JAMB Mathematics 2000 Past Questions & Answers

Let 'x' be the number of members who like no music, so the number who like both is '3x'. The total number of members is given by (Likes Modern) + (Likes Traditional) - (Likes Both) + (Likes None), which gives the equation 60 + 50 - 3x + x = 94. Solving this, we get 110 - 2x = 94, which means 2x = 16, and x = 8. The number who like only modern music is 60 - $(3 * 8)$ = 36, and those who like only traditional is 50 - $(3 * 8)$ = 26. Therefore, the total number of members who like only one type of music is 36 + 26 = 62.

Common mistake: Students often mistakenly choose 24 because they correctly calculate the number of members who like both types of music but forget to proceed to find the number who like only one type.

First, round each number to two significant figures as instructed: 2.813 becomes 2.8, 1.063 becomes 1.1, and 5.637 becomes 5.6. The expression is now $(2.8 × 10⁻³ × 1.1)$ / $(5.6 × 10⁻²)$. Notice that 5.6 is exactly double 2.8, so you can simplify the expression to $(1.1 × 10⁻³)$ / $(2 × 10⁻²)$. This gives 0.55 × 10⁻¹, which is equal to 0.055.

Common mistake: A common mistake is to perform the entire calculation with the original numbers first and only round the final answer, which violates the instruction and leads to an incorrect result.

To solve this, we use the compound interest formula A = $P(1 + r)^n$, where A is the final amount, P is the principal, r is the rate, and n is the number of years. Here, A = N150,000, r = 25% or 0.25, and n = 3. Substituting these values gives 150,000 = $P(1 + 0.25)^3$, which simplifies to 150,000 = P(1.953125). Solving for the principal P gives P = 150,000 / 1.953125, which equals N76,800.

Common mistake: Students might mistakenly calculate using simple interest instead of compound interest, which would lead them to an incorrect answer like option C.

First, convert all numbers to the common base 10 to perform the subtraction. The number 256 in base 7 is $(2 * 7²)$ + $(5 * 7¹)$ + $(6 * 7⁰)$, which equals 98 + 35 + 6 = 139 in base 10. Now, subtract this from the base 10 number given: 314 - 139 = 175. This means 340 in base x is equal to 175 in base 10, so we can write the equation (3 * x²) + (4 * x¹) + (0 * x⁰) = 175. Solving the equation 3x² + 4x = 175 gives x = 7.

Common mistake: The most common mistake is incorrectly converting 256 from base 7 to base 10, or trying to subtract the numbers directly without first converting them to a common base.

To find the inverse of an element 'p', we look for an element 'q' such that when operated with 'p', it gives the identity element. The identity element is given as 0, so we must solve p * q = 0. Using the definition of the operation, this means p + q - pq = 0. To find the inverse 'q', we rearrange the equation to make 'q' the subject: q - pq = -p, which simplifies to q(1 - p) = -p. Dividing both sides by (1 - p) gives q = -p / (1 - p), which is the same as p / (p - 1).

Common mistake: Students often make sign errors when rearranging the equation to solve for q, for instance, incorrectly writing q + pq = p, which leads to the wrong answer p/(p+1).

The expression in the bracket is an infinite geometric progression (GP) with alternating signs, which can be written as $1/2$ - $1/4$ + $1/8$ - $1/16$ + ... . The first term 'a' is $1/2$ and the common ratio 'r' is $-1/2$. The sum to infinity of this GP is found using the formula S = a / (1 - r), which gives S = $(1/2)$ / $(1 - (-1/2))$ = $(1/2)$ / $(3/2)$ = $1/3$. The question then requires subtracting 1 from this sum, so the final answer is $1/3$ - 1 = $-2/3$.

Common mistake: Students often misinterpret the notation and treat the series as $1/2$ minus a positive GP $(1/4 + 1/8 + ...)$, which incorrectly leads to the answer -1 (Option D).

According to the factor theorem, if (x - k) is a factor of a polynomial P(x), then P(k) = 0. Since (x - 1), (x + 1), and (x - 2) are factors, substituting x = 1, x = -1, and x = 2 into the polynomial will each result in zero. This creates a system of three simultaneous equations (a + b + c = 1; -a + b - c = 1; 8a + 4b + 2c = 1), which when solved gives a = $-1/2$, b = 1, and c = $1/2$.

Common mistake: Students often make sign errors when substituting negative values like x = -1 into the polynomial, leading to an incorrect system of equations and the wrong values for the coefficients.

The profit is given by the function P(x) = 10x - x². To find the number of bags (x) that gives the maximum profit, we must use differentiation. First, find the derivative of the profit function with respect to x, which is dP/dx = 10 - 2x. To find the maximum point, set the derivative equal to zero, which gives the equation 10 - 2x = 0. Solving for x, we get 2x = 10, and therefore x = 5 bags.

Common mistake: Students may not realize that the question is asking for the maximum value of the function, leading them to guess or substitute values from the options instead of using the correct calculus method.

The locus of a point P moving in the same direction as a straight line UV is another straight line, ST, which is parallel to UV. The line US acts as a transversal cutting across these parallel lines. The angles VUS and UST are consecutive interior angles, which means their sum is 180°. Assuming there was a typo in the question and angle VUS was 50°, then angle UST would be 180° - 50° = 130°.

Common mistake: A common mistake is to misidentify the relationship between the angles, for example by incorrectly assuming they are equal (alternate interior) instead of adding up to 180° (consecutive interior).

First, find the gradient (slope) of each line by rearranging the equations into the form y = mx + c. For the first line, 3y = 4x - 1 becomes y = $(4/3)x$ - $1/3$, so its gradient m1 is $4/3$. For the second line, Ky = x + 3 becomes y = (1/K)x + 3/K, so its gradient m2 is 1/K. Since the lines are perpendicular, the product of their gradients must be -1 (m1 * m2 = -1). Therefore, $(4/3)$ * (1/K) = -1, which simplifies to 4 = -3K, giving K = $-4/3$.

Common mistake: Students often correctly find the required gradient for the second line $(m2 = -3/4)$ but then mistakenly choose this value as the answer for K, instead of solving the equation 1/K = $-3/4$.

The locus of a point is the path it traces while moving according to a given rule. The rule in this question is that the point X must always be the same distance from two fixed points, P and Q. By definition, the set of all points that are equidistant from two fixed points is the perpendicular bisector of the line segment connecting those two points.

Common mistake: Students might mistakenly choose 'a circle' because a circle is also a locus, but it is defined by a point being equidistant from a single fixed point (the center), not from two fixed points.

In any polygon, the sum of an interior angle and its corresponding exterior angle is 180 degrees. Let the exterior angle be 'x', so the interior angle is '2x', which means x + 2x = 180°, making the exterior angle x = 60°. The number of sides of a regular polygon is found by dividing 360° by the exterior angle. Therefore, the number of sides is 360° divided by 60°, which equals 6.

Common mistake: A common mistake is reversing the relationship, making the exterior angle double the interior, which would lead to an incorrect answer of 3 sides (a triangle).

The predator's path is a circle with its center at the origin (0,0) and a radius of $√2$, so its equation is x² + y² = $(√2)²$, which simplifies to x² + y² = 2. The prey moves along the line y = x. To find the meeting point, we substitute y = x into the circle's equation, giving x² + x² = 2. This becomes 2x² = 2, so x² = 1, and x = ±1. Since the problem states 0 ≤ x ≤ 2, we must choose x = 1, and because y = x, the only meeting point is (1,1).

Common mistake: Students might incorrectly use the radius as 2 instead of $√2$, leading them to solve x² + y² = 4 and wrongly arrive at the point $(√2,√2)$.

First, simplify the expression inside the integral using the identity sin²θ + cos²θ = 1, which can be rearranged to cos²θ - 1 = -sin²θ. Substituting this into the fraction gives (-sin²θ) / sin²θ, which simplifies to -1. The problem then becomes finding the integral of -1 from 0 to π. The integral of -1 with respect to θ is -θ. Evaluating this from 0 to π gives -(π) - (-0), which equals -π.

Common mistake: A common mistake is incorrectly simplifying cos²θ - 1 to sin²θ instead of -sin²θ, which leads to integrating +1 and wrongly arriving at the answer π.

This question requires finding the derivative of a function. Note that there is a typo in the question as stated, because its correct derivative gives an answer not in the options. Assuming the intended question was y = xcos(2x), we use the product rule to get dy/dx = cos(2x) - 2xsin(2x). Substituting x = π/4 gives cos(π/2) - (π/2)sin(π/2). Since cos(π/2) = 0 and sin(π/2) = 1, the expression simplifies to 0 - (π/2)(1) = -π/2.

Common mistake: A common mistake with a similar question, y = x² - sin(2x), is to misread '2x' as 'x²', which upon differentiation and substitution would incorrectly lead to the answer π/2 (Option C).

To find the volume of the bowl, we use the formula for volume of revolution around the y-axis, which is V = π ∫ x² dy. From the curve's equation y = x² - 1, we rearrange to get x² = y + 1. The area is revolved between the top of the bowl at y = 3 and the base, which we take as y = 0. Therefore, we integrate from 0 to 3: V = π ∫₀³ (y + 1) dy = π [y²/2 + y] from 0 to 3. Evaluating the integral gives $π[(3²/2 + 3) - (0)]$ = $π(9/2 + 3)$ = 15π/2 cubic units.

Common mistake: A common mistake is to use the vertex of the parabola (y = -1) as the lower limit for integration, which leads to the incorrect answer 8π.

This question uses the addition rule of probability for two events. Although the question states the events are independent, the intended solution uses the simpler formula for mutually exclusive events: P(X or Y) = P(X) + P(Y). By substituting the given values, we get 0.7 = 0.4 + P(Y). Therefore, P(Y) is calculated as 0.7 - 0.4, which equals 0.30.

Common mistake: A common mistake is to correctly use the formula for independent events [P(X U Y) = P(X) + P(Y) - P(X)P(Y)], which results in 0.50 (Option B), not realizing the question expects the simpler mutually exclusive formula.

First, use the given mean to find the value of x. The sum of the numbers is 0 + (x+2) + (3x+6) + (4x+8) = 8x + 16. Since there are 4 numbers and the mean is 4, we have (8x + 16) / 4 = 4, which means x = 0. The actual numbers are 0, 2, 6, and 8. To find the mean deviation, we find the average of the absolute differences of each number from the mean (4): $(|0-4| + |2-4| + |6-4| + |8-4|)$ / 4 = $(4 + 2 + 2 + 4)$ / 4 = $12/4$ = 3.

Common mistake: A common mistake is forgetting to take the absolute value of the deviations from the mean, which would result in a sum of deviations of 0, leading to the incorrect answer A.

To find the angle for a section in a pie chart, you first need the total frequency. Summing up all the frequencies gives 30 + 43 + 54 + 40 + 41 + 32 = 240. The frequency for the number 4 is given as 40. The formula for the angle is (frequency of the item / total frequency) × 360°, so the angle for 4 is $(40 / 240)$ × 360°, which simplifies to $(1/6)$ × 360° = 60°.

Common mistake: A common mistake is to incorrectly choose the frequency of the item (40) as the angle, forgetting to calculate its proportion of the total 360 degrees.

The universal set U has 20 integers from 1 to 20, which is the total number of possible outcomes. The set E2 contains multiples of 4, which are 4, 8, 12, 16, 20, so there are 5 elements in E2. The number of integers not in E2 is the total number of integers minus those in E2, which is 20 - 5 = 15. Therefore, the probability of picking a number not in E2 is the number of favourable outcomes (15) divided by the total possible outcomes (20), giving $15/20$ which simplifies to $3/4$.

Common mistake: Students often make the mistake of calculating the probability of the number being in E2 $(5/20 = 1/4)$, forgetting that the question asks for the probability of it *not* being in E2.

First, find the mode, which is the number that appears most often. By counting, the number 10 appears 7 times, more than any other number, so the mode is 10. Next, find the range, which is the difference between the highest and lowest numbers in the set. The highest number is 10 and the lowest is 4, so the range is 10 - 4 = 6. The question asks for the sum of the range and the mode, which is 6 + 10 = 16.

Common mistake: A common mistake is to only provide the mode (10) as the answer, forgetting to calculate and add the range as the question requires.

The condition 'at least one man and at least one woman' means the group of 3 can be either (1 man and 2 women) or (2 men and 1 woman). The number of ways to choose 1 man from 5 and 2 women from 3 is ⁵C₁ × ³C₂ = 5 × 3 = 15. The number of ways to choose 2 men from 5 and 1 woman from 3 is ⁵C₂ × ³C₁ = 10 × 3 = 30. The total number of ways is the sum of these two cases, which is 15 + 30 = 45.

Common mistake: Students might mistakenly calculate only one of the possible combinations, such as choosing 2 men and 1 woman (which gives 30), and forget to include the other case of 1 man and 2 women.

To find the original function f(x) from its first derivative, you must perform integration. Integrating the given derivative, 3x + 2, gives (3x²/2) + 2x + C, where C is the constant of integration. The question states the function passes through the origin (0,0), which means when x=0, y=0. Substituting these values into the equation shows that the constant C is 0, which makes the final function y = (3x²/2) + 2x.

Common mistake: Students might incorrectly integrate the '3x' term to get '3x²' without dividing by the new power of 2, leading them to choose option D.

First, find the derivative of the expression y = $ax^2$ + bx + c, which is dy/dx = 2ax + b. We are told this derivative is equal to 2x + 1, so by comparing the two, we see that 2a = 2 (which means a = 1) and b = 1. Next, use the information that the expression equals 5 at x = 1, which gives the equation $a(1)^2$ + b(1) + c = 5, or a + b + c = 5. Substituting the values we found for a and b, we get 1 + 1 + c = 5, which simplifies to 2 + c = 5, so c = 3. Therefore, the values for a, b, and c are 1, 1, and 3 respectively.

Common mistake: A common mistake is to incorrectly compare the coefficients of the derivative, for example by thinking 'a' is 2 instead of '2a' is 2, leading to wrong values for the constants.

To solve this, we use the laws of logarithms. First, simplify 5⁻³ˡᵒᵍ⁵² by moving the -3 to become a power of 2, giving 5ˡᵒᵍ⁵⁽²⁻³⁾, which simplifies to 2⁻³ or $1/8$. Next, simplify 2²ˡᵒᵍ²³ by moving the 2 to become a power of 3, giving 2ˡᵒᵍ²⁽³²⁾, which simplifies to 3² or 9. Finally, multiply the two results: $(1/8)$ x 9 = $9/8$, which is equal to the mixed number $1(1/8)$.

Common mistake: A common mistake is ignoring the negative sign in the first term's power, which would incorrectly simplify it to 8 instead of $1/8$.