The value of \(\int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}}\) will be

• (A) 0
• (B) \(\frac{\pi}{2}\)
• (C) \(\frac{\pi}{4}\)
• (D) \(\frac{\pi}{8}\)

The degree of differential equation \(9\frac{d^{2}y}{dx^{2}} = \left\{1 + \left(\frac{dy}{dx}\right)^{2}\right\}^{\frac{1}{3}}\) is

• (A) 1
• (B) 6
• (C) 3
• (D) 2

The value of expression \(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\) is

• (A) 0
• (B) 1
• (C) 2
• (D) 3

The modulus function \(f : \mathbb{R} \to \mathbb{R}^{+}\) given by \(f(x) = |x|\) is

• (A) one-one and onto
• (B) many-one and onto
• (C) one-one but not onto
• (D) neither one-one nor onto

A relation \(R = \{(a, b) : a = b - 1, b \geq 3\}\) is defined on set \(N\), then

• (A) \((2, 4) \in R\)
• (B) \((4, 5) \in R\)
• (C) \((4, 6) \in R\)
• (D) \((1, 3) \in R\)

Prove that the function \(f(x) = |x|\), is continuous at \(x = 0\).

Find the degree of the differential equation \(xy \frac{d^{2}y}{dx^{2}} + x \left(\frac{dy}{dx}\right)^{2} - y \left(\frac{dy}{dx}\right) = 2\)

If \(P(A) = 0.12, P(B) = 0.15\) and \(P(B/A) = 0.18\), then find the value of \(P(A \cap B)\).

Find the angle between the vectors \(-2\hat{i} + \hat{j} + 3\hat{k}\) and \(3\hat{i} - 2\hat{j} + \hat{k}\).

If \(f : \mathbb{R} \to \mathbb{R}\) and \(g : \mathbb{R} \to \mathbb{R}\) be functions defined by \(f(x) = \cos x\) and \(g(x) = 3x^{2}\) respectively, then prove that \(g \circ f \neq f \circ g\).

Find the general solution of differential equation \(ydx + (x - y^{2})dy = 0\).

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

If three vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) satisfying the condition \(\vec{a} + \vec{b} + \vec{c} = 0\). If \(|\vec{a}| = 3, |\vec{b}| = 4\) and \(|\vec{c}| = 2\), then find the value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\).

The radius of an air bubble is increasing at the rate of \(\frac{1}{2}\) cm/s. At what rate is the volume of the bubble increasing while the radius is 1 cm?

Show that the function \(f(x) = 7x^{2} - 3\) is an increasing function when \(x > 0\).