\(\frac{25}{4} - m\)
\(\frac{25}{4} - 2m\)
\(\frac{25}{4} + m\)
\(\frac{25}{4} + 2m\)
Correct answer is A
\(2x^{2} - 5x + m = 0\)
\(a = 2, b = -5, c = m\)
\(\alpha + \beta = \frac{-b}{a} = \frac{5}{2}\)
\(\alpha \beta = \frac{c}{a} = \frac{m}{2}\)
\(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta\)
= \((\frac{5}{2})^{2} - 2(\frac{m}{2}) \)
= \(\frac{25}{4} - m\)
Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.
\(6 + \sqrt{7}\)
\(3 + \sqrt{7}\)
\(3 - \sqrt{7}\)
\(6 - \sqrt{7}\)
Correct answer is B
Rationalizing \(\frac{2}{3 - \sqrt{7}}\) by multiplying through with \(3 + \sqrt{7}\),
\(\frac{2}{3 - \sqrt{7}} \frac{(3 + \sqrt{7})}{(3 + \sqrt{7})} = \frac{6 + 2\sqrt{7}}{9 - 7}\)
= \(\frac{6 + 2\sqrt{7}}{2} = 3 + \sqrt{7}\)
Which of the following binary operations is not commutative?
\(a * b = \frac{1}{a} + \frac{1}{b}\)
\(a * b = a + b - ab\)
\(a * b = 2a + 2b + ab\)
\(a * b = a - b + ab\)
Correct answer is D
All other options given are commutative i.e. \(a * b = b * a\), except option D.
\(a * b = a - b + ab\)
\(b * a = b - a + ba\)
\(a - b = -(b - a) \neq b - a\)
Find the coefficient of \(x^{4}\) in the binomial expansion of \((2 + x)^{6}\)
120
80
60
15
Correct answer is C
\((2 + x)^{6} \)
\(x^{4} = ^{6}C_{2}(2^{2})(x^{4}) = 15 \times 4 = 60\)
\(- \sqrt{3}\)
\(-\frac{\sqrt{3}}{2}\)
\(\frac{\sqrt{3}}{2}\)
\(\sqrt{3}\)
Correct answer is A
\(\sin \theta = \frac{\sqrt{3}}{2} \implies opp = \sqrt{3}; hyp = 2\)
\(adj^{2} = 2^{2} - (\sqrt{3})^{2} = 1 \implies adj = 1\)
\(\cos \theta = \frac{1}{2}\)
\(\sin 2\theta = \sin (180 - \theta) = \sin \theta = \frac{\sqrt{3}}{2}\)
\(\cos 2\theta = \cos (180 - \theta) = -\cos \theta = -\frac{1}{2}\)
\(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\)
= \(- \sqrt{3}\)