CBSE - XII Mathematics 2007 Question Paper

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Mathematics

Time allowed : 3 hours

Maximum Marks : 100

General Instructions :

The question paper consists of three sections A,B and C. Section A is compulsory for all students. In addition to Section A, every student has to attempt either Section B OR Section C.
For Section A
Question numbers 1 to 8 are of 3 marks each.
Question numbers 9 to 15 are of 4 marks each.
Question numbers 16 to 18 are of 6 marks each.
For Section B/Section C
Question numbers 19 to 22 are of 3 marks each.
Question numbers 23 to 25 are of 4 marks each.
Question number 26 is of 6 marks.
All questions are compulsory.
Internal choices have been provided in some questions. You have to attempt only one of the choices in such questions.
Use of calculator is not permitted. However, you may ask for logarithmic and statistical tables, if required.

Section - A

If show that . Hence find $A^{-1}$ .
An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting (a) 2 red balls (b) 2 blue balls© one red and one blue ball.
Using the properties of determinants, prove that following :
A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is neither a ace nor a king.
Evaluate :

$\int\cfrac {1+x^2}{1+x^4}dx$
Solve the following differential equation :

$x\cos\operatorname{y}dy=(xe^x\log x+e^x)dx$
Form the differential equation of the family of curves $y=A\cos^2x+B\sin^2x,$ where A and B are constants.

or

Solve the following differential equation :

$\cfrac {dy}{dx}+2y=6e^x$
Evaluate :

$\int \cos 4x\cos \operatorname{3x}dx$
Using properties of definite integrals, prove the following :

$\int_0^{\pi}$ $\cfrac{x\tan x}{\sec x \quad cosec \quad x}$ $dx=\cfrac {\pi^2}{4}$
Evaluate :

$\int \cfrac {\sin x}{(1-\cos x)(2-\cos x)}dx$