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**Question 1**
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In the diagram, PQRS is a circle center O. PQR is a diameter and ∠PRQ = 40o. Calculate ∠QSR.

**Question 2**
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A binary operation * is defined by a*b = ab. If a*2 = 2-a, find the possible values of a.

**Answer Details**

a * b = ab

a * 2 = a2 = 2 - a

a2 + 2a - a - 2 = 0

a(a+2) - 1(a+2) = 0

(a-1)(a+2) = 0

a = 1, -2

**Question 3**
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The cumulative frequency curve represents the ages of ages of students in a school. What age group do 70% of the students belongs?

**Question 4**
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A predator moves in a circle of radius √2 centre (0,0), while a prey moves along the line y = x. If 0 ≤ $\le $ x ≤ $\le $ 2, at which point(s) will they meet?

**Answer Details**

The predator moves in a circular path with a radius of √2 centered at the origin (0,0), while the prey moves along the line y = x. The prey's movement is restricted between x = 0 and x = 2. At some point, the predator will catch up to the prey, and they will meet. The predator will catch the prey when the distance between them is less than or equal to √2. We can find the equation of the circle centered at (0,0) with a radius of √2: x^2 + y^2 = (√2)^2 x^2 + y^2 = 2 We can substitute y = x into the equation to get: x^2 + x^2 = 2 2x^2 = 2 x^2 = 1 x = ±1 Therefore, the possible points of intersection are (1,1) and (-1,-1), but we know that the prey can only be at x values between 0 and 2, so the only possible point of intersection is (1,1). Therefore, the answer is (1,1) only.

**Question 5**
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If the mean of the numbers 0, (x+2), (3x+6), and (4x+8) is 4, find their mean deviation.

**Answer Details**

The mean deviation of a set of numbers is the average of the absolute deviations from their mean. To find the mean deviation of the given set of numbers, we need to first find their mean. Mean = (0 + (x+2) + (3x+6) + (4x+8))/4 Mean = (8x + 16)/4 Mean = 2x + 4 We are given that the mean is 4. Therefore, 2x + 4 = 4 2x = 0 x = 0 Substituting x = 0 in the set of numbers, we get: 0, 2, 6, 8 Mean = (0+2+6+8)/4 = 4 To find the mean deviation, we need to find the deviation of each number from the mean and then take the average of their absolute values. Deviations from the mean: 0-4 = -4 2-4 = -2 6-4 = 2 8-4 = 4 Mean deviation = (|-4| + |-2| + |2| + |4|)/4 Mean deviation = (12/4) Mean deviation = 3 Therefore, the answer is 3.

**Question 6**
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P is a point on one side of the straight line UV and P moves in the same direction as UV. If the straight line ST is on the locus of P and angle VUS = 60°, find angle UST.

**Answer Details**

Make a good sketch of the question, and follow this steps.

Since ST is parallel to UV => angle USR = 50°(alternate to VUS.

Angle UST = 180° - 50° = 130° (angle on a straight line)

**Question 7**
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In how many ways can the word MATHEMATICS be arranged?

**Answer Details**

Number of letter in MATHEMATICS = 11

Number of letter M = 2

Number of letter E = 2

Number of letter A = 2

Arrangement = 11!/(2! 2! 2!) ways

**Question 8**
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Find the value of ∫π0cos2θ−1sin2θdθ

**Question 9**
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In the diagram, POS and ROT are straight lines, OPQR is a parallelogram. |OS| = |OT| and ∠OST = 50o. Calculate ∠OPQ.

**Question 10**
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In how many ways can a delegation of 3 be chosen from 5 men and 3 women. If at least 1 man and 1 woman must be included?

**Question 11**
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3y = 4x - 1 and Ky = x + 3 are equations of two straight lines. If the two lines are perpendicular to each other, find K.

**Answer Details**

Grad of 3y = 4x - 1

y = 4x/3 - 1/3

Grad = 4/3

Grad of Ky = x + 3

y = x/k + 3/4

Grad = 1/k

Since two lines are perpendicular,

1/k = -3/4

-3k = 4

k = -4/3

**Question 12**
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In a regular polygon, each interior angle doubles its corresponding exterior angle. Find the number of sides of the polygon.

**Answer Details**

2x + x = 180°, => 3x = 180°, and thus x = 60°

Each exterior angle = 60° but size of ext. angle = 360°/n

Therefore 60° = 360°/n

n = 360°/60° = 6 sides

**Question 13**
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A man wishes to keep his money in a savings deposit at 25% compound interest so that after three years he can buy a car for ₦150,000. How much does he need to deposit?

**Answer Details**

Amount A = P(1+r)n;

A = ₦150,000, r = 25%, n = 3.

150,000 = P(1+0.25)3 = P(1.25)3

P = 150,000/1.253 =₦76,800.00

**Question 14**
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Evaluate 5-3log52 x 22log23

**Answer Details**

5-3log52 x 22log23*i* Let -3log52 = p => log52-3 = p

∴2-3 = 5p

∴5-3log52 = 5log52-3 = 5p*ii* 22log23 = q => log532 = q

∴32 = 2q

∴222log23 = 2q

= 32 = 9
*i x ii* = 2-3 x 9 = 1/23 x 9

= 1/8 x 9

= 9/8 = 11/8

**Question 15**
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If (2√3−√2)(√3+2√2)=m+n√6 $\frac{(2\sqrt{3}-\sqrt{2})}{(\sqrt{3}+2\sqrt{2})}=m+n\sqrt{6}$, find the values of m and n respectively.

**Answer Details**

Rationalize (2√3−√2)(√3+2√2) $\frac{(2\sqrt{3}-\sqrt{2})}{(\sqrt{3}+2\sqrt{2})}$ and equate to m+n√6 $m+n\sqrt{6}$. Such that m = -2, and n = 1.

**Question 16**
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Simplify 11+√5 $\frac{1}{1+\sqrt{5}}$ - 11−√5

**Answer Details**

11+√5
$\frac{1}{1+\sqrt{5}}$ - 11−√5
$\frac{1}{1-\sqrt{5}}$

= 3−√5−3−√5(3+√5)(3−√5
$\frac{3-\sqrt{5}-3-\sqrt{5}}{(3+\sqrt{5})(3-\sqrt{5}}$

= −2√59−5
$\frac{-2\sqrt{5}}{9-5}$

= −2√54
$\frac{-2\sqrt{5}}{4}$

= - 12√5

**Question 17**
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If P3446 - 23P26 = 2PP26, find the value of the digit P.

**Answer Details**

Covert everything to base 10 and collect like terms, such that:

210P - 42P = 434 + 406

168P = 840

P = 840/168 = 5

**Question 18**
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Simplify 3(2n+1) - 4(2n-1) 2n+1 - 2n

**Answer Details**

Start by expanding 3(2n+1) - 4(2n-1) 2n+1 - 2n:

3 x 2n x 21 - 22 x 2n x 2-1 2n x 2 -2n

Solving the equation above gives;

2n x 21 x 2 2n = 2 x 2 = 4

**Question 19**
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A ship sails a distance of 50km in the direction S50°E and then sails a distance of 50km in the direction ₦40°E. FInd the bearing of the ship from the original position.

**Question 20**
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Evaluate (1/2 - 1/4 - 1/8 - 1/16 + ...) - 1

**Answer Details**

The given expression is a geometric series with the first term as 1/2 and the common ratio as -1/2. Using the formula for the sum of an infinite geometric series with |r| < 1, we can find that the sum of the series is: S = a/(1-r) = (1/2)/(1-(-1/2)) = (1/2)/(3/2) = 1/3 Therefore, the expression (1/2 - 1/4 - 1/8 - 1/16 + ...) - 1 simplifies to: 1/3 - 1 = -2/3 So, the answer is -2/3.

**Question 22**
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The variance of x, 2x, 3x, 4x and 5x is

**Answer Details**

Hint: prepare a three-columned table, one for (x), another for the (deviation), and the last for the (squared-deviation).

Sum of (x) = 15x, algebraic sum of (deviation) = zero (0)

Sum of (squared-deviation) = 10x2

Variance = ∑(squared-deviation)/n = 10x2/5 = 2x2

**Question 23**
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if (x - 1), (x + 1) and (x - 2) are factors of the polynomial ax3 + bx2 + cx - 1, find a, b, c in that order.

**Answer Details**

This is a polynomial of the 3rd order, thus x should have three answers. Use the factors given to get values of x as 1, -1 and -2.

Form three equations, and carry out elimination and subsequent substitution to get a = -1/2, b = 1, and c = 1/2

**Question 24**
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A bowl is designed by revolving completely the area enclosed by y = x2 - 1, y = 3 and x ≥ 0 around the axis. What is the volume of this bowl?

**Answer Details**

∫30 π(y+1) dy = π[y2 + y30

= π(9/2 + 3) = 15π/2

**Question 25**
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In a youth club with 94 members, 60 like modern music, and 50 like traditional music. The number of members who like both traditional and modern music is three times those who do not like any type of music. How many members like only one type of music?

**Answer Details**

Use a venn diagram:

60-3x+3x+50-3x = 94-x.

110-3x+x = 94

-2x = 94-110

=>-2x = -16, this x = -8.

Members that like only one game:

= 60 - 3x + 50 - 3x

= 60 - 3x8 + 50 - 3x8

= 60 - 24 + 50 - 24

= 36 + 26 = 62

**Question 26**
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Solve the inequality 2 - x > x2.

**Question 27**
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If 31410 - 2567 = 340x, find x.

**Answer Details**

31410 - 2567 = 340x,

Convert 2567 and 340x to base 10, such that:

314 - 139 = 3x2 + 4x

=> 3x2 + 4x - 175 = 0 (quadratic)

Factorising, (x - 7) (3x + 25) = 0,

either x = 7 or x = -25/3 ( but x cannot be negative)

Therefore, x = 7.

**Question 28**
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If α and β are the roots of the equation 3x2 + 5x - 2 = 0, find the value of 1/α + 1/β

**Answer Details**

We know that if α and β are the roots of the quadratic equation ax^2 + bx + c = 0, then α + β = -b/a and α × β = c/a. In this equation, 3x^2 + 5x - 2 = 0, we have a = 3, b = 5, and c = -2. So, α + β = -b/a = -5/3 And, α × β = c/a = -2/3 We need to find the value of 1/α + 1/β = (α + β)/(α × β) = (-5/3) / (-2/3) = 5/2. Therefore, the value of 1/α + 1/β is 5/2.

**Question 29**
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Given that the various faces of a fair dice 1, 2, 3, 4, 5, 6 appeared 30, 43, 54, 40, 41, 32 times respectively in a single toss. Picture the figures as being represented in a simple table with number (X) against frequency (f).

If a pie chart is used to depict the data, the angle corresponding to 4 is?

**Answer Details**

To find the angle corresponding to 4 in the pie chart, we need to calculate the percentage of times 4 appeared in the toss. Total frequency = 30 + 43 + 54 + 40 + 41 + 32 = 240 Frequency of 4 = 40 Percentage of times 4 appeared = (Frequency of 4 / Total frequency) x 100% = (40/240) x 100% = 16.67% The pie chart represents a circle of 360°. Therefore, to find the angle corresponding to 4 in the pie chart, we need to calculate 16.67% of 360°. Angle corresponding to 4 = (16.67/100) x 360° = 60° Therefore, the angle corresponding to 4 in the pie chart is 60°. Option (d) is the correct answer.

**Question 30**
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If the diagram is the graph of y = x2, the shaded area is

**Answer Details**

A = ∫x=4x=0 ${\int}_{x=0}^{x=4}$ ydx = ∫40 ${\int}_{0}^{4}$ x2dx = [x33+c]40 $\frac{{x}^{3}}{3}+c{]}_{0}^{4}$

= 643 $\frac{64}{3}$ square units

**Question 31**
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The 3rd term of an A.P is 4x - 2y and the 9th term is 10x - 8y. Find the common difference.

**Answer Details**

In an Arithmetic Progression (A.P), the difference between any two adjacent terms is constant. Let d be the common difference of the A.P, then: - The 3rd term is the first term plus two common differences, i.e. a + 2d = 4x - 2y - The 9th term is the first term plus eight common differences, i.e. a + 8d = 10x - 8y We can now solve for d by subtracting the first equation from the second: a + 8d - (a + 2d) = 10x - 8y - (4x - 2y) 6d = 6x - 6y d = (x - y) Therefore, the common difference is x - y, which is option (C).

**Question 32**
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Two chords PQ and RS of a circle when produced meet at K. If ∠KPS = 31o and ∠PKR = 42o, find ∠KQR

**Answer Details**

QPS - QRK = 31o

QRK + RKQ + KQR = 180

31 + 42 + KQR = 180o

KQR = 180 - 73 = 107o

**Question 33**
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In the diagram above, EFGH is a circle center O. Fh is a diameter and GE is a chord which meets FH at right angle at the point N. If NH = 8cm and EG = 24cm, calculate FH.

**Answer Details**

To solve this problem, we need to use the property of a circle that states that a diameter is twice the radius of the circle. Since FH is a diameter, then FH = 2OH, where OH is the radius of the circle. Since GE is a chord that meets FH at right angle at the point N, then N is the midpoint of FH. Therefore, NH = NF = 8 cm. Let's use the Pythagorean theorem to find OH. Since GE is a chord of the circle, we can draw radii from the center O to the endpoints of the chord so that we have a right triangle. Let x be the length of OE. Then, HG = 2x, and OH = x + 12 (since FH = 2OH = 2(x + 12) = 2x + 24). Using the Pythagorean theorem, we have: x^2 + 12^2 = (2x)^2 x^2 + 144 = 4x^2 3x^2 = 144 x^2 = 48 x = sqrt(48) = 4sqrt(3) Therefore, OH = x + 12 = 4sqrt(3) + 12. Finally, we can calculate FH: FH = 2OH = 2(4sqrt(3) + 12) = 8sqrt(3) + 24 ≈ 32.39 cm. Therefore, the answer is 32 cm.

**Question 34**
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A trader realizes 10x - x2 naira profit from the sale of x bags on corn. How many bags will give him the desired profit?

**Answer Details**

The trader's profit function can be represented as P(x) = 10x - x^2, where x is the number of bags sold and P(x) is the profit obtained from selling x bags of corn. To find the number of bags that will give him the desired profit, we need to maximize the profit function P(x). We can do this by finding the derivative of P(x) with respect to x, and then setting it to zero to find the critical point(s) of the function. P'(x) = 10 - 2x Setting P'(x) to zero gives: 10 - 2x = 0 Solving for x gives: x = 5 Therefore, the trader will make the maximum profit when he sells 5 bags of corn. This means that the answer is, 5.

**Question 35**
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if P and Q are fixed points and X is a point which moves so that XP = XQ, the locus of X is

**Answer Details**

Given two fixed points P and Q, the locus of points X which are equidistant from P and Q is the perpendicular bisector of PQ. This is because any point on the perpendicular bisector of PQ will be equidistant from both P and Q by definition. Therefore, if X is a point which moves so that XP = XQ, it must lie on the perpendicular bisector of PQ. Therefore, the correct answer is "the perpendicular bisector of PQ".

**Question 36**
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The expression ax2 + bx + c equals 5 at x = 1. If its derivative is 2x + 1, what are the values of a, b, c respectively?

**Answer Details**

At x = 1, substituting x = 1 in the equation: ax2 + bx + c = 5;

f(1) => a + b + c = 5 .....(1)

Taking the first derivative of f(x) in the original equation gives dy/dx = 2ax + b = 2x + 1 (given)....(2)

From (2),=> b = 1, and 2ax = 2x, => a = 1.

substituting into (1) 1 + 1 + c = 5, => c = 5 - 2 = 3

Thus a = 1, b = 1 and c = 3

**Question 37**
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If the population of a town was 240,000 in January 1998 and it increased by 2% each year, what would be the population of the town in January, 2000?

**Answer Details**

To solve this problem, we can use the formula for exponential growth, which is: A = P(1 + r)^t where A is the final amount, P is the initial amount, r is the annual growth rate as a decimal, and t is the time in years. In this problem, we are given P = 240,000, r = 0.02, and t = 2 (since we want to find the population after two years). We can plug these values into the formula and simplify: A = 240,000(1 + 0.02)^2 A = 240,000(1.02)^2 A = 240,000(1.0404) A = 249,696 Therefore, the population of the town in January 2000 would be 249,696. So the answer is 249,696.

**Question 38**
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Find the sum of the range and the mode of the set of numbers 10, 9, 10, 9, 8, 7, 7, 10, 8, 10, 8, 4, 6, 9, 10, 9, 10, 9, 7, 10, 6, 5

**Answer Details**

To find the range, we need to subtract the smallest number from the largest number in the set. The smallest number in the set is 4 and the largest number is 10. So, the range is 10 - 4 = 6. To find the mode, we need to determine the number that occurs the most frequently in the set. In this case, the number 10 occurs the most frequently, appearing 7 times. So, the mode is 10. To find the sum of the range and the mode, we simply add the two values together. Thus, the sum of the range and the mode is 6 + 10 = 16. Therefore, the answer is option (A) 16.

**Question 39**
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Let P = {1, 2, u, v, w, x}; Q = {2, 3, u, v, w, 5, 6, y} and R = {2, 3, 4, v, x, y}.

Determine (P-Q) ? R

**Answer Details**

To determine (P-Q) ? R, we need to first find the set difference (P-Q), which is the set of elements that are in P but not in Q. So, we need to remove all the elements in Q from P. (P-Q) = {1, w, x} Then, we take the intersection of (P-Q) and R, which means we need to find all the elements that are common to both (P-Q) and R. (P-Q) ? R = {x} Therefore, the answer is {x}.

**Question 40**
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Find the inverse of p under the binary operation * defined by p*q = p + q - pq, where p and q are real numbers and zero is the identity

**Answer Details**

If P-1 is the inverse of P and O is the identity, Then P-1 * P = P * P-1 = 0

i.e. P-1 + P - P-1.P = 0

P-1 - P-1.P = -P

P-1(1 - P) = -P

P-1 = -P/(P-1)

= P/(P-1)

**Question 41**
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An equilateral triangle of side √3cm is inscribed in a circle. Find the radius of the circle.

**Answer Details**

In an equilateral triangle, all the sides are of equal length and each angle is 60 degrees. Let's draw an equilateral triangle with side √3 cm inscribed in a circle. Since the triangle is equilateral, we know that the circumcenter (center of the circle that passes through all the vertices of the triangle) is also the centroid (point of intersection of the medians). The median of an equilateral triangle is the line segment from a vertex to the midpoint of the opposite side. Therefore, the circumcenter is also the midpoint of any side of the triangle. Let's choose one side of the triangle and label its midpoint as point M. We can draw a perpendicular line from M to the opposite vertex of the triangle, which will bisect the side and form a right triangle with one leg being half of the side of the triangle and the other leg being the radius of the circle. Using the Pythagorean theorem, we can solve for the radius: (radius)^2 = (√3/2)^2 + (1/2)^2 (radius)^2 = 3/4 + 1/4 (radius)^2 = 1 radius = 1 cm Therefore, the radius of the circle is 1 cm. Answer: 1 cm.

**Question 42**
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X and Y are two events. The probability of X or Y is 0.7 and that of X is 0.4. If X and Y are independent, find the probability of Y.

**Answer Details**

P (X or Y) = P(X) + P(Y), when they are independent as given.

0.7 = 0.4 + P(Y)

P(Y) = 0.7 - 0.4 = 0.30

**Question 43**
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If y = 2x - sin2x, find dy/dx when x = π/4

**Answer Details**

y = 2x cos2x - sin2x

dy/dx = 2 cos2x +(-2x sin2x) - 2 cos2x

= 2 cos2x - 2x sin2x - cos2x

= -2x sin2x

= -2 x (π/4) sin2 x (π/4)

= -(π/2) x 1 = -(π/2)

**Question 44**
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If U = {x : x is an integer and 1 ≤
$\le $ x ≤
$\le $ 20

E1 = {x : x is a multiple of 3}

E2 = {x : x is a multiple of 4}

and an integer is picked at random from U, find the probability that it is not in E2

**Answer Details**

U = {1, 2, 3, 4, 5,..., 20}

E1 = {3, 6, 9, 12, 15, 18}

E2 = {4, 8, 12, 16, 20}

P(E1) = 5/20

P(not E1) = 1 - (5/20) = 15/20 = 3/4

**Question 45**
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A function f(x) passes through the origin and its first derivative is 3x + 2. What is f(x)?

**Answer Details**

Hints:

1. Integrate the given first derivative of f(x) at the boundaries, (0,0)

Then solve accordingly to get f(x) = y = (3x2/)2 + 2x

**Question 46**
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A frustrum of pyramid with square base has its upper and lower sections as squares of sizes 2m and 5m respectively and the distance between them 6m. Find the height of the pyramid from which the frustrum was obtained.

**Question 47**
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If the volume of a hemisphere is increasing at a steady rate of 18π m/s, at what rate is its radius changing when its is 6m?

**Answer Details**

We can solve this problem using the formula for the volume of a hemisphere, which is given by V = (2/3)πr^3. To find the rate of change of the radius, we need to differentiate this equation with respect to time t. Thus, we have: dV/dt = (2/3)π(3r^2)(dr/dt) where dV/dt is the rate of change of volume (18π m/s), and r = 6 m is the radius of the hemisphere. Solving for dr/dt, we get: dr/dt = (3dV/dt)/(4πr^2) = (3(18π))/(4π(6)^2) = 0.25 m/s Therefore, the rate of change of the radius when the volume of the hemisphere is increasing at a steady rate of 18π m/s and its radius is 6 m is 0.25 m/s. Thus, the correct option is: 0.25 m/s.