List of top Questions asked in ICAR AIEEA (PG)- 2020

Integrating factor for the differential equation \((x^2 + y^2 + x) \, dx + xy \, dy = 0\) is:

Number of first order equations in a system equivalent to the equation \(x''' + 3x'' + 2x' - 5x = \sin 2t\) is:

Solution of the differential equation \(x \sin \frac{y}{x} \, dy = \left( y \sin \frac{y}{x} - x \right) dx\) is:

A solution to the differential equation \(\left(y - x \frac{dy}{dx}\right) = a \left(y^2 + \frac{dy}{dx}\right)\) is:

Which of the following equations is exact?

The equations of asymptotes for the hyperbola \(xy - 3x - 2y = 0\) are:

A solution to the difference equation \(u_{x+3} - 3u_{x+1} - 2u_x = 0\) is of the form:

For the differential equation \(x^{\alpha} y^{\beta} (m y \, dx + n x \, dy) = 0\), the integrating factor, for any value of \(k\) is:

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

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

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

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

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

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 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:

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

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

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

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:

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

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