Class 12th

Let A₁, A₂, A₃,…. be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = $\frac{1}{1296}$  and A₂ + A₄ = $\frac{7}{36},$  then, the value of A₆ + A₈ + A₁₀ is equal to

A triangular shaped wire carrying 10 A current is a uniform magnetic field of 0.5T, as shown in figure.

The magnetic force on segment CD is (Given BC = CD = BD = 5cm).

Given below are two statements:

Statement I : A uniform wire of resistance 80 $\Omega$ is cut into four equal parts. These parts are now connected in parallel. The equivalent resistance of the combination will be 5 $\Omega .$

Statement II : Two resistances 2R and 3R are connected in parallel in a electric circuit. The value of thermal energy developed in 3R and 2R will be in the ratio 3 : 2.

In the light of the above statements, choose the most appropriate answer from the option given below

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is

Starting from the origin at time $t=0$ , with initial velocity $5\hat{j}{\mathrm{m}\mathrm{s}}^{-1}$ , a particle moves in the $x$$-y$ plane with a constant acceleration of $(10\hat{i}+4\hat{j}){\mathrm{m}\mathrm{s}}^{-2}$ . At time $t$ , its coordinates are $(20$ $\left.m,y_{0}m\right)$ . The values of $t$ and $y_0$ are, respectively:

A uniform electric field E = (8m/e) V/m is created between two parallel plate of length 1m as shown in figure, (where m = mass of electron and e = charge of electron). An electron enters the field symmetrically between the plates with a speed of 2 m/s. The angle of the deviation ( θ) of the path of the electron as it comes out of the field will be________.

Let A be matrix of order 3 * 3 and det(A) = 2. Then $det\left(det\left(A\right)adj\left(5adj\left(A^3\right)\right)\right)$ is equal to

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