Without expanding prove that det    =  (1 + a² + b²)³

Q.1. Find the equation of line which is tangent at one point and normal at another ponit of the curve y = 8t³- 1, x = 4t² + 3.

Q.2.  Find the minimum value of (x₁ - x₂)² +( x₁²/4 - (16 - (x₂ - 12)²)^1/2 , x₁R, x2[8,16].

Q.1. If the normal to the curve y=f(x) at x=0 can be given by the equation 3x-y+3=0, then find the value of  x²[(f(x²) - 5f(4x²) + 4f(7x²)]^-1.

Q.2. If the tangent at any point of the curve x^2/3 + y^2/3 = a^2/3 meets the axes of the coordinates in A and B then show that the locus of the mid-point of AB is a circle.

Q.3. If the curve x = a(cost + log(tan(t/2))), y = asint, show that the portion of the tangent between the point of contactb and the x-axis is of constant length.

you are asking about 15 days? i did it in a single day for my school exam. 15 days are more than enough to prepare a topic provided you are consistent. what i think is that you are giving too much time for these topics.

mam,  this is a comprehension passage on which these five questions were based. how can i post these questions on different posts?

Totally Challanging questions.

Questions 1-5

  Particle storage rings faciliate collisions b/w elctronns and positrons (positively charged electrons). When electrons and positrons collide at high energies, they can annihilate each other and produce a variety of elementary particles, including photons. In these reactions, momentum is always conserved. Powerful magents are placed at various points along the ring to create a force directed towards the centre, thereby guiding the paticles in a circular motion. In a scattering exp., the particle beams circle in opposite directions and collide head-on at the intersection points, which are surrounded by particle detectors. The particles in the ring are acceletrated, they radiate electromagnetic energy. Radio frequecy power is continually fed into the storage ring to compensate for the energy loss.

  The reaction rate, R is the no. of particle scattered per second in the storage ring. It is given by the formula: R + L, where L is the luminousity and is the cross-section of the rewaction. The cross-section is a quantity that depends only on the particular type of reaction being considered. The luminousity contains all of the information about the initial conditions for a reaction and is given by:

                                   L = (N_e-N_e+)f / A

where N_e- refers to the no. of electrons, N_e+ referes to the no. of positrons, A is the cross-sectional area of the storage ring, and f is the no. of revolutions per second made by the particles. Electron-positron reaction rates for modern particle storage rings are typically on the order of 10^-3 s^-1.

 Q.1. What percentage of the wotk done on the circulating paticles is done by the magnetic field?

(a) 0%, because the direction of the magnetic froce is perpendicular to the direction in which the particles travel.

(b) 0%, because the magentic foeld doesn't exert a force on the particles.

(c) 50%, because the magenetic and electric field provide equal amounts to the paricles.

(d) 100%, because the magnetic field is the source for the centripetal force that accelerates the particles.

Q.2. Which of the following would increase the reaction rate in an electron-positron storage ring?

 (I) Decreasing the cross-sectional area of the storage ring

 (II) Increasing the energy of the particles

 (III) Increasing the no. of positrons in the storage ring.

(a) I only                            (b) III only                              (c) II and III only                                 (d) I, II and III

Q.3. The work done on the particles by the gravitational field of the earth is not considered when figuring their energy loss per revolution. This is because

(a) the gravitationl force is perpendicular to the gravitational ecceleration

(b) the energy lost is due to gravity is equal to the energy gained from the magnetic fields

(c) the particles do not experience a sigmificant gravitational force

(d) the luminosity is not dependant on gravitational acceleration

Q.4. Two electrons with equal speed collide head-on at the total energuy of 180 GeV. If the speed of the first electron after the collision is 0.9c, what is the speed of the second electron after collision ( c = 3 x 10⁸ ms^-1is the speed of light)

(a) 0.45c                              (b) 0.7c                           (c) 0.8c                               (d) 0.9c

Q.5. An electron and positron are held in a storage ring, and they each lose 260 MeV of energy per revolution in the form of electromagnetic radiation. If the frequency of revolution is 10,000 Hz, how much power must be supplied to keep the total energy constant at 180 GeV?

 (a) 1.8 x 10⁶ MeV/s                                                                       

 (b) 2.6 x 10⁶ MeV/s

 (c) 5.2 x 10⁶ MeV/s

 (d) 9 x 10⁸ MeV/s

Please explain the answer with solution. Rates assured...