Class 12th

If the system of linear equations

2x + 3y – z = 2

x + y + z = 4

$x-y+\left|\lambda \right|z=4\lambda -4$ where $\lambda \in R,$ has no solution, then

A slab of dielectric constant K has the same cross-sectional area as the plates of a parallel plate capacitor and thickness $\frac{3}{4}d,$ where d is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be:

(Given C₀ = capacitance of capacitor with air as medium between plates.)

Let a function f : N → N be defined by

$f\left(n\right)=[\begin{array}{ccc}\hfill 2n\hfill & \hfill ,\hfill & \hfill n=2,4,6,8....\hfill \\ \hfill n-1\hfill & \hfill ,\hfill & \hfill n=3,7,11,15,....\hfill \\ \hfill \frac{n+1}{2}\hfill & \hfill ,\hfill & \hfill n=1,5,9,13,.....\hfill \end{array}$

then, f is

If  then the value of 16α is equal to

A vessel contains 14 g of nitrogen gas at a temperature of 27°C. The amount of heat to be transferred to the gas to double the r.m.s speed of its molecules will be:

Take R = 8.32 J mole1 k1.

Consider a cylindrical tank of radius 1m is filled with water. The top surface of water is at 15m front the bottom of the cylinder. There is a hole on the wall of a cylinder at a height of 5m from the bottom. A force of 5 * 10⁵N is applied an the top surface of water using a piston. The speed of ifflux from the hole will be:

(given atmospheric pressure P_A = 1.01 * 10⁵Pa, density of water ${\rho }_w=1000kg/m^3$  and gravitational acceleration g = 10m/s²)

Assume there are two identical simple pendulum clocks. Clock – 1 is placed on the earth and Clock – 2 is placed on a space station located at a height h above the earth surface. Clock – 1 and Clock – 2 operate at time periods 4s and 6s respectively. Then the value of h is – (consider radius of earth R_E = 6400km and g on earth 10 m/s²)

A bullet of mass 200g having initial kinetic energy 90 J is shot inside a long swimming pool as shown in the figure. If it's kinetic energy reduces to 40 J within 1s, the minimum length of the pool, the bullet has to travel so that it completely comes to rest is

A uniform metal chain of mass m length 'L' passes over a massless and frictionless pulley. It is released from rest with a part of its length $\text{'}\mathcal{l}\text{'}$ is hanging on one side and rest of its length $\text{'}L-\mathcal{l}\text{'}$ is hanging on the other side of the pully. At a certain point of time, when $\mathcal{l}=\frac{L}{x},$ the acceleration of the chain is $\frac{g}{2}.$ The value of x is__________.