Column-I Column-II (I) The – InterviewSolution

Column-I Column-II (I) The curve C which passes through (1,1) and has differential equation as (2x2y−2y4)dx+(2x3+3xy3)dy=0 is given by αln|x|+βln|y|+y3x2=γ, then (α+β+γ) equals- (P) 1 (II) A curve y=f(x) passes through (2,0) and slope of tangent at any point P(x,y) on it equals (x+1)2+(y−3)(x+1), then f(3) is- (Q) 3 (III) Let f:R+→R satisfies the functional equation f(xy)=exy−y−x(eyf(x)+exf(y)) for all x,y∈R+. If f′(1)=e, then ln(ln(f(e))) equals- (R) 5 (IV) A curve y=f(x) passing through the point (12,14) satisfies the differential equation xydydx=3x2−y2y−x2 is a conic whose length of latus rectum is- (S) 2 (T) 4 Which of the following is only correct combination?

Answer»

Column-I	Column-II
$(I)$ The curve $C$ which passes through $(1, 1)$ and has differential equation as $(2 x^{2} y - 2 y^{4}) d x + (2 x^{3} + 3 x y^{3}) d y = 0$ is given by $α ln \| x \| + β ln \| y \| + \frac{y^{3}}{x^{2}} = γ$ , then $(α + β + γ)$ equals-	(P) $1$
$(I I)$ A curve $y = f (x)$ passes through $(2, 0)$ and slope of tangent at any point $P (x, y)$ on it equals $\frac{(x + 1)^{2} + (y - 3)}{(x + 1)}$ , then $f (3)$ is-	(Q) $3$
$(I I I)$ Let $f : R^{+} \to R$ satisfies the functional equation $f (x y) = e^{x y - y - x} (e^{y} f (x) + e^{x} f (y))$ for all $x, y \in R^{+}$ . If $f^{'} (1) = e,$ then $ln (ln (f (e)))$ equals-	(R) $5$
$(I V)$ A curve $y = f (x)$ passing through the point $(\frac{1}{2}, \frac{1}{4})$ satisfies the differential equation $\frac{x}{y} \frac{d y}{d x} = \frac{3 x^{2} - y}{2 y - x^{2}}$ is a conic whose length of latus rectum is-	(S) $2$
	(T) $4$

Which of the following is only correct combination?