Let the maximum and minimum value of the expression 2 ^ 2 θ + cos θ...

2\cos^{2} \theta + cos \theta + 1 = 2\left(\cos^{2} \theta + \dfrac{cos \theta}{2} + \dfrac{1}{2}\right)$<br /><br />$2\left\{{\left(cos \theta + \dfrac{1}{4}\right)}^{2} + \dfrac{7}{16}\right\}$<br /><br />Given expression is maximum when$cos \theta = 1$and minimum when$cos \theta = - \dfrac{1}{4}$<br /><br />$⇒ M = 2\left({\left(\dfrac{5}{4}\right)}^{2} + \dfrac{7}{16}\right) = 2\left(\dfrac{32}{16}\right) = 4$</p><p>and$m = 2\left(\dfrac{7}{16}\right) = \dfrac{7}{8}$</p><p>Hence, \left[\dfrac{M}{m}\right] = \left[\dfrac{32}{7}\right] = 4$