Posted on 03/13/2006 8:40:36 PM PST by USMMA_83
Fellow Freepers, last week and this week is class 12 exam time in India. There have already been a handful of suicides due to the pressure to pass the final year exam. It's compulsory to pass this exam to get into college in India. Have a look at the sample question papers, and you'll see how far our education is falling behind.
MATHEMATICS CLASS- XII
Matrices
1. Find a, b, c when f(x) = ax2 + bc + c, f(2) = 11 and f(-3) = 6 = f(0) Determine the quadratic function f(x) and find its value when x = 1.
2. Using determinants solve the following system of equations : (i) 2x – 4y = -3 (b) 4x + 3y = 3 4x + 2y = 9 8x – 9y = 1.
3. Solve the following system of equations using Cramer’s rule : (i) x + 2y = 1 (b) 9x + 5y = 10 3x + y = 4 3y – 2x = 8
4. Solve the following system of equations by using Cramer’s rule : (a) x + y + z = 6 (b) 3x + y + z = 10 x – y + z = 2 x + y – z = 0 2x + y – z = 1 5x – 9y = 1
(c) 2x – y + 3z = 9 (d) 3x + y + 2z = 3 x + y + z = 6 x + y – z = -3 x – y + z = 2 x – 2y + z = 4
5. Solve the following system of equations by using Cramer’s rule : (a) x – y + z – 4 = 0 (b) x + y + z = 1 2x + y – 3z = 0 3x + 5y + 6z = 4 x + y + z – 2 = 0 9x + 2y – 36z = 17
6. Solve the following system of equations by using Cramer’s rule : (a) 5t – s + 4u = 5 (b) x + y + z + w = 1 2t + 3s + 5u = 2 x – 2y + 2z + 2w = -6 5t – 3s + 6u = -1 2x + y + 2z – 2w = -5 3x – y + 3z – 3w = -3. 7. Adjoint of a Square Matrix : The adjoint of a square matrix is the transpose of the matrix obtained by replacing each element of A by its co-factor in | A |.
8. Theorem : If A be any n-rowed square matrix : then (Adj. A) A = A(Adj. A) = | A | ln where ln is the n-rowed matrix.
9. For the following matrix A ; prove that A (Adj. A) = 0 1 -1 1 A = 2 3 0 18 2 10
10. Find the adjoint of the matrix 1 0 -1 A = 3 4 5 0 -6 -7
11. Singular Matrix : A square A is called a singular matrix of a non-singular matrix according as | A | or | A | 0, respectively.
12. Theorem: If A, B, be two n-rowed non-singular matrices, then A B is also non-singular and (AB) –1 = b –1 A –1 i.e. the inverse of a product is the product of the inverses taken in the order.
3 8 13. Let A be the matrix Find A –1 and verify that A –1 = 1/13 A – 4/13 I 2 1 where I is 2 2 unit matrix.
3 1 4 0 14. If A = and B = verify that (AB) –1 = B –1 A –1 4 0 2 5
1 2 15. Find the adjoint of the matrix A = and verify A (Adj.A) A = | A | I2 3 -5 a b 16. If A = , find Adj. A. c d
2 -3 17. Given A = , compute A –1 and show that 2A –1 9I – A. -4 7
1 0 0 18. Find Adj. A and A –1, if it exits where A = 3 3 0 5 2 -1
1 -1 1 19. If A = 2 -1 0 , find A2 and show that A2 = A –1 1 0 0
3 -1 2 1 20. If A = -4 0 and B = -1 -2 . Find (A’B) –1 2 1 1 1
1 2 5 21. Compute the inverse of the matrix A = 2 3 1 and verify that A-1 A = 1 -1 1 1
1 2 2 22. Let A = 2 1 2 . Prove that A2 - 4A – 5I = 0, Hence obtain A –1 2 2 1
2 0 -1 23. If A = 5 1 0 Prove that A –1 = A2 – 6A + 11I. 0 1 3
-4 -3 -3 24. If A = 1 0 1 Show that Adj. A = A 4 4 3
1 1 1 24. If A = 1 2 -3 Verify the theorem A (Adj. A) = (Adj. A ) A = | A | I. 2 -2 1
1 -2 3 25. Find A (Adj. A) for the matrix A = 0 2 -1 -4 5 2
26. Compute the inverse of each of the following matrices. 1 2 3 cos -sin 0 (i) 2 3 2 (ii) sin cos 0 3 3 4 0 0
27. Verify that (A B) –1= B –1 A –1 for the matrices A and B
2 1 4 5 Where A = and B = 5 3 3 4
2 0 0 1 28. Where A = and B = Verify that (AB) –1= B –1 A –1 5 3 2 4
2 5 29. If A = , find A-1 and verify that A –1 = -1/7 A + 8/7 I. 1 6
1 1 2 1 2 0 30. If A = 1 9 3 and B = 1 3 -1 , verify that (AB) –1 = B –1 A –1 1 4 2 1 -1 3
4 5 31. If A = then, show that A – 3I = 2[I + 3A –1] 2 1 32. Find the inverse of each of the following matrices and verify : A –1 A = I
2 0 -1 2 3 1 (i) 5 1 0 (ii) 3 4 1 0 1 3 3 7 2
-8 1 4 33. (a) If A = 1/9 4 4 7 Prove that A –1= A’. 1 -8 4
0 -1 2 0 1 (b) Given A = , B –1 1 0 2 -2 0 1 1
From the product C = AB and find C –1. What is the matrix BA? cos x -sin x 0 34. (a) If F(x) = sin x cos x 0 0 0 0
then show that F(x)F(y) = F(x + y), Hence prove that [F(x)] –1 = F(-x).
5 0 4 1 3 3 (b) Given A = 2 3 2 , B –1 = 1 4 3 compute (AB) -1 1 2 1 1 3 4
cos sin 35. If A = , verify that (i) (A –1) –1 = A (ii) (A’) –1 = (A –1)’ sin cos
PHYSICS Class XII
EMI, AC and OPTICS
Q.1> Define mutual inductance. Write its SI unit and give 2 factors on which it depends.
Q.2> Two coherent sources whose intensity ratio is 81:1 produce interference fringes. Find the ratio of intensity of maxima to that of minima
Q.3> A concave mirror and a concave lens are held in water. What changes, if any, do you expect in their focal lengths.
Q.4> You are given two convex lenses of focal length 80mm and 800mm. Which one will you use as an objective and which one as an eyepiece in an astronomical telescope. Draw the ray diagram for image formation and write the formula for magnification.
Q.5> With the help of a labeled diagram, explain the construction and working of an AC generator
Q.6> An LCR circuit has L = 4H, C = 0.1mF and R = 40ohm connected across a variable frequency 220V supply. Calculate a.> Resonance frequency b.> Impedance of circuit and amplitude of current at resonance c.> Rms potential drop across L, C and R at resonance d.> Phase difference b/w Current and Voltage at resonance
Q.7> A choke coil and a bulb are connected in series to a DC source. How does the brightness of the bulb change when an iron core is inserted into the choke.
Q.8> Define Self inductance. Give its SI unit and state 2 factors on which it depends
Q.9> The two slits in YDSE are separated by 0.03mm and the screen is kept 1.5m away. The 4th bright fringe is at a distance of 1cm from the central maxima. Calculate the wavelength of light used.
Q.10> Explain why white light is dispersed while passing through a prism.
Q.11> With the help of a ray diagram, illustrate the formation of image in a compound microscope. Derive the expression for magnifying power. How can the magnifying power be increased.
Q.12> Explain, with the help of a labeled diagram, the principle, construction and working of a step up transformer. Why is its core laminated?
Q.13> A 25μF capacitor, 0.1H inductor and 25ohm Resistor are connected across an AC source given by E = 310sin( 314t ). Find (a) Frequency of AC (b) Reactance of the circuit (c) Impedance (d) RMS current
Q.14> If a rate of change of current of 2A/s induces an emf of 10mV in a solenoid then find its inductance
Q.15> Give 3 differences b/w the fringes obtained in single slit experiment and YDSE
Q.16> A ray of light travelling from a denser to a rarer medium undergoes total internal reflection. Derive the expression for critical angle in terms of speed of light in the two media.
Q.17> The radius of curvature of each face of a biconvex lens of refractive index 1.5 is 30cm. Calculate the focal length in air and in water.
Q.18> A capacitor C, Resistor R and a 40mH inductor are connected across 60Hz AC. Calculate the capacitance if current is in phase with the voltage
Q.19> Prove that an ideal Inductor or Capacitor does not dissipate power in an AC circuit
Q.20> How can one distinguish b/w polarized and unpolarized light?
Q.21> A resistor R is connect across AC supply Eosinωt. Show that the power dissipated in Eo2/2R
Q.22> What is the difference b/w resistance, reactance and impedance?
Q.23> An astronomical telescope consist of 2 thin lenses 36cm apart and has a magnifying power of 8. Calculate the focal length of the lenses.
Q.24> Why is diffraction of sound waves easier to observe than that of light waves. What major changes would you expect in the diffraction experiment if white light is used instead of monochromatic light?
Q.25> Give reasons for the following observations on the MOON (a) Sun rise and sun set are abrupt (b) Sky appears dark (c) Rainbow is never formed
Q.26> An LR circuit is connected across 12V, 50Hz AC supply. The current drawn if 0.5A at an angle of π/3 with the voltage. Calculate the value of L and R.
Q.27> Find the ratio of velocities of red and blue light in air.
Q.28> When a capacitor is connected in series to a LR circuit the current increases. Explain
Q.29> State Lenz law and show that it obeys energy conservation.
Q.30> When an AC of 200V is applied across a device X, a current of 0.5A flows through the circuit and is in phase with the applied voltage. When the same source is applied across another device Y, the same current flows through the circuit but it leads the applied voltage by π/2. Name the devices X and Y (b) Calculate the current if the same source is applied across a combination of X and Y
Q.31> Derive the expression for the fringe width in YDSE
Q.32> Verify Snell’s law of refraction using Huygen’s wave theory
Q.33> A double convex lens made of glass (1.5) has both radii of curvature 20cm. An object 2cm high is placed at 10cm from the lens. Find the position, nature and size of the image.
Q.34> Draw the variation of the following with frequency of AC (a) Reactance of inductor (b) Reactance of capacitor (c) Resistance (d) Inductance
Q.35> An LR circuit draws a power of 560W from a 210V, 60Hz AC source. The power factor of the circuit Is 0.8, Find the value of L and R
Q.36> The image of an object formed by a lens on a screen is not in sharp focus. Suggest a suitable method to get the correct focus without disturbing the object, screen or lens
Q.37> Derive an expression for the width of the central maxima for diffraction of light at a single slit.
Q.38> Find the ratio of current flowing if an inductor is put across 200V,50Hz and 200V,100Hz AC supply. R P
Q.39> An inductor L and a resistor R are connected in parallel to A battery. The resistance of R is same as that of the inductor. Two L Q Identical bulbs are connected in each arm. When the switch is closed Which of the 2 bulbs lights up earlier. Justify.
Q.40> What are coherent sources of light? Why no interference is observed when the 2 sources are placed infinitely close to each other
I probably could have passed this in 12th grade when all this was fresh in my brain. I didn't go into engineering or science. Hence, my mind is now full of publishing knowledge--which is my profession.
Two other points.
1) They still have 300 million-400 million people living in poverty (roughly $1 / day).
2) Supply and demand. IF you are faced with the above, and these classes are your ticket out...
But the paradox is if too many people are crowding for the exits (so to speak) the price that such learning commands starts to drop. Hasn't happened yet, but give it time.
Oh, by the way, if they are such world beaters, why are publications like Outsourcing Business Week reporting that the Indian IT firms are outsourcing to China...?
Full Disclosure: What are the requirements to get into MIT or CalTech in the US?
Cheers!
I'm 65 but I think I could handle it OK but I would need to review about a month. However I am a Chemical Engineer, GPA 3.68. PBK. I also have 4 other degrees and experience and I worked the entire time I attended collage. Never had to spend a dime. My company paid for it all. I have a problem with spelling.
Where are the questions about how to fit a condom over a cucumber?
Most are making money in China running IT courses. Like NIIT.
Cheers!
These are just simple differential equations. Cramer's rule!! You're right, the formatting is crap, as well as some of the questions don't make sense. Who are they trying to bulldoze!! ROFLMAO!!
The physics questions are from a freshman engineering class! ???
That's a very important point that is never brought up when tests are shown comparing "high school seniors" in the U.S. to other countries. For instance, in Germany, a decision is made at about 14 as to whether a student will go on an academic pre-college track ("Gymnasium") or not. Almost half don't go to Gymnasium. In England, only the college-bound stay in school past age 16 (for "A-levels").
In the U.S., you have a small minority of kids who "drop out" before finishing high school. But in most other countries you have large percentages (40% in many countries) who branch off into non-academic tracks and are thus not included in comparisons to our HS seniors.
I'll bet if you just compared U.S. kids who take A.P. or Honors classes (which is probably a similar percentage of the U.S. high-school age population) to the academic-track, college-bound kids in other countries, U.S. kids would not fare any worse.
"Pigs get slaughtered!" "SELL SELL SELL!"
from Jim Cramer's Mad Money on MSNBC ;-)
PhD in molecular collision theory here...
Cheers!
Some fake poster for the Chinese website, I guess.
Here's the Indian site:
http://www.niit.com/tech/
Never saw the show. Working on Phd in engineering here.
I have a theory that can account for part of this. At least in Canada, someone with a science or math degree in the private sector can easily make 2-3x as much outside of public school teaching. Private school teachers also make much more than high school teachers (and the quality of the education recieved in especially math and science, by private school students, is much higher than that of public schoolers - I see the difference every year in freshman course performance).
You are still going to have incredible teachers in the public system, don't get me wrong. BUT...a lot of the best prospects for teaching are going to head for greener pastures (either industry or private schooling).
Contrast that to India. I read (and wish I could find the source again) that teaching math or science in an Indian school grants a very high pay relative to other jobs. You have a larger number of qualified personnel applying for the same ratio of teaching jobs, simply because the pay and benefits are better than elsewhere. From that larger pool you get better teachers. And MORE teachers.
It is a bit of a silly question- we are seeing the test cold, and I bet the majority of posters have been out of school 10+ years. However, if we had the time that Indian students have to study for the exam, I think many American students would be able to pass it.
Thank you. Thank you. Your post is perfect. I was feeling bad for a minute, but you are right. We had this a while back and could do it then and maybe even do some of it now. Like some have said, they are making two cents a day, but SOME of them can do that math and physics. Americans in general are intelligent, just a little lazy with math and science myself included. When needed, I passed both.
There are many much more important things to live for.
Of course no one wants to live in poverty...understood...but a lot of the pressure to perform is unnessesary.
I would rather not know math and be happy that be 'ahead' and miserable.
What are these guys going to do? Write a math book for a living? Why not do something practical?
That's true. The kids in other countries like India, Japan and China spend much more time studying than in the U.S. Our kids are more into sports and extra curricular activities and being real human beings.
That's a genius! Can't blame him.
Sometimes, the simplest solution is the right one. Occam's Razor?
It tends to be "process oriented" and not "creativity oriented"--which may be why India's so in love with it. That, and it gives management thw warm fuzzies to hear it.
(Tends to work best for large, hidebound bureaucracies that have the turning radius of an aircraft carrier, rather than startups and other nimble organizations...)
Cheers!
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