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**Question 1**
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The bar chart above shows the distribution of marks in a class test. If the pass mark is 5, what percentage of students failed the test?

**Answer Details**

**Question 2**
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If y = x2 - 1x $\frac{1}{x}$, find δyδx

**Answer Details**

To find δyδx (the derivative of y with respect to x), we need to apply the power rule of differentiation, which states that if y = x^n, then δyδx = n*x^(n-1). Applying this rule to y = x^2 - 1/x, we get: δyδx = 2x + 1/x^2 Therefore, the answer is not one of the options provided. Option A (2x - 1/x^2) is close but has a minus sign instead of a plus sign before the second term. Option B (2x + x^2) and option C (2x - x^2) are incorrect because they don't take into account the derivative of the second term (-1/x). Option D (2x + 1/x^2) is the correct answer based on the power rule of differentiation.

**Question 3**
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The sum to infinity of a geometric progression is −110
$-\frac{1}{10}$ and the first term is −18
$-\frac{1}{8}$. Find the common ratio of the progression.

**Answer Details**

Sr = a1−r
$\frac{a}{1-r}$

−110
$-\frac{1}{10}$ = 18×11−r
$\frac{1}{8}\times \frac{1}{1-r}$

−110
$-\frac{1}{10}$ = 18(1−r)
$\frac{1}{8(1-r)}$

−110
$-\frac{1}{10}$ = 18−8r
$\frac{1}{8-8r}$

cross multiply...

-1(8 - 8r) = -10

-8 + 8r = -10

8r = -2

r = -1/4

**Question 4**
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U is inversely proportional to the cube of V and U = 81 when V = 2. Find U when V = 3

**Answer Details**

When two quantities, U and V, are inversely proportional, it means that as one of them increases, the other decreases, and vice versa, in such a way that their product remains constant. In mathematical terms, we can express this relationship as U*V^3 = k, where k is a constant. In this problem, we are told that U is inversely proportional to the cube of V. Therefore, we can write U*V^3 = k, where k is a constant of proportionality that we need to find. We are also given that U equals 81 when V equals 2. Substituting these values into the equation, we get: 81*2^3 = k k = 648 Now that we have the constant of proportionality, we can use the equation U*V^3 = 648 to find U when V equals 3: U*3^3 = 648 U*27 = 648 U = 648/27 Simplifying this expression, we get U = 24. Therefore, the answer is 24.

**Question 5**
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Evaluate 219 $\frac{21}{9}$ to 3 significant figures

**Answer Details**

To evaluate 21/9 to 3 significant figures, we need to round the result to the nearest thousandth. The result of 21 divided by 9 is 2.33333333333... The third digit after the decimal point is 3, which is greater than or equal to 5. Therefore, we need to round up the second digit after the decimal point, which is 3. Therefore, rounding 2.33333333333... to 3 significant figures gives us 2.33. Therefore, the correct option is 2.33.

**Question 6**
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Given that I3 is a unit matrix of order 3, find |I3|

**Answer Details**

A unit matrix is a square matrix in which all the diagonal elements are equal to 1, and all the other elements are equal to 0. The symbol I3 represents the unit matrix of order 3, which is a 3x3 matrix. So, the matrix I3 can be written as: |1 0 0| |0 1 0| |0 0 1| To find the determinant of I3, we can use the formula for the determinant of a 3x3 matrix: |a b c| |d e f| |g h i| = a(ei - fh) - b(di - fg) + c(dh - eg) Applying this formula to I3, we get: |1 0 0| |0 1 0| |0 0 1| = 1(1*1 - 0*0) - 0(0*1 - 0*0) + 0(0*1 - 0*0) = 1 Therefore, the determinant of I3 is 1, and the correct answer is option (C) 1. Options (A) -1, (B) 0, and (D) 2 are incorrect.

**Question 7**
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Evaluate n+1Cn-2 If n =15

**Answer Details**

n+1(n−2)(n+1)!
$\frac{n+1(n-2)}{(n+1)!}$

(n+1)+(n−2)!(n−2)!(n+1)!
$\frac{(n+1)+(n-2)!(n-2)!}{(n+1)!}$

(n+1)(n+1−1)(n+1−2)(n+1−3)!3!(n−2)!
$\frac{(n+1)(n+1-1)(n+1-2)(n+1-3)!}{3!(n-2)!}$

(n+1)(n)(n−1)(n−2)!3!(n−2)!
$\frac{(n+1)(n)(n-1)(n-2)!}{3!(n-2)!}$

(n+1)(n)(n−1)3!
$\frac{(n+1)(n)(n-1)}{3!}$

Since n = 15

(15+1)(15)(15−1)3!
$\frac{(15+1)(15)(15-1)}{3!}$

16×15×143×2×1
$\frac{16\times 15\times 14}{3\times 2\times 1}$

= 560

**Question 8**
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In the diagram above, |PQ| = |QR|, |PS| = |RS|, ∠PSR = 30o and ∠PQR = 80o. Find ∠SPQ.

**Answer Details**

Join PR

QRP = QPR

= 180 - 80 = 100/20 = 50o

SRP = SPR

= 180 - 30 = 150/2 = 75o

∴ SPQ = SPR - QPR

= 75 - 50 = 25o

**Question 9**
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If angle ? $?$ is 135o, evaluate cos?

**Answer Details**

θ
$\theta $ = 135o

Cos 135o = Cos(90 + 45)o

= cos90ocos45o - sin90osin45o

= 0cos45o - (1 x √22
$\frac{\sqrt{2}}{2}$)

= −√22

**Question 10**
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A man earns ₦3,500 per month out of which he spends 15% on his children's education. If he spends additional ₦1,950 on food, how much does he have left?

**Answer Details**

The man earns ₦3,500 per month and spends 15% of it on his children's education. We can calculate the amount he spends on his children's education as: 15% of ₦3,500 = (15/100) x ₦3,500 = ₦525 So, his total expenses on education and food are: ₦525 + ₦1,950 = ₦2,475 Subtracting his total expenses from his monthly income, we get his savings: ₦3,500 - ₦2,475 = ₦1,025 Therefore, the man has ₦1,025 left after spending on his children's education and food. Hence, the answer to the question is ₦1,025.

**Question 11**
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If 27x + 2 ÷ $\xf7$ 9x + 1 = 32x, find x

**Answer Details**

27x+2÷9x+1=32x
${27}^{x+2}\xf7{9}^{x+1}={3}^{2x}$

Reduce 27 and 9 to have the same base as 3

33(x+2)÷32(x+1)=32x
${3}^{3(x+2)}\xf7{3}^{2(x+1)}={3}^{2x}$

Note, - is ÷ and + is × for powers of indices of the same bases.

33(x+2)−2(x+1)=32x
${3}^{3(x+2)-2(x+1)}={3}^{2x}$

Equating the exponentials

3(x+2)−2(x+1)=2x
$3(x+2)-2(x+1)=2x$

Clear brackets

3x+6−2x−2=2x
$3x+6-2x-2=2x$

Collect like terms

3x−2x−2x=−6+2
$3x-2x-2x=-6+2$

x−2x=−4
$x-2x=-4$

−x=−4
$-x=-4$

Divide both sides by −
$-$

x=4
$x=4$

Answer is B

**Question 12**
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Find the equation of the line through the points (-2, 1) and (-12
$\frac{1}{2}$, 4)

**Answer Details**

y−y1x+x1
$\frac{y-{y}_{1}}{x+{x}_{1}}$ = y2−y1x−x1
$\frac{{y}_{2}-{y}_{1}}{x-{x}_{1}}$

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

= y−1x+2
$\frac{y-1}{x+2}$ = 332
$\frac{3}{\frac{3}{2}}$

y = 2x + 5

**Question 13**
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Find the median of 2,3,7,3,4,5,8,9,9,4,5,3,4,2,4 and 5

**Answer Details**

Arrange all the values in ascending order,

2,2,3,3,3,4,4,__4,4__,5,5,5,7,8,9,9

**Question 14**
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If log3x2 = -8, what is x?

**Answer Details**

We can start by using the definition of logarithms to rewrite the equation: log3(x^2) = -8 This means that 3 raised to the power of -8 is equal to x^2: 3^(-8) = x^2 To solve for x, we can take the square root of both sides: sqrt(3^(-8)) = sqrt(x^2) On the left side, we can simplify the expression using the rule that says sqrt(a^b) = a^(b/2): 3^(-8/2) = x Simplifying the exponent, we get: 3^(-4) = x Recall that a negative exponent means the reciprocal of the corresponding positive exponent. So: 3^(-4) = 1/3^4 Using the exponent rule that says a^b = a*a*a*...*a (b times), we get: 1/81 = x Therefore, the correct answer is option (D) 181.

**Question 15**
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Convert 726 to a number in base three

**Answer Details**

To convert 726 to base three, we need to find the largest power of three that is less than or equal to 726. In this case, 3^6 = 729, which is greater than 726, so we need to use powers of three lower than 729. First, we divide 726 by 243, which is 3^5. The quotient is 2 and the remainder is 240. Next, we divide the remainder 240 by 81, which is 3^4. The quotient is 2 and the remainder is 78. Continuing in this way, we divide 78 by 27, which is 3^3. The quotient is 2 and the remainder is 24. Then, we divide 24 by 9, which is 3^2. The quotient is 2 and the remainder is 6. Finally, we divide 6 by 3, which is 3^1. The quotient is 2 and the remainder is 0. Therefore, the base-three representation of 726 is 222202, which corresponds to the last option (1122).

**Question 17**
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The gradient of the straight line joining the points P(5, -7) and Q(-2, -3) is

**Answer Details**

PQ = y1−y0x1−x0 $\frac{{y}_{1}-{y}_{0}}{{x}_{1}-{x}_{0}}$ = −3−(−7)−2−5 $\frac{-3-(-7)}{-2-5}$ = −3+7−2−5 $\frac{-3+7}{-2-5}$ = 4−7

**Question 18**
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The grades of 36 students in a test are shown in the pie chart above. How many students had excellent?

**Answer Details**

Angle of Excellent

= 360 - (120+80+90)

= 360 - 290

= 70∘
$\circ $

If 360∘
$\circ $ represents 36 students

1∘
$\circ $ will represent 36/360

50∘
$\circ $ will represent 36/360 * 70/1

= 7

**Question 19**
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Evaluate ∫31(X2−1)dx

**Answer Details**

∫31(x2−1)dx=[13x2−x]31=(9−3)−(13−1)=6−(−23)=6+23=623

**Question 20**
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In the diagram above, PQR is a circle centre O. If < QPR is XO, find < QRP.

**Answer Details**

**Question 21**
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Calculate the volume of a cuboid of length 0.76cm, breadth 2.6cm and height 0.82cm.

**Answer Details**

Volume of cuboid = L x b x h

= 0.76cm x 2.6cm x 0.82cm

= 1.62cm3

**Question 22**
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The mean of seven numbers is 96. If an eighth number is added, the mean becomes 112. Find the eighth number.

**Answer Details**

The mean of seven numbers is the sum of all seven numbers divided by 7. If the mean is 96, then the sum of the seven numbers is 96 * 7 = 672. When an eighth number is added, the mean becomes 112, which means that the sum of all eight numbers is 112 * 8 = 896. To find the eighth number, we can subtract the sum of the first seven numbers from the sum of all eight numbers: Eighth number = 896 - 672 = 224. So, the eighth number is 224.

**Question 23**
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Make 'n' the subject of the formula if w = v(2+cn)1-cn

**Answer Details**

w = v(2+cn)1−cn
$\frac{v(2+cn)}{1-cn}$

2v + cnv = w(1 - cn)

2v + cnv = w - cnw

2v - w = -cnv - cnw

Multiply through by negative sign

-2v + w = cnv + cnw

-2v + w = n(cv + cw)

n = −2v+wcv+cw
$\frac{-2v+w}{cv+cw}$

n = 1c−2v+wv+w
$\frac{1}{c}\frac{-2v+w}{v+w}$

Re-arrange...

n = 1cw−2vv+w

**Question 24**
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If cos(x + 40)o = 0.0872, what is the value of x?

**Answer Details**

cos(x + 40)o = 0.0872

x + 40 = cos-10.0872

x + 40o = 84.99o

x = 84.99o - 40o

x = 44.99

x = 45o

**Question 25**
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If P is a set of all prime factors of 30 and Q is a set of all factors of 18 less than 10, find P ∩ $\cap $ Q

**Answer Details**

The set P contains all the prime factors of 30, which are 2, 3, and 5. The set Q contains all the factors of 18 less than 10, which are 1, 2, and 3. Therefore, the intersection of sets P and Q (denoted by the symbol ∩) is the set of factors that are both prime factors of 30 and less than 10, which is {2, 3}. Thus, the answer is: {2, 3}.

**Question 27**
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Find the remainder when 2x3 - 11x2 + 8x - 1 is divided by x + 3

**Answer Details**

To find the remainder when a polynomial is divided by another polynomial, we can use the polynomial long division method. The steps for polynomial long division are as follows: 1. Divide the highest degree term of the dividend (2x^3 in this case) by the highest degree term of the divisor (x+3) and write the result above the long division bracket. 2. Multiply the divisor (x+3) by the quotient obtained in step 1 and write the result below the dividend. 3. Subtract the result obtained in step 2 from the dividend and write the remainder below. 4. Bring down the next term of the dividend and repeat steps 1-3 until there are no more terms to bring down. Using this method, we can divide 2x^3 - 11x^2 + 8x - 1 by x + 3 to find the remainder: 2x^2 - 17x + 51 ______________________ x + 3 | 2x^3 - 11x^2 + 8x - 1 2x^3 + 6x^2 ____________ -17x^2 + 8x -17x^2 - 51x ____________ 59x - 1 59x + 177 ________ -178 Therefore, the remainder when 2x^3 - 11x^2 + 8x - 1 is divided by x + 3 is -178.

**Question 28**
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The angles of a polygon are given by x, 2x, 3x, 4x and 5x respectively. Find the value of x.

**Answer Details**

The sum of the interior angles of a polygon can be found using the formula: (n - 2) * 180°, where n is the number of sides of the polygon. In this case, the polygon has 5 sides, so the sum of its interior angles is (5 - 2) * 180° = 540°. Let's call the value of x "a". The sum of the angles of the polygon is given by: x + 2x + 3x + 4x + 5x = 15x Since the sum of the angles is 540°, we have: 15a = 540 Dividing both sides by 15, we get: a = 36 So, the value of x, or the common difference between the angles of the polygon, is 36°

**Question 29**
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Find the range of 4,9,6,3,2,8,10 and 11

**Answer Details**

The range of a set of numbers is the difference between the largest and smallest values in the set. To find the range of the set {4, 9, 6, 3, 2, 8, 10, 11}, we need to find the largest and smallest values in the set and then subtract the smallest value from the largest value. The smallest value in the set is 2 and the largest value is 11. So, the range is 11 - 2 = 9. So, the range of the set {4, 9, 6, 3, 2, 8, 10, 11} is 9.

**Question 30**
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In how many ways can the letters of the word TOTALITY be arranged?

**Answer Details**

8!3!
$\frac{8!}{3!}$

8×7×6×5×4×3×2×13×2×1