List of practice Questions

An external pressure \(P\) is applied on a cube at \(0^{\circ}C\) so that it is equally compressed from all sides.\(K\) is the bulk modulus of the material of the cube and \(\alpha\) is its coefficient of linear expansion.Suppose we want to bring the cube to its original size by heating it. The temperature should be raised by :

$C_p$ and $C_v$ are specific heats at constant pressure and constant volume respectively. It is observed that $C_p-C_v$ = a for hydrogen gas $C_p-C_v$ = b for nitrogen gas The correct relation between $a$ and $b$ is :

A car is standing $200\, m$ behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2 \, m/s^2$ and the car has acceleration $4 \, m/s^2$. The car will catch up with the bus after a time of :

Two particles $A$ and $B$ of equal mass $M$ are moving with the same speed $v$ as shown in the figure. They collide completely inelastically and move as a single particle $C$. The angle $\theta$ that the path of $C$ makes with the $X-axis$ is given by :

A diverging lens with magnitude of focal length $25\, cm$ is placed at a distance of $15\, cm$ from a converging lens of magnitude of focal length $20\, cm$. A beam of parallel light falls on the diverging lens. The final image formed is :

For any three positive real numbers a, b and c, $9(25a^2 + b^2) + 25 (c^2 - 3ac) = 15b (3a + c)$. Then :

Let $a, b, c \, \in \, R$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f (x + y) = f (x) + f (y) + xy, \forall \, x, y \, \in \, R,$ then $\displaystyle\sum^{10}_{n = 1} f(n)$ is equal to :

$\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{\left(\pi - 2x\right)^{3}} $ equals :

If for $x \epsilon \left(0, \frac{1}{4}\right) ,$ the derivative of $ \tan^{-1} \left(\frac{6x \sqrt{x}}{1-9x^{3}}\right) $ is $\sqrt{x} . g(x)$, then $g(x)$ equals :

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in $s m$) of the flower-bed, is :

Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :

The integral $\int\limits^{\frac{3\, \pi}{4}}_{\frac{\pi}{4}} \frac{dx}{ 1 + \cos \, x}$ is equal to :

The area (in s units) of the region $\{ (x , y) : x \geq 0 , x + y \leq 3, x^2 \leq 4 y$ and $y \leq 1 + \sqrt{x} \}$ is

If $(2 + \sin \, x ) \frac{dy}{dx} + (y + 1) \cos \, x = 0$ and $y(0) = 1,$ then $y \left( \frac{\pi}{2} \right)$ is equal to :

A steel rail of length $5\,m$ and area of cross section 40 $cm^2$ is prevented from expanding along its length while the temperature rises by $10^{\circ}C$. If coefficient of linear expansion and Young?s modulus of steel are $1.2 \times 10^{-5} \, K^{-1}$ and $2 \times 10^{11} \, Nm^{-2}$ respectively, the force developed in the rail is approximately :

Addition of sodium hydroxide solution to a weak acid (HA) results in a buffer of pH 6. If ionisation constant of HA is $10^{-5}$, the ratio of salt to acid concentration in the buffer solution will be :

An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency $10\, GHz$. What is the frequency of the microwave measured by the observer ? (speed of light $= 3 \times 10^8 \, ms^{-1}$)

Both lithium and magnesium display several similar properties due to the diagonal relationship; however, the one which is incorrect, is :

If $(27)^{999}$ is divided by $7$, then the remainder is :

If, for a positive integer n, the quadratic equation, $x(x+1)+(x+1)(x+2)+....+(x + \overline{ n - 1}) (x+ n)=10n$ has two consecutive integral solutions, then $n$ is equal to :

If two different numbers are taken from the set $\{0,1,2,3, \ldots \ldots, 10\}$then the probability that their sum as well as absolute difference are both multiple of $4$, is :

In a coil of resistance $100 \, \Omega$, a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is :

In a physical balance working on the principle of moments, when $5\,mg$ weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct ?

In the above circuit the current in each resistance is

Let a vertical tower $AB$ have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that $AP = 2AB$. If $\angle BPC = \beta $, then tan $\beta$ is equal to :