VITEEE 2017 :: Mathematics :: General questions

• Mathematics - General questions

The function  f (x) = sin x - kx - c where k and c are constants, decreases always when

Answer:

Explanation:

Let f (x) = sin x - kx - c where k and c are constants

f'(x) = cos x - k

.'. f decreases if cos x ≤ k

Thus, f (x) = sin x - kx - c decrease always when k ≥1

The equation $z\overline{z} + (2 -3i)z + (2+3i)\overline{z} + 4 = 0$ represents a circle of radius

Answer:

Explanation:

Consider the equation

$z\overline{z} + (2 -3i)z + (2+3i)\overline{z} + 4 = 0$  ....(1)

$Let z= x + iy and \overline{z} = x - iy, z\overline{z} = x^{2} + y^{2}$

Put values of z, $\overline{z}$ and $z\overline{z}$ in equation 1, we get

(x² + y²) + (2 - 3i) (x + iy) + (2 + 3i) (x - iy) + 4 = 0

4x + 6y + 4 + x² + y² = 0

Now, we make it perfect square

x² + y² + 4x + 6y + 4 + 4 + 9 = 4+9

(x+2)² + (y+3)² = 9

This represents a circle of radius 3

Negation of the proposition : If we control population growth, we prosper

Answer:

Explanation:

p: we control population, q: we prosper

.'. we have p → q

Its negation is ˜ (p → q) i.e. p ^ ˜ q

i.e., we control population but we do not prosper

A value of c for which conclusion of Mean Value Theorem holds for the function f (x) = log_ex on the interval [1, 3] is

Answer:

Explanation:

Using Lagrange's Mean Value Theorem Let f(x) be a function defined on [a, b]

then f'(c) = $\frac{f(b) - f(a)}{b - a}$   ...(1)

$c\epsilon[a, b]$

$\therefore Given f(x) =\log_{e}{x}$                         $\therefore  f(x) =\frac{1}{x}$

equation (1) become

$\frac{1}{c}=\frac{f(3) - f(1)}{3 - 1}$

$\Rightarrow\frac{1}{c}=\frac{\log_{e}{3} - \log_{e}{1}}{2}$ = $\frac{\log_{e}{3}}{2}$

c = $2\log_{3}{e}$

sin^-1(sin 5) > x² - 4x holds if

Answer:

Explanation:

$\frac{3\pi}{2}<5<\frac{5\pi}{2}$

$\Rightarrow \sin^{-1}(\sin5)=5-2\pi$

$Given \sin^{-1}(\sin5)>x^{2}-4x$

$\Rightarrow x^{2}-4x+4<9-2\pi$

$\Rightarrow (x-2)^{2}<9-2\pi$

$\Rightarrow -\sqrt{9-2\pi}<x-2<\sqrt{9-2\pi}$

$\Rightarrow 2-\sqrt{9-2\pi}<x<2+\sqrt{9-2\pi}$

• Mathematics - General questions