JEE Main 2022 · 24 Jun · Shift 1 · Mathematics Q11

MathematicsSingle correct+4 / -1Question 11 of 90

If the tangent at the point $(x_1,y_1)$ on the curve $y=x^3+3x^2+5$ passes through the origin, then $(x_1,y_1)$ does NOT lie on the curve :

x^2+\dfrac{y^2}{81}=2

\dfrac{y^2}9-x^2=8

y=4x^2+5

\dfrac x3-y^2=2

Answer and worked solution

Official answer: D

Tangent through the origin. $y_1=x_1\,y'(x_1)$ : $x_1^3+3x_1^2+5=3x_1^3+6x_1^2$ , i.e. $2x_1^3+3x_1^2-5=0$ .

Root. $(x_1-1)(2x_1^2+5x_1+5)=0$ and the quadratic has no real roots, so the point is $(1,9)$ .

Test. (A) $1+1=2$ ✓; (B) $9-1=8$ ✓; (C) $4+5=9$ ✓; (D) $\frac13-81\ne2$ .

(D)

Topics:Tangents and normals

JEE Main 2022 · 24 Jun · Shift 1, Mathematics question 11