Class 12th

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, x²/4 + y²/2 = 1 from any of its foci?

Let M = {A = [a,b;c,d] : a,b,c,d ∈ {±3, ±2, ±1, 0}}. Define f: M → Z, as f(A) = det(A), for all A ∈ M, where Z is set of all integers. Then the number of A ∈ M such that f(A) = 15 is equal to………

Let y = y(x) be solution of the following differential equation e?(dy/dx) - 2e?sinx + sinxcos²x = 0, y(π/2) = 0. If y(0) = log?(α + βe?²), then 4(α+β) is equal to…….

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:

Let p = 2i + 3j + k and q = i + 2j + k be two vectors. If a vector r = (αi + βj + γk) is perpendicular to each of the vectors (p + q) and (p - q), and |r| = √3, then |α| + |β| + |γ| is equal to……

Charge Q? and Q? are at point A and B of a right angle triangle OAB (see figure). The resultant electric field at point O is perpendicular to the hypotenuse, then Q?/Q? is proportional to:

An AC circuit has R = 100Ω, C = 2µF and L = 80mH, connected in series. The quality factor of the circuit is:

If f(x+y)=f(x)f(y) and Σₓ₌₁^∞ f(x)=2, x,y∈N where N is the set of all natural numbers, then the value of f(4)/f(2) is:

In the figure below, P and Q are two equally intense coherent sources emitting radiation of wavelength 20 m. The separation between P and Q is 5 m and the phase of P is ahead of that of Q by 90°. A, B and C are three distinct points of observation, each equidistant from the midpoint of PQ. The intensities of radiation at A, B, C will be in the ratio:

The general solution of the differential equation √(1+x²+y²+x²y²) + xy(dy/dx)=0 is: (where C is a constant of integration)