List of practice Questions

If \(\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}} = m\) and  \(\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n,\) then the point \( (m, n) \) lies on the line

Suppose \( \theta \in \left[ 0, \frac{\pi}{4} \right] \) is a solution of \( 4 \cos \theta - 3 \sin \theta = 1 \). Then \( \cos \theta \) is equal to:

Consider the following two statements:
Statement I: For any two non-zero complex numbers \( z_1, z_2 \),
\((|z_1| + |z_2|) \left| \frac{z_1}{|z_1|} + \frac{z_2}{|z_2|} \right| \leq 2 (|z_1| + |z_2|)\)
Statement II: If \( x, y, z \) are three distinct complex numbers and \( a, b, c \) are three positive real numbers such that  
\(\frac{a}{|y - z|} = \frac{b}{|z - x|} = \frac{c}{|x - y|},\)
then  
\(\frac{a^2}{y - z} + \frac{b^2}{z - x} + \frac{c^2}{x - y} = 1.\)
Between the above two statements,

Let the line 2x + 3y – k = 0, k > 0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is x² + y² – 3x – 2y = 0 and the length of the latus rectum of the ellipse x² + 9y² = k² is m n , where m and n are coprime, then 2m + n is equal to

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :

Let \( A \) and \( B \) be two square matrices of order 3 such that \( |A| = 3 \) and \( |B| = 2 \). Then \(|A^\top A (\text{adj}(2A))^{-1} (\text{adj}(4B)) (\text{adj}(AB))^{-1} A A^\top|\) is equal to:

The coefficients a, b, c in the quadratic equation ax² + bx + c = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is :

The integral \(\int_{0}^{\pi/4} \frac{136 \sin x}{3 \sin x + 5 \cos x} \, dx\) is equal to

If the function \(f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3},\) \(x \in \mathbb{R} \), is continuous at \( x = 0 \), then \( f(0) \) is equal to:

Let \( A = \{1, 3, 7, 9, 11\} \) and \( B = \{2, 4, 5, 7, 8, 10, 12\} \). Then the total number of one-one maps \( f: A \to B \), such that \( f(1) + f(3) = 14 \), is:

If the system of equations \(11x + y + \lambda z = -5,\) \(2x + 3y + 5z = 3,\) \(8x - 19y - 39z = \mu\) has infinitely many solutions, then \( \lambda^4 - \mu \) is equal to:

If \( y = y(x) \) is the solution of the differential equation \(\frac{dy}{dx} + 2y = \sin(2x), \quad y(0) = \frac{3}{4},\)then \(y\left(\frac{\pi}{8}\right)\) is equal to:

Let two straight lines drawn from the origin \( O \) intersect the line \(3x + 4y = 12\) at the points \( P \) and \( Q \) such that \( \triangle OPQ \) is an isosceles triangle and \( \angle POQ = 90^\circ \).  If \( l = OP^2 + PQ^2 + QO^2 \), then the greatest integer less than or equal to \( l \) is:

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)² is equal to :

Let \( d \) be the distance of the point of intersection of the lines \(\frac{x+6}{3} = \frac{y}{2} = \frac{z+1}{1}\) and \(\frac{x-7}{4} = \frac{y-9}{3} = \frac{z-4}{2}\) from the point \((7, 8, 9)\). Then \( d^2 + 6 \) is equal to:

The number of halobenzenes from the following that can be prepared by Sandmeyer's reaction is .......

9.3 g of pure aniline is treated with bromine water at room temperature to give a white precipitate of the product 'P'. The mass of product 'P' obtained is 26.4 g. The percentage yield is ......... %.

In the lewis dot structure for \(NO^{-}_{2}\), total number of valence electrons around nitrogen is .......

During Kinetic study of reaction 2A + B \(\mapsto \) C + D, the following results were obtained:

	A[M]	B[M]	initial rate of formation of D
I	0.1	0.1	6.0 × 10-3
II	0.3	0.2	7.2 × 10-2
III	0.3	0.4	2.88 × 10-1
IV	0.4	0.1	2.40 × 10-2

Based on above data, overall order of the reaction is ..........

An artificial cell is made by encapsulating \(0.2 \, \text{M}\) glucose solution within a semipermeable membrane. The osmotic pressure developed when the artificial cell is placed within a \(0.05 \, \text{M}\) solution of NaCl at 300 K is ______ \( \times 10^{-1} \, \text{bar} \). (Nearest Integer)
Given: \(R = 0.083 \, \text{L bar mol}^{-1} \text{K}^{-1}\)
Assume complete dissociation of NaCl.

In a borax bead test under hot condition, a metal salt (one from the given) is heated at point B of the flame, resulted in green colour salt bead. The spin-only magnetic moment value of the salt is ............. BM (Nearest integer)
[Given atomic number of Cu = 29, Ni = 28,Mn = 25, Fe = 26]

The spin-only magnetic moment value of the ion among \( \text{Ti}^{2+}, \text{V}^{2+}, \text{Co}^{3+}, \text{and} \, \text{Cr}^{2+} \), that acts as a strong oxidizing agent in aqueous solution is:\(\dots\) BM (Nearest integer).
Given atomic numbers: Ti: 22, V: 23, Cr: 24, Co: 27

The heat of combustion of solid benzoic acid at constant volume is \(-321.30 \, \text{kJ}\) at \(27^\circ \text{C}\). The heat of combustion at constant pressure is \((-321.30 - xR)\, \text{kJ}\). The value of \(x\) is:

Consider the given chemical reaction sequence :

Total sum of oxygen atoms in Product A and Product B are .........

\(C_{11}H_{18}O_{12}\)The value of the Rydberg constant (\(R_H\)) is \(2.18 \times 10^{-18} \, \text{J}\). The velocity of an electron having mass \(9.1 \times 10^{-31} \, \text{kg}\) in Bohr's first orbit of the hydrogen atom is:\(\dots \times 10^5 \, \text{ms}^{-1}\) (nearest integer).