Answer the following by appropriately matching the lists based on the information given in the paragraph Let $$f(x)=$$$$sin$$⁡$$($$π$$cos$$⁡$$x)$$ and $$g(x)=$$$$cos$$⁡(2$$$$π$$$$sin$$⁡$$x) be two functions defined for x$$0$$ . Define the following sets whose elements are written in the increasing order : $$X=x$$:$$fx=0$$,$$Y=x$$:$$f1x=0$$,$$Z=x$$:$$gx=0$$,$$W=x$$:$$g1x=0List$$ – I contains the sets X, Y, Z and W. List – II contains some information regarding these sets.Which of the following is the only CORRECT combination?

Question

Answer the following by appropriately matching the lists based on the information given in the paragraph Let $$f(x)=$$$$sin$$⁡$$($$π$$cos$$⁡$$x)$$ and $$g(x)=$$$$cos$$⁡(2$$$$π$$$$sin$$⁡$$x) be two functions defined for x>$$0$$ .  Define the following sets whose elements are written in the increasing order : $$X=x$$:$$fx=0$$,$$Y=x$$:$$f1x=0$$,$$Z=x$$:$$gx=0$$,$$W=x$$:$$g1x=0List$$ – I contains the sets X, Y, Z and W. List – II contains some information regarding these sets.Which of the following is the only CORRECT combination?

Infinity Learn · Accepted Answer

For x,$$__f(x)=0$$⇒$$sin$$⁡$$($$π$$cos$$⁡$$x)=0$$⇒π$$cos$$⁡$$x=n$$π⇒$$cos$$⁡$$x=0($$ or $$)1(or)$$−$$1$$⇒$$x=n$$$$π$$$$2$$,n∈N For Y,$$f1(x)=0$$⇒−π$$cos$$⁡$$($$π$$cos$$⁡$$x)$$$$sin$$⁡$$x=0$$⇒$$sin$$⁡$$x=0($$ or $$)$$$$cos$$⁡$$($$π$$cos$$⁡$$x)=0$$⇒$$x=n$$π,n∈N (or) π$$cos$$⁡$$x=(2n$$−$$1)$$π$$2$$,n∈N⇒$$cos$$⁡$$x=$$−$$12$$,$$12$$⇒$$x=2n$$π±π$$3$$,$$2n$$π±$$2$$$$π$$$$3$$,n∈I,x>$$0$$ For Z,$$g(x)=0=$$$$cos$$⁡(2$$$$π$$$$sin$$$$)=0$$⇒$$2$$$$π$$$$sin$$⁡$$x=(2n$$−$$1)π$$2$$,n∈Nsin⁡$$x=2n$$−$$14=$$−$$14$$,$$14$$,$$34$$,−$$34$$ For W,$$g1(x)=0$$⇒−$$2$$$$π$$$$sin$$⁡(2$$$$π$$$$sin$$$$)$$cos$$⁡$$x=0$$⇒$$sin$$⁡(2$$$$π$$$$sin$$⁡$$x)=0(or)$$cos$$⁡$$x=0$$ ⇒$$x=(2n$$−$$1)$$π$$2$$,n∈$$N(or)2$$$$π$$$$sin$$⁡$$x=n$$π⇒$$sin$$⁡$$x=n2=$$±$$1$$,±$$12$$,$$0$$ $$x=(2n$$−$$1)$$π$$2$$,nπ,n∈N nπ,nπ$$+($$−$$1)n$$$$π$$$$6$$,nπ$$+($$−$$1)n$$−π$$6$$

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App