Find the fourth term of the binomial expansion of \((x - k)^{5}\) in descending powers of x.<
\(10x^{3}k^{2}\)
\(5x^{3}k^{2}\)
\(-5x^{2}k^{3}\)
\(-10x^{2}k^{3}\)
Correct answer is D
\((x - k)^{5} = ^{5}C_{0}x^{5}(-k)^{0} + ^{5}C_{1}x^{4}(-k)^{1} + ...\)
The fourth term in the expansion = \(^{5}C_{4 - 1}(x)^{5 - 3}(-k)^{3 = 10 \times x^{2} \times -k^{3}\)
= \(-10x^{2}k^{3}\)
\(\frac{2x}{x - 3}, x \neq 3\)
\(\frac{2x}{x + 3}, x \neq -3\)
\(\frac{3x}{x - 3}, x \neq 3\)
\(\frac{3x}{x + 3}, x \neq -3\)
Correct answer is A
From the rules of binary operation, \(x * e = x\)
\(\implies x * e = 3x + 3e - xe = x\)
\(3e - xe = x - 3x = -2x\)
\(e = \frac{2x}{x - 3}, x \neq 3\)
Simplify \(\frac{\tan 80° - \tan 20°}{1 + \tan 80° \tan 20°}\)
\(3\sqrt{2}\)
\(2\sqrt{3}\)
\(\sqrt{3}\)
\(\sqrt{2}\)
Correct answer is C
\(\tan (x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}\)
\(\implies \frac{\tan 80 - \tan 20}{1 + \tan 80 \tan 20} = \tan (80 - 20) = \tan 60°\)
\(\tan 60 = \frac{\sin 60}{\cos 60} = \frac{\sqrt{3}}{2} ÷ \frac{1}{2}\)
= \(\sqrt{3}\)
Simplify \(\frac{\sqrt{3}}{\sqrt{3} - 1} + \frac{\sqrt{3}}{\sqrt{3} +1}\)
\(\frac{1}{2}\)
\(\frac{1}{2}\sqrt{3}\)
\(3\)
\(2\sqrt{3}\)
Correct answer is C
\(\frac{\sqrt{3}}{\sqrt{3} - 1} + \frac{\sqrt{3}}{\sqrt{3} + 1}\)
= \(\frac{\sqrt{3}(\sqrt{3} + 1) + \sqrt{3}(\sqrt{3} - 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}\)
= \(\frac{6}{3 - 1} \)
= 3
-4
0
4
12
Correct answer is D
\(Gradient = \frac{y_{1} - y_{2}}{x_{1} - x_{2}} \)
\(\frac{1}{2} = \frac{5 - 9}{4 - x}\)
\(\frac{1}{2} = \frac{-4}{4 - x} \implies -8 = 4 - x\)
\(x = 4 + 8 = 12\)