Question

Assertion (A): $3\sin x+4\cos x=7$ has no solution
Reason (R): $a\cos x+b\sin x=c$ has no solution if $|c|>\sqrt{a^2+b^2}$.

• A. Both A and R are true and R is correctexplanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true and R is correctexplanation of A

A function of type $a\sin x+b\cos x$ has the range of $[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$

$⟹c\in [-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$

Here , The Function is $3\sin x+4\cos x$ So the range is $[\sqrt{3^2+4^2},\sqrt{3^2+4^2}]$

$⟹[-5,5]$

$7$ does not lie in the given range .So, The equation has NO Solution.

Both A and R are true and R is correctexplanation of A.