MCQ
The set of all values of $\lambda$ for which the equation $\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x=\lambda$
- A$[-2,-1]$
- B$\left[-2,-\frac{3}{2}\right]$
- C$\left[-1,-\frac{1}{2}\right]$
- ✓$\left[-\frac{3}{2},-1\right]$
✓
Answer
Correct option: D.
$\left[-\frac{3}{2},-1\right]$
d
$\lambda=\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x$
$\lambda=\cos ^2 2 x-2 \sin ^4 x-2 \cos ^2 x$
$\text { convert all in to } \cos x \text {. }$
$\lambda=\left(2 \cos ^2 x-1\right)^2-2\left(1-\cos ^2 x\right)^2-2 \cos ^2 x$
$=4 \cos ^4 x-4 \cos ^2 x+1-2\left(1-2 \cos ^2 x+\cos ^4 x\right)-$
$2 \cos ^2 x$
$=2 \cos ^4 x-2 \cos ^2 x+1-2$
$=2 \cos ^4 x-2 \cos ^2 x-1$
$=2\left[\cos ^4 x-\cos ^2 x-\frac{1}{2}\right]$
$=2\left[\left(\cos ^2 x-\frac{1}{2}\right)^2-\frac{3}{4}\right]$
$\left.\lambda_{\max }=2\left[\frac{1}{4}-\frac{3}{4}\right]=2 \times\left(-\frac{2}{4}\right)=-1 \text { (max Value }\right)$
$\left.\lambda_{\min }=2\left[0-\frac{3}{4}\right]=-\frac{3}{2} \text { (MinimumValue }\right)$
$\text { So, Range }=\left[-\frac{3}{2},-1\right]$
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
If $m$ is chosen in the quadratic equation $\left( {{m^2} + 1} \right)\,{x^2} - 3x + {\left( {{m^2} + 1} \right)^2} = 0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is
→The area of the region bounded by $x = 0, y = 0,x = 2, y = 2, y \leq\ e^x$ and $y\geq\ log\ x$ is
→The value of $\int_{}^{} {\left( {1 + \frac{1}{{{x^2}}}} \right)\;{e^{\left( {x - \frac{1}{x}} \right)}}} \;dx$ equals
→Consider a square matrix of order $5$ such that ${a_{ij}} = 0\,\,\forall \,\,i + j\, = n + 1,\,a_{ij}\, \in \left\{ {0,1} \right\}\,\,\forall \,\,i,j$ . In each row as well as in each column there in only one non-zero element. Then number of such matrices is
→If a chord, which is not a tangent, of the parabola $y^2=16 x$ has the equation $2 x+y=p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p, h$ and $k$ ?
→In the four numbers first three are in $G.P.$ and last three are in $A.P.$ whose common difference is $6$. If the first and last numbers are same, then first will be
→The circles $x^2 + y^2 + 2x -2y + 1 = 0$ and $x^2 + y^2 -2x -2y + 1 = 0$ touch each other :-
→The value of $k$ for which one of the roots of ${x^2} - x + 3k = 0$ is double of one of the roots of ${x^2} - x + k = 0$ is
→A board has 16 squares as shown in the figure :
out of these 16 squares, two squares are chosen at random. the probality that they have no side in common is :
out of these 16 squares, two squares are chosen at random. the probality that they have no side in common is :
Let $a$ be the largest real root and $b$ be the smallest real root of the polynomial equation $x^6-6 x^5+15 x^4-20 x^3+15 x^2-6 x+1=0$ Then $\frac{a^2+b^2}{a+b+1}$ is
→