List of practice Questions

Let \( f(x) = x^2 + \frac{1}{x^2} \) and \( g(x) = x - \frac{1}{x} \) for \( x \in \mathbb{R} \setminus \{-1, 0, 1\} \). Then, the local minimum of \( \frac{f(x)}{g(x)} \) is:
The range of the function \( f(x) = \frac{x^2 + x + 1}{x^2 - x + 1} \) is:
Which of the following quadratic equations whose real roots \( x_1, x_2 \) satisfy the conditions \( x_1^2 + x_2^2 = 5 \), \( 3(x_1^5 + x_2^5) = 11(x_1^3 + x_2^3) \)?
The remainder when the polynomial \( 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) is divided by \( x^2 - x - 3 \) is:
If \( ^nC_{r-1} = 36 \), \( ^nC_r = 84 \), and \( ^nC_{r+1} = 126 \), then the value of \( nr^2 \) is:
The total number of positive integral solutions \((x, y, z)\) of \( xyz = 24 \) is:
If \(A = \begin{bmatrix}x & 2 & 1 \\ 2 & x & 1 \\ 2 & 1 & 0\end{bmatrix}\) and \(\det(A) = 125\), then \(x =\)
Let \(A = \begin{bmatrix}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{bmatrix}\) and \(B = \begin{bmatrix}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{bmatrix}\). Then \(A^2 + B^2 + AB =\)
Let \( A = \begin{bmatrix} n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0 \end{bmatrix} \). Then \( A^2 + B^2 + AB = \)
Let \( z \) and \( w \) be two complex numbers such that \( \bar{z} + i\bar{w} = 0 \) and \( \text{Arg}(zw) = \pi \). Then \( \text{Arg}(z) = \)
If the complex numbers \( z_1, z_2, 0 \) are vertices of an equilateral triangle, then \( z_1^2 + z_2^2 = \)
Let \( z = x + iy \) be a complex number with \( x, y \in \mathbb{Z} \). Then the area (in square units) of the rectangle whose vertices are the roots of the equation \( \bar{z} \cdot z^3 + z \cdot \bar{z}^3 = 350 \) is:
If a function \( f \) satisfies \( f(x+1) + f(x-1) = \sqrt{2}f(x) \), then \( f(x+2) + f(x-2) = \)
The domain of the real-valued function \(f(x) = \frac{\sqrt{\log_{0.5}(x-3)}}{\sqrt{x-1}}\) is:
If the system of equations \(x+y-z=6\), \(3x-y+z=2\), and \(x+ky+z=-8\) has a unique solution \(x=2\), \(y=0\), \(z=-4\), then the value of \(k\) satisfies which quadratic equation?
A plane electromagnetic wave of wave interface 10W/m2 strikes a small mirror of area 20 cm2, held perpendicular to the approaching wave. The radiation force on the mirror will be:
Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is 0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.
 
Measurement conditionMain scale readingCircular scale reading
Two arms of gauge touching each other without wire0 division4 divisions
Attempt-1: With wire4 divisions
20 divisions
Attempt-2: With wire
4 divisions
16 divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?
 
The maximum value of \( f(x) = \frac{x}{1 + 4x + x^2} \) is
The minimum value of \( f(x) = x + \frac{4}{x + 2} \) is
If \( f(x) = \max \{3 - x, 3 + x, 6\} \) is not differentiable at \( x = a \) and \( x = b \), then \( |a| + |b| = \)
What are the oxidation states of the three Br atoms in the molecule \( \mathrm{Br_3O_6} \)?
The bond lengths of \( \mathrm{C_2} \), \( \mathrm{N_2} \), and \( \mathrm{B_2} \) molecules are \( X_1 \), \( X_2 \), and \( X_3 \) pm respectively. The correct order of their bond lengths is:
Arrange the following in decreasing order of their pK$_b$ values:
Arrange the following in decreasing order of their pKb values:
Identify the major product formed from the following:
Nai/dry acetone
When 1-chlorobutane is treated with aqueous KOH it gives P. However, when treated with alcoholic KOH it gives Q. Identify P and Q respectively.