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Solve the inequality \(3 + 2x - x^2 \geq 0\).
Find the domain and range of the function \(f(x) = \sqrt{6 - x - x^2}\).
If a, b, c, and d are in G.P., prove that \(a^2 - b^2, b^2 - c^2, c^2 - d^2\) are also in G.P.
Prove that \[ \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \\ \end{vmatrix} = xyz \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \]
Find the equation of the ellipse whose latus rectum is 5 and the eccentricity is \(\frac{1}{\sqrt{2}}\).
If \(\vec{a} = \sqrt{3} \hat{i} + \hat{j}\) and \(\vec{a} \times \vec{b} = (1, 2, 2)\), find the angle between \(\vec{a}\) and \(\vec{b}\).
How many numbers of three different digits less than 500 can be formed from the integers 1, 2, 3, 4, 5, and 6?
Attempt any TWO questions
[2x10=20]Prove that \[ \frac{3 + 4i}{1 - i} + \frac{3 - 4i}{1 + i} \] is a real number.
b. If \(x^2 + y^2 = 11xy\), prove that \[ \log\left(\frac{x - y}{3}\right)^2 = \frac{1}{2} (\log x + \log y) \]
a.Find the Maclaurin series of the function \(f(x) = \cos x\).
b.Take any matrix of order 3 × 3 and express it as a sum of symmetric and skew-symmetric matrix.
a.Find the equation of a hyperbola in standard form having focus (-2, 0) and Directrix \(x = -\frac{1}{2}\).
b.In an examination paper on mathematics, 20 questions are set. In how many different ways you can choose 18 questions to answer?