List of practice Questions

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as \( \frac{1}{3} \) and \( \frac{2}{3} \), respectively. Let \( x \) be the number of matches that the team wins, and \( y \) be the number of matches that the team loses. If the probability \( P(|x - y| \leq 2) \) is \( p \), then \( 3^9 p \) equals .
The coefficients a, b, c in the quadratic equation ax2 + bx + c = 0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216p equals :
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is
Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability that Ajay will appear in the exam and Vijay will not appear is:
How is Mukesh different from other boys of his age ? (Lost Spring)
Which of the following combinations is correct?
If \( \log_e y = 3 \sin^{-1}x \), then \( (1 - x)^2 y'' - xy' \) at \( x = \frac{1}{2} \) is equal to:
For a differentiable function \( f : \mathbb{R} \to \mathbb{R} \), suppose \[ f'(x) = 3f(x) + \alpha, \] where \( \alpha \in \mathbb{R} \), \( f(0) = 1 \), and \[ \lim_{x \to -\infty} f(x) = 7. \] Then \( 9f(-\log_2 3) \) is equal to __________ .
The temperature \( T(t) \) of a body at time \( t = 0 \) is \( 160^\circ \)F and it decreases continuously as per the differential equation \[ \frac{dT}{dt} = -K(T - 80), \] **where \( K \) is a positive constant. If \( T(15) = 120^\circ \)F, then \( T(45) \) is equal to
Let \( y = y(x) \) be the solution of the differential equation \(\sec^2 x \, dx + \left( e^{2y} \tan^2 x + \tan x \right) dy = 0,\)\( 0 < x < \frac{\pi}{2} \), \( y \left( \frac{\pi}{4} \right) = 0 \). If \( y \left( \frac{\pi}{6} \right) = \alpha \), then \( e^{8\alpha} \) is equal to\(\_\_\_\_\_\).
If the solution \( y = y(x) \) of the differential equation \( \left( x^4 + 2x^3 + 3x^2 + 2x + 2 \right) dy - \left( 2x^2 + 2x + 3 \right) dx = 0 \)
satisfies \( y(-1) = -\frac{\pi}{4} \), then \( y(0) \) is equal to:
Let the solution \( y = y(x) \) of the differential equation \[\frac{dy}{dx} - y = 1 + 4 \sin x\] satisfy \( y(\pi) = 1 \). Then \( y\left( \frac{\pi}{2} \right) + 10 \) is equal to _____
If \( y = y(x) \) is the solution of the differential equation \(\frac{dy}{dx} + 2y = \sin(2x), \quad y(0) = \frac{3}{4},\)then \(y\left(\frac{\pi}{8}\right)\) is equal to:
The solution curve, of the differential equation \( 2y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx} \), passing through the point \( (0, 1) \), is a conic, whose vertex lies on the line:
The solution of the differential equation \( (x^2 + y^2) dx - 5xy \, dy = 0, \, y(1) = 0 \), is:
Let \(y = y(x)\) be the solution curve of the differential equation \[ \sec y \frac{dy}{dx} + 2x \sin y = x^3 \cos y, \] \(y(1) = 0\). Then \(y\left(\sqrt{3}\right)\) is equal to:
Let \(\alpha |x| = |y| e^{xy - \beta}\), \(\alpha, \beta \in \mathbb{N}\) be the solution of the differential equation \[ xdy - ydx + xy(xdy + ydx) = 0, \quad y(1) = 2. \] Then \(\alpha + \beta\) is equal to _.
Let \( f(x) \) be a positive function such that the area bounded by \( y = f(x) \), \( y = 0 \), from \( x = 0 \) to \( x = a>0 \) is \[ \int_0^a f(x) \, dx = e^{-a} + 4a^2 + a - 1. \] Then the differential equation, whose general solution is \[ y = c_1 f(x) + c_2, \] where \( c_1 \) and \( c_2 \) are arbitrary constants, is:
Let $y = y(x)$ be the solution of the differential equation $(1 + y^2)e^{\tan x} dx + \cos^2 x (1 + e^{2\tan x}) dy = 0$, $y(0) = 1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to:
Let \( y = y(x) \) be the solution of the differential equation: \[ (x^2 + 4)^2 \, dy + \left( 2x^3 y + 8xy - 2 \right) dx = 0. \] If \( y(0) = 0 \), then \( y(2) \) is equal to:
Let \( y = y(x) \) be the solution of the differential equation \[ (x + y + 2)^2 \, dx = dy, \quad y(0) = -2. \] Let the maximum and minimum values of the function \( y = y(x) \) in \( \left[ 0, \frac{\pi}{3} \right] \) be \( \alpha \) and \( \beta \), respectively. If \[ (3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}, \quad \gamma, \delta \in \mathbb{Z}, \] then \( \gamma + \delta \) equals \( \dots \).
The differential equation of the family of circles passing the origin and having center at the line y = x is:
Let \( y = y(x) \) be the solution of the differential equation\[\frac{dy}{dx} + \frac{2x}{\left( 1 + x^2 \right)^2} y = x e^{\frac{1}{1+x^2}}, \quad y(0) = 0. \] Then the area enclosed by the curve \[ f(x) = y(x) e^{\frac{1}{1+x^2}} \]and the line \( y - x = 4 \) is _______.
Suppose the solution of the differential equation \[\frac{dy}{dx} = \frac{(2 + \alpha)x - \beta y + 2}{\beta x - 2\alpha y - (\beta \gamma - 4\alpha)}\]represents a circle passing through the origin. Then the radius of this circle is: