MCQ
If $\tan\text{X}=\frac{\text{a}}{\text{b}},$ then $\text{b}\cos2\text{x}+\text{a}\sin2\text{x}$ is equal to:
- A$\text{a}$
- ✓$\text{b}$
- C$\frac{\text{a}}{\text{b}}$
- D$\frac{\text{b}}{\text{a}}$
✓
Answer
Correct option: B.
$\text{b}$
Givan: $\tan\text{x}=\frac{\text{a}}{\text{b}}$
Now,
$=\cos2\text{x}+\alpha\sin2\text{x}$
$=\text{b}\Big(\frac{1-\tan^2\text{x}}{1+\tan^2\text{x}}\Big)+\text{a}\Big(\frac{2\tan\text{x}}{1+\tan^2\text{x}}\Big)$
$=\text{b}\Bigg(\frac{1-\frac{\text{a}^2}{\text{b}^2}}{1+\frac{\text{a}^2}{\text{b}^2}}\Bigg)+\text{a}\Bigg(\frac{1\times\frac{\text{a}}{\text{b}}}{1+\frac{\text{a}^2}{\text{b}^2}}\Bigg)$
$=\frac{\text{b}(\text{b}^2-\text{a}^2)}{\text{a}^2+\text{b}^2}+\frac{2\text{a}^2\text{b}^2}{\text{a}^2+\text{b}^2}$
$=\frac{\text{b}^3-\text{a}^2\text{b}+2\text{a}^2\text{b}}{\text{a}^2+\text{b}^2}$
$=\frac{\text{b}^3+\text{a}^2\text{b}}{\text{a}^2+\text{b}^2}$
$=\frac{\text{b}(\text{b}^2+\text{a}^2)}{\text{a}^2+\text{b}^2}$
$=\text{b}$
Now,
$=\cos2\text{x}+\alpha\sin2\text{x}$
$=\text{b}\Big(\frac{1-\tan^2\text{x}}{1+\tan^2\text{x}}\Big)+\text{a}\Big(\frac{2\tan\text{x}}{1+\tan^2\text{x}}\Big)$
$=\text{b}\Bigg(\frac{1-\frac{\text{a}^2}{\text{b}^2}}{1+\frac{\text{a}^2}{\text{b}^2}}\Bigg)+\text{a}\Bigg(\frac{1\times\frac{\text{a}}{\text{b}}}{1+\frac{\text{a}^2}{\text{b}^2}}\Bigg)$
$=\frac{\text{b}(\text{b}^2-\text{a}^2)}{\text{a}^2+\text{b}^2}+\frac{2\text{a}^2\text{b}^2}{\text{a}^2+\text{b}^2}$
$=\frac{\text{b}^3-\text{a}^2\text{b}+2\text{a}^2\text{b}}{\text{a}^2+\text{b}^2}$
$=\frac{\text{b}^3+\text{a}^2\text{b}}{\text{a}^2+\text{b}^2}$
$=\frac{\text{b}(\text{b}^2+\text{a}^2)}{\text{a}^2+\text{b}^2}$
$=\text{b}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The equation of the locus of a point equidistant from the point $A (1, 3)$ and $B (-2, 1)$ is:
→$5$ boys and $5$ girls are sitting in a row randomly. The probability that boys and girls sit alternatively is
→A box contains $10$ mangoes out of which $4$ are rotten. $2$ mangoes are taken out together. If one of them is found to be good, the probability that the other is also good is
→is $ \text{f(x)} = \displaystyle \frac {\text{x}^2+6\text{x}}{\sin \text{x}}$ then $\lim_\limits{\text{x} \rightarrow 0} \text{f(x)=}$
→In the figure given below, if the areas of the two regions are equal then which of the following is true?
The straight line $x + 2y = 1$ meets the coordinate axes at $A$ and $B$. A circle is drawn through $A, B$ and the origin. Then the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is
→If a straight line passing through the point $P(-3, 4)$ is such that its intercepted portion between the coordinate axes is bisected at $P,$ then its equation is
→One card is drawn from a pack of $52$ cards. The probability that it is the card of a king or spade is:
→Answer the following by appropriately matching the lists based on the information given in the paragraph
→Let $f( x )=\sin (\pi \cos x )$ and $g ( x )=\cos (2 \pi \sin x )$ be two functions defined for $x >0$. Define the following sets whose elements are written in the increasing order :
$X =\{ x : f( x )=0\}, Y =\left\{ x : f^{\prime}( x )=0\right\}$
$Z =\{ x : g ( x )=0\}, W =\left\{ x : g ^{\prime}( x )=0\right\}.$
$List-I$ contains the sets $X , Y , Z$ and $W$. $List -II$ contains some information regarding these sets.
| $List-I$ | $List-II$ |
| $(I)$ $X$ | $(P)$ $\supseteq\left\{\frac{\pi}{2}, \frac{3 \pi}{2}, 4 \pi, 7 \pi\right\}$ |
| $(II)$ $Y$ | $(Q)$ an arithmetic progression |
| $(III)$ $Z$ | $(R)$ $NOT$ an arithmetic progression |
| $(IV)$ $W$ | $(S)$ $\supseteq\left\{\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{13 \pi}{6}\right\}$ |
| $(T)$ $\supseteq\left\{\frac{\pi}{3}, \frac{2 \pi}{3}, \pi\right\}$ | |
| $( U )$ $\supseteq\left\{\frac{\pi}{6}, \frac{3 \pi}{4}\right\}$ |
($1$) Which of the following is the only $CORRECT$ combination?
$(1) (II), (R), (S)$ $(2) (I), (P), (R)$ $(3) (II), (Q), (T)$ $(4) (I), (Q), (U)$
($2$) Which of the following is the only $CORRECT$ combinations?
$(1) (IV), (Q), (T)$ $(2) (IV), (P), (R), (S)$ $(3) (III), (R), (U)$ $(4) (III), (P), (Q), (U)$
Give the answer the quetion ($1$) and ($2$)
If $\frac{\pi }{2} < \alpha < \pi ,\,{\rm{ }}\pi < \beta < \frac{{3\pi }}{2};$ $\sin \alpha = \frac{{15}}{{17}}$ and $\tan \beta = \frac{{12}}{5}$, then the value of $\sin (\beta - \alpha )$ is
→