Question

Find the equations of the tangent and normal to the curve ${\frac{x^{2}}{a^{2}}}-{\frac{y^{2}}{b^{2}}}=1$ at the point $({\sqrt{2}}a,b)$ .

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

Using properties of determinants, prove that $\begin{vmatrix}x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x\end{vmatrix}$ = x³

In a school, there are 1,000 students, out of which 380 are girls. Out of 380 girls, 10% of the girls scored highest in GS. What is the probability that a student chosen randomly scored highest in GS given that the chosen student is a girl ?

Show that the relation S in the set R of real numbers defined as S = {(a, b) : a, b ∈ R and a ≤ b³} is neither reflexive nor symmetric nor transitive.

Find the particular solution of the differential equation satisfying the given condition ${\frac{d y}{d x}}=y\tan x,$ given that y = 1, when x = 0.

Find the equation of tangent to the curve y = cos (x + y), -2π ≤ x ≤ 0, that is parallel to the line x + 2y = 0.