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**Question 1**
**Report**

Factorize a2−b2−4a+4

**Answer Details**

The trinomial = a2−4a+4
${a}^{2}-4a+4$

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

(a2−2a−2a+4)−b2
$({a}^{2}-2a-2a+4)-{b}^{2}$

a(a−2)−2(a−2)−b2
$a(a-2)-2(a-2)-{b}^{2}$

(a−2)2−b2
$(a-2{)}^{2}-{b}^{2}$

(a−2+b)(a−2−b)

**Question 2**
**Report**

How many sides has a regular polygon whose interior angles are 120° each?

**Answer Details**

A regular polygon is a polygon with equal sides and equal angles. The formula to find the interior angles of a regular polygon is: Interior Angle = (n - 2) x 180 / n where n is the number of sides of the polygon. We are given that the interior angles of the regular polygon are 120°. Substituting this value into the formula, we get: 120 = (n - 2) x 180 / n Simplifying this equation, we get: 120n = 180n - 360 360 = 60n n = 6 Therefore, the regular polygon has 6 sides.

**Question 3**
**Report**

If y = (x2+3x?1)÷(3x+4). $({x}^{2}+3x?1)\xf7(3x+4).$ Find dy/dx

**Answer Details**

dy/dx (4x3+3x2+2x+1)
$(4{x}^{3}+3{x}^{2}+2x+1)$

= 12x2+6x+2
$12{x}^{2}+6x+2$

⇒ 12x2+6+2
$12{x}^{2}+6+2$

Divide both sides by 2

12x22−6x2−22
$\frac{12{x}^{2}}{2}-\frac{6x}{2}-\frac{2}{2}$

= 6x2+3x+1
$6{x}^{2}+3x+1$

dy/dx

= 6x2+3x+1

**Question 5**
**Report**

Calculate the value of x and y if 27x ÷ 81x+2y = 9, x + 4y = 0

**Answer Details**

27x÷81(x+2y)=9
${27}^{x}\xf7{81}^{(x+2y)}=9$

(27)x=9×81(x+2y)
$(27)x=9\times {81}^{(x+2y)}$

(33)x=32×34(x+2y)
$({3}^{3}{)}^{x}={3}^{2}\times {3}^{4(x+2y)}$

=3(2+4x+8y)
$={3}^{(2+4x+8y)}$

33x=3(2+4x+8y)
${3}^{3x}={3}^{(2+4x+8y)}$

3x=2+4x+8y
$3x=2+4x+8y$

3x−4x−8y=2........(1)
$3x-4x-8y=2........(1)$

x+4y=0........(2)
$x+4y=0........(2)$

−4y=2
$-4y=2$

y=(−2)÷4=−12
$y=(-2)\xf74=-\frac{1}{2}$

y=−12
$y=-\frac{1}{2}$

Substitute the value of y into equation (2)

i.e x+4y=0
$x+4y=0$

x+4(−12)=0
$x+4(-\frac{1}{2})=0$

x−2=0
$x-2=0$

x=2
$x=2$

∴x=2,y=−12
$\therefore x=2,y=-\frac{1}{2}$

Method II

27x÷31(x+2y)=9
${27}^{x}\xf7{31}^{(x+2y)}=9$

33x×3(−4x−8y)=32
${3}^{3x}\times {3}^{(-4x-8y)}={3}^{2}$

3(3x−8y)=32
${3}^{(3x-8y)}={3}^{2}$

−x−8y=2........(1)
$-x-8y=2........(1)$

x+4y=0........(2)
$x+4y=0........(2)$

−4y=2
$-4y=2$

y=−24=−12
$y=-\frac{2}{4}=-\frac{1}{2}$

y=−12
$y=-\frac{1}{2}$

Substitute the value of y into equation 2

x+4y=0
$x+4y=0$

x+4(−1)÷2)=0
$x+4(-1)\xf72)=0$

x−2=0
$x-2=0$

x=2
$x=2$

x=2,y=−12

**Question 6**
**Report**

Simplify (0.09)2 and give your answer correct to 4 significant figures

**Answer Details**

(0.09)2 = 0.09 × 0.09

= 0.0081

= 0.008100 to 4 significant figures

Please, start counting first from the non-zero digits i.e. 8

**Question 7**
**Report**

The area of an ellipse is 132cm2. The length of its major axis is 14cm. Find the length of it minor axis.

**Answer Details**

The area of an ellipse (e) = πab4
$\frac{{\textstyle \pi ab}}{{\textstyle 4}}$

Let a rep. the length of its major axis = 14cm

Let b rep. the length of its minor axis = ?

π = 3.142

Area of an ellipse = 132cm2

132 = (π14 × b) ÷ 4

132 = (3.142 × 14b) ÷ 4

3.142 × 14b =132 × 4

b = (1324 × 4) ÷ (3.142 × 14)

= 528/43.988

= 12cm

The length of its minor axis (b) = 12cm

**Question 9**
**Report**

Solve x2 − 2x − 3 = 0

**Answer Details**

To solve the equation x^2 - 2x - 3 = 0, we can use the quadratic formula which is: x = (-b ± sqrt(b^2 - 4ac)) / 2a where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In this case, a = 1, b = -2, and c = -3. Substituting these values into the formula, we get: x = (-(-2) ± sqrt((-2)^2 - 4(1)(-3))) / 2(1) Simplifying this expression, we get: x = (2 ± sqrt(16)) / 2 x = (2 ± 4) / 2 So, x can be either (2 + 4)/2 = 3 or (2 - 4)/2 = -1. Therefore, the answer is: x = 3 or -1.

**Question 11**
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If an investor invest ₦450,000 in a certain organization in order to yield X as a return of ₦25,000. Find the return on an investment of ₦700,000 by Y in the same organization.

**Answer Details**

To solve the problem, we need to use the concept of proportionality. We can set up a proportion between the return and the investment amount: Return/Investment = X/₦450,000 We can rearrange this to solve for X: X = (Return * ₦450,000) / Investment Now, we can use this equation to find the return Y on an investment of ₦700,000: Y = (X * ₦700,000) / ₦450,000 Substituting the given values, we get: X = (₦25,000 * ₦450,000) / ₦450,000 = ₦25,000 Y = (₦25,000 * ₦700,000) / ₦450,000 = ₦38,888.89 (rounded to ₦38,888.90K) Therefore, the answer is option D: ₦38,888.90K.

**Question 12**
**Report**

Factorize x2−2x−15

**Answer Details**

The expression **x^2 - 2x - 15** can be factored into the product of two binomials. To find these binomials, we can use the "**ac method**," where we try to find two numbers that multiply to -15 and add up to -2. These numbers are **-5** and **3**. We can then write the expression as:

```
(x - 5)(x + 3)
```

So, the expression **x^2 - 2x - 15** can be factored as **(x - 5)(x + 3)**.

**Question 13**
**Report**

Integral ∫(5x3+7x2−2x+5) $(5{x}^{3}+7{x}^{2}-2x+5)$dx

**Answer Details**

∫[(x2+4x2+1)÷x2]dx=∫(x3/x2)dx+∫(1/x2)dx
$\int [({x}^{2}+4{x}^{2}+1)\xf7{x}^{2}]dx=\int ({x}^{3}/{x}^{2})dx+\int (1/{x}^{2})dx$

⇒∫xdx+∫4−1or+∫(1/x2)dx
$\int xdx+\int {4}^{-1}or+\int (1/{x}^{2})dx$

= x2/2−4x−1/x2+C

**Question 14**
**Report**

Simplify 4 √[(390695T− 8)½)]

**Answer Details**

4√(390625T − 8)½

[(390625T − 8)½]¼

[((390625)½ × T − 8)½]¼

(√ 390625 × T − 4 )¼

(625 × T −¼)¼

(54 × T −4)¼

= (54)¼ × (T − 4 )¼

= 5 × T − 1

= 5 × 1/T

= 5/T

[(390625T − 8)½]¼

[((390625)½ × T − 8)½]¼

(√ 390625 × T − 4 )¼

(625 × T −¼)¼

(54 × T −4)¼

= (54)¼ × (T − 4 )¼

= 5 × T − 1

= 5 × 1/T

= 5/T

**Question 15**
**Report**

Find the simple interest on ₦325 in 5years at 3% per annum.

**Answer Details**

The formula for finding the simple interest on a principal amount **P** for a period of **t** years at an interest rate of **r** per annum is:

```
simple interest = P * r * t / 100
```

In this case, **P** = 325, **t** = 5, and **r** = 3, so we can plug in the values into the formula:

```
simple interest = 325 * 3 * 5 / 100
simple interest = 48.75
```

So, the simple interest on ₦325 in 5 years at 3% per annum is ₦48.75.

**Question 16**
**Report**

A man sells his new brand car for ₦420,000 at a gain of 15%. What did it cost him?

**Answer Details**

If the man sold the car for ₦420,000 at a gain of 15%, then we can find the cost price of the car using the following formula: Selling price = Cost price + Profit We know the selling price is ₦420,000 and the profit is 15% of the cost price. Let's call the cost price "x." Profit = 15% of x = 0.15x Substituting the values in the formula: ₦420,000 = x + 0.15x ₦420,000 = 1.15x To solve for x, we need to divide both sides by 1.15: x = ₦420,000 / 1.15 x = ₦365,217.39 Therefore, the cost price of the car was ₦365,217.39. So, option B - ₦365,217 is the correct answer.

**Question 17**
**Report**

Simplify 6112−2¾+1½

**Answer Details**

6½ − 2¾ + 1½

73/12 − 11/4 + 3/2

[(73 − 33 + 18) ÷ (12)]

58/12

29/6

456

**Question 18**
**Report**

If duty is levied at 25%, find the duty to be added to a bill of ₦80.

**Answer Details**

The correct answer is ₦20. To find the duty to be added to a bill of ₦80, we need to calculate 25% of ₦80. 25% means 25 out of 100, so we can calculate 25% of ₦80 by multiplying ₦80 by 25/100. ₦80 * 25/100 = ₦20 So the duty to be added to a bill of ₦80 is ₦20.

**Question 19**
**Report**

Find x if 132x = 70 eight.

**Answer Details**

132x = 708

1×x2+3×x1+2×x0
$1\times {x}^{2}+3\times {x}^{1}+2\times {x}^{0}$

7×81+0×80
$7\times {8}^{1}+0\times {8}^{0}$

x2+3x+2−56=0
${x}^{2}+3x+2-56=0$

x2+3x−54=0
${x}^{2}+3x-54=0$

x(x + 9)−6(x + 9) = 0

(x + 9)(x − 6) = 0

Either (x + 9) = 0 or (x − 6) = 0

x = − 9 or +6

The positive values for x = 6

The base number for 132x = 1326

**Question 20**
**Report**

The area of a circle of radius 4cm is equal to (Take π = 3.142 )

**Answer Details**

Area of a circle (A) = πr2

Radius = 4cm

π = 3.142

A = 3.142 × (4)2

= 3.142 × 16

= 50.272

A = 50.3cm2 (1dp)

**Question 22**
**Report**

A man bought a car for ₦800 and sold it for ₦520. Find his loss per cent

**Answer Details**

The loss percentage can be calculated as follows: Loss percentage = (Loss / Cost price) x 100 Here, the cost price of the car is ₦800, and the selling price is ₦520. Since the selling price is less than the cost price, the man has incurred a loss. Loss = Cost price - Selling price Loss = ₦800 - ₦520 Loss = ₦280 Substituting the values in the formula: Loss percentage = (280 / 800) x 100 Loss percentage = 0.35 x 100 Loss percentage = 35% Therefore, the man incurred a loss of 35%.

**Question 23**
**Report**

If √(2x + 2) − √x = 1, find x.

**Answer Details**

To solve for x in the equation √(2x + 2) − √x = 1, we can use algebraic manipulation to isolate x on one side of the equation. Here's how: 1. Start by adding √x to both sides of the equation: √(2x + 2) = √x + 1 2. Square both sides of the equation to eliminate the square root: (√(2x + 2))^2 = (√x + 1)^2 Simplifying the left side: 2x + 2 = x + 1 + 2√x + 1 3. Rearrange terms: x - 2√x + 1 = 0 4. Factor the left side of the equation: (√x - 1)^2 = 0 5. Solve for x by taking the square root of both sides: √x - 1 = 0 √x = 1 x = 1^2 = 1 Therefore, the solution to the equation √(2x + 2) − √x = 1 is x = 1.

**Question 24**
**Report**

A pipe made of metal 10cm thick has an external radius of 11cm. find the area of metal in making 2.4cm of pipe

**Answer Details**

The external radius = 11cm

The internal radius = 10cm

The area of cross section = π(π2 − 102)

= π (11 + 10)(11 − 10)

= π(21)(1)

= 21πcm2

The internal radius = 10cm

The area of cross section = π(π2 − 102)

= π (11 + 10)(11 − 10)

= π(21)(1)

= 21πcm2

**Question 25**
**Report**

The probability of an event A is 1/5. The probability of B is 1/3 . The probability both A and B is 1/15. What is the probability of either event A or B or both?

**Answer Details**

The probability of either event A or B or both can be calculated using the formula for the union of two events: P(A or B) = P(A) + P(B) - P(A and B) where P(A) is the probability of event A, P(B) is the probability of event B, and P(A and B) is the probability of both events happening at the same time. Plugging in the given probabilities, we get: P(A or B) = 1/5 + 1/3 - 1/15 = 7/15 So, the probability of either event A or B or both is 7/15.

**Question 26**
**Report**

The sum of two numbers is 5; their product is 14. Find the numbers.

**Answer Details**

Let x represent the first number;

Then, the other is (5 − x) , since their sum is 5 and their product is 14

x(5 − x) = 14

5x − x2 = 14

x2 − 5x − 14 = 0

(x2 − 7x) + (2x − 14) = 0

x(x − 7) + 2(x − 7) = 0

(x − 7)(x + 2) = 0

Either (x − 7) = 0 or (x + 2) = 0

x = 7 or x = − 2

The two numbers are 7 and − 2

**Question 27**
**Report**

A public car dealer marked up the cost of a car at 30% in an attempt to make 20% gross profit. Due to the value of dollar, he now placed 20% discount on the car. What profit or loss will he make?

**Answer Details**

Let assume the cost price is 100%

Marked up price + cost price = 20 + 100 = 120%

Discount at 20% = 20/100 × 120 % of cost price

Selling price = cost price − gain

= (120 − 24)% of cost price

= 96% of cost price

Loss = (100− 96)% of cost price

= 4% of cost price

∴ He will make a 4% loss.

**Question 28**
**Report**

Simplify 2.04 × 3.7 (Leave your answer in 2 decimal place)

**Answer Details**

To simplify the expression 2.04 × 3.7, we can use the distributive property of multiplication over addition. That is: 2.04 × 3.7 = (2 + 0.04) × 3.7 We can now expand this expression as: (2 + 0.04) × 3.7 = 2 × 3.7 + 0.04 × 3.7 Simplifying further: 2 × 3.7 = 7.4 0.04 × 3.7 = 0.148 So, substituting these values back into the expression: 2.04 × 3.7 = 7.4 + 0.148 = 7.548 Rounding this answer to two decimal places, we get: 2.04 × 3.7 = 7.55 Therefore, the answer is option D - 7.55.

**Question 29**
**Report**

Find the area of the curved surface of a cone whose base radius is 3cm and whose height is 4cm (π = 3.14)

**Answer Details**

To find the curved surface area of a cone, we need to use the formula: `Curved Surface Area = πrs` where `r` is the radius of the base of the cone and `s` is the slant height of the cone. To find the slant height of the cone, we can use the Pythagorean theorem, which tells us that: `h^2 + r^2 = s^2` where `h` is the height of the cone. Substituting `r = 3cm` and `h = 4cm`, we get: `4^2 + 3^2 = s^2` `16 + 9 = s^2` `25 = s^2` `sqrt(25) = s` `s = 5cm` Now, substituting `r = 3cm` and `s = 5cm` into the formula for curved surface area, we get: `Curved Surface Area = πrs` `= 3.14 × 3 × 5` `= 47.1cm^2` Therefore, the curved surface area of the cone is `47.1cm^2`. Hence, option C, "47.1cm^2" is the correct answer.

**Question 30**
**Report**

Given that S and T are sets of real numbers such that S = {x : 0 ≤ $\le $ x ≤ $\le $ 5} and T = {x :− 2 < x < 3} Find S ∪ $\cup $ T

**Answer Details**

S = {0, 1, 2, 3, 4, 5}

T = {− 1, 0, 1, 2}

S ∪
$\cup $ T = {− 1, 0, 1, 2, 3, 4, 5 }

⇒ − 2 < x ≤ 5

**Question 31**
**Report**

The volume of a cone (s) of height 6cm and base radius 5cm is

**Answer Details**

The formula for the volume of a cone is given by: V = (1/3)πr²h where V is the volume, r is the radius of the base of the cone, h is the height of the cone, and π is a constant value of approximately 3.14. Substituting the given values into the formula, we have: V = (1/3)π(5cm)²(6cm) Simplifying this expression, we get: V = (1/3)π(25cm²)(6cm) V = (1/3)π(150cm³) V = 50π cm³ Using a calculator to approximate π to two decimal places (π ≈ 3.14), we have: V ≈ 157 cm³ Therefore, the correct option is: 157cm3

**Question 32**
**Report**

Determine the third term of a geometrical progression whose first and second term are 2 and 14 respectively

**Answer Details**

1st G.P. = a =2

2nd G.P. = ar − 1 = 54

2(r) = 54

r = 54/2 = 27

r = 27

3rd term = ar2 = (2) (27)2

2 × 27 × 27

= 1458

**Question 33**
**Report**

Simplify 1(x+1)+1(x−1)

**Answer Details**

[1 ÷ (x+1)] + [1 ÷ (x − 1)]

= ((x − 1) + [(x + 1)) ÷ (x+1)(x − 1)]

Using the L.C.M.

= (x − 1 + x + 1) ÷ (x + 1)(x − 1)

= (x + 2 − 1 + 1) ÷ (x + 1)(x − 1)

= 2x ÷ (x + 1)(x − 1) =2x ÷ (x + 1)(x − 1)

**Question 34**
**Report**

Simplify √30 × √40

**Answer Details**

√30 × √40

= √(30 × 40)

= √(3 × 10 × 4 × 10)

= √(400 × 3)

= √400 × √3

= 20 × √3

= 20√3

**Question 35**
**Report**

Simplify [1÷(x2+3x+2)]+[1÷(x2+5x+6)]

**Answer Details**

[1÷(x2+3x+2)]+[1÷(x2+5x+6)]
$[1\xf7({x}^{2}+3x+2)]+[1\xf7({x}^{2}+5x+6)]$

= 1÷(x2+3x+2)+[1÷(x2+5x+6)]
$1\xf7({x}^{2}+3x+2)+[1\xf7({x}^{2}+5x+6)]$

= [1÷((x2+x)+(2x+2))]+[1÷((x2+3x)+(2x+6))]
$[1\xf7(({x}^{2}$