values = (a - b)2 + b2 - 2ab
Substituting the given values = (a - b)2
= 16 - 7
= 9
(a - b) = $\pm$9
= $\pm$3
OR a - b = 3, -3

Find the sum of the first twenty terms of the progression log a, log a2, log a3.....

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The H.C.F. of a2bx + ab2x and a2b - b2 is

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To find the H.C.F. of the given terms, we need to factorize them first: a²bx + ab²x = abx(a + b) a²b - b² = b(a - b) The H.C.F. of two or more terms is the product of their common factors raised to the lowest power. Both the terms have a common factor of 'b', so the H.C.F. will have 'b' as a factor. Also, the first term has a common factor of 'abx', and the second term does not have any factor of 'abx'. So, the H.C.F. will not have any factor of 'abx'. Therefore, the H.C.F. of a²bx + ab²x and a²b - b² is 'b'. Hence, the correct answer is: b.

What is the equation of the quadratic function represented by the graph?

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The required equation is y = x2 - x - 2

i.e. B where the graph touches the graph touches the x-axis y = 0

x2 - x - 2 = 0 = (x + 1)(x - 2) = 0

Hence roots of the equation are -1 and 2 as shown in the graph

Simplify $\sqrt{27}$ + $\frac{3}{\sqrt{3}}$

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Each of the interior angles of a regular polygon is 140o. How many sides has the polygon?

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To find out the number of sides of a regular polygon when given the measure of its interior angles, we can use the formula: Interior angle = (n-2) x 180 / n Where 'n' is the number of sides of the polygon. In this case, we know that each interior angle of the polygon measures 140 degrees. Substituting this value into the formula, we get: 140 = (n-2) x 180 / n Simplifying this equation, we get: 140n = (n-2) x 180 140n = 180n - 360 360 = 40n n = 9 Therefore, the regular polygon has 9 sides. Thus, the correct answer is 9.

Three children shared a basket of mangoes in such a way that the first child took $\frac{1}{4}$ of the mangoes and the second $\frac{3}{4}$ of the remainder. What fraction of the mangoes did the third child take?

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You can use any whole numbers (eg. 1. 2. 3) to represent all the mangoes in the basket.
If the first child takes $\frac{1}{4}$ it will remain 1 - $\frac{1}{4}$ = $\frac{3}{4}$
Next, the second child takes $\frac{3}{4}$ of the remainder
which is $\frac{3}{4}$ i.e. find $\frac{3}{4}$ of $\frac{3}{4}$
= $\frac{3}{4}$ x $\frac{3}{4}$
= $\frac{9}{16}$
the fraction remaining now = $\frac{3}{4}$ - $\frac{9}{16}$
= $\frac{12?9}{16}$
= $\frac{3}{16}$

Below are the scores of a group of students in a test:

$\begin{array}{ccccccc}Scores& 1& 2& 3& 4& 5& 6\\ \text{No. of students}& 1& 4& 5& 6& x& 2\end{array}$

If the average scores is 3.5, find the value of x.

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$\begin{array}{ccccccc}Scores& 1& 2& 3& 4& 5& 6\\ \text{No. of students}& 1& 4& 5& 6& x& 2\end{array}$
Average = 3.5
3.5 = $\frac{(1\times 1)+(2\times 4)+(3\times 5)+(4\times 6)+5x+(6\times 2)}{1+4+5+6+x+2}$
$\frac{3.5}{1}$ = $\frac{1+8+15+24+5x+12}{18+x}$
$\frac{3.5}{1}$ = $\frac{60+5x}{18+x}$
60 + 5x = 3.5(18 $÷$ x)
60 + 5x = = 63 + 1.5x
5x - 1.5x = 63 - 60
1.5x = 3
x = $\frac{3}{1.5}$
$\frac{30}{15}$ = 2

Simplify and express in standard form $\frac{0.00275\times 0.0064}{0.025\times 0.08}$

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To simplify and express in standard form the expression: $$\frac{0.00275 \times 0.0064}{0.025 \times 0.08}$$ We can start by simplifying the numerator and denominator: $$\frac{0.00275 \times 0.0064}{0.025 \times 0.08} = \frac{0.0000176}{0.002}$$ Then, we can simplify the fraction by dividing both the numerator and denominator by the greatest common factor of the two: $$\frac{0.0000176}{0.002} = \frac{8.8}{1000} = \frac{22}{2500}$$ Finally, we can express this fraction in standard form by writing it as a number between 1 and 10 multiplied by a power of 10: $$\frac{22}{2500} = 0.0088 = 8.8 \times 10^{-3}$$ Therefore, the expression $\frac{0.00275 \times 0.0064}{0.025 \times 0.08}$ simplified and expressed in standard form is $8.8 \times 10^{-3}$.

An arc of length 22cm subtends an angle of θ at the center of the circle. What is the value of θ if the radius of the circle is 15cm?[Take π = 22/7]

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We can use the formula for the length of an arc of a circle given by: L = rθ where L is the length of the arc, r is the radius of the circle, and θ is the angle subtended by the arc at the center of the circle in radians. In this problem, we are given L = 22cm and r = 15cm, and we need to find θ. We are also given that π = 22/7. Using the formula above and substituting the given values, we can solve for θ: 22 = 15θ θ = 22/15 radians To convert this to degrees, we can multiply by 180/π: θ = (22/15) * (180/22/7) θ = 84 degrees (rounded to the nearest degree) Therefore, the answer is: - 84o

Simplify $\frac{1}{x-2}$ + $\frac{1}{x+2}$ + $\frac{2x}{x^2-4}$

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$\frac{1}{x-2}$ + $\frac{1}{x+2}$ + $\frac{2x}{x^2-4}$
= $\frac{(x+2)+(x-2)+2x}{(x+2)(x-2)}$
= $\frac{4x}{x^2-4}$

At what rate would a sum of ₦100.00 deposited for 5 years raise an interest of ₦7.50?

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To calculate the interest rate that would raise a sum of ₦100.00 deposited for 5 years to an interest of ₦7.50, we need to use the formula: Simple Interest = (Principal * Rate * Time) / 100 Where: - Principal = ₦100.00 - Simple Interest = ₦7.50 - Time = 5 years By substituting the values in the formula, we get: ₦7.50 = (₦100.00 * Rate * 5) / 100 Simplifying the equation, we get: Rate = (₦7.50 * 100) / (₦100.00 * 5) Rate = 1.5% Therefore, the interest rate that would raise a sum of ₦100.00 deposited for 5 years to an interest of ₦7.50 is 1.5%.

The mean of ten positive numbers is 16. When another number is added, the mean becomes 18. Find the eleventh number

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The mean of ten positive numbers is given to be 16. This means that the sum of these ten numbers is 10 times 16, which is 160. When an eleventh number is added, the mean becomes 18. Let's call this number "x". Now we have a total of eleven numbers, and their sum is 11 times 18, which is 198. We can find the eleventh number by subtracting the sum of the first ten numbers (160) from the sum of all eleven numbers (198): x = 198 - 160 = 38 Therefore, the eleventh number is 38, which is.

At what value of x is the function x2 + x + 1 minimum?

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To find the value of `x` that minimizes the function `x^2 + x + 1`, we need to find the vertex of the parabola represented by this function. The vertex of a parabola of the form `ax^2 + bx + c` is given by `(-b/2a, f(-b/2a))`. In this case, `a = 1`, `b = 1`, and `c = 1`, so the vertex is located at `(-1/2, f(-1/2))`. To find `f(-1/2)`, we substitute `-1/2` for `x` in the function: `f(-1/2) = (-1/2)^2 + (-1/2) + 1 = 3/4` Therefore, the vertex of the parabola is `(-1/2, 3/4)`. Since the parabola opens upwards (the coefficient of `x^2` is positive), the value of `x` that minimizes the function is the x-coordinate of the vertex, which is `-1/2`. Therefore, the answer is `1`.

If sin x = cos 50o, then x equals

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To solve this problem, we need to use the fact that the sine and cosine functions are related to each other through the unit circle. First, we need to find the angle whose cosine is equal to cos 50°. We can do this by drawing a unit circle and marking the angle 50° on the circle. Then, we draw a line from the center of the circle to the point on the circumference that corresponds to the angle 50°. This line will intersect the x-axis at a point that has the same x-coordinate as the cosine of 50°. Next, we need to find the angle whose sine is equal to this x-coordinate. We can do this by drawing a perpendicular line from the point on the circumference to the y-axis. This line will intersect the y-axis at a point that has the same y-coordinate as the sine of the angle we are looking for. Finally, we can find the angle by using the inverse sine function (also known as arcsine) to find the angle whose sine is equal to this y-coordinate. Using this method, we find that the angle x is approximately 40.5°, which is closest to option (A) 40°. Therefore, the answer is (A) 40°.

Simplify 3 log69 + log612 + log664 - log672

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3 log69 + log612 + log664 - log672
= log693 + log612 + log664 - log672
log6729 + log612 + log664 - log672
log6(729 x 12 x 64) = log6776
= log665 = 5 log66 = 5
N.B: log66 = 1

Find the sum of the first 18 terms of the progression 3, 6, 12......

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3 + 6 + 12 + .....18thy term
1st term = 3, common ratio $\frac{6}{3}$ = 2
n = 18, sum og GP is given by Sn = a$\frac{(r^n-1)}{r-1}$
s18 = 3$\frac{(2^{18}-1)}{2-1}$
= 3(217 - 1)

A sector of a circle of radius 7cm has an area of 44cm2. Calculate the angle of the sector correct to the nearest degree [Take π = 22/7]

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The area of a sector of a circle is given by the formula A = (θ/360)πr^2, where A is the area of the sector, r is the radius of the circle, and θ is the central angle of the sector in degrees. We are given that the radius of the circle is 7cm and the area of the sector is 44cm^2, so we can plug these values into the formula: 44 = (θ/360)(22/7)(7)^2 Simplifying this expression, we get: 44 = (θ/360)(22)(7) 44 = (77θ/180) θ = (180/77)(44) θ ≈ 102.6 degrees Therefore, the angle of the sector is approximately 103 degrees to the nearest degree, which is.

In the figure, YXZ = 30${}^{\circ }$, XYZ = 105${}^{\circ }$ and XY = 8cm. Calculate YZ

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To find the length of YZ, we can use the sine rule which relates the sides and angles of a triangle. The sine rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle. In other words: a b c --- --- --- = 1 sinA sinB sinC where a, b, and c are the lengths of the sides of the triangle opposite the angles A, B, and C, respectively. Using the sine rule, we can find the length of YZ as follows: YZ XY XY -- = --- = --- sinX sinZ sinY Substituting the given values, we get: YZ 8 8 -- = --- = --- sin30 sin105 sin45 Using a calculator to evaluate the sines, we get: YZ 8 8 -- = --- = --- × (2 + √2) ≈ 8.485 0.5 0.966 0.707 Therefore, YZ is approximately equal to 8.485 cm, which is equivalent to 4√2 cm to two decimal places. Therefore, the correct option is: 4√2 cm.

Factorize 9(x + y)2 - 4(x - y)2

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9(x + y)2 - 4(x - y)2
Using difference of two squares which says
a2 - b2 = (a + b)(a - b) = 9(x + y)2 - 4(x - y)2
= [3(x + y)]2 - [2(x - y)]-2
= [3(x + y) + 2(x - y) + 2(x - y)][3(x + y) - 2(x - y)]
= [3x +3y + 2x - 2y][3x + 3y - 2x + 2y]
= (5x + y)(x + 5y)

If f(x - 4) = x2 + 2x + 3, Find, f(2)

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f(x - 4) = x2 + 2x + 3
To find f(2) = f(x - 4)
= f(2)
x - 4 = 2
x = 6
f(2) = 62 + 2(6) + 3
= 36 + 12 + 3
= 51

In triangles XYZ and XQP, XP = 4cm, XQ = 5cm and PQ = QY = 3cm. Find ZY

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