List of practice Questions

If the equation \(x^4 + 7x^3 + 18x^2 + 20x + 8 = 0\) has a repeated root, then that repeated root is:
If the equation \(2x^3 + 5x^2 - 4x - 12 = 0\) has a repeated root, then the constant term of the quadratic equation whose roots are the distinct roots of the given equation is:
The greatest integer \( r \) such that \( 30^r \) divides \( 30! \) is:
A matrix whose elements \( a_{ij} \) are defined by \( a_{ij} = \frac{1}{3}|i - 5j|, \; i, j = 1, 2, 3 \) is
If \( P, Q \) and \( R \) are \( 3 \times 3 \) matrices such that} \[ P = \begin{bmatrix} 3x^2 + x + 3 & 2x^2 - x + 4 & 7x^2 + 8x + 5 \\ 5x^2 + 3x + 2 & 4x^2 - 2x - 1 & 7x^2 + 5x + 8 \\ 3x^2 + 2x + 5 & 4x^2 - x - 2 & 3x^2 + 8x + 7 \end{bmatrix} = Px^2 + Qx + R \] then det \( R = \) ?
If \( C \) is a point on the straight line joining the points \( A = -2+i \) and \( B = 3 - 4i \) in the Argand plane and } \[ \frac{AC}{CB} = \frac{1}{2}, \] then the argument of \( C \) is
If \( \alpha \) is the modulus of \( z_1 = 4 + 3i \), then a point that does not lie in the region represented by } \[ |z - z_1| \leq \alpha \] is
If \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle and \( z \) is its circumcentre, then
If \( a \) and \( b \) are the roots of the equation \( y^2 + y + 1 = 0 \), then the value of \( a^4 + b^4 + a^{-1}b^{-1} \) is
Let \( A \subseteq \mathbb{R}, B \subseteq \mathbb{R} \) and \( f : A \to B \) be defined by \( f(x) = x^2 - 3x + 2 \). If \( f \) is a bijection, then
If \( f:\mathbb{R} \to \mathbb{R} \) is defined by \( f(x+y) = f(x) + f(y) \ \forall x, y \in \mathbb{R} \) and \( f(1) = 7 \), then \(\sum\limits_{r=1}^n f(r) =\)
Let \( A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \), where \( \theta \in \mathbb{R} \). Then \( A^n \), for any positive integer \( n \), is equal to:
A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight line on the horizontal ground. The lamp post and the person are always vertical. If the speed of the person is 60 cm/sec, then find the speed of tip point of its shadow with respect to the person in cm/sec.
Arrange the following stations of Trance Siberian Railway from East to West direction.
A. Chita
B. Vladivostak
C. Moscow
D. Omsk
Choose the correct answer from the options given below :
Resistance of a carbon resistor determined from colour codes is (22000±5%)Ω. The colour of third band must be:

Identify correctly matched pairs: \[ \text{I. Borax} \quad \text{Na}_2\text{B}_4\text{O}_7 \cdot 10\text{H}_2\text{O} \quad \text{II. Kernite} \quad \text{Na}_2\text{B}_4\text{O}_7 \cdot 7\text{H}_2\text{O} \] \[ \text{III. Bauxite} \quad \text{Al}_2\text{O}_3 \cdot \text{H}_2\text{O} \quad \text{IV. Cryolite} \quad \text{Na}_3\text{AlF}_6 \]

The correct value for log\(_{10}\) \(K_c\) (where \(K_c\) is the equilibrium constant) for the equilibrium reaction at 298 K \[ \text{2Fe}^{3+} + 3\text{I}^- \rightleftharpoons 2\text{Fe}^{2+} + \text{I}_3^- \quad (\text{E}^\circ_{\text{cell}} = x \, \text{V}) \]

The reaction between ethyl bromide and ethanolic solution of ammonia (excess) is a

What is the correct boiling point order of the following haloalkanes? \[ \text{i) 2-chloro 2-methylpropane} \quad \text{ii) 1-Cholobutane} \quad \text{iii) 2-Chlorobutane} \]

Para-nitro phenol has higher boiling point than ortho-nitrophenol. This is due to
Which of the following carbonyl compounds reacts with HCN at a faster rate?

In the following reaction sequence, A and B respectively are

Sodium in liquid ammonia initiates the polymerization of which of the following monomers?
Which of the following represents α-D-fructofuranose?
Which of the following conversions indicate the presence of –CH\(_2\)OH group in glucose?