List of top Statistical Science Questions asked in ICAR AIEEA (PG)

If $f$ is continuous on $[a, b]$, then there is at least one number $c$ between $a$ and $b$ such that

For the function $f(x) = \sin x + \cos x$ , the point of relative maxima on $[0, 2\pi]$ is given by

Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . If $f(a) = f(b)$ , then there exists at least one number $c$ between $a$ and $b$ such that

If $f$ and $g$ are integrable on $[a, b]$ and $f(x) \geq g(x)$ for all $x$ in $[a, b]$, then

The value of $\lim_{x \to +\infty} \frac{\ln x}{x^2}$ is given by}

Suppose that a particle moves on a coordinate line so that its velocity at time $t$ is $v(t) = t^2 - 2t \text{ m/s}$ , then the displacement of the particle during the time interval $0 \leq t \leq 3$ is given by}

If $\Delta$ and $\nabla$ be the first forward difference and first backward difference operator respectively of a function $f(x)$ then the expression $(\Delta - \nabla)$ is equivalent to

If a continuous function $f$ has a relative extremum at $c$ , then $c$ must be

The solution of the differential equation $\frac{d^3 y}{dx^3} + \frac{d^2 y}{dx^2} + 4 \frac{dy}{dx} + 4y = 0$ is given by}

While applying Simpson's three eighth rule for numerical integration, the number of sub-intervals should be taken as multiple of

Suppose the function $f$ is continuous at the point $P(c, f(c))$. Then, the graph of $f$ has a cusp at $P$ if

The technique of estimating the value of a function for any intermediate value of the given independent variable is

The solution of the differential equation $x \frac{dy}{dx} + y = x^3 y^6$ is given by}

The solution of the differential equation $(1 + y e^{xy}) \, dx + (2y + x e^{xy}) \, dy = 0$ is given by

The value of $\int_0^{\pi} \int_0^{\cos y} x \sin y \, dx \, dy$ is given by}

The value of $\int_0^{\pi/2} \sin^2 \frac{x}{2} \cos^2 \frac{x}{2} \, dx$ is given by}

The value of $\int x \tan^{-1}x \, dx$ is given by}

Using chain rule, the derivative of $\omega = x^2 y - y^2$ with respect to t along the path $x = \sin t$ and $y = e^t$ at $t = 0$ is given by

The value of \(\lim_{x \to 1^-} \frac{x-1}{|x^2-1|}\) is given by

If $u = \sin^{-1}\left( \frac{x+2y+3z}{x^8+y^8+z^8} \right)$, then the value of $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} + z\frac{\partial u}{\partial z}$ is given by}

The horizontal asymptote for \(y = \frac{-8}{x^2-4}\) is given by

If \(x = a\sin^3 t\) and \(y = b\cos^3 t\), then the value of \(\frac{d^2y}{dx^2}\) is given by

Roots of the equation \(x^3 - 23x^2 + 142x - 120 = 0\) are

From point Q which is outside the circle, the length of the tangent to the circle is 24 cm and the distance of Q from the centre is 25 cm, then the radius of the circle is given by

The sum and product of the roots of the polynomial \(x^2 - 2x - 8\) are respectively