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Consider f(x)=x21+x2 and g(x)=px where pR. Then which of the following statements is (are) CORRECT?

A
If f(x)=g(|x|) has exactly three distinct solutions, then number of all possible real values of p is 1.
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B
If f(x)=g(|x|) has exactly five distinct solutions, then range of all possible real values of p is (0,12).
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C
Number of distinct tangents that can be drawn to the curve y=f(x) from the origin is 3.
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D
Number of distinct tangents that can be drawn to the curve y=f(x) from the origin is 1.
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Solution

The correct options are
A If f(x)=g(|x|) has exactly three distinct solutions, then number of all possible real values of p is 1.
B If f(x)=g(|x|) has exactly five distinct solutions, then range of all possible real values of p is (0,12).
C Number of distinct tangents that can be drawn to the curve y=f(x) from the origin is 3.
f(x)0
limxf(x)=1
So, range of f is [0,1)

Clearly, f(x)=g(|x|) has x=0 as one solution.
f(x)=g(|x|) has three solutions iff each branch of g(|x|)=p|x|, p>0 is tangent to y=f(x)
f(x)=p
2x(1+x2)2=p

Intersection point of f(x)=g(|x|):
x21+x2=2x2(1+x2)2
1+x2=2
x=±1
p=12 as p>0

If p>12, y=f(x) will not intersect y=g(|x|) and hence, there will be no solution.
For five distinct solutions, p(0,12)

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