\[ \Delta_1 = \begin{vmatrix} m & b \\ n & d \\ \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} a & m \\ c & n \\ \end{vmatrix}, \quad \Delta_3 = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} \]

Then the values of $ x $ and $ y $ respectively (where $ e $ is the base of the natural logarithm) are:

If the roots of the equation $x^2 + 2ax + b = 0$ are real, distinct and differ utmost by $2m$, then b lies in the interval

Evaluate: \[ \sin^3 10^\circ + \sin^3 50^\circ - \sin^3 70^\circ = ? \]

G(1,0) is the centroid of the triangle ABC. If A = (1, -4, 2) and B = (3, 1, 0), then AG$^2$ + CG$^2$ =
Identify the correct option from the following:

If $ x \ne (2n+1)\frac{\pi}{4} $, then the general solution of $ \cos x + \cos 3x = \sin x + \sin 3x $ is

If \[ \frac{x^2}{(x^2 + 2)(x^4 - 1)} = \frac{A}{x^2 - 1} + \frac{B}{x^2 + 1} + \frac{C}{x^2 + 2}, \text{ then } A + B - C =\ ? \]

If the probability distribution of a random variable $X$ is as follows, then $P(X \leq 2) =$

$ \int \frac{x}{(1-x^2)\sqrt{2 - x^2} dx = $}

Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]

If $ \log_2 3 = p $, express $ \log_8 9 $ in terms of $ p $:

Evaluate the infinite series: \[ \left|\begin{array}{ccc} 2 & 1 & \frac{1}{3}
3 & 1 & 1
\end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{3} & \frac{1}{2}
3 & 1 & 1
\end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{9}
3 & 1 & 1
\end{array}\right| + \left|\begin{array}{ccc} 1 & \frac{1}{4} & \frac{1}{27}
3 & 1 & 1
\end{array}\right| + \cdots = ? \]

If the volume of a sphere is increasing at the rate of 12 $ \text{cm}^3/\text{sec} $, then the rate (in $ \text{cm}^2/\text{sec} $) at which its surface area is increasing when the diameter of the sphere is 12 cm is:

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