Statement 1: There are 8 integral values of λ for which 7 cos x + 5 sin x = 2λ + 1 has a solution. Statement 2: a cos θ + b sin θ = c has at least one solution if |c|

Question

Statement 1: There are 8 integral values of λ for which 7 cos x + 5 sin x = 2λ + 1 has a solution. Statement 2: a cos θ + b sin θ = c has at least one solution if |c| < √(a² + b²).

TestHub · Accepted Answer

The left side ranges over [-√74, √74]. Since 2λ + 1 is an integer and √74 is irrational, the endpoints cannot occur. Statement 2 gives |2λ + 1| < √74, yielding λ = -4, -3, -2, -1, 0, 1, 2, 3: eight integers. Both are true and Statement 2 explains Statement 1.

Mathematics - Trigonometric Equation Question with Solution | TestHub

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