List of practice Questions

The value of \(\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}\) is :
The sum of solutions of the equation \(\frac{\cos x}{1 + \sin x} = |\tan 2x|\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) - \left\{\frac{\pi}{4}, -\frac{\pi}{4}\right\}\) is :
Let \(ABC\) be a triangle with \(A(-3, 1)\) and \(\angle ACB = \theta, 0<\theta \leq \frac{\pi}{2}\). If the equation of the median through \(B\) is \(2x + y - 3 = 0\) and the equation of angle bisector of \(C\) is \(7x - 4y - 1 = 0\), then \(\tan \theta\) is equal to :
If a line along a chord of the circle \(4x^2 + 4y^2 + 120x + 675 = 0\), passes through the point \((-30, 0)\) and is tangent to the parabola \(y^2 = 30x\), then the length of this chord is :
On the ellipse \(\frac{x^2}{8} + \frac{y^2}{4} = 1\) let \(P\) be a point in the second quadrant such that the tangent at \(P\) to the ellipse is perpendicular to the line \(x + 2y = 0\). Let \(S\) and \(S'\) be the foci of the ellipse and \(e\) be its eccentricity. If \(A\) is the area of the triangle \(SPS'\) then, the value of \((5 - e^2) \cdot A\) is :
A plane \(P\) contains the line \(x + 2y + 3z + 1 = 0 = x - y - z - 6\), and is perpendicular to the plane \(-2x + y + z + 8 = 0\). Then which of the following points lies on \(P\) ?
Let \(y = y(x)\) be a solution curve of the differential equation \((y + 1) \tan^2x dx + \tan x dy + y dx = 0, x \in \left(0, \frac{\pi}{2}\right)\). If \(\lim_{x \to 0^+} xy(x) = 1\), then the value of \(y\left(\frac{\pi}{4}\right)\) is :
Let \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{j} - \hat{k}\). If \(\vec{c}\) is a vector such that \(\vec{a} \times \vec{c} = \vec{b}\) and \(\vec{a} \cdot \vec{c} = 3\), then \(\vec{a} \cdot (\vec{b} \times \vec{c})\) is equal to :
If the truth value of the Boolean expression \(((p \vee q) \wedge (q \to r) \wedge (\sim r)) \to (p \wedge q)\) is false, then the truth values of the statements \(p, q, r\) respectively can be :
Let \(A\) and \(B\) be independent events such that \(P(A) = p, P(B) = 2p\). The largest value of \(p\), for which \(P(\text{exactly one of } A, B \text{ occurs}) = \frac{5{9}\), is :}
The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If \(\alpha\) and \(\sqrt{\beta}\) are the mean and standard deviation respectively for correct data, then \((\alpha, \beta)\) is :
The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is _________.
Let \( a, b \in \mathbb{R}, b \neq 0 \). Define a function \[ f(x) = \begin{cases} a \sin \frac{\pi}{2}(x - 1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases} \] If \( f \) is continuous at \( x = 0 \), then \( 10 - ab \) is equal to ________.
The sum of all integral values of $k$ ($k \neq 0$) for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is _________.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left( \frac{4}{\pi} + 1 \right)k$ is equal to _________.
The area of the region $S = \{(x, y) : 3x^2 \leq 4y \leq 6x + 24\}$ is _________.
If $y = y(x)$ is an implicit function of $x$ such that $\log_e (x + y) = 4xy$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is equal to _________.
If ${}^1P_1 + 2 \cdot {}^2P_2 + 3 \cdot {}^3P_3 + \dots + 15 \cdot {}^{15}P_{15} = {}^qP_r - s$, where $0 \leq s \leq 1$, then $q + s + {}^qC_{r - s}$ is equal to _________.
Let the line $L$ be the projection of the line \[ \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2} \] in the plane $x - 2y - z = 3$. If $d$ is the distance of the point $(0, 0, 6)$ from $L$, then $d^2$ is equal to _________.
Let \( z = \frac{1 - i\sqrt{3}}{2}, \, i = \sqrt{-1} \). Then the value of

\[ 21 + \left( z + \frac{1}{z} \right)^3 + \left( z^2 + \frac{1}{z^2} \right)^3 + \left( z^3 + \frac{1}{z^3} \right)^3 + \dots + \left( z^{21} + \frac{1}{z^{21}} \right)^3 \] is _________.

The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter $d$. Then $d^2$ is equal to _________.
In Millikan's oil drop experiment, what is viscous force acting on an uncharged drop of radius \(2.0 \times 10^{-5}\) m and density \(1.2 \times 10^3\) kgm\(^{-3}\)? Take viscosity of liquid \(= 1.8 \times 10^{-5}\) Nsm\(^{-2}\). (Neglect buoyancy due to air).
Moment of inertia of a square plate of side l about the axis passing through one of the corner and perpendicular to the plane of square plate is given by :

The resultant of these forces \(\vec{OP}, \vec{OQ}, \vec{OR}, \vec{OS}\) and \(\vec{OT}\) is approximately ________ N. 
[Take \(\sqrt{3} = 1.7, \sqrt{2} = 1.4\). Given \(\hat{i}\) and \(\hat{j}\) unit vectors along x, y axis] 

 

Which of the following is not a dimensionless quantity ?