\(\left[3+\left|5-7 \sin ^{2} x\right|\right]^{2}\) lies in the interval (a) \([9,64]\) (b) \([3,8]\) (c) \([0,25]\) (d) \([9,25]\)

The inner absolute function is \(\left|5-7 \sin ^{2} x\right|\). To find the interval that this function lies in, we need to find its minimum and maximum values. We know that \(\sin^2 x\) ranges from \(0\) to \(1\), i.e., \(0 \leq \sin^2 x \leq 1\). Now, let's find the minimum and maximum values of 5-7 \(\sin^2 x\): Minimum value: If \(\sin^2 x\) is equal to its maximum value i.e., 1, then the minimum value of \(5-7 \sin^2 x\) will be: \(\min(5-7 \sin^2 x) = 5 - 7(1) = -2\). Maximum value: If \(\sin^2 x\) is equal to its minimum value i.e., 0, then the maximum value of \(5-7 \sin^2 x\) will be: \(\max(5-7 \sin^2 x) = 5 - 7(0) = 5\). Now we have this range for the term \((5-7 \sin^2 x)\): \(-2 \leq 5-7 \sin^2 x \leq 5\).

Since we now know that \(-2 \leq 5-7 \sin^2 x \leq 5\), we can find the interval of the absolute function \(\left|5-7 \sin ^{2} x\right|\) by taking the absolute value for the minimum and maximum values. For the minimum value, \(\min(\left|5-7 \sin ^{2} x\right|) = \left|-2\right| = 2\). For the maximum value, \(\max(\left|5-7 \sin ^{2} x\right|) = \left|5\right| = 5\). Therefore, the interval of the absolute function is \(2 \leq \left|5-7 \sin ^{2} x\right| \leq 5\).

Now we have to find the interval of the whole expression, which is \(\left[3+\left|5-7 \sin ^{2} x\right|\right]^{2}\). Add 3 to the absolute function interval: \(3+2 \leq 3+\left|5-7 \sin ^{2} x\right| \leq 3+5\), which is: \(5 \leq 3+\left|5-7 \sin ^{2} x\right| \leq 8\). Now, square the interval: \(\min(\left[3+\left|5-7 \sin ^{2} x\right|\right]^{2}) = 5^2 = 25\), \(\max(\left[3+\left|5-7 \sin ^{2} x\right|\right]^{2}) = 8^2 =64\). So the interval of the whole expression is \(25 \leq \left[3+\left|5-7 \sin ^{2} x\right|\right]^{2} \leq 64\). Comparing our result with the given options, we conclude that the function lies in the interval: \(Answer: (a) [9,64]\).