Let the set of all a R such that the equation 2 x+a x=2 a-7 has a… - Vidyadip

MCQ

Let the set of all $a \in R$ such that the equation $\cos 2 x+a \sin x=2 a-7$ has a solution be $[p, q]$ and $\mathrm{r}=\tan 9^{\circ}-\tan 27^{\circ}-\frac{1}{\cot 63^{\circ}}+\tan 81^{\circ}$, then $pqr$ is equal to ....................

A
$62$
B
$55$
✓
$48$
D
$45$

✓

Answer

Correct option: C.

$48$

c
$\cos 2 x+a \cdot \sin x=2 a-7 $

$ a(\sin x-2)=2(\sin x-2)(\sin x+2) $

$ \sin x=2, \quad a=2(\sin x+2) $

$ \Rightarrow a \in[2,6] $

$ p=2 \quad q=6$

$ r=\tan 9^{\circ}+\cot 9^{\circ}-\tan 27-\cot 27 $

$ r=\frac{1}{\sin 9 \cdot \cos 9}-\frac{1}{\sin 27 \cdot \cos 27} $

$=2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right] $

$ r=4 $

$ \text { p.q. } r=2 \times 6 \times 4=48$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from 3. trignometrical ratios,functions and identities→3. trignometrical ratios,functions and identities — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The expression,$\frac{{\tan \,\left( {{\textstyle{{3\,\pi } \over 2}}\,\, - \,\,\alpha } \right)\,\,\,\cos \,\left( {{\textstyle{{3\,\pi } \over 2}}\,\, - \,\,\alpha } \right)}}{{\cos \,(2\,\pi \,\, - \,\alpha )}}$ $+ cos \left( {\alpha \,\, - \,\,\frac{\pi }{2}} \right) \,sin (\pi -\alpha ) + cos (\pi +\alpha ) sin \,\left( {\alpha \,\, - \,\,\frac{\pi }{2}} \right)$ when simplified reduces to :

The latus-rectum of the hyperbola $16{x^2} - 9{y^2} = $ $144$, is

Given that $n$ A.M.'s are inserted between two sets of numbers $a,\;2b$and $2a,\;b$, where $a,\;b \in R$. Suppose further that ${m^{th}}$ mean between these sets of numbers is same, then the ratio $a:b$ equals

A chord $AB$ drawn from the point $A(0,3)$ on circle ${x^2} + 4x + {(y - 3)^2} = 0$ meets to $M$ in such a way that $AM = 2AB$, then the locus of point $M$ will be

If $\text{f(z)}=\frac{7-\text{z}}{1-\text{z}^2},$ where $\text{z}=1+2\text{i},$ then $|\text{f(z)}|$ is:

Choose the correct answer: The negation of the statement. “Rajesh or Rajni lived in Bangalore” is.

Circle$(s)$ touching $x$-axis at a distance $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on $y$-axis is (are)

$(A)$ $x^2+y^2-6 x+8 y+9=0$

$(B)$ $x^2+y^2-6 x+7 y+9=0$

$(C)$ $x^2+y^2-6 x-8 y+9=0$

$(D)$ $x^2+y^2-6 x-7 y+9=0$

Let $P \left( x _0, y _0\right)$ be the point on the hyperbola $3 x ^2-4 y ^2$ $=36$, which is nearest to the line $3 x+2 y=1$. Then $\sqrt{2}\left( y _0- x _0\right)$ is equal to :

$ABC$ is an isosceles triangle inscribed in a circle of radius $r$ . If $AB = AC $ $\& h$ is the altitude from $A$ to $BC$ and $P$ be the perimeter of $ABC$ then $\mathop {lim}\limits_{h \to 0} $ $\frac{\Delta }{{{P^3}}}$ equals (where $\Delta$ is the area of the triangle)

Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1 d_2=10$. For each $i=1,2, \ldots, 100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$. If $A_{51}-A_{50}=1000$, then the value of $A_{100}-A_{90}$ is. . . . .