List of practice Questions

If \( X \sim B(4, p) \) and \( 2 P(X = 3) = 3 P(X = 2) \), then the value of \( p \) is:

If \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) represents a joint equation of directrices of the hyperbola \( 16x^2 - 9y^2 = 144 \), then \( g + f - c = \)

If \( A = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}, B = \begin{bmatrix} 4 & 3 \\ 1 & 1 \end{bmatrix} \), then \( (A + B)^{-1x} = \)

Evaluate \( \displaystyle \int_{-\pi}^{\pi} \frac{2x}{1+\cos^2 x}\,dx \)

Evaluate \( \displaystyle \int_{0}^{1} \tan^{-1}\!\left(\frac{2x}{1-x^2}\right)\,dx \)

The probability that a person who undergoes a certain operation will survive is 0.2. If 5 patients undergo similar operations, find the probability that exactly four will survive.

If one of the lines given by the equation \( x^2 + kxy + 2y^2 = 0 \) is \( x + 2y = 0 \), then the value of \( k \) is

If \( x^2 + y^2 = 1 \), then find \( \dfrac{d^2x}{dy^2} \).

Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sin x \cos x}{1 + \sin^4 x} \, dx = \]

If \[ A = \{2, 4\}, \quad B = \{3, 4, 5\}, \quad \text{then} \quad (A \cap B) \times (A \cup B) = \]

Evaluate the integral: \[ \int e^{\cos^{-1}x} \left[ x - \sqrt{1 - x^2} \right] \frac{dx}{\sqrt{1 - x^2}}. \]

If the Cartesian equation of the line is \[ x - 1 = 2y + 3 = 3 - z, \] then its vector equation is

If \(\vec a, \vec b, \vec c, \vec d\) are the position vectors of the points \(A,B,C,D\) respectively such that \[ 3\vec a - \vec b + 2\vec c - 4\vec d = \vec 0, \] then the position vector of the point of intersection of the line segments \(AC\) and \(BD\) is

If $\omega$ is a complex cube root of unity and $A=\begin{pmatrix}\omega & 0\\0 & \omega\end{pmatrix}$, then $A^{-1}=$

If the points $A(2,1,-1)$, $B(0,-1,0)$, $C(4,0,4)$ and $D(2,0,x)$ are coplanar, then $x=$

The equation of a plane containing the lines \( \vec{r} = (i + 2j - 4k) + \lambda (2i + 3j + 6k) \) and \( \vec{r} = (i + 3j + 4k) + \mu (i + j - k) \) is

If \( y\sqrt{1-x^2} + x\sqrt{1-y^2} = 1 \), then \( \dfrac{dy}{dx} = \)

The centre and radius of a circle \[ x = 4a \left( \frac{1 - t^2}{1 + t^2} \right), \quad y = \frac{8at}{1 + t^2} \] are respectively

If the tangent to the curve given by \( x = t^2 - 1 \) and \( y = t^2 - t \) is parallel to the x-axis, then the value of \( t \) is

Evaluate \( \int_{-4}^{4} \log \left( \frac{8 - x}{8 + x} \right) dx \)

If \[ A = \{x, y, z\}, \quad B = \{1, 2\}, \text{ then the total number of relations from set } A \text{ to set } B \text{ are} \]

If the line \[ \vec{r} = (i - 2j + 3k) + \lambda(2i + j + 2k) \quad \text{is parallel to the plane} \quad \vec{r} \cdot (3i - 2j + mk) = 10, \] then the value of \( m \) is

Evaluate the integral \[ \int \frac{x^2+1}{x^4+x^2+1}\,dx \]

The cartesian co-ordinates of the point whose polar co-ordinates are \( \left( \frac{1}{2}, 120^\circ \right) \text{ are}