List of top Questions asked in KEAM- 2015

The range of the function \( f(x) = \frac{1}{2 - \cos 3x} \) is

If \( f(z) = \frac{1 - z^3}{1 - z} \), where \( z = x + iy \) with \( z \neq 1 \), then \( \mathrm{Re}(f(z)) = 0 \) reduces to

Let \( S_1 \) be a square of side 5 cm. Another square \( S_2 \) is drawn by joining midpoints of the sides of \( S_1 \). Square \( S_3 \) is drawn similarly and so on. Then \( \text{Area}(S_1) + \cdots + \text{Area}(S_{10}) \) is

The power of \( x \) in the term with the greatest coefficient in the expansion of \( \left(1 + \frac{x}{2}\right)^{10} \) is

Sum of coefficients of the last 6 terms in the expansion of \( (1 + x)^{11} \) when the expansion is in ascending powers of \( x \), is

If \( f(x) = \sin^{-1} \left( \frac{1 - \cos 2x}{2 \sin x} \right) \), then \( |f'(x)| \) is equal to

If \( f(x) = 3x^2 - 7x + 5 \), then \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} \) is equal to

\( \int \frac{(x+1)^2}{x^2+1} \, dx = \)

\( \int \frac{(x+1)^2}{x^2+1} \, dx = \)

\( \int_0^1 \frac{1}{(x^2+16)(x^2+25)} \, dx = \)

The order and degree of the differential equation

\[ \left[2\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 \right]^{\frac{3}{2}} = \frac{d^3y}{dx^3} \]

The solution of the differential equation \( y'(y^2 - x) = y \) is

The slope of a curve at any point \((x,y)\), other than origin, is \( y + \frac{y}{x} \). Then the equation of the curve is

\( \int_{0}^{\frac{\sqrt{\pi}}{2}} 2x^3 \sin(x^2)\, dx = \)

If \( x^2 + y^2 = 1 \), then

\( \int_{-\pi}^{\pi} \frac{\sin^2 x}{1+7^x}\, dx \) is equal to

The area bounded by \( y = x^2 + 3 \) and \( y = 2x + 3 \) is (in sq. units)

\( \int_{-1}^{1} x(1-x)(1+x)\, dx = \)

\( \int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx = \)

\( \int \frac{x^{n-1}}{x^n + a^n} dx = \)

\( \int \frac{4e^x}{2e^x - 5e^{-x}} dx = \)

The function \( f(x) = \sin x - kx - c \), where \( k \) and \( c \) are constants, decreases always when

The chord joining the points \( (5,5) \) and \( (11,227) \) on the curve \( y = 3x^2 - 11x - 15 \) is parallel to tangent at a point on the curve. The abscissa of the point is

If \( y = m\log x + nx^2 + x \) has its extreme values at \( x = 2 \) and \( x = 1 \), then \( 2m + 10n = \)