Consider the curve represented parametrically by the equation x=t3-4 t2-3 t text and y=2 t2+3 t-5 text where t ∈ R . If H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then

Question

Consider the curve represented parametrically by the equation x=t3-4 t2-3 t text and y=2 t2+3 t-5 text where t ∈ R . If H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then (A) H =2 and V =1 (B) H =1 and V =2 (C) H =2 and V =2 (D) H =1 and V =1

Tardigrade Admin · Accepted Answer

( dx / dt )=3 t 2-8 t -3 ; ( dy / dt )=4 t +3 ∴ ( dy / dx )=(4 t +3/3 t 2-8 t -3) Hence H means 4 t +3=0 ⇒ t =-3 / 4 ⇒ H =1 V means 3 t2-8 t-3=0 ⇒ t=3 t=-1 / 3 ⇒ V=2

Q. Consider the curve represented parametrically by the equation
$x = t^{3} - 4 t^{2} - 3 t and$
$y = 2 t^{2} + 3 t - 5 where t \in R .$
If $H$ denotes the number of point on the curve where the tangent is horizontal and $V$ the number of point where the tangent is vertical then

$H = 2$ and $V = 1$

$H = 1$ and $V = 2$

$H = 2$ and $V = 2$

$H = 1$ and $V = 1$

$\frac{d x}{d t} = 3 t^{2} - 8 t - 3; \frac{d y}{d t} = 4 t + 3$
$∴ \frac{d y}{d x} = \frac{4 t + 3}{3 t ^{2} - 8 t - 3}$
Hence $H$ means $4 t + 3 = 0 \Rightarrow t = - 3/4 \Rightarrow H = 1$
$V$ means $3 t^{2} - 8 t - 3 = 0 \Rightarrow t = 3& t = - 1/3 \Rightarrow V = 2$

Q. Consider the curve represented parametrically by the equation x=t3−4t2−3t and y=2t2+3t−5 where t∈R. If H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then

H=2 and V=1

H=1 and V=2

H=2 and V=2

H=1 and V=1