If x²y⁵ = (x + y)⁷, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• y/x²

• x/y

• 1

• 0

If x = $\mathrm{x} = \sec \left(\mathrm{\theta }\right), \mathrm{y} = \tan \left(\mathrm{\theta }\right)$, then the value of $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ at $\mathrm{\theta } = \frac{\mathrm{\pi }}{4}$ is

• 0

• 1

• - 1

• 2

 If x = f(t) and y =g(t), then the value of $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• $\frac{\mathrm{f}\text{'}\left(\mathrm{t}\right)\mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\left(\mathrm{t}\right)\mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left\{\mathrm{f}\text{'}\left(\mathrm{t}\right)\right\}}^3}$

• $\frac{\mathrm{f}\text{'}\left(\mathrm{t}\right)\mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\left(\mathrm{t}\right)\mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left\{\mathrm{f}\text{'}\left(\mathrm{t}\right)\right\}}^2}$

• $\frac{\text{'}\left(\mathrm{t}\right)\mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right)\mathrm{f}\text{'}\left(\mathrm{t}\right)}{{\left\{\mathrm{f}\text{'}\left(\mathrm{t}\right)\right\}}^2}$

• $\frac{\mathrm{g}\text{'}\left(\mathrm{t}\right)\mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right)\mathrm{f}\text{'}\left(\mathrm{t}\right)}{{\left\{\mathrm{f}\text{'}\left(\mathrm{t}\right)\right\}}^3}$

The value of a and b such that the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{- 2\sin \left(\mathrm{x}\right), - \mathrm{\pi } \le \mathrm{x} \le - \frac{\mathrm{\pi }}{2} \\ \mathrm{asin}\left(\mathrm{x}\right) + \mathrm{b}, - \frac{\mathrm{\pi }}{2} < \mathrm{x} < \frac{\mathrm{\pi }}{2} \\ \cos \left(\mathrm{x}\right), \frac{\mathrm{\pi }}{2} \le \mathrm{x} \le \mathrm{\pi }\right\} \end{gathered}$$

is continuous in $\left[- \mathrm{\pi }, \mathrm{\pi }\right]$, are

• - 1, 0

• 1, 0

• 1, 1

• - 1, 1

5.

The equation of tangent to the curve y² = ax² + b at point (2, 3) is y = 4x - 5, then the values of a and b are

• 3, - 5

• 6, - 5

• 6, 15

• 6, - 15

Let A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\end{array}\right]$, then the inverse of A is

• $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}- \cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}- \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\\ - \cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\end{array}\right]$

The value of f(4) - f(3) is

• $∆\mathrm{f}\left(2\right) + {∆}^2\mathrm{f}\left(1\right) + {∆}^3\mathrm{f}\left(1\right)$

• $∆\mathrm{f}\left(3\right) + {∆}^2\mathrm{f}\left(2\right) + {∆}^3\mathrm{f}\left(1\right)$

• $∆\mathrm{f}\left(2\right) + {∆}^2\mathrm{f}\left(1\right) + {∆}^3\mathrm{f}\left(0\right)$

• None of these

${\left(1 + ∆\right)}^{\mathrm{n}}\mathrm{f}\left(\mathrm{a}\right)$ is equal to

• f(a + h)

• f(a + 2h)

• f(a + nh)

• f(a + (n - 1)h)

Simplify the Boolean function (x · y) + [(x + y') - y]'.

• 0

• 1

• x + y

• xy

If matrix $\mathrm{A} = \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$, then ${\left|\mathrm{A}\right|}^{- 1}$ is equal to

• ab - bc

• $\frac{1}{\mathrm{ab} - \mathrm{bc}}$

• $\frac{1}{\mathrm{ab} - \mathrm{bc}}\left[\begin{array}{cc}\mathrm{d}& - \mathrm{b}\\ - \mathrm{c}& \mathrm{a}\end{array}\right]$

• None of these