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Question

# If the given points (a, −5, a−3),(a+1, a−2, 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 zeroFor collinear points: ⇒x3−x1x2−x1=y3−y1y2−y1=z3−z1z2−z1 ⇒−1−a1=−85+5a+3=16−a+33 ⇒−1−a1=19−a3 ⇒−3−3a=19−a ⇒2a=−22 ⇒a=−11 satisfy all three conditions. Now, →limx→2(ax2+22x+1)1/(2a+11x) =elimx→2ax2+22x2a+11x =elimx→211x(−x+2)11(x−2)=e−2 →limx→ax2+8x−33x2+x−110 =limx→a(x+11)(x−3)(x+11)(x−10) =−14−21=23 →x2+ax+2021=x2−11x+2021 Let f(x)=x2−11x+2021 f′(x)=2x−11 ∴f′(x)=0⇒x=112 ∴ minimum of f(x) is equal to f(112)=2021−1214=79634>1 But g(x)=sinx≤1 Hence, f(x)=g(x) has no real solutions.

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