List of practice Questions

The particular solution of the differential equation \( x \, dy + 2y \, dx = 0 \), when \( x = 2 \) and \( y = 1 \) is

The statement pattern \( \sim (p \vee q) \vee (\sim p \wedge q) \) is equivalent to

The value of \( m \) if the vectors \[ \mathbf{A} = i - j - 6k, \quad \mathbf{B} = i - 3j + 4k, \quad \mathbf{C} = 2i - 5j + mk \] are coplanar, is

A random variable \( X \) takes the values 0, 1, 2. Its mean is 1.2. If \( P(X = 0) = 0.3 \), then \( P(X = 1) = \)

With usual notations, in \( \triangle ABC \), if \( a = 2 \), \( b = 3 \), \( c = 5 \) and \[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{k}{7} + \frac{30}{30}, \] then \( k = \)

If \(\vec u = \hat i - 2\hat j + \hat k\), \(\vec v = 3\hat i + \hat k\) and \(\vec w = \hat j - \hat k\), then the volume of the parallelepiped with \(\vec u \times \vec v\), \(\vec u + \vec w\) and \(\vec v + \vec w\) as coterminous edges is

If \(2\sin^2 x + 7\cos x = 5\), then the permissible value of \(\cos x\) is

The differential equation obtained by eliminating the arbitrary constant from the equation \[ y^2 = (2x+c)^5 \] is

If \(P(3,2,6)\), \(Q(1,4,5)\) and \(R(3,5,3)\) are the vertices of \(\triangle PQR\), then the measure of \(\angle PQR\) is

The function $f(x) = (x + 2)e^{-x}$ is

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is given by $f(x)=7x+8$ and $f^{-1}(12)=\dfrac{k}{7}$, then the value of $k$ is

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

If \( \tan \theta + \sin \theta = a \) and \( \tan \theta - \sin \theta = b \), then the values of \( \cot \theta \) and \( \csc \theta \) are respectively:

If \( x = a(1 - \cos\theta) \), \( y = a(\theta - \sin\theta) \), then \( \frac{d^2y}{dx^2} = \)

If \( a \), \( b \), \( c \) are non-negative distinct numbers and \( ai + aj + ck \), \( i + j + k \), and \( ci + cj + bk \) are coplanar vectors, then

The shortest distance between the lines \( 1 + x = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) is

The domain of the function \( f(y) = \frac{\cos^{-1(y - 5)}{\sqrt{25 - y^2}} \) is}

If \( \int_{0}^{1} (5x^2 - 3x + k) \, dx = 0 \), then \( k = \)

If \( CP \) and \( CD \) is a pair of semi-conjugate diameters of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), then \( CP^2 + CD^2 = \)

If \( A + B + C = 180^\circ \), then the value of \( \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right) + \tan \left( \frac{B}{2} \right) \tan \left( \frac{C}{2} \right) + \tan \left( \frac{C}{2} \right) \tan \left( \frac{A}{2} \right) \) is

The differential equation of all lines perpendicular to the line \[ 5x + 2y + 7 = 0 \text{ is} \]

Evaluate the integral \[ \int \frac{1 + 2e^{-x}}{1 - 2e^{-x}} \, dx \]

In \( \triangle ABC \) with usual notations, \( a = 4 \), \( b = 3 \), \( \angle A = 60^\circ \), then \( c \) is a root of the equation

If \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \[ f(x) = \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}, \] then \( f \) is

The L.P.P. to maximize \( z = x + y \), subject to \[ x + y \le 30,\; x \le 15,\; y \le 20,\; x + y \ge 15,\; x \ge 0,\; y \ge 0 \] has