List of practice Questions

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :
Let \(B_i\) (\(i=1, 2, 3\)) be three independent events in a sample space. The probability that only \(B_1\) occurs is \(\alpha\), only \(B_2\) occurs is \(\beta\) and only \(B_3\) occurs is \(\gamma\). Let \(p\) be the probability that none of the events \(B_i\) occurs and these 4 probabilities satisfy the equations \((\alpha - 2\beta) p = \alpha\beta\) and \((\beta - 3\gamma) p = 2\beta\gamma\) (All the probabilities are assumed to lie in the interval (0, 1)). Then \(\frac{P(B_1)}{P(B_3)}\) is equal to __________.
Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
A student appeared in an examination consisting of 8 true - false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least 'n' correct answers is less than $\frac{1}{2}$, is :
Let 9 distinct balls be distributed among 4 boxes, $B_1, B_2, B_3$ and $B_4$. If the probability that $B_3$ contains exactly 3 balls is $k \left(\frac{3}{4}\right)^9$ then $k$ lies in the set :
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :
Let \(y(x)\) be the solution of the differential equation \(2x^2 dy + (e^y - 2x)dx = 0, x>0\). If \(y(e)=1\), then \(y(1)\) is equal to :
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 :
A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, -3) form the line 3x + 4y = 5, is given by :
If the solution curve of the differential equation \((2x - 10y^3)dy + y dx = 0\), passes through the points (0, 1) and (2, \(\beta\)), then \(\beta\) is a root of the equation :
If \(x \phi(x) = \int_{5}^{x} (3t^2 - 2\phi'(t))dt\), \(x>-2\), and \(\phi(0)=4\), then \(\phi(2)\) is ___________
If the curve, y=y(x) represented by the solution of the differential equation $(2xy^2-y)dx+xdy=0$, passes through the intersection of the lines, 2x$-$3y=1 and 3x+2y=8, then $|y(1)|$ is equal to ________ .
Let $y=y(x)$ be solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 3x+4y$, with $y(0)=0$. If $y\left(-\frac{2}{3}\log_e 2\right) = \alpha \log_e 2$, then the value of $\alpha$ is equal to :
Let F : [3, 5] $\rightarrow$ R be a twice differentiable function on (3, 5) such that
$F(x) = e^{-x} \int_3^x (3t^2 + 2t + 4F'(t))dt$.
If $F'(4) = \frac{\alpha e^\beta - 224}{(e^\beta-4)^2}$, then $\alpha+\beta$ is equal to _________.
If $y=y(x)$, $y \in [0, \pi/2)$ is the solution of the differential equation $\sec y \frac{dy}{dx} - \sin(x+y) - \sin(x-y) = 0$, with $y(0)=0$, then $5y'(\pi/2)$ is equal to _________.
If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is (x² - 4x + y + 8)/(x - 2), then this curve also passes through the point :
The population $P = P(t)$ at time 't' of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$, then the time at which population becomes zero is :
Let y = y(x) be the solution of the differential equation dy/dx = (y + 1) ((y + 1)e^{x²/2 - x} - 1), 0<x<2.1, with y(2) = 0. Then the value of dy/dx at x=1 is equal to :
Let y=y(x) be the solution of the differential equation x dy - y dx = √(x² - y²) dx, x ≥ 1, with y(1) = 0. If the area bounded by the line x=1, x=e^{π}, y=0 and y=y(x) is α e^{2π} + β, then the value of 10(α + β) is equal to ________.
Let $y=y(x)$ be the solution of the differential equation $(x-x^3)dy = (y+yx^2-3x^4)dx$, $x>2$. If $y(3)=3$, then $y(4)$ is equal to :
Let $y=y(x)$ be the solution of the differential equation $dy = e^{\alpha x+y}dx; \alpha \in \mathbb{N}$. If $y(\log_e 2) = \log_e 2$ and $y(0)=\log_e(\frac{1}{2})$, then the value of $\alpha$ is equal to ________.
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} = 1 + xe^{y-x}, -\sqrt{2}<x<\sqrt{2}, y(0) = 0$. Then, the minimum value of $y(x), x \in (-\sqrt{2}, \sqrt{2})$ is equal to :
Let \(y=y(x)\) be solution of the following differential equation \(e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0, y(\pi/2) = 0\). If \(y(0) = \log_e(\alpha + \beta e^{-2})\), then \(4(\alpha + \beta)\) is equal to __________.
The differential equation satisfied by the system of parabolas y² = 4a(x + a) is :
Let f be a twice differentiable function defined on ℝ such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x ∈ ℝ. If $| f(x) \ f'(x) | \ | f'(x) \ f''(x) | = 0$, then the value of f(1) lies in the interval :