List of practice Questions

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:
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 \(A^2 + B^2 + AB =\)
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:
The domain of the real-valued function \(f(x) = \frac{\sqrt{\log_{0.5}(x-3)}}{\sqrt{x-1}}\) is:
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 \( |a| + |b| = \)
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
Arrange the following in increasing order of their reactivity for nucleophilic addition reaction:
Arrange the following in increasing order of their reactivity for nucleophilic addition reaction:
The diameter range of colloidal particles is approximately:
Photographic plates are prepared by coating emulsion of which of the following in gelatin?
What are x and y in the following reaction?
$x \text{Pb}_3\text{O}_4 \rightarrow y \text{PbO} + \text{O}_2$
The catalyst used in the manufacture of polyethylene is a mixture of
Which of the following is correct related to the colours of TiCl$_3$ (X) and [Ti(H$_2$O)$_6$]Cl$_3$ (Y)?
The compound or ion which is not aromatic in the following is
If molten NaCl contains SrCl$_2$ as impurity, crystallization can generate
Identify the correct statements about Boron:
I. It has high melting point.
II. It has high density.
III. It has high electrical conductivity.
IV. B-10 isotope of it has high ability to absorb neutrons