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Question

If the given points (a, 5, a3),(a+1, a2, a),(1,85,16) are collinear, then which of the following is/are correct ?

A
limx2(ax2+22x+1)1/(2a+11x)=e2
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B
limx2(ax2+22x+1)1/(2a+11x)=e2
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C
limxax2+8x33x2+x110=23
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D
The number of real solutions of the equation x2+ax+2021=sinx is zero
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Solution

The correct option is D The number of real solutions of the equation x2+ax+2021=sinx is zero
For collinear points:
x3x1x2x1=y3y1y2y1=z3z1z2z1
1a1=85+5a+3=16a+33
1a1=19a3
33a=19a
2a=22
a=11
satisfy all three conditions.
Now,
limx2(ax2+22x+1)1/(2a+11x)
=elimx2ax2+22x2a+11x
=elimx211x(x+2)11(x2)=e2

limxax2+8x33x2+x110
=limxa(x+11)(x3)(x+11)(x10)
=1421=23

x2+ax+2021=x211x+2021
Let f(x)=x211x+2021
f(x)=2x11
f(x)=0x=112
minimum of f(x) is equal to f(112)=20211214=79634>1
But g(x)=sinx1
Hence, f(x)=g(x) has no real solutions.

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