List of top Questions asked in KEAM

If \( a_1, a_2, a_3, \dots \) are in A.P., then the value of \( \begin{vmatrix} a_1 & a_2 & 1 \\ a_2 & a_3 & 1 \\ a_3 & a_4 & 1 \end{vmatrix} \) is equal to:

The value of \( {}^2P_1 + {}^3P_1 + \dots + {}^nP_1 \) is equal to:

The middle term in the expansion of \( \left(\frac{10}{x} + \frac{x}{10}\right)^{10 \) is:}

The coefficient of \( x^{49} \) in the product \( (x-1)(x-2)(x-3) \dots (x-50) \) is:

The sum of the coefficients in the binomial expansion of \( \left( \frac{1}{x} + 2x \right)^6 \) is equal to:

Let \( x_1, x_2, \dots, x_n \) be in an A.P. If \( x_1 + x_4 + x_9 + x_{11} + x_{20} + x_{22} + x_{27} + x_{30} = 272 \), then \( x_1 + x_2 + x_3 + \dots + x_{30} \) is equal to:

If the sum of first 75 terms of an A.P. is 2625, then the \( 38^{th} \) term of the A.P. is:

If \( -5, k, -1 \) are in A.P., then the value of \( k \) is equal to:

Let \( T_n \) denote the number of triangles which can be formed by using the vertices of a regular polygon of \( n \) sides. If \( T_{n+1} - T_n = 36 \), then \( n \) is equal to:

If the second and fifth terms of a G.P. are 24 and 3 respectively, then the sum of first six terms is:

If two positive numbers are in the ratio \( 3 + 2\sqrt{2} : 3 - 2\sqrt{2} \), then the ratio between their A.M. and G.M. is:

The sum of the series \( \sum_{n=8}^{17} \frac{1}{(n+2)(n+3) \) is equal to:}

If the roots of \( x^2 - ax + b = 0 \) are two consecutive odd integers, then \( a^2 - 4b \) is:

If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + 3x - 4 = 0 \), then \( \frac{1}{\alpha} + \frac{1}{\beta} \) is equal to:

If the roots of the equation \( x^2 + 2bx + c = 0 \) are \( \alpha \) and \( \beta \), then \( b^2 - c = \)

The equation whose roots are the squares of the roots of the equation \( 2x^2 + 3x + 1 = 0 \) is:

If \( \alpha \) and \( \beta \) are the roots of \( x^2 - ax + b^2 = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:

The value of \( x \) such that \( 3^{2x} - 2(3^{x+2}) + 81 = 0 \) is:

If \( z = \frac{(\sqrt{3} + i)^3 (3i + 4)^2{(8 + 6i)^2} \), then \( |z| \) 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 \( z = e^{2\pi i / 3} \), then \( 1 + z + 3z^2 + 2z^3 + 2z^4 + 3z^5 \) is equal to:

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 the complex numbers \( z_1, z_2 \) and \( z_3 \) denote the vertices of an isosceles triangle, right angled at \( z_1 \), then \( (z_1 - z_2)^2 + (z_1 - z_3)^2 \) is equal to:

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

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