Answer

Verified

457.5k+ views

**Hint:**Solve the equation by using trigonometric formulae like $sin 2x= 2 sin x cos x$ and $cos 2x= 1- 2 sin ^2x$ and find the range of k .

**Complete step-by-step answer:**Given , the equation $sin x(sin x+cos x)=k$ has real solutions .

By solving the equation we get ,

$sin x(sin x+cos x)=k$

$\Rightarrow sin^2 x + sinx cos x=k……(i)$

We know that ,

$sin 2x= 2 sin x cos x$

$\Rightarrow sin x cos x = \dfrac{sin 2x}{2}…..(ii)$

Again ,

$cos 2x=1-2sin^2x$

$\Rightarrow 2sin^2x =1-cos 2x$

$\Rightarrow sin^2x = \dfrac{1}{2}-\dfrac{cos 2x}{2}…..(iii)$

Substituting the values of (ii) and (iii) in the equation (i) we get ,

$\dfrac{1}{2}-\dfrac{cos 2x}{2}+ \dfrac{sin 2x}{2}=k$

$\Rightarrow 1-cos 2x+sin2x=2k $

$\Rightarrow sin2x -cos 2x =2k-1……(iv)$

Now , let $f(x)=sin x- cos x$

$=\sqrt{2} \times \dfrac{1}{ \sqrt{2}}(sinx-cosx)$

$=\sqrt{2} (\dfrac{1}{ \sqrt{2}}sinx- \dfrac{1}{ \sqrt{2}}cosx$

$=\sqrt{2} (cos 45 ^{\circ} sin x - sin 45 ^{\circ} cos x )$

$=\sqrt{2}sin(x-\dfrac{\pi}{4})$

Range of the sin function is [ -1 , 1 ]

The range of f(x) will be $[- \sqrt{2}, \sqrt{2}]$----(v)

Putting the range in equation (iv) we get ,

$ sin2x -cos 2x =2k-1$

$\Rightarrow - \sqrt{2}\leq 2k-1 \leq \sqrt{2} $

$\Rightarrow (1- \sqrt{2})\leq 2k \leq (1+\sqrt{2}) $

Dividing both sides by 2 we get ,

$\Rightarrow (\dfrac{1- \sqrt{2}}{2}) \leq k \leq (\dfrac{1+ \sqrt{2}}{2})$

The range of k is $ (\dfrac{1- \sqrt{2}}{2}) \leq k \leq (\dfrac{1+\sqrt{2}}{2})$

**Thus , the correct option is d) $ (\dfrac{1- \sqrt{2}}{2}) \leq k \leq (\dfrac{1+ \sqrt{2}}{2})$.**

**Note:**Before starting the problem , the student needs to be properly acquainted with the trigonometric formulae . Now , after using the formulae , the equation can be easily solved . The student needs to find the range from the trigonometric functions in the equation (iv) . After that , they can put the values of the range and find the range of k . Students go wrong in understanding the problem and choosing the required formulae . Once they understand the concept properly , the sum can be solved easily .

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Derive an expression for drift velocity of free electrons class 12 physics CBSE

Which are the Top 10 Largest Countries of the World?

Write down 5 differences between Ntype and Ptype s class 11 physics CBSE

The energy of a charged conductor is given by the expression class 12 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Derive an expression for electric field intensity due class 12 physics CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Derive an expression for electric potential at point class 12 physics CBSE