MCQ
If $\sin\alpha+\sin\beta=\text{a}$ and $\cos\alpha-\cos\beta=\text{b}$ then $\tan\frac{\alpha-\beta}{2}=$
- A$-\frac{\text{a}}{\text{b}}$
- B$-\frac{\text{b}}{\text{a}}$
- C$\sqrt{\text{a}^2+\text{b}^2}$
- DNone of these
Solution:
Given:
$\sin\alpha+\sin\beta=\text{a}$
$\Rightarrow2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha+\beta}{2}=\text{a}\ .....(1)$
Also,
$\cos\alpha+\cos\beta=\text{b}$
$\Rightarrow-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha+\beta}{2}=\text{b}\ .....(2)$
On dividing (1) by (2), we get
$\frac{-\cos\frac{\alpha+\beta}{2}}{\sin\frac{\alpha-\beta}{2}}=\frac{\text{a}}{\text{b}}$
$\Rightarrow\frac{-\sin\frac{\alpha+\beta}{2}}{\cos\frac{\alpha-\beta}{2}}=\frac{\text{b}}{\text{a}}$
$\Rightarrow\tan\frac{\alpha-\beta}{2}=-\frac{\text{b}}{\text{a}}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random. What is the probability that they are of different colours?
If $\cos\text{x}=\frac{1}{2}\Big(\text{a}+\frac{1}{\text{a}}\Big),$ and $3\text{x}=\lambda\Big(\text{a}^3+\frac{1}{\text{a}^3}\Big),$ then $\lambda=$
A real value of x satisfies the equation $\Big(\frac{3-4\text{ix}}{3+4\text{ix}}\Big)=\alpha-\text{i}\beta(\alpha,\beta\in\text{R})$ if $\alpha^2+\beta^2=$
$\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}^{\text{m}}-1}{\text{x}^{\text{n}}-1}$ is equal to:
$1$
$\frac{\text{m}}{\text{n}}$
$\frac{-\text{m}}{\text{n}}$
$\text{m}^{2}\text{n}^{2}$