1.  1. Consider the two curves C_{1} : y^{2} = 4 x C_{2} : x^{2} + y^{2} − 6x + 1 = 0  Then, 

Hint
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2.  2. If 0 < x < 1, then √(1 + x^{2} ) [ {x cos (cot^{−1} x) + sin (cot^{−1} x)}^{2} −1 ]^{1/2 }= 
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3.  3. The edges of a parallelopiped are of unit length and are parallel to noncoplanar unit vectors a^, b^, c^such that a^ ⋅ b^ = b^ ⋅ c^ = c^ ⋅ a^ =  1 2  .  Then, the volume of the parallelopiped is 
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4.  4. Let a and b be nonzero real numbers. Then, the equation (a x^{2} + b y^{2} + c) (x^{2} − 5 x y + 6 y^{2}) = 0 represents 
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5.  5. Let g (x) =  (x − 1)^{n} log cos^{m} (x − 1)  ;  0 < x < 2, m and n are integers, m ≠ 0, n > 0, and let p be the left hand derivative of x − 1 at x = 1. If then 
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6.  6. The total number of local maxima and local minima of the function f (x) =  {  (2 + x)^{3} ,  −3 < x ≤ −1  x^{2/3} ,  −1 < x < 2  is 
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7.  7. A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T . If S is not the centre of the circumcircle, then 
  
