List of practice Questions

If \( x \) is a real number and \( \lfloor x \rfloor \) is the greatest integer \( \leq x \), then \[ 3\lfloor x \rfloor + 2 - \lfloor x \rfloor = 0 \] Will the above equation have any real root?

If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \[ x^2 - 10x + 15 = 0, \] then find the quadratic equation whose roots are \( \left( \alpha + \frac{\alpha}{\beta} \right) \) and \( \left( \beta + \frac{\beta}{\alpha} \right) \).

Let \( P = \{2, 3, 4, \ldots, 100\} \) and \( Q = \{101, 102, 103, \ldots, 200\} \). How many elements of \( Q \) are there such that they do not have any element of \( P \) as a factor?

What is the sum of all the 2-digit numbers which leave a remainder of 6 when divided by 8?

Which of the terms \( 2^{1/3}, 3^{1/4}, 4^{1/6}, 6^{1/8}, 10^{1/12} \) is the largest?

If the roots of the equation \[ (a^2 + b^2)x^2 + 2(b^2 + c^2)x + (b^2 + c^2) = 0 \] are real, which of the following must hold true?

Find the remainder when \( 2^{1040} \) is divided by 131.

In the figure below, \( \angle MON = \angle MPO = \angle NQO = 90^\circ \), \( OQ \) is the bisector of \( \angle MON \), and \( QN = 10 \), \( OR = 40/\sqrt{7} \). Find \( OP \).

Consider a sequence \( S \) whose \( n \)th term \( T_n \) is defined as \( T_n = 1 + \frac{3}{n} \), where \( n = 1, 2, \ldots \). Find the product of all the consecutive terms of \( S \) starting from the 4th term to the 60th term.

100 mL of N/5 HCl was added to 1 g of pure $ {CaCO_3}$ What would remain after the reaction?

Juxtaglomerular apparatus is made up of

The ratio of number of oxygen atoms (O) in 16.0 g ozone $(O_3), \,28.0\, g$ carbon monoxide $(CO)$ and $16.0$ oxygen $(O_2)$ is (Atomic mass: $C = 12,0 = 16$ and Avogadro?? constant $N_A = 6.0 x 10^{23}\, mol^{-1}$)

In the Wheatstone's network given, \(P = 10\,\Omega , Q = 20\,\Omega , R = 15 \,\Omega , S = 30 \,\Omega\), the current passing through the battery (of negligible internal resistance) is

A solid body of constant heat capacity $1 \,J/?C$ is being heated by keeping it in contact with reservoirs in two ways : (i) Sequentially keeping in contact with $2$ reservoirs such that each reservoir supplies same amount of heat. (ii) Sequentially keeping in contact with $8$ reservoirs such that each reservoir supplies same amount of heat. In both the cases body is brought from initial temperature $100^{\circ}C$ to final temperature $200^{\circ}C$. Entropy change of the body in the two cases respectively is :

Rearrange the jumbled alphabets in the following four options and find the odd word among them.

The equation of straight line through the intersection of the lines $x - 2y = 1 $ and $x + 3y = 2$ and parallel $3x + 4y = 0$ is

A man of mass \(100 \,kg\). is standing on a platform of mass \(200 \,kg\). Which is kept on a smooth ice surface. If the man starts moving on the platform with a speed \(30 \,m/sec\) relative to the platform then calculate with what velocity relative to the ice the platform will recoil?

The length of the sub-tangent, ordinate and the sub-normal are in

$100\, mL$ of $0.1\, M$ acetic acid is completely neutralized using a standard solution of $NaOH$.The volume of Ethane obtained at $STP$ after the complete electrolysis of the resulting solution is

The instantaneous values of alternating current and voltages in a circuit are given as $i=\frac{1}{\sqrt2}\sin(100\pi t)$ ampere $e=\frac{1}{\sqrt2}\sin(100\pi t+\frac{\pi}{3})$ Volt . The average power in watts consumed in the circuit is

The motion of a particle along a straight line is described by equation $ x = 8 + 12t - t^3 $ where $x$ is in metre and $t$ in second. The retardation of the particle when its velocity becomes zero is

Two sources of sound placed close to each other, are emitting progressive waves given by $y _{1}=4 \sin 600\, \pi t$ and $y _{2}=5 \sin 608\, \pi t :$ An observer located near these two sources of sound will hear

If mean and variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value atleast one is

The value of $a$ for which the volume of parallelepiped formed by the vectors $\hat {i}+a\hat{j} + \hat{k} \,\, \hat{j}+a\hat{k} $ and $ a\hat{i}+\hat{k} $ is minimum, is

Select a suitable name for this process : ${C_6H_{12}O_6 + 2 A D P + 2 P i -> 2C_2H_5OH + 2 ATP + 2CO_2\uparrow } $