List of practice Questions

If the shortest distance between the line joining the points $(1,2,3)$ and $(2,3,4)$, and the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}$ is $\alpha$, then $28 \alpha^2$ is equal to

Two parallel plate capacitors of capacity $C$ and $3 C$ are connected in parallel combination and charged to a potential difference $18\, V$ The battery is then disconnected and the space between the plates of the capacitor of capacity $C$ is completely filled with a material of dielectric constant $9$ The final potential difference across the combination of capacitors will be______ $V$

Let $a \in R$ and let $\alpha, \beta$ be the roots of the equation $x^2+60^{\frac{1}{4}} x+a=0$. If $\alpha^4+\beta^4=-30$, then the product of all possible values of $a$ is_____

For the two positive numbers $a, b$, if $a, b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a}, 10$ and $\frac{1}{b}$ are in an arithmetic progression, then $16 a+12 b$ is equal to

A hole is drilled in a metal sheet At $27^{\circ} C$, the diameter of hole is $5 cm$ When the sheet is heated to $177^{\circ} C$, the change in the diameter of hole is $d \times 10^{-3} cm$ The value of $d$ will be ___ if coefficient of linear expansion of the metal is $16 \times 10^{-5} { }^{\circ} C$

A spherical body of mass $2\, kg$ starting from rest acquires a kinetic energy of $10000\, J$ at the end of $5^{\text {th }}$ second The force acted on the body is ___$N$

A hollow cylindrical conductor has length of $314\, m$, while its inner and outer diameters are $4 \, mm$ and $8 \, mm$ respectively The resistance of the conductor is $n \times 10^{-3} \Omega$.If the resistivity of the material is $24 \times 10^{-8} \Omega m$ The value of $n$ is ___

Assume that protons and neutrons have equal masses Mass of a mucleon is $16 \times 10^{-27} kg$ and radius of nucleus is $15 \times 10^{-15} A ^{1 / 3} m$ The approximate ratio of the nuclear density and water density is $n \times 10^{13}$ The value of $n$ is ___

Vectors $a \hat{i}+b \hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+4 \hat{k}$ are perpendicular to each other when $3 a+2 b=7$, the ratio of $a$ to $b$ is $\frac{x}{2}$ The value of $x$ is __

As shown in the figure, a combination of a thin plano concave lens and a thin plano convex lens is used to image an object placed at infinity The radius of curvature of both the lenses is $30 \,cm$ and refraction index of the material for both the lenses is $175$ Both the lenses are placed at distance of $40 \,cm$ from each other Due to the combination, the image of the object is formed at distance $x=$__ $cm$, from concave lens.

A stream of a positively charged particles having $\frac{q}{ m }=2 \times 10^{11} \frac{ C }{ kg }$ and velocity $\vec{v}_0=3 \times 10^7 \hat{i} m / s$ is deflected by an electric field $18 kV / m$ The electric field exists in a region of $10 cm$ along $x$ direction Due to the electric field, the deflection of the charge particles in the $y$ direction is___ $mm$

Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed The resultant amplitude is equal to the $\sqrt{3}$ times of amplitude of individual motions The phase difference between the two motions is _______(degree)

In the circuit shown in the figure, the ratio of the quality factor and the band width is__s

Two ideal diodes are connected in the network as shown in figure The equivalent resistance between $A$ and $B$ is ______$\Omega$

Solid sphere $A$ is rotating about an axis $PQ$ If the radius of the sphere is $5 cm$ then its radius of gyration about $PQ$ will be $\sqrt{x} cm$ The value of $x$ is __

Solid sphere A is rotating about an axis PQ.If the radius of the sphere is 5cm then its radius of gyration about PQ will be √xcm.The value of x is

In a potentiometer arrangement, a cell of emf $1.20\, V$ gives a balance point at $36 \, cm$ length of wire This cell is now replaced by another cell of emf $1.80\, V$. The difference in balancing length of potentiometer wire in above conditions will be _______$cm$

A block of a mass $2\, kg$ is attached with two identical springs of spring constant $20\, N / m$ each The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure) When the mass is displaced from its equilibrium position, it executes a simple harmonic motion The time period of oscillation is $\frac{\pi}{\sqrt{x}}$ in SI unit The value of $x$ is

$\frac{x}{x+4}$ is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) third permitted energy level to the second level and (ii) the highest permitted energy level to the second permitted level The value of $x$ will be

Uracil is a base present in RNA with the following structure $\%$ of $N$ in uracil is ___

A block of ice of mass $120\, g$ at temperature $0^{\circ} C$ is put in $300\, gm$ of water at $25^{\circ} C$ The $xg$ of ice melts as the temperature of the water reaches $0^{\circ} C$ The value of $x$ is [Use: Specific heat capacity of water $=4200\,Jkg ^{-1} K ^{-1}$, Latent heat of ice = 3.5 $\times$ 10⁵JKg^-1]

Three identical spheres each of mass $M$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $3 m$ each Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $\sqrt{ x } m$ The value of $x$ is

A particle is moving in a straight line such that its velocity is increasing at $5 \,ms ^{-1}$ per meter. The acceleration of the particle is _____$ms ^{-2}$ at a point where its velocity is $20 \,ms ^{-1}$

Let $S=\{\theta \in[0,2 \pi): \tan (\pi \cos \theta)+\tan (\pi \sin \theta)=0\}$ Then $\displaystyle\sum_{\theta \in s } \sin ^2\left(\theta+\frac{\pi}{4}\right)$ is equal to _______

If $\frac{1^3+2^3+3^3+\ldots \text { up to } n \text { terms }}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \text { up to } n \text { terms }}=\frac{9}{5}$, then the value of $n$ is

If the shortest between the lines $\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and $\frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{z+2 \sqrt{6}}{5}$ is $6$ , then the square of sum of all possible values of $\lambda$ is