The slope of the tangent at (x, y) to a curve passing through $\left(1, \frac{\mathrm{\pi }}{4}\right)$ is given by $\frac{\mathrm{y}}{\mathrm{x}} - {\cos }^2\left(\frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}\right)$, then the equation of the curve is

• $\mathrm{y} = {\tan }^{-1}\left[\log \left(\frac{\mathrm{e}}{\mathrm{x}}\right)\right]$

• $\mathrm{y} = {\mathrm{xtan}}^{-1}\left[\log \left(\frac{\mathrm{x}}{\mathrm{e}}\right)\right]$

• $\mathrm{y} = {\mathrm{xtan}}^{-1}\left[\log \left(\frac{\mathrm{e}}{\mathrm{x}}\right)\right]$

• None of the above

Let f(x) = $\frac{\mathrm{ax} +\hspace{0.17em}\mathrm{b}}{\mathrm{cx} +\hspace{0.17em}\mathrm{d}}$. Then, fof(x) = x provided that

• d = - a

• d = a

• a = b = c =  d = 1

• a = b = 1

If A = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\alpha }\right)\\ \cos \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\alpha }\right)& {\sin }^2\left(\mathrm{\alpha }\right)\end{array}\right]$ and B = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\beta }\right)& \cos \left(\mathrm{\beta }\right)\sin \left(\mathrm{\beta }\right)\\ \cos \left(\mathrm{\beta }\right)\sin \left(\mathrm{\beta }\right)& {\sin }^2\left(\mathrm{\beta }\right)\end{array}\right]$ are two matrices such that the product AB is null matrix, then $\mathrm{\alpha } - \mathrm{\beta }$ is

• 0

• multiple of $\mathrm{\pi }$

• an odd multiple of $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• None of these

If A is a square matrix of order n x n, then adj (adj A) is equal to

• ${\left|\mathrm{A}\right|}^{\mathrm{n}}\mathrm{A}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 1}\mathrm{A}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 2}\mathrm{A}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 3}\mathrm{A}$

If f(x) = ${\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)$, then f(x) is differentiable on

• [- 1, 1]

• R - {- 1, 1}

• R - (- 1, 1)

• None of these

The function f (x) = ${\mathrm{e}}^{- \left|\mathrm{x}\right|}$ is

• continuous everywhere but not differentiable at x = 0

• continuous and differentiable everywhere

• not continuous at x = 0

• None of the above

For the equations x + 2y + 3z = 1, 2x + y + 3z = 2 and 5x + 5y + 9z = 4

• there is only one solution

• there exists infinitely many solution

• there is no solution

• None of the above

8.

If y² = ax² + bx + c, where a, b, c are constants, then ${\mathrm{y}}^3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• a constant

• a function of x

• a function of y

• a function of x and y both

If  the function f(x) = 2x³ - 9ax² + 12a²x + 1 attains its maximum and minimum at p and q respectively such that p²  = q, then a equals 

• 0

• 1

• 2

• None of these

On the interval [0, 1] the function x²⁵(1 - x)⁷⁵ takes its maximum value at the point

• 0

• 1/4

• 1/2

• 1/3