Mathematics 2012 Jamb Examination

To find the remainder when a polynomial \(f(x)\) is divided by a linear factor \((x - a)\), we use the Remainder Theorem, which states that the remainder is equal to \(f(a)\). In this case, the divisor is \(x + 3\), which can be written as \(x - (-3)\), so we need to evaluate the polynomial at \(x = -3\).
Let \(f(x) = 2x^3 - 11x^2 + 8x - 1\). Substituting \(x = -3\) into the expression: \[f(-3) = 2(-3)^3 - 11(-3)^2 + 8(-3) - 1\]\[f(-3) = 2(-27) - 11(9) - 24 - 1\]\[f(-3) = -54 - 99 - 24 - 1\]Summing these values together:\[-54 - 99 = -153\]\[-153 - 24 = -177\]\[-177 - 1 = -178\]The calculated value represents the remainder of the division process.

We are given a system of equations consisting of a quadratic and a linear equation. The first equation is a difference of two squares: \(x^2 - y^2 = 4\). This can be factored as follows: \[(x - y)(x + y) = 4\]We are already given the value of the second expression in the second equation: \(x + y = 2\). Substituting this value into our factored equation: \[(x - y)(2) = 4\]Dividing both sides by 2, we find:\[x - y = 2\]Now we have a simple system of linear equations:\[1) \; x + y = 2\]\[2) \; x - y = 2\]Adding both equations together to eliminate \(y\):\[(x + x) + (y - y) = 2 + 2\]\[2x = 4 \implies x = 2\]Substituting \(x = 2\) back into the equation \(x + y = 2\):\[2 + y = 2 \implies y = 0\]The solution set is \(x = 2\) and \(y = 0\).

Since \(y\) varies directly as the square root of \(n\), we can express this relationship with a constant of variation \(k\): \[y = k\sqrt{n}\]First, we use the given initial conditions (\(y = 4\) when \(n = 4\)) to find the value of \(k\):\[4 = k\sqrt{4}\]\[4 = k(2)\]\[k = \frac{4}{2} = 2\]Now that we have the constant, the specific equation for this variation is \(y = 2\sqrt{n}\). We need to find \(y\) when \(n = \frac{17}{9}\):\[y = 2\sqrt{\frac{17}{9}}\]Using the property of square roots where \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\):\[y = 2 \cdot \frac{\sqrt{17}}{\sqrt{9}}\]\[y = 2 \cdot \frac{\sqrt{17}}{3}\]\[y = \frac{2\sqrt{17}}{3}\]Checking the options, it appears there might be a typo in the original question's options or provided text, but based on the standard calculation, the result is \(\frac{2\sqrt{17}}{3}\). However, if we re-evaluate the source data, the most logical intended answer for a simplified radical form is \(\sqrt{17}\) if the constant were different, but mathematically with the provided values, the steps above are absolute.

The problem states that \(U\) is inversely proportional to the cube of \(V\). This relationship is defined by the formula: \[U = \frac{k}{V^3}\]where \(k\) is the constant of variation. First, we determine \(k\) using the values \(U = 81\) and \(V = 2\):\[81 = \frac{k}{2^3}\]\[81 = \frac{k}{8}\]\[k = 81 \times 8 = 648\]Now we use the constant \(k = 648\) to find \(U\) when \(V = 3\):\[U = \frac{648}{3^3}\]\[U = \frac{648}{27}\]Performing the division:\[U = 24\]Therefore, when \(V\) increases from 2 to 3, the value of \(U\) decreases to 24 due to the inverse cubic relationship.

To solve the inequality \(\frac{1}{5}y + \frac{1}{5} < \frac{1}{2}y + \frac{2}{5}\), we first eliminate the fractions by multiplying the entire inequality by the Lowest Common Multiple (LCM) of the denominators (2 and 5), which is 10: \[10 \left( \frac{1}{5}y + \frac{1}{5} \right) < 10 \left( \frac{1}{2}y + \frac{2}{5} \right)\]\[2y + 2 < 5y + 4\]Next, we collect the like terms by moving all \(y\) terms to one side and constants to the other. Subtract \(2y\) from both sides:\[2 < 3y + 4\]Now, subtract 4 from both sides:\[2 - 4 < 3y\]\[-2 < 3y\]Divide both sides by 3 to isolate \(y\):\[-\frac{2}{3} < y\]This can be rewritten as \(y > -\frac{2}{3}\).

The inequality \((m - 3)(m - 4) < 0\) is a quadratic inequality in factored form. The critical values (roots) where the expression equals zero are \(m = 3\) and \(m = 4\). These points divide the number line into three intervals: \[1) \; m < 3\]\[2) \; 3 < m < 4\]\[3) \; m > 4\]We test a value from each interval to see where the product is negative:
For \(m < 3\) (e.g., \(m = 0\)): \((0 - 3)(0 - 4) = (-3)(-4) = 12\), which is \(> 0\).
For \(3 < m < 4\) (e.g., \(m = 3.5\)): \((3.5 - 3)(3.5 - 4) = (0.5)(-0.5) = -0.25\), which is \(< 0\).
For \(m > 4\) (e.g., \(m = 5\)): \((5 - 3)(5 - 4) = (2)(1) = 2\), which is \(> 0\).
The inequality is satisfied when the product is less than zero, which occurs in the middle interval. Thus, the range of values is \(3 < m < 4\).

The formula for the nth term is given as \(T_n = n^2 - 6n - 4\). We need to calculate the 3rd term (\(T_3\)) and the 4th term (\(T_4\)) and then find their sum.
First, calculate \(T_3\) by substituting \(n = 3\):\[T_3 = (3)^2 - 6(3) - 4\]\[T_3 = 9 - 18 - 4 = -13\]Next, calculate \(T_4\) by substituting \(n = 4\):\[T_4 = (4)^2 - 6(4) - 4\]\[T_4 = 16 - 24 - 4 = -12\]Finally, find the sum of these two terms:\[ ext{Sum} = T_3 + T_4\]\[ ext{Sum} = -13 + (-12) = -25\]The combined value of the two specific terms in the sequence is -25.

The formula for the sum to infinity \(S_{\infty}\) of a geometric progression (GP) is given by: \[S_{\infty} = \frac{a}{1 - r}\]where \(a\) is the first term and \(r\) is the common ratio. We are given \(S_{\infty} = -\frac{1}{10}\) and \(a = -\frac{1}{8}\). Substituting these values into the formula: \[-\frac{1}{10} = \frac{-\frac{1}{8}}{1 - r}\]To solve for \(r\), we can first multiply both sides by -1 to simplify: \[\frac{1}{10} = \frac{1}{8(1 - r)}\]Now, cross-multiply: \[8(1 - r) = 10\]\[8 - 8r = 10\]Rearrange the equation to isolate the term with \(r\):\[8 - 10 = 8r\]\[-2 = 8r\]\[r = -\frac{2}{8} = -\frac{1}{4}\]The common ratio of the progression is \(-\frac{1}{4}\).

To evaluate the expression \(2 * (3 * 4)\), we must follow the order of operations and solve the term inside the parentheses first. The rule for the operation is \(p * q = pq + p - q\).
Step 1: Evaluate \((3 * 4)\) where \(p=3\) and \(q=4\):\[3 * 4 = (3)(4) + 3 - 4\]\[3 * 4 = 12 + 3 - 4 = 11\]Step 2: Now substitute this result back into the main expression, which becomes \(2 * 11\):\[2 * 11 = (2)(11) + 2 - 11\]\[2 * 11 = 22 + 2 - 11\]\[2 * 11 = 24 - 11 = 13\]The result of the nested binary operation is 13.

In a binary operation, let \(e\) be the identity element and \(a^{-1}\) be the inverse of an element \(a\). The property of the inverse states that: \[a * a^{-1} = e\]We are given the operation \(m*n = \frac{mn}{2}\), the identity element \(e = 2\), and we need to find the inverse of \(a = -5\). Let the inverse be \(x\). \[-5 * x = 2\]Applying the definition of the operation to the left side:\[\frac{(-5)(x)}{2} = 2\]To solve for \(x\), multiply both sides by 2:\[-5x = 4\]Divide both sides by -5:\[x = -\frac{4}{5}\]Therefore, the inverse of -5 under this specific operation is \(-\frac{4}{5}\).