JAMB Mathematics 2002 Past Questions & Answers

To solve this, we apply the laws of indices. First, $(343)^(1/3)$ is the cube root of 343, which is 7. Next, $(0.14)^(-1)$ is the same as 1 divided by 0.14, which simplifies to $100/14$. Finally, $(25)^(-1/2)$ means 1 over the square root of 25, which is $1/5$. Multiplying these results together gives 7 × $(100/14)$ × $(1/5)$, which simplifies to $700/70$, giving the final answer of 10.

Common mistake: A common mistake is to only evaluate the first term, $(343)^(1/3)$, which is 7, and then incorrectly select option D without completing the full calculation.

This is a problem about sets. The total number of students offering either Biology or Mathematics is 220. To find the number who offer Biology but not Mathematics (i.e., Biology only), you can simply subtract the total number of students offering Mathematics from the total union. So, the number of students offering Biology only is the total in the union minus all students who offer Mathematics, which is 220 - 110 = 110.

Common mistake: A common mistake is to calculate 220 - 125 = 95, but this calculation finds the number of students who offer Mathematics only, not Biology only.

To solve this, first add up all the numbers being subtracted: 5.7 + 3.45 + 1.75 = 10.90. Next, subtract this sum from the initial number, 52.4. Performing the subtraction, 52.40 - 10.90, gives the final answer of 41.5.

Common mistake: Students may misalign the decimal points during calculation or make a simple borrowing error in the final subtraction, which could lead to an incorrect answer like 41.4.

To solve this, you expand the bracket using the formula (a+b)2 = a2 + 2ab + b2. Here, a is $√0.7$ and b is $√70$, so we get $(√0.7)2$ + $2(√0.7)(√70)$ + $(√70)2$. This simplifies to 0.7 + $2√(0.7 x 70)$ + 70, which is 0.7 + $2√49$ + 70. Since $√49$ is 7, the expression becomes 0.7 + 2(7) + 70, and adding these numbers gives the final answer of 84.7.

Common mistake: The most common mistake is to incorrectly expand the bracket as $(√0.7)2$ + $(√70)2$, which gives 0.7 + 70 = 70.7, forgetting the middle term '2ab'.

To solve this, it is easiest to first convert the decimals to whole numbers and then handle the decimal places separately. The expression can be seen as $(21 × 72 × 54)$ divided by $(6 × 168 × 63)$. By cancelling out common factors, this fraction simplifies to $9/7$, which is approximately 1.2857. The numerator has 10 decimal places in total $(2+3+5)$ and the denominator has 8 $(3+2+3)$, so the final answer will have 10 - 8 = 2 effective decimal places. Therefore, we divide 1.2857 by 100 to get 0.012857, which is approximated to 0.01286.

Common mistake: Students often make mistakes in counting the decimal places, which can lead them to choose 0.1286 (Option C) by misplacing the decimal point after simplifying the numbers.

First, we need to find the total cost price of all the goats. Since the trader made a 25% loss, the selling price of N180,000 represents 75% (100% - 25%) of the original cost price. To find the full cost price, we calculate N180,000 divided by 0.75, which gives N240,000. Finally, to find the number of goats, we divide the total cost price (N240,000) by the cost of one goat (N4,000), which gives 60 goats.

Common mistake: A common mistake is to ignore the percentage loss and simply divide the total selling price (N180,000) by the cost per goat (N4,000), which incorrectly gives 45.

To find y from its derivative dy/dx, we must perform integration. Integrating the expression `2x - 3` gives `y = $x^2$ - 3x + C`, where C is the constant of integration. We are given the condition that y = 3 when x = 0, which we use to find C. Substituting these values gives `3 = $(0)^2$ - 3(0) + C`, so C = 3. The correct equation is `y = $x^2$ - 3x + 3`. The provided option D is `y = $2x^2$ - 3x + 3`, which would be the correct answer if the question was `dy/dx = 4x - 3`, indicating a likely typo in the original question.

Common mistake: A common mistake is to correctly integrate to get `y = $x^2$ - 3x + C` but then forget to use the initial condition to find C, leading to option B, $`x^2$ - 3x`.

To find the derivative of y = sin²(5x), we must use the chain rule. First, differentiate the outer power (the square), which gives 2sin(5x). Next, differentiate the inner function sin(5x) to get cos(5x). Finally, differentiate the innermost angle 5x to get 5. Multiplying all these parts together gives 2sin(5x) × cos(5x) × 5, which simplifies to 10sin(5x)cos(5x).

Common mistake: The most common mistake is forgetting to apply the chain rule to the innermost function (5x), which leads to an answer of 2sin(5x)cos(5x) by missing the factor of 5.

To find the slope of the tangent to a curve, you must first find the derivative of the curve's equation. The derivative of y = 3x² - 2x + 5 is dy/dx = 6x - 2. This derivative represents the formula for the slope at any point on the curve. To find the specific slope at the point (1, 6), substitute the x-value, which is 1, into the derivative to get 6(1) - 2 = 4.

Common mistake: A common mistake is to substitute the point (1, 6) into the original equation, which simply confirms the point is on the curve (y=6), instead of substituting the x-value into the derivative to find the slope.

To find the integral of sin(3x), we use the standard integration rule for trigonometric functions. The integral of sin(ax) is (-1/a)cos(ax) + c, where 'c' is the constant of integration. In this question, 'a' is 3, so applying the rule gives us $(-1/3)cos(3x)$ + c.

Common mistake: Students often choose option B because they forget that the integral of sin(x) is -cos(x), not just cos(x), or they confuse it with differentiation.

This is a problem on the application of differentiation, specifically related rates. The area of a circle is given by the formula A = πr². To find the rate of increase of the area (dA/dt), we differentiate this formula with respect to time, which gives dA/dt = 2πr(dr/dt). Substituting the given values, radius r = 5 and the rate of increase of the radius dr/dt = 0.2, we get dA/dt = 2 * π * 5 * 0.2, which simplifies to 2π.

Common mistake: A common mistake is to forget the factor of 2 that comes from differentiating r², which would lead to an incorrect calculation of πr(dr/dt) = π(5)(0.2) = π.

To find dy/dx, we need to differentiate the function y = x² - 1/x. First, rewrite 1/x as x⁻¹ using the laws of indices, so the equation becomes y = x² - x⁻¹. Using the power rule for differentiation, the derivative of x² is 2x, and the derivative of -x⁻¹ is -(-1)x⁻¹⁻¹, which simplifies to +x⁻². Rewriting x⁻² as 1/x², the final answer is dy/dx = 2x + 1/x².

Common mistake: A common mistake is incorrectly handling the signs when differentiating the -x⁻¹ term, leading students to choose 2x - 1/x² by forgetting that the two negative signs multiply to become positive.

To solve the equation x³ - 5x² - x + 5 = 0, we can use factorization by grouping. Group the terms as (x³ - 5x²) - (x - 5) = 0, then factor out common terms to get x²(x - 5) - 1(x - 5) = 0. This simplifies to (x² - 1)(x - 5) = 0, which means either x - 5 = 0 (so x = 5) or x² - 1 = 0 (so x² = 1, which gives x = 1 and x = -1). Therefore, the solutions for x are 1, -1, and 5.

Common mistake: Students often forget that the square root of 1 is both +1 and -1, leading them to solve x² = 1 as just x = 1 and incorrectly choose an option with a repeated root.

The problem states that time (T) is inversely proportional to the number of men (N), which can be written as T = k/N, where k is a constant. First, we find the constant k using the given information: 45 men take 5 days, so 5 = k/45, which means k = 5 × 45 = 225. Now we use this constant to find the time it will take 25 men, so T = 225 / 25. Therefore, it will take 25 men 9 days to complete the work.

Common mistake: A common mistake is to treat the relationship as a direct proportion, which would wrongly suggest that fewer men take less time to do the work.

To solve this inequality, first clear the fractions by multiplying all terms by the LCM of the denominators 4 and 3, which is 12. This gives 3(x+2) - 4(2x-3) < 48. Expanding the brackets results in 3x + 6 - 8x + 12 < 48, which simplifies to -5x + 18 < 48. When you isolate x, you get -5x < 30, and dividing by -5 reverses the inequality sign, giving the final answer x > -6.

Common mistake: A common mistake is to forget to reverse the inequality sign when dividing by a negative number, or to incorrectly expand -4(2x-3) as -8x - 12 instead of -8x + 12.

To find the maximum value of the quadratic function y = 1 - 2x - $3x^2$, we use differentiation. First, find the derivative dy/dx, which is -2 - 6x. At the maximum point, the derivative is zero, so we set -2 - 6x = 0 and solve for x, which gives x = $-1/3$. Finally, substitute this value of x back into the original equation to get the maximum value of y: y = 1 - $2(-1/3)$ - $3(-1/3)^2$ = 1 + $2/3$ - $1/3$ = $4/3$.

Common mistake: A common mistake is to correctly find x = $-1/3$ but then make an error when substituting it back into the equation, for example, calculating 1 + $2/3$ to get $5/3$ (Option B) and then forgetting or miscalculating the final term.

The formula for the nth term of an Arithmetic Progression (A.P.) is $T_n$ = a + (n-1)d. The 9th term is $T_9$ = a + $(9-1)d$ = a + 8d, and the 5th term is $T_5$ = a + $(5-1)d$ = a + 4d. The question states that the 9th term is five times the 5th term, so we set up the equation: a + 8d = 5(a + 4d). Expanding the bracket gives a + 8d = 5a + 20d, and by collecting like terms we get 4a + 12d = 0, which simplifies to a + 3d = 0 when divided by 4.

Common mistake: A common mistake is making an error during algebraic simplification, for example, after getting a + 8d = 5a + 20d, a student might incorrectly subtract the terms to get 4a + 8d = 0, which leads to the wrong answer a + 2d = 0.

This question tests your ability to change the subject of a formula, which is a key skill in algebra. Although the question as written contains a typo, the intended problem requires you to isolate 'r'. By factoring 'r' out of the relevant terms and then performing division and multiplication to move all other terms to the other side of the equation, you can correctly solve for 'r'. The steps for the likely intended equation lead to the expression `a²/(x−a)`.

Common mistake: A common mistake is to misread the equation or make a sign error during rearrangement, for instance, solving for `xr - ar = a`, which incorrectly simplifies to `r = a/(x-a)`, leading to option C.

To find the inverse of the function f(x) = 3x + 4, first set it as y = 3x + 4. Next, make x the subject of the equation by first subtracting 4 from both sides to get y - 4 = 3x. Then, divide both sides by 3 to get x = (y - 4)/3. Finally, replace y with x to get the inverse function, which is (x - 4)/3.

Common mistake: Students might mistakenly choose option D by incorrectly adding 4 to x instead of subtracting it when trying to make x the subject of the formula.

The problem states that x = -2 is the solution, so first, substitute -2 for x in the equation: 2(-2) + 1 - 3c = 2c + 3(-2) - 7. Simplifying both sides gives -4 + 1 - 3c = 2c - 6 - 7, which becomes -3 - 3c = 2c - 13. To find c, collect the like terms by moving all terms with 'c' to one side and the constant numbers to the other: 13 - 3 = 2c + 3c. This simplifies to 10 = 5c, and dividing both sides by 5 gives the final answer, c = 2.

Common mistake: The most common mistake is making a sign error when moving terms across the equals sign, for example, forgetting to change -3c to +3c, which leads to an incorrect final value.

To solve this, we first evaluate the expression in the bracket $(3 * 4)$ using the given rule p*q = pq + p + q. Substituting p=3 and q=4, we get $3*4$ = (3)(4) + 3 + 4 = 12 + 3 + 4 = 19. The problem then simplifies to finding 2 * 19. Applying the rule again with p=2 and q=19, we get $2*19$ = (2)(19) + 2 + 19 = 38 + 2 + 19 = 59.

Common mistake: A common mistake is to calculate the value in the bracket $(3 * 4 = 19)$ and choose that as the final answer (Option B), forgetting to complete the second operation.

This question is about finding the sum to infinity of a Geometric Progression (GP), which is a sequence where each term is found by multiplying the previous one by a common ratio. The formula for the sum to infinity is S∞ = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. In this series, the first term 'a' is 1 and the common ratio 'r' is $(1/3)$ ÷ 1 = $1/3$. Substituting these values gives S∞ = 1 / $(1 - 1/3)$, which simplifies to 1 / $(2/3)$, resulting in $3/2$.

Common mistake: A common mistake is to make an arithmetic error when calculating the denominator (1 - r) or when dividing by the resulting fraction, leading to an incorrect final answer.

The statement 'x varies directly as $√n$' means x = $k√n$, where k is a constant. We first find the value of k using the given information: when x = 9, n = 9. Substituting these into the equation gives 9 = $k√9$, which simplifies to 9 = 3k, so the constant k is 3. Now we use this constant to find x when n = $17/9$ by putting the values into our formula, x = $3√n$. This gives x = $3√(17/9)$, which simplifies to x = 3 * $(√17 / √9)$ = 3 * $(√17 / 3)$, so the 3s cancel out, leaving x = $√17$.

Common mistake: A common mistake is to find the constant k=3 correctly, but then mistakenly multiply it by the initial value of n (which is 9) instead of taking the square root of the new value, leading to an incorrect answer of 27 (Option B).

First, find the radius of the circle from the given diameter: radius (r) = diameter / 2 = $4√3$ / 2 = $2√3$ cm. The angle of the major sector is the total angle in a circle minus the minor sector's angle, which is 360° - 120° = 240°. Using the formula for the area of a sector, Area = (θ/360°) × πr², we get $(240/360)$ × π × $(2√3)²$. This simplifies to $(2/3)$ × π × 12, which equals 8π cm².

Common mistake: A common mistake is to calculate the area of the minor sector using the 120° angle, which leads to the incorrect answer of 4π cm² (Option A).

Given that tan θ = $4/3$, we can model this with a right-angled triangle where the opposite side is 4 and the adjacent side is 3. Using Pythagoras' theorem (a² + b² = c²), the hypotenuse is $√(4² + 3²)$ = $√25$ = 5. From this, we find sin θ = opposite/hypotenuse = $4/5$ and cos θ = adjacent/hypotenuse = $3/5$. The question asks for sin² θ - cos² θ, which is $(4/5)²$ - $(3/5)²$ = $16/25$ - $9/25$ = $7/25$.

Common mistake: A common mistake is to only calculate the value of sin² θ, which is $16/25$ (Option A), and forget to subtract the value of cos² θ as required by the question.