Maths Sp Dalvir( 2009 )

SAMPLE PAPER ( 2009) Subject : Mathematics

CLASS- XII CLASS- XII

Subject teacher:Mr. Dalvir Singh Mobile : 9810-246577 Email : dalvir_dav@yahoo. co. in

Subject Teacher : Mr. Dalvir SinghMOBILE 9810246577 Subject Teacher : Mr. Dalvir SinghMOBILE 9810246577 Subject Teacher : Mr. Dalvir SinghMOBILE 9810246577 Subject Teacher : Mr. Dalvir SinghMOBILE 9810246577 Subject Teacher : Mr. Dalvir SinghMOBILE 9810246577 Subject Teacher : Mr. Dalvir singh

Time: 3 Hours

Max. marks : 100

GENERAL INSTRUCTIONS

1. 2.

3. 4. 5.

All Question are compulsory. The Question paper consist of 29 questions divided into three sections A,B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 qustions of four marks each and section C comprises of 07 questions of six marks each. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question There is no overall choice. However, Internal choice has provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions. Use of calculators is not permitted. You may ask for logarithmic tables, if required.

SECTION –A Q.NO.1:

Find the value of tan – 1 ( 1 ) + cos – 1 ( – ½ ) + sin – 1 ( – ½ )

Q.NO.2:

Examine which of the following is a binary operation f : R Æ R :f(x)=8x3

&

g : R Æ R : g (x ) = x 1 / 3

Find g o f . Q.NO.3:

A matrix A of order 3 x 3 has determinant 20 , what is the value of | 3A |

Q.NO.4:

If A is a square matrix of order 3 such that | adj A | = 441 , find | A |.

Q.NO.5:

Find the value of x , If 2

3

x

3

2x

5

= 4 Q.NO.6:

5

Evaluate :

∫ Q.NO.7:

Q.NO.8: Q.NO.9:

dx

Find the order & degree of the differential equation 2 d y

dy

x y ---------- + x

-----------

2 d x

dx

2 –

dy y

---------- = 0 dx

If a is a unit vector &(x–a ).(x+a)=8

then find | x |

If θ is the angle between any two vectors a & b and

Q.NO.10:

√ x 2 + 121

|a.b| = |axb|

If the lines x–1

y– 2

z–3

----------- = -------------- = ------------–3

x–1

2k

y– 1

2

z–6

----------- = -------------- = ------------3k

1

–5

are perpendicular , find the value of k.

then find θ.

SECTION –B Q.NO.11:

Prove that sin –1 ( 8 / 17 ) + sin –1 ( 3 / 5 ) = tan – 1 ( 77 / 36 ) (OR) Prove that √ 1 + sin x – √ 1 – sin x ----------------------------------√ 1 + sin x + √ 1 – sin x

tan –1

Q.NO.12:

=x/2

Examine which of the following is a binary operation (i)

a*b= ab+1

(ii)

a+b a * b = ----------2

, a,bε Q , a,bε Q

Also check the commutativity of the above operations ? (OR) Show that the Modulus function f : R Æ R given by f ( x ) = | x | is neither one-one nor onto. Q.NO.13.

By using elementary transformations find the inverse of the matrix: 2

3

5

–2

A= (OR) Prove the following identity. 2x

2x

2y

y–z–x

2y

2z

2z

z–x–y

x–y–z

= ( x + y + z )3

Q.NO.14.

Determine whether the function is continuous ?

f(x) =

if x ≠ 0

|X| -------x 2

Q.NO.15.

If

y = a cos ( logx ) + b sin ( logx ) x2

Q.NO.16.

Prove that :

dy d2y --------- + x --------- + y = 0 dx2 dx √1–x2 + √1–y2 = a(x–y)

If

Prove that

Q.NO.17:

if x = 0

dy -------- = dx

√1 – y2 -----------√ 1 – x2

A man is known to speak truth 5 out of 6 times. He draws a ball from a bag containing 4 white and 6 black and reports that it is white. Find the probability that it was actually a white ball. (OR) A card from a pack of 52 cards is dropped. From the remaining cards of the pack, two cards drawn and are found to be clubs. Find the probability that dropped card is a club.

Q.NO.18:

Find the value of λ so that the vectors 3 i + 2 j + 9k & i + λj + 3 k are (i) parallel (ii) perpendicular ……….…..to each other.

Q.NO.19:

Find the equation of plane passing through three points ( 2,2,–1) ,

Q.NO.20:

(3,4,2) , ( 7,0, 6),

Evaluate 4x–2

∫

------------------------------ dx 2

(x+2) ( x + 9 ) Q.NO.21:

Solve the initial value problem dy 2 ------- + y tan x = 2 x + x tan x dx

,

y(0)=1

Q.NO.22:

Find the intervals in which the function f ( x ) = – 2x

3

– 9x

2

– 12 x + 2008

is strictly increasing or strictly decreasing

SECTION –C Q.NO.23.

If A=

1 1 2

2 −1 3

5 −1 −1

Find A−1 & hence solve the following system of equations: x + 2 y + 5 z = 10 ;

Q.NO.24:

x − y − z = −2 ;

2 x + 3y − z = −11

Using integration find the area of the region bounded by triangle ABC whose vertices are ( 1 , 0 ) , ( 2 , 2 ) and ( 3 , 1 ). (OR) Evaluate 3

∫

( x 2 + x + 1 ) dx

as the limit of a sum.

0 Q.NO.25.

A window is in the form of a rectangle surmounted by a semi-circular opening. If the perimeter of the window is 30 m , find the dimensions of the window so that the maximum possible light is admitted through the whole opening.

Q.NO.26.

Find the shortest distance between the lines whose vector equations are : r = 2 i – 5 j + k + λ ( 3 i + 2 j + 6k) &

r = 7 i – 6 k + μ ( i + 2j + 2 k )

Q.NO.27.

Two cards are drawn successively with replacement from a wellshuffled deck of 52 cards. Find the probability distribution of the number of aces . Also find the mean , variance & standard deviation of the number of aces. (OR) Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. find the probability that (i) both balls are red (ii) first ball is black and second is red (iii) one of them is black and other is red.

Q.NO.28.

Evaluate : x +

∫ Q.NO.29.

3

------------------------- dx √ 5–4x – x2

If a youngman rides his motor cycle at 25 km per hour, he has to spend Rs. 2 per km on petrol, if he rides at a faster speed of 40 km/hr, the petrol cost increases at Rs. 5 per km. He has Rs.100 to spend on petrol and wishes to find , what is the maximum distance he can travel within one hour ? Solve the problem graphically.

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