List of top Questions asked in ICAR AIEEA (PG)

Equations of the circle circumscribing the triangle formed by the lines \(x + y = 6\), \(2x + y = 4\), \(x + 2y = 5\) is:

For the parabola \(4(y - 1)^2 = -7(x - 3)\), the coordinates of vertex, focus and length of the latus rectum are respectively:

If the angle between the lines joining the foci of an ellipse to an extremity of the minor axis is \(90^\circ\), and the major axis is \(2\sqrt{2}\), then the equation of ellipse and its eccentricity is:

The value of \(\int \frac{2x - 9}{\sqrt{x^2 - 9x + 1}} \, dx\) is:

The value of \(\int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx\) is:

The value of \(\int_{0}^{1} \frac{x^2}{\sqrt{1 - x^2}} \, dx\) is:

The value of \(\iint_R x^2 y^5 \, dx \, dy\), where R is the rectangle with vertices (1, 0), (1, 1), (2, 1) and (2, 0) is:

The value of \(\int_{0}^{4} \int_{0}^{4-x} x y \, dx \, dy\) is:

For a continuous function \(f(r, \theta)\) defined on a region D in \((r, \theta)\) plane, the area of the domain D is given by:

The value of the integral \(\int \frac{dx}{1 + e^x}\) is equal to:

For the sequence \((x_n)\), if \(x_1 > 1\) and \(x_{n+1} = 2 - \frac{1}{x_n}\), \(n \geq 1\), which of the following is true:

For the series with \(n^{th}\) term \(x_n = \frac{n}{(n+1)(n+2)\), the following is true:}

Vertical and horizontal asymptotes of the curve \(y = \frac{3x + 5}{7 - x}\) are respectively:

The infinite series \(3 + \frac{1}{6}x - \frac{1}{216}x^2 - \frac{1}{3888}x^3 + \cdots\) represents the Maclaurin series for the function:

First four terms of the Taylor series of the function \(f(x) = x^3 + 2x^2 + x - 1\) at \(c = 3\) are:

For function \(f(x) = x^{2/3(2x + 5)\), if:}
a) \(x = 0\) is a horizontal tangent
b) The curve has no asymptote
c) The curve has a cusp at (0, 0)
d) The curve has no vertical tangent
Then:

The value of \(\lim_{x \to 0^+} \left(\frac{1}{x} - \frac{1}{\sin x}\right)\) is:

For the sequence \(s_n = \frac{b^n}{n^2}\), which of the following is true?

For the function \(f(x) = x^3 - 3x\) on \([-1, 3]\), the points of maxima and minima are respectively:

The value of \(\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sin x}\) is:

The value of \(\lim_{x \to \infty} \sqrt{\frac{3x - 5}{x - 2}}\) is:

For \(f(x) = [x^2 + 1]^{[x + 1]\), where \([x]\) denotes the greatest integer less than or equal to \(x\), which of the following is not true:}

A particle moving on the x-axis has position \(x(t) = 2t^3 + 3t^2 - 36t + 40\) feet after an elapsed time of t seconds. Its velocity after 3 seconds is:

If A is an n x n real matrix, then:

If \(A = \begin{pmatrix} 2 & 3 1 & 2 \end{pmatrix}\), then \(A^{-1}\) is: