List of top Questions asked in KEAM- 2014

If \( z_1 = 2\sqrt{2}(1 + i) \) and \( z_2 = 1 + i\sqrt{3} \), then \( z_1^2 z_2^3 \) is equal to:

If \( z = e^{2\pi i / 3} \), then \( 1 + z + 3z^2 + 2z^3 + 2z^4 + 3z^5 \) is equal to:

Let \( w \neq \pm 1 \) be a complex number. If \( |w| = 1 \) and \( z = \frac{w - 1}{w + 1} \), then \( \text{Re}(z) \) is equal to:

If \( z = \frac{(\sqrt{3} + i)^3 (3i + 4)^2{(8 + 6i)^2} \), then \( |z| \) is equal to:}

If \( f(x) = \sqrt{x \) and \( g(x) = 2x - 3 \), then \( (f \circ g)(x) \) is:}

The range of the function \( f(x) = x^2 + 2x + 2 \) is:

If $A = \{1, 3, 5, 7\}$ and $B = \{1, 2, 3, 4, 5, 6, 7, 8\}$, then the number of one-to-one functions from $A$ into $B$ is:}

Let $f(x) = |x - 2|$, where $x$ is a real number. Which one of the following is true?

The number of students who take both the subjects mathematics and chemistry is 30. This represents 10% of the enrolment in mathematics and 12% of the enrolment in chemistry. How many students take at least one of these two subjects?

If the operation $\oplus$ is defined by $a \oplus b = a^2 + b^2$ for all real numbers $a$ and $b$, then $(2 \oplus 3) \oplus 4 =$

An integrating factor of the differential equation \( \sin x \frac{dy}{dx} + 2y \cos x = 1 \) is:

The order and degree of the differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{\frac{3}{2}} = y$ are respectively:

The solution of the differential equation \( \frac{dy}{dx} = e^x + 1 \) is:

The solution of the differential equation \( (kx - y^2) dy = (x^2 - ky) dx \) is:

The value of the integral \( \int_{2}^{4} \left( \frac{\log t}{t} \right) \, dt \) is equal to:

The value of the integral \( \int_{0}^{1} \frac{x^3}{1 + x^8} \, dx \) is equal to:

The value of the integral \( \int_{1}^{e} \frac{1 + \log x}{3x} \, dx \) is equal to:

Area bounded by the curves \( y = e^x \), \( y = e^{-x} \) and the straight line \( x = 1 \) is (in sq. units):

The value of \( \int_{0}^{1} \frac{dx}{e^x + e} \) is equal to:

\( \int \frac{\sqrt{5 + x^2}}{x^4} \, dx \) is equal to:

\( \int \frac{\sqrt{x^2 - 1}}{x} \, dx \) is equal to:

\( \int \sqrt{1 + \cos x} \, dx \) is equal to:

\( \int e^x (\sin x + 2 \cos x) \sin x \, dx \) is equal to:

$\int \frac{x e^x}{(1+x)^2} \, dx$ is equal to:

\( \int \frac{(1 + x) e^x{\sin^2}(x e^x)} dx \) is equal to: