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Guess Paper – 2013Class – XII

Mathematics

Time -3 Hours Max. Marks-100

SECTION-A

1. Prove that the function f: N → N, defined by f(x) = x3 +x2 +1 is one –one.2. Find the principal value of sec−1¿3. Find the point on the curve y =x2 -4x +5, where the tangent do not intersect X –axis at any

point.4. Let A be any non singular square matrix of order 3 with |A| =p ( p is any integer ) then

find the value of |Adj . A|5. If A and Bare two independent events, such that P ( A ) =0.3 , and P ( B ) =0.6 ,

find P ( not A and B )

6. Evaluate ∫ 1sin2 xcos2 x

dx

7. Find the value of λ ,if the vector a =2 i +λ j+k , and b = i -2 j + 3k are perpendicular

8. Evaluate | a+ib c+id−c+id a−ib|

9. Find ∫ logx dx

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GENERAL INSTRUCTIONS:

1. All questions are compulsory.2. The question paper consists of 29 questions divided in to three sections A, B and C, section A

contains 10 questions of one mark each, section B contains 12 questions of 4 marks each and section C contains 7 questions of 6 marks each.

3. All the questions of section A are to answered in one word, one sentence or as per the exact requirement of the question.

4. There is no overall choice .However, internal choices have been provided in 4 questions of 4 marks each and in 2 questions of 6 marks each.

5. Use calculator is not permitted.



10. If A =[2 31 −4] ,and B =[ 1 −2

−1 3 ], then find ( AB )-1

SECTION-B

11. Let f ; N → N ,be defined as f ( n ) ={n+12, If nis odd .

n2, If n iseven

, state whether the function is

bijective.

12. Show that sin−1 1213 +cos−1 4

5 +tan−1 6316 = π

OR

Prove that cot−1( √1+sinx+√1−sinx√1+sinx−√1−sinx ) = x2 , x ∈ (0, π2 )

13. Differentiate xsinx +sinxcosx 14. Find the interval in which the function is given by f(x) = sinx + cosx , 0≤ x≤ 2π ,is

strictly increasing and strictly decreasing .OR

Prove that the curve x =y2 and x.y =K cut at right angle if 8 k2 =1.

15. Show that |1+a 1 11 1+b 11 1 1+c| =abc (1+

1a+ 1b +

1c )

16. Evaluate ∫ (x2+1 )( x2+2)(x2+3 )(x2+4)

dx

OR

Evaluate ∫ ex( 1+sinx1+cosx

)dx

17. Show that the general solution of the differential equation dydx +

y2+ y+1x2+x+1

, is given

by (x+y+1) =A (1 –x-y-2xy), where A is a parameter.OR

Show that the given differential equation is homogeneous and solve it

(1 +exy) dx +e

xy( 1-

xy )dy =0

18. Solve the differential equation is (1 +e2x ) dy + ( 1+y2) ex dx =0 given that x=0 and y=1.

19. Evaluate ∫ 5 x+3√ x2+4 x+10

dx





20. If with reference the right handed system of mutually perpendicular unit vectors i , j and k ,α =3 i - j , β =2 i+ j -3k ,then express β in the form of β =β 1+β 2 ,where β 1 is

parallel to α and β 2 is perpendicular to α21. A box of oranges inspected by examining three randomly selected oranges drawn

without replacement ,if all three oranges are good , the box is approved for sale, otherwise it is rejected .find the probability that a box contains 15 oranges out of which 12 are good and only 3 are bad ones will be approved for sale.

22. Find the equation of plane passing through the point P ( -1 , 3 , 2 ) , and perpendicular to each of the planes x +2y +3z =5 and 3x+ 3y +z =0

SECTION-C23. Evaluate the integration as limit of sums ∫

0

4

(x+ex)dx

24. Solve the following system of equation using matrix method X +2y +z =7, x +3z =11, 2x -3y =1

OR

Using elementary row transformation, find the inverse of matrix [ 1 2 −2−1 3 00 −2 1 ]

25. Show that the semi vertical angle of the right circular cone of given surface area

and maximum volume is sin−1( 13)

26. Find area lying above x- axis and included between the circle x2 +y2 =8x and the

parabola y2 =4x.

OR

Find area of the region enclosed between two circles x2 +y2 =4 and (x-2)2 +y2=4.

27. Show that the lines r = ( i + j - k ) + λ (3i- j ), and r = ( 4 i - k ) + μ( 2i +3k), are coplanar .Also find the plane which contains these two lines.

28. There are two factories located one at place P and other at place Q .From these locations, a certain commodity is to be delivered to three depots situated at A, B, and C .The weekly requirement of the depots are 5, 5, and 4 units of commodity while the production capacities of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given as





How many units should be transported from each factory to each depot in order that the transportation cost is minimum .what will be the minimum transportation cost?



From/To Cost (in Rs.)

A B C

P 160 100 150

Q 100 120 100



29. Assuming that the chances of a patient having a heart attack is 40% .It is also assumed that a meditation and yoga reduce the risk of heart attack by 30% and prescription of certain drug reduce its chances by 25% .At a time patient can chose one of the two options with equal probabilies .It is given that after going through one of the two options the patient selected at random suffers a heart attack .Find the probability that the patient followed a course of meditation and yoga.

ANSWERS -1

2. π6 3. ( 2 , 1 )

4. P2 5. 0.426. Tanx – cotx +c 7. λ =

52

8. a2+b2+c2+d2 9. x logx-x+c

10. A-1 =111 [14 5

5 1] 11. No.

13. xsinx ¿ sinxx

+¿cosx.logx] +sinxcosx

[ cosx.cotx –sinx.logsinx ]

14. Increase [ 0,π4 ) ∪ (

5π4 , 2π

] ,decreasing ( π4, 5 π

4¿

16. x +2√3

tan−1 x√3

-3 tan−1 x2 +c 16. (OR) ex tan x

2 +c

17. ( OR) yexy +x =c 18. y =

1ex

19. 5 √ x2+4 x+10 -7 log |x+2+√ x2+4 x+10|+c

20. β 1 =32i-

12 j ,β 2 =

12i+

32 j -3k

21.4491 22. 7x -8y +3z +25 =0

23. 15+e8

224. X=2, y=1, z=3

24. ( OR ) A-1 =[3 2 61 1 22 2 5] 26.

43(8+3π )

26. ( OR ) 8π3 – 2√3 27. r . ( 3 i +9 j -2k )=14

28. (0,5,3)units from P ,(5,0,1)units from to A,B, and C minimum cost =Rs 1550 29.

1429





TIPS:

Always Hope for the best ……… ………….. YOU are simply advised to attempt section B first then section C and in last section A. However you can also decide this order in first 15 minutes (reading time), that which

section you should attempt first and why? In first hour you must try to attempt such questions which are easy to solve to you.

DEV ADVANCE SAMPLE PAPER -2 (Most expected questions for Board Exam -2011)

MATHEMATICSTime -3 Hours CLASS -XII Max. Marks-100 GENERAL INSTRUCTIONS: Same as above dev advance -1

SECTION-A1. Let ⨂ be a binary operation on N given by a ⨂ b =L.c.m of a and b. Is ⨂ a

commutative binary operation?2. Find the value of tan−1¿¿ )]

3. If A =[ cosα −sinαsinαα cosα ] and A +AT =I , then find the value of reflex ‘ α ’

4. Find the value of x if | x 218 x| =| 6 2

18 6|5. Find the value of the determinants |2x b x+b

2 y c y+c2 z d z+d|

6. Find a point on the curve y= ( x-2 )2 at which the tangent is parallel to x-axis .

7. Integrate the function √1−4 x2

8. If a unit vector a makes angles π3 with i ,

π4 with j and an acute angle θ with k .find

θ9. Find λ ,if ( 2 i +6 j+14 k) × (i –λ +7k ) =010. Find the distance between two planes 2x +3y +4z =4 and 4x +6y +8z =12

SECTION-B





11. If R1 and R2 are two equivalence relations, then show that their intersection is also an equivalence relation.

12. Prove that tan−1 √x =12 cos−1¿¿ ) , x ∈ [ 0,1 ]

OR

If cos−1 xa +cos−1 y

b=α , then show that x

2

a2 - 2xyab

cosα+ y2

b2 =sin2α

13. Use the product [1 −1 20 2 −33 −2 4 ] [−2 0 1

9 2 −36 1 −2] , to solve the following system of

equations x – y +2z =1, 2y – 3z = 1 , 3x – 2y +4z =2.OR

If A = [ 0 − tan α2

tan α2

0 ] , and I be the identity matrix of second order, then show that (

I – A ) [cosα −sinαsinα cosαα ] =I + A

14. If f (X) ={sin (a+1 ) x+sinx

x , x<0

c , x=0√ x+b x2−√x

bx√ x, x>0

is continuous at x =0, find the values of a, b ,c.

15. Two numbers are selected at random (without replacement), from the first six positive integers. Let X denotes the larger of the two numbers obtained. Find expectation of x.

16. If x √1+ y +y √1+x =0 , then prove that dydx =

−1(x+1)2

OR

If y = [logx +√1+x2] 2, then show that (1+x2)d2 ydx2 +x

dydx -2 =0

17. Evaluate ∫π6

π3

11+√ tanx

dx

18. Evaluate ∫ 1

√sin3 x . sin (x+α )dx





19. Solve the differential equation dy =(1- x+ y – xy)dxOR

Solve the differential equationdxdy +y cotx =2x +x2cotx(x≠0 ¿, given that y=0 at x=

π2 .

20. Solve the differential equation x cos (yx

¿ dydx =y cos (

yx

¿+x

21. Let a ,band c are three vectors such that |a| =3 ,|b| =4 and |c| =5, and each one of them is perpendicular to sum of other two, find |a+b+c|

22. If a plane has the intercepts a ,b and c and it is at a ‘p’ unit distance from the origin

then show that 1a2 +

1b2 +

1c2 =

1p2.

SECTION-C

23. If x , y and z are different and A =|x x2 1+x3

y y2 1+ y3

z z2 1+z3| =0 ,then show that xyz = - 1

OR

Using the properties of determinants show that |1 x x2

x2 1 xx x2 1 | = (1- x3 ) 2

24. A square piece of tin of side 18 cm is to be cut in to a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of square to cut off so that the volume of box is maximum?ORShow that the height of cylinder of greatest volume which can be inscribed in a right circular cone of height ‘h’ and semi-vertical angle α is one third that of the cone and

greatest volume of cone is 4

27 πh3tan2α

25. Evaluate the integration as limit of sums ∫1

3

(x¿¿2+5x )dx¿

26. Find the area of the region { ( x, y ) : x2+y2 ≤ 2ax ,y2 ≥ax , x ≥0 , y≥0}27. Find the equation of the plane which contains the line of intersection of the planes r.

( i+ 2 j+3 k ) - 4 =0, r.( 2 i+ j−k ) + 5=0, and which is perpendicular to the plane r.(5 i+ 3 j−6 k ) + 8 =0,





28. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods

F1 and F2 are available, food F1 and F2 costs per unit are Rs 4 and Rs 6 respectively.

Food F1 contains 3 units of vitamin A and 4 units of minerals and food F2 contains 6 units of vitamin A and 3 unit of minerals .Formulate it as L.P.P and find the minimum cost for diet that consists of mixture of two foods and also meet the minimal nutritional requirement.

29. A man is known to speak truth 3 out of 4 times. He throws a die and report that it is six. Find the probability that it is actually six.ANSWERS-2

2.π4 3.

5π3

4. X =± 6 5. Zero6. ( 2 ,0 ) 7.

14 sin−12 x +

12 x √1−4 x2 +c

8.π3

9. λ = - 3

10.2

√29 units

13. X =0 , y =5 , z =3

14. a =32 ,b =R – {0} , c =

12 15.

143

17.π12 18. −2

sinα √ sin (x+α)sinx

+c

19. log (1+y) = x - x2

2+c 19. ( OR ) y =x2 - π2

4 sinx

20. Sin(yx

¿ =log |e x| 21. |a+b+c| =5√2

24. Side of square = 3 cm. 25.863

26. ( π4−2

3¿a2 27. 33x + 45y +50z – 41 =0

28. Minimum cost = Rs. 104 29.38

DEV ADVANCE SAMPLE PAPER -3 (Most expectable questions for Board Exam -2011)

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Time -3 Hours CLASS -XII Max. Marks-100 GENERAL INSTRUCTIONS: Same as above dev advance -1

SECTION-A

1. If f(x) = x2 – 1 , g(x) =2x +5 , then find fog(x)

2. Evaluate integration of sin3(2 x+1)

3. Find the value of cos−1 12 +2 sin−1 1

24. What kind of angle does the slope of tangent to the curve y = 3x2 + 1makes in

anticlockwise direction of x- axis at x =−12 ?

5. Find a unit vector perpendicular to vectors a =i+ j−kand 2i - j +k

6. Find the determinant value of the matrix, which is the product of given matrix [ 3 0 4 ] and its transpose.

7. Write the vector equation of normal to the given plane 2x – 3y +5z = 9.

8. Evaluate ∫ ( 4 x+2 ) √x2+ x+1dx

9. Find the value of x ,such that |3 xx 1| = |3 2

4 1|10.Find k, if the matrix [k 2

3 4 ] has no inverse.

SECTION-B

11.Show that the relation R defined in the set A of all triangles as R = { ( T1 , T2 ) : T1 is

similar to T2) , is an equivalence relation .Consider the three right angled triangles T1

with sides 5 ,12 ,13 , T2 with 3 , 4 ,5 and T3 with sides 6 , 8, 10 .which triangles among

T1 ,T2 and T3 are related.

12.If sin ( sin−1 15 +cos−1 x ) = 1, then find the value of x.

13.Using properties of determinants ,show that

|1 a2+bc a3

1 b2+ca b3

1 c2+ab c3|=- ( a-b)(b-c)(c-a)(a2+b2+c2)

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If |3 x−8 3 33 3 x−8 33 3 3x−8| = 0then solve for x

14.If y = [logx +√1+x2] 2, then show that (1+x2)d2 ydx2 +x

dydx -2 =0

OR

If x y = ex− y ,then prove that dydx =

logx(1+logx)2

15.If f (x) = {x2+ax+b ,0≤ x<23 x+2,2≤x ≤4

2ax+5b ,4<x ≤8 , if f(x) is continuous on [0, 8 ] ,find a and b.

16.Evaluate ∫ 15+4 sinx

dx

OR

Evaluate∫ 1x (x8+1)

dx

17.Evaluate ∫1

4

{|x−1|+|x−2|+|x−3|}dx

18.Solve (1+x2) dy+2xydx =cotx dx , ( x ≠0 ¿

OR

Find the equation of the curve passing through the origin, given that the slope of the tangent to the curve at any point (x, y) is equal to sum of co-ordinates of the point.

19. Solve the differential equation dydx = x

2+ y2

2xy20.If a ,band c are three mutually perpendicular vectors of equal magnitude, Prove that

the angle which ( a+ b+c ) makes with any of three vectors a ,band c is cos−1 1√3

21.Show that the lines x−1

3=1+ y

2= z−1

5 , and x+24

= y−13

= z+1−2 do not intersect.

22.Two cards are drawn successively with replacement from a well shuffled deck of 52 cards .Find the probability distribution of the number of aces.

SECTION-C





23.Find A -1 ,if A = [0 1 21 2 33 1 1] ,Hence solve the system of linear equations ,y+2z+8=0 ,

x+2y +3z +14 =0, 3x+y+z +8=0

OR

Using the elementary transformation find inverse of [2 0 −15 1 00 1 3 ]

24.Evaluate, ∫0

3

(2x2+5 x)dx , as a limit of sums.

25.Show that the height of the cylinder of maximum volume that can be inscribed in a

sphere of radius R is 2R√3

. Also find the maximum volume.

ORShow that the semi-vertical angle of the cone of maximum volume and given slant height istan−1 √2.

26.Find the area of the region { (x,y) : y2 ≤ 4x , 4x2 +4 y2 ≤9 }

27.Find the length and foot of perpendicular from the point ( 7 ,14,5) to the plane 2x+4y –z =2

28.An aero plane can carry a maximum 200 passengers. A profit of Rs 1000 is made on each executive class and a profit of Rs 600 is made on each economy class ticket. The airlines reserve at least 20 seats for executive class, however at least 4 times as many passengers prefer to travel by economy class than by executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. Also find the maximum profit.

29.A doctor is visit a patient .From the past experience, it is known that the probability that he will come by train ,bus ,scooter or by other means of transport are respectively

310, 15,

110

∧2

5 .The probabilities that he will be late are 14

, 13, 112 ,If he comes by

train ,bus and scooter ,but if he comes by other means of transport ,then he will not be late. When he arrives, he is late .Find the probability that he comes by train.





ANSWERS-3

1. fog(x) =4x2+20x+24 2. 14

3. 2π3

4. Obtuse.

5. −3 j−3 k3√2

6. 5

7. 2i−3 j+5 k 8. 43 (x2+ x+1)

32

9. X =±2√2 10.K =32

11.T2 is similar to T3 12.x =15

13.(OR) x = 23, 11

315.A=3 , b = - 2

16. 23 (

13 tan−1 x

2 ) +c 16.(OR) 18 log (

x8

x8+1¿+c

17. 192

18.Y = (1+x)-2log|sinx| +c (1+x2¿-1

18.(OR) x+y+1 =ex 19.x2− y2=cxX 0 1 2P(x) 144

16924169

2169

22.





23. A-1 =−12 [−1 1 1

8 −6 2−5 3 −1] ,x=- 1,

y=- 2, z=- 3

23.(OR) A-1 =[ 3 −1 1−15 6 −5

5 −2 2 ] 24. 812

26.[ √26 + 9π

8 - 94

sin−1 13

¿¿ sq. unit 27.Foot of ⊥ = (1,2 ,8) ⊥ distance =3√21units

28. 12

29.40tickets of executive class 160 of economy ,maxi profit =Rs 1,36,000

DEV ADVANCE SAMPLE PAPER -1




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