If the equation cos^{4} theta + sin^{4} theta | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the equation $\cos^{4} 	heta + \sin^{4} 	heta + \lambda = 0$,we can write $\lambda = -(\sin^{4} 	heta + \cos^{4} 	heta)$.Using the ident

Vedclass · Accepted Answer

$[-1, -\frac{1}{2}]$

Vedclass · Answer

(D) Given the equation $\cos^{4} 	heta + \sin^{4} 	heta + \lambda = 0$,we can write $\lambda = -(\sin^{4} 	heta + \cos^{4} 	heta)$.Using the identity $\sin^{4} 	heta + \cos^{4} 	heta = (\sin^{2} 	heta + \cos^{2} 	heta)^{2} - 2 \sin^{2} 	heta \cos^{2} 	heta$,we get $\sin^{4} 	heta + \cos^{4} 	heta = 1 - \frac{1}{2} \sin^{2} 2	heta$.Thus,$\lambda = -(1 - \frac{1}{2} \sin^{2} 2	heta) = \frac{1}{2} \sin^{2} 2	heta - 1$.Since $0 \le \sin^{2} 2	heta \le 1$,we have $0 \le \frac{1}{2} \sin^{2} 2	heta \le \frac{1}{2}$.Subtracting $1$ from all parts,we get $-1 \le \frac{1}{2} \sin^{2} 2	heta - 1 \le -\frac{1}{2}$.Therefore,$\lambda \in [-1, -\frac{1}{2}]$.

If the equation $\cos^{4} \theta + \sin^{4} \theta + \lambda = 0$ has real solutions for $\theta$,then $\lambda$ lies in the interval

Similar Questions

Two cells $A$ and $B$ are connected in the secondary circuit of a potentiometer one at a time,and the balancing lengths are $400 \ cm$ and $440 \ cm$ respectively. The emf of cell $A$ is $1.08 \ V$. The emf of the second cell $B$ in volts is:

The negation of the statement ''If $I$ become a teacher,then $I$ will open a school'',is

Which of the following is a homosporous pteridophyte?

Which oxide of manganese is amphoteric?

Vedclass Test Series

Exam Paper Generator

Online Exam Module