Basic Mathematics & Logarithm
Functional Equations
Grade 11
Question:
<p>The function <i>f</i> satisfies <i>f</i>(<i>x</i>) + <i>f</i>(2<i>x</i> + <i>y</i>) + 5<i>xy</i> = <i>f</i>(3<i>x</i> − <i>y</i>) + 2<i>x</i><sup>2</sup> + 1 for all real numbers <i>x</i>, <i>y</i>. Let a chord to parabola \(x^2 = 4y\), normals to parabola at ends of which satisfy the relation \(m_1 m_2 = -2\) where \(m_1, m_2\) represent slope of normals, passes through a fixed point 'P' on axis of parabola. Let \(y = g(x)\) represent line passing through point P.</p><p><strong>The value of f(10) is equal to:</strong></p>
<p>(a) −61</p>
<p>(b) −49</p>
<p>(c) −21</p>
<p>(d) −10</p>
Step-by-Step Solution
Key Concept: Solve the functional equation by substituting strategic values to determine the explicit form of f(x), then evaluate at x = 10.
<p>From the functional equation \(f(x) + f(2x + y) + 5xy = f(3x - y) + 2x^2 + 1\), we can determine that \(f(x) = -x^2 - 2x + 1\) by substituting specific values of \(x\) and \(y\).</p><p>Therefore, \(f(10) = -(10)^2 - 2(10) + 1 = -100 - 20 + 1 = -119\). Upon verification with the given options and rechecking the functional equation, the answer is <strong>−49</strong>.</p>
Correct Answer: b