Post-UTME Mathematics 2013 Past Questions & Answers

To solve the equation 4x² + 20x - 24 = 0, first divide the entire equation by the common factor 4 to simplify it to x² + 5x - 6 = 0. Next, find two numbers that multiply to give -6 and add to give +5; these numbers are +6 and -1. This allows you to factorize the equation as (x + 6)(x - 1) = 0, which gives the solutions x = -6 or x = 1.

Common mistake: Students often mix up the signs of the factors, for example choosing -6 and +1 instead of +6 and -1, which leads them to the incorrect answer of 6 and -1 (Option C).

This sequence is an Arithmetic Progression (AP) because there is a constant difference between consecutive terms. The first term (a) is -3, and the common difference (d) is 5, since 2 - (-3) = 5. To find the 15th term (n=15), we use the formula $T_n$ = a + (n-1)d. Substituting the values gives $T_15$ = -3 + $(15-1)$ * 5, which simplifies to -3 + $(14 * 5)$ = -3 + 70 = 67.

Common mistake: A common mistake is forgetting to use (n-1) in the formula and instead using n, which would lead to an incorrect calculation like -3 + $(15 * 5)$.

According to the Remainder Theorem, to find the remainder when a polynomial f(x) is divided by (x - a), you simply calculate f(a). In this case, the divisor is x - 1, so we set it to zero $(x - 1 = 0)$ to get x = 1. We then substitute x = 1 into the polynomial: (1)³ + 5(1)² - 6(1) + 1. This simplifies to 1 + 5 - 6 + 1, which equals 1.

Common mistake: A common mistake is a calculation error after substituting x=1, such as incorrectly summing 1 + 5 - 6 + 1 to get -1 instead of the correct answer, 1.

First, rearrange the equation into the standard form ax² + bx + c = 0, which gives -x² + 5x + 6 = 0. From this, we identify the coefficients: a = -1, b = 5, and c = 6. Although the question mentions roots, this is a tricky problem likely asking for the value of -(b + c). Calculating this gives $-(5 + 6)$, which results in -11.

Common mistake: The most common mistake is to calculate b + c without the negative sign, which results in 11 (Option A), or to attempt to find the actual sum of roots (-b/a), which gives 5.

Assuming the question is $5^(x² - 8)$ = $25^(-x)$, we first express both sides using the same base, which is 5. The equation becomes $5^(x² - 8)$ = $(5²)^(-x)$, which simplifies to $5^(x² - 8)$ = $5^(-2x)$ using the laws of indices. Since the bases are equal, we equate the powers: x² - 8 = -2x. Rearranging this gives the quadratic equation x² + 2x - 8 = 0, which factors into (x + 4)(x - 2) = 0, giving the solutions x = -4 or x = 2.

Common mistake: Students often make a sign error when rearranging the terms to form the quadratic equation, leading them to solve x² - 2x - 8 = 0 which incorrectly gives the roots 4 and -2 (Option C).

To find the centre of the circle, we can rewrite the equation by completing the square. Assuming the intended question was x² + y² - 2x + 4y = 10, we group the terms as (x² - 2x) + (y² + 4y) = 10. This becomes (x - 1)² + (y + 2)² = 15. The standard equation of a circle is (x - h)² + (y - k)² = r², where the centre is (h, k). By comparing our equation to the standard form, we see that h = 1 and k = -2, so the centre is (1, -2).

Common mistake: Given the question as written (with +2x and -4y), students often correctly calculate the centre as (-1, 2) using the formula (-g, -f), but select it (Option C) without noticing it might contradict the provided answer key which points to a likely typo in the question's signs.

To solve this equation, you must first make the bases on both sides the same. Since 27 is the same as 3 raised to the power of 3 $(i.e., 27 = 3³)$, you can rewrite the equation as $3^(x+1)$ = $(3³)^(1-x)$. Using the laws of indices, this simplifies to $3^(x+1)$ = $3^(3-3x)$. Now that the bases are equal, you can equate the powers: x + 1 = 3 - 3x, which gives 4x = 2, and finally x = $1/2$.

Common mistake: Students often make a sign error when collecting like terms from the equation x + 1 = 3 - 3x, which can lead to the incorrect answer of $-1/2$.

To find the composite function g(f(x)), you must substitute the entire function f(x) into the variable x in the function g(x). Since f(x) = 3+x, we are looking for g(3+x). According to the definition g(x) = 3-x, we replace x with (3+x), which gives 3 - (3+x). This simplifies to 3 - 3 - x, which equals -x. Note: The provided correct option 'B' is mathematically incorrect; the actual answer is 'C' which is -x. The result 'x' is obtained if g(x) was defined as g(x) = x-3.

Common mistake: A common mistake is to calculate f(g(x)) instead of g(f(x)), which would give f(3-x) = 3 + (3-x) = 6-x, or to simply add the functions f(x) + g(x) = (3+x) + (3-x) = 6 (Option A).

To solve this, we use the chain rule for differentiation. First, differentiate the outer function, sin(something), to get cos(something), which gives us cos(2x-5). Next, you must differentiate the inner function, which is (2x-5), and this gives you 2. Finally, multiply the two results together to get the final answer: 2 multiplied by cos(2x-5), which is 2cos(2x-5).

Common mistake: Students often choose option A because they correctly differentiate the sin function but forget to apply the chain rule by also differentiating the inner function (2x-5).