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(i) The points \((-1,0),(4,-2)\) and \((\cos 2 \theta, \sin 2 \theta)\) are collinear (ii) The points \((-1,0),(4,-2)\) and $\left(\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}, \frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)$ are collinear (a) both statements are equivalent (b) statement (i) has more solution than statement (ii) for \(\theta\) (c) statement (ii) has more solution than statement (i) for \(\theta\) (d) none of the above

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Option (c) statement (ii) has more solution than statement (i) for \( \theta \)
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Step 1: Collinearity Condition

Three points \( A(x_1,y_1), B(x_2,y_2) \) and \( C(x_3,y_3) \) are collinear if the determinant formed by their coordinates is zero, according to the formula \(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0\).

Step 2: Apply Collinearity Condition to Statement (i)

Apply the collinearity condition to the coordinates of the points given in the first statement. After substituting the values and simplifying, we get the equation \( -2 \cdot \cos 2\theta - 4(1 - \sin 2\theta) = 0 \). Solving it gives \(\theta = n \pi\), where n is any integer.

Step 3: Apply Collinearity Condition to Statement (ii)

Use the same collinearity condition for the second statement. After substituting values and simplifying, we get the equation \( -4(1 - \tan^2\theta) + 2(1 + \tan^2\theta) \cdot \tan\theta = 0 \). Upon solving, we get \(\theta = n \pi, (2n+1) \frac{\pi}{4}\), where n is an integer.

Step 4: Compare the Solutions of the Statements

Compare the solutions of the two statements. It is evident that the second statement \( (ii) \) has more solutions for \( \theta \) compared to statement \( (i) \). Therefore, the answer is option (c).

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