List of practice Questions

If \( A(0, 4, 0) \), \( B(0, 0, 3) \), and \( C(0, 4, 3) \) are the vertices of \( \Delta ABC \), then its incenter is:

If the angle between the lines whose direction ratios are \( 4, -3, 5 \) and \( 3, 4, k \) is \( \frac{\pi}{3} \), then \( k = \)

If the symbolic form of the switching circuit is \( \sim p \vee ( p \wedge \sim q) \vee q \), then the current flows through the circuit only if:

The probability distribution of a discrete r.v. \( X \) is: \[ \begin{array}{c|ccccc} X = x & 0 & 1 & 2 & 3 & 4 \\ P(X = x) & k & 2k & 4k & 2k & k \\ \end{array} \] Then, the value of \( P(X \leq 2) \) is:

The rate of decay of mass of a certain substance at time \( t \) is proportional to the mass at that instant. The time during which the original mass of \( m_0 \) gram will be left to \( m_1 \) gram is
\[ k \text{ is constant of proportionality} \]

If the error involved in making a certain measurement is continuous random variable \( X \) with probability density function \( f(x) = k (4 - x^2) \) for \( -2 \leq x \leq 2 \), and \( f(x) = 0 \) otherwise, then \[ P(|-1<X<1|) \]

The equation of the line passing through the points \( (3, 4, -7) \) and \( (6, -1, 1) \) is:

If \( | \vec{a} \, \vec{b} \, \vec{c} | = 3 \), then the volume of the parallelepiped with \( 2\vec{a} + \vec{b} \), \( 2\vec{b} + \vec{c} \), \( 2\vec{c} + \vec{a} \) as coterminous edges is:

If \( \frac{x}{x-y} = \log \left( \frac{a}{x-y} \right) \), then \( \frac{dy}{dx} = \)

Evaluate the integral: \[ \int_{-1}^{1} \left[ \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right] dx \]

Evaluate the integral: \[ \int \left[ \frac{1 - \log x}{1 + (\log x)^2} \right]^2 dx \]

The equation of a circle passing through the origin and making x-intercept 3 and y-intercept -5 is:

The co-ordinates of the foot of the perpendicular from the point \( (0, 2, 3) \) on the line
\[ \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \]

If \(A(3,2,-1)\) and \(B(1,4,3)\), then the equation of the plane which bisects the segment \(AB\) perpendicularly is

ABCD is a parallelogram. \(P\) is the midpoint of \(AB\). If \(R\) is the point of intersection of \(AC\) and \(DP\), then \(R\) divides \(AC\) internally in the ratio

Two dice are thrown together. The probability that the sum of the numbers is divisible by 2 or 3 is

Evaluate the sum of the series: \[ \frac{1^2}{2} + \frac{1^2 + 2^2}{3} + \frac{1^2 + 2^2 + 3^2}{4} + \frac{1^2 + 2^2 + 3^2 + 4^2}{5} + \cdots \quad \text{upto 8 terms} \]

Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \cos x} \]

Which of the following have the same value:
(a) \( \sin 120^\circ \)
(b) \( \cos 930^\circ \)
(c) \( \tan 840^\circ \)
(d) \( \cot(-1110^\circ) \)

The approximate value of \( \log_{10}99 \) is (Given \( \log_{10}e = 0.4343 \))

If the function \[ f(x)= \begin{cases} \dfrac{\log 10 + \log(0.1+2x)}{2x}, & x \neq 0 \\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then \(k+2=\)

\(y = mx + \dfrac{2}{m}\) is the general solution of

In a triangle \(ABC\) with usual notations, \[ \frac{\cos A - \cos C}{a - c} + \frac{\cos B}{b} = \]

The maximum value of \(Z = 10x + 25y\) subject to \[ 0 \le x \le 3,\; 0 \le y \le 3,\; x + y \le 5,\; x \ge 0,\; y \ge 0 \] is

The equation of a line passing through the point of intersection of the lines \[ x + 2y + 8 = 0 \quad \text{and} \quad 3x - y + 4 = 0 \] and having x– and y–intercepts zero is