List of top Questions asked in VITEEE- 2013

Which of these ions is expected to be coloured in aqueous solution?

Which of the following ligands is tetradentate?

What is the EAN of \( [\text{Al}(\text{C}_4\text{O}_4)_3]^{3-} \)?

The solution of the differential equation \[ (1 + y^2) \, \frac{dy}{dx} = e^{-(x - y)} \] is

If the gradient of the tangent at any point \( (x, y) \) of a curve passing through the point \( (1, \frac{\pi}{4}) \) is \[ \left| \frac{dy}{dx} \right| = \frac{1}{x} \cdot \left| \log \left( \frac{y}{x} \right) \right| \] then the equation of the curve is

In the expansion of \( a + bx \), the coefficient of \( x^r \) is

If \( n = 1999 \), then \( \sum_{i=1}^{1999} \log x_i \) is equal to

\( P \) is a fixed point \( (a, a, a) \) on a line through the origin equally inclined to the axes, then any plane through \( P \) perpendicular to \( OP \), makes intercepts on the axes, the sum of whose reciprocals is equal to

For which of the following values of \( m \), the area of the region bounded by the curve \( y = x - x^2 \) and the line \( y = mx \) equals 5?

If \( R \to R \) be such that \( f(1) = 3 \) and \( f'(1) = 6 \), then \( f(x) \) is equal to

If \( f(x) = \left\{ \begin{array}{ll} 1 + \left| \sin x \right|, & \text{for } -\pi \leq x<0
e^{x/2}, & \text{for } 0 \leq x<\pi
\end{array} \right. \) then the value of \( a \) and \( b \), if \( f \) is continuous at \( x = 0 \), are respectively

The domain of the function \[ f(x) = \frac{1}{\log(1 - x)} + \sqrt{x + 2} \] is

\( \int (x + 1)(x - x^2) e^x \, dx \) is equal to

If \( f(x) = x - \lfloor x \rfloor \), for every real number \( x \), where \( \lfloor x \rfloor \) is the integral part of \( x \), then \[ \int f(x) \, dx \] is equal to

The value of the integral \[ \int_1^\infty \frac{x+1}{|x-1|} \left( \frac{x-1}{x+1} \right)^{1/2} \, dx \] is

If a tangent having slope \( \frac{-4}{3} \) to the ellipse \[ \frac{x^2}{18} + \frac{y^2}{32} = 1 \] intersects the major and minor axes in points A and B respectively, then the area of \( \triangle OAB \) is equal to

The locus of mid points of tangents intercepted between the axes of ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] is

If \( P \) is a double ordinate of hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] such that \( OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola, then the eccentricity \( e \) of the hyperbola satisfies

The sides \( AB \), \( BC \), and \( CA \) of a triangle \( \triangle ABC \) have respectively 3, 4, and 5 points lying on them. The number of triangles that can be constructed using these points as vertices is

The vector \( \mathbf{r} = 3\hat{i} + 4\hat{k} \) can be written as the sum of a vector \( \mathbf{v} \), parallel to \( \hat{i} + \hat{k} \), and a vector \( \mathbf{u} \), perpendicular to \( \hat{i} + \hat{k} \). Then, the value of \( \mathbf{v} \) is

If the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are collinear, then the rank of the matrix \[ \begin{bmatrix} x_1 & y_1 & 1
x_2 & y_2 & 1
x_3 & y_3 & 1 \end{bmatrix} \]

The value of the determinant \[ \begin{vmatrix} \cos(\alpha - \beta) & \cos \alpha & \cos \beta
\cos(\alpha - \beta) & 1 & \cos \beta
\cos \alpha & \cos \beta & 1 \end{vmatrix} \]

The number of integral values of \( K \), for which the equation \( 7 \cos x + 5 \sin x = 2K + 1 \) has a solution, is

The line joining two points \( A(2,0) \), \( B(3,1) \) is rotated about \( A \) in anti-clockwise direction through an angle of \( 15^\circ \). The equation of the line in the new position is

The line \( 2x + \sqrt{6}y = 2 \) is tangent to the curve \( x^2 - 2y^2 = 4 \). The point of contact is