List of top Questions asked in KEAM- 2017

The roots of \( ax^2 + x + 1 = 0 \), where \( a \neq 0 \), are in the ratio \( 1:1 \). Then \( a \) is equal to:

If \( z^2 + z + 1 = 0 \) where \( z \) is a complex number, then the value of \( \left( z + \frac{1}{z} \right)^2 + \left( z^2 + \frac{1}{z^2} \right)^2 + \left( z^3 + \frac{1}{z^3} \right)^2 \) equals:

The equations \( x^5 + ax + 1 = 0 \) and \( x^6 + ax^2 + 1 = 0 \) have a common root. Then \( a \) is equal to:

Let \( x_1 \) and \( x_2 \) be the roots of the equation \( x^2 + px - 3 = 0 \). If \( x_1^2 + x_2^2 = 10 \), then the value of \( p \) is equal to:

If the product of roots of the equation \( mx^2 + 6x + (2m - 1) = 0 \) is \( -1 \), then the value of \( m \) is:

If \( x \) and \( y \) are the roots of the equation \( x^2 + bx + 1 = 0 \), then the value of \( \frac{1}{x+b} + \frac{1}{y+b} \) is:

The value of \( 8^{2/3} - 16^{1/4} - 9^{1/2} \) is:

If \( A = \begin{pmatrix} 1 & 5 \\ 0 & 2 \end{pmatrix} \), then which of the following equations is satisfied by matrix \( A \)?

If \( \begin{pmatrix} 2x+y & x+y \\ p-q & p+q \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \), then \( (x, y, p, q) \) equals:

If \( x_1 \) and \( x_2 \) are the roots of \( 3x^2 - 2x - 6 = 0 \), then \( x_1^2 + x_2^2 \) is equal to:

Let \( x=2 \) be a root of \( y = 4x^2 - 14x + q = 0 \). Then \( y \) is equal to:

The value of \( \left| \sqrt{4+2\sqrt{3}} \right| - \left| \sqrt{4-2\sqrt{3}} \right| \) is:

If \( \begin{pmatrix} 1 & 2 & 4 \\ 1 & 3 & 5 \\ 1 & 4 & a \end{pmatrix} \) is singular, then the value of \( a \) is:

If \( \begin{pmatrix} x+y & x-y \\ 2x+z & x+z \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} \), then the values of \( x, y \) and \( z \) are respectively:

The value of \( \begin{pmatrix} 7 & 1 & 5 \\ 8 & 0 & 0 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} + 5 \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) is equal to:

If \( A = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} \), then \( A^{-1} \) is:

If \( \begin{pmatrix} 1 & 2 & -3 \\ 0 & 4 & 5 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \), then \( (x, y, z) \) is equal to:

If \( U = \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \), then \( U^{-1} \) is:

The condensation reaction between one equivalent of acetone and two equivalents of benzaldehyde in presence of dilute alkali leads to the formation of

The value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ p & q & r \\ p & q & r+1 \end{vmatrix} \) is equal to:

Let \( A = \begin{bmatrix} 5 & 0 \\ 1 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \). If \( 4A + 5B - C = 0 \), then the matrix \( C \) is:

The strongest base among the following is

The product Y for the below reaction is

The product formed in the following reaction is

X and Y in the below reaction are ----------- and ------------, respectively.