THE LUCKNOW PUBLIC SCHOOL

( a c.p. singh foundation)

A Leading Chain Of C.B.S.E. Schools in U.P.

HOME

ASSIGNMENT

XII

MATHEMATICS

2019 - 2020

PREFACE

The home assignment , being given to the

students , aims at strengthening the concepts of

each student through reinforcement of topics

covered in class lectures by practicing the

questions given in the assignment.

Some sample papers in each subject have been

given.

The maximum benefit can be gained only if the

student goes through all the topics of fragment 1

thoroughly before attempting the sample paper.

The assignment will not only act as a

reinforcement tool but will also help the students

acquire the tendency of solving the entire paper

in one sitting, just like he/she sits in an

examination for three hours. Hence , improving

student’s time management skill in examination.

This assignment will also fetch the student 5

marks in practical. The solutions of all the

question papers given in the home assignment

have to be written on separate A-4 sheets.

For all the subjects , the assignments have to be

submitted in different folders respectively.

Last date to submit the assignment for each

subject 5th

July ’19.

- PRINCIPAL

ASSIGNMENT 1 : 2019-20

TIME:3 Hrs. MATHEMATICS : XII MARKS: 80

I. All questions are compulsory.

II. This question paper consist of 36 questions divided into four sections A, B, C and D.

Section A comprise of 20 questions of one mark each, Section B comprises of 6

questions of 2 marks each, Section C comprises of 6 questions of 4 marks each and

section D comprises of 4 questions of 6 marks each.

III. Use of calculator is not permitted.

SECTION –A From question 1 to question 10 write the correct answer:

1. The principal value of cos�� �cos �� + sin�� �sin �� .

a) �� b)� c)

� d)−�

2. The value of tan�� �2 sin �2 cos�� √� �. a)

� b)− � c)� d) 0

3. If tan�� � + tan�� � = �� , �� < 1, then the value of x + y + xy.

a)0 b)1 c)2 d)��

OR The range of one branch of sin-1x, other than the principal branch.

a)(0,�) b) ��� , �� c)��� , �� � d) none of these

4. If sin �sin�� �� + cos�� � = 1, then the value of x.

a)0 b)5 c)�� d) 1

OR

The value of cot�tan�� � + cot�� � . a)0 b)1 c)-1 d)not defined

5. If !� + 3 4� − 4 � + �$ = �5 43 9� , then x and y are:

a)2,5 b)7,2 c)2,-5 d)2,7

6. If A = �cos ' − sin 'sin ' cos ' �, then for what values of ', A is an identity matrix?

a)�� b)� c)0 d)− ��

OR

The order of the product matrix :)123* +2 3 4, will be.

a)1 X 1 b)3 X 3 c) 1 X 3 d)does not exist

7. If � � − � -2� − � .� = �−1 40 5�, find the value of x + y.

a)0 b)1 c)2 d)3

8. For what values of x, is the matrix A= ) 0 1 −2−1 0 3� −3 0 * a skew-symmetric matrix?

a)-2 b)2 c)1 d)0 OR

For what value of x the equation +� 1, � 1 0−2 0� = 0

a)1 b)0 c)2 d)-1 9. For what vaue of x the function

0�� = 1� + 3, � ≠ 01, � = 0 is not continuous.

a)0 b)1 c)-1 d)2

10. For what value of k the function 0�� = 13�� , � ≤ 23 , � > 2 , is continuous at x = 2.

a)4 b)� c)2 d)

�

OR

The number of points at which the function 0�� = ��67�6�68 is discontinuous are:

a)1 b)2 c)3 d)none of these 11. Write the condition of continuity of a function f(x) at a point c in its domain.

12. Write the set of all points at which the function f(x) = tan x is discontinuous.

OR Write the set of all points at which |sin x| is not differentiable.

13. Evaluate 9cos : − sin :sin : cos : 9. 14. If 9� �1 �9 = 93 41 29, write the positive value of x.

OR

Expand the determinant using second row ;5 3 82 0 11 2 3; 15. If A is a matrix of order 3 X 3 then write the number of minors in determinant of A

16. If a is a square matrix of order 3 X 3 such that |A| = 5, then find the value of |A-1|.

OR If A and B are matrices of order 3 and |A| = 5, |B|= 3 , then write the value of |3AB|.

17. Write the inequality for non-negative constraints.

18. Write the meaning of optimal value of objective functions.

19. Define feasible region.

20. Write the region in which the optimal value of objective function may or may not exist.

SECTION-B

21. Solve for x, if tan�� ���6�=6 = �� tan�� � , � > 0

OR

If 3tan�� � + cot�� � = � , find the value of x.

22. If sin�� � + sin�� � = ��, then find the value of cos�� � +cos�� �.

23. If A and B are symmetric matrices of the same order then prove that AB-BA is a skew

symmetric matrix.

OR

If A= �� >sin����� tan�� �6�

sin�� �6� cot����� ? , B = �� >cos����� tan�� �6�

sin�� �6� − tan����� ?, then find the value of A-B.

24. Show that the value of determinant

; � sin � cos �− sin � −� 1cos � 1 � ; is independent of x.

25. Discuss the continuity of the function 0�� = sin �@AB �.

26. Find CDC6 , if � = sin�� � �√�=67

OR

Find CD C6 , if � = EAFG�log �

SECTION-C

27. Solve tan�� � ��6=� + tan�� � ��6=� = tan�� � �67.

OR

Solve sin�� � + sin���1 − � = cos�� �

28. If tan�� � + tan�� � + tan�� - = �� , show that �� + �- + -� = 1.

29. Find the matrix a satisfying matrix equation

�2 13 2� J �−3 25 −3� = �1 00 1� 30. Using the properties of determinant prove that

;1 + � 1 11 1 + K 11 1 1 + @; = �K@ L1 + 1� + 1K + 1@M = �K@ + K@ + @� + �K

OR Using the properties of determinants

Solve for x, if ;� + � � − � � − �� − � � + � � − �� − � � − � � + �; = 0

31. Differentiate sin�� ��NOP�=�N with respect to x.

OR Find the value of constant k so that the function

0�� = Q√�=R6�√��R66 , S0 − 1 ≤ � < 0�6=�6�� , S0 0 ≤ � < 1 is continuous at x = 0.

32. Minimise and maximise - = 3� + 9�.

Subject to constraints: � + � ≥ 10, � + 3� ≤ 60, � ≤ �, � ≥ 0, � ≥ 0. VWXYZ[\ − ]

33. If A = � 2 3−1 2� , then show that A2 + 4A + 7I = 0. Using this result calculate A5 also.

OR

if possible, using elementry row transformations, find the inverse of A = ) 2 −1 3−5 3 1−3 2 3* 34. Given P = ) 2 2 −4−4 2 −42 −1 5 * and Q = )1 −1 02 3 40 1 2* ,find P.Q and use this to solve the system of

equations: � + 2- = 7, � − � = 3, 2� + 3� + 4- = 17.

35. If the function defined by

0�� =_ab��cde86fgc76 , S0 � < ��� , S0 � = ��h���ijk 6 ����6 7 , S0 � > ��

is continuous at � = ��, then find the value of a & b.

lm

if � = sin n, � = sin on, then prove that �1 − �� C7DC67 − � CDC6 + o�� = 0

36. A youngman rides his motorcycle 25 km/h he has to spend ₹ 2 per kilometer on petrol, if he

rides at a faster speed of 40 km/h the petrol cost increases to ₹ 5 per kilometer. He has ₹ 100 to

spend on petrol and wishes to find the maximum distance he can travel within one hour. Express

this as a linear programming problem and then solve it.

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ASSIGNMENT 2 : 2019-20

TIME:3 Hrs. MATHEMATICS : XII MARKS: 80

I. All questions are compulsory.

II. This question paper consist of 36 questions divided into four sections A, B, C and D.

Section A comprise of 20 questions of one mark each, Section B comprises of 6

questions of 2 marks each, Section C comprises of 6 questions of 4 marks each and

section D comprises of 4 questions of 6 marks each.

III. Use of calculator is not permitted.

SECTION -A

1. � − sin�� �− ��

a)0 b)�� c)

�� d)�

2. Range of cos�� � is

a)[0,�] b)�0, ��� c)�− �� , ��� d)none of these.

3. If A = �cos ' − sin 'sin ' cos ' �, ‘'’ when J + Jp = q

a)�� b)

� c)�� d)

�r

4. |adj A|

a)|a|n-2 b)|A|n-1 c)|A|n d)none of these

5. ;1! 2! 3!2! 3! 4!3! 4! 5!; a)2 b)6 c)24 d)120

6. If A = �cos 15° sin 15°sin 15° cos 15°� a)1 b)

�� c)√� d)none of these

7. If 0�� = 13�� , � ≤ 23 , � > 2 , is continuous, K will be

a)k=� b)k=

� c)k=3 d)k=�

8. 0�� = �6�� is continuous in

a)R b)R-{0} c)R-{5} d none of these. 9. Objective function is a

a)linear b)quadratic c)cubic d)none of these.

10. �, � ≥ 0 define that the region lying in

a)second quadrant b)first quadrant c)third quadrant d)fourth quadrants.

11. Evaluate sin u� − sin�� �− ��v.

12. Evaluate sin�cot�� � .

13. Let A = �2 3 −10 −5 7 �, varify that (A’)’ = A.

14. Find � + � − - .ℎxy �5 3� 7� = �� -1 7� 15. Evaluate 92 cos : −2 sin :sin : cos : 9. 16. Find x, if 92� 58 �9 = 96 −27 3 9. 17. Find ‘k’ if 0�� = z 67�{6� , � ≠ 33 − 1 , � = 3 is continuous at x=3

18. Discuss the continuity of the function 0�� = z 6|6| , � ≠ 00 , S0 � = 0.

19. Graph the solution sets � + � ≥ 4. 20. Find the corner points of 2� + � ≥ 1 and 2� − � ≥ −3

SECTION-B

21. Solve for x, if 2 tan���cos � = tan���@ABx@ �

22. Find the value of tan�� √3 − sec���−2

23. If J = � 3 −5−4 2 � show that A2-5A-14I=0 .

24. For what value ‘x’,the given matrix J = �3 − 2� � + 12 4 � is a singular .

25. For what value of ‘a’ and ‘b’ such that the following fuction is continuous:

0�� = � 5 , � ≤ 2�� + K, 2 ≤ � < 1021, � ≥ 10

26. Find the value of ‘k’ for which the function

0�� = �R ��i 6���6 , � ≠ ��3 , � = �� is continuous at x =

��

SECTION-C

27. Prove that tan�� L√�=67=√��67√�=67�√��67M = �� + �� cos�� ��.

28. Solve for ‘x’ sin���1 − � − 2 sin�� � = ��.

29. If J = � 0 − tan ' 2�tan ' 2� 0 � and I is idendity matrixs of order 2, show that

�q + J = �q − J �cos ' − sin 'sin ' cos ' � 30. Prove that ; �� K@ �@ + @��� + �K K� �@�K K� + K@ @� ; = 4��K�@�

31. Differentiate cot�� �√�=ijk 6=√��ijk 6√�=ijk 6�√��ijk 6 with respect to x.

32. Minimise P = 3� + 9�. Such that

� + 3� ≥ 3, � + � ≥ 2, � ≥ 0, � ≥ 0. VWXYZ[\ − ]

33. Use the product )1 −1 20 2 −33 −2 4 * )−2 0 19 2 −36 1 −2* to solve thefollowing system of equations

� − � + 2- = 1, 2� − 3- = 1, 3� − 2� + 4- = 2. lm

By using elementry operations, find thevinvrse of the matrix J = �3 −1 −22 0 −13 −5 0 � 34. In triangle ABC, if ; 1 1 11 + sin J 1 + sin � 1 + sin �sin J + BSy�J sin � + BSy�� sin � + BSy��; = 0 then prove that ∆J�� is an

isosceles triangle.

35. If � = ���i 6 + cos �ijk 6 , 0Sy� CDC6

36. An areoplane carries a maximum number of two hundred passengers. A profit of ₹1000 is

made on each executive class ticket and profit of ₹600 is made on each economy class ticket.

The airline reserves atleast 20 seats for executive class however atleast 4 times as many

passengers prefer to travel by economy class than by the executive class. Determine how

many tickets of each type must be sold in order to maximise the profit for the airline, what is

maximum profit?

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ASSIGNMENT 3 : 2019-20

TIME:3 Hrs. MATHEMATICS : XII MARKS: 80

1. All questions are compulsory.

2. This question paper consist of 36 questions divided into four sections A, B, C and D.

Section A comprise of 20 questions of one mark each, Section B comprises of 6

questions of 2 marks each, Section C comprises of 6 questions of 4 marks each and

section D comprises of 4 questions of 6 marks each.

3. Use of calculator is not permitted.

1. Write the value of tan(2tan-1 ��).

2. If �� + 2� 5−� 3�=� 7 5−2 3� , then find the value ofy.

3. Find the derivative of log10x with respect to x.

4. If 9� + 2 3� + 5 49 =3, then find the value of x.

5. Shade the region, for inequality x+y≤6.

6. Find the cofactor of a12in the following:;1 − 3 56 0 41 5 − 7; 7. What is the value of |J| ,if

A=)4 � K + @4 K @ + �4 @ � + K* 8. Find

CDC6 , if x-y=π.

9. Write the principal value of cos-1���+2sin-1���.

10. If sin{ sin-1�� + cos-1 x}=1,then find the value of x.

11. Differentiate w.r.t. x: 2�cot ���

12. Write the value of |J|,if A=)� − K K − @ @ − �K − @ @ − � � − K@ − � � − K K − @* 13. Express in simplest form : tan-1���i 6�ijk 6��i 6=ijk 6 ,-�� <x<

�� 14. Shade the region, for inequality y≥ 2�. 15. Find the value of x, y and z if )� + � + -� + -� + - *=)957* 16. If A=�@AB' −BSy'BSy' @AB' � , then for what value of ',A is an identity matrix?

17. The region represented by the inequation system x,y≥0 , y≤6 , x+y≤3

18. From the following matrix equation, find the value of x: �1 34 5� �x2� = �56�

19. Shade the region, for inequality y≤3x.

20. Evaluate:9 � + SK @ + S�−@ + S� � − SK9

SECTION –B

Questions numbers from 21 to 26 carry 2 marks each.

21. Evaluate:;1 � K + @1 K @ + �1 @ � + K; 22. Find

CDC6 ,if y=sin-1� �6�=67

23. If y=xsinx ,x>0,then find CDC6.

24. Write the value of tan-1[2sin(2cos-1 √� )].

25. Solve for x, +1 �, �2 −11 2 � �13� = +0,. 26. Write the simplest form of tan-1!���=67 ��6 $.

SECTION –C Questions numbers from 27 to 32 carry 4marks each.

27. If A =) 2 −1 1−1 2 −11 −1 2 *,then verify that A3 -6A2 +9A-4I=0 and hence find A-1 .

28. Prove that: tan-1 ��+ tan-1 �G + tan-1 � + tan-1�� =��.

29. Using properties of determinants , prove the following:

;1 1 1a b cabc ;=(a-b)(b-c)(c-a)(a+b+c).

30. Solve the following linear programming problem graphically:

Minimise Z=x-5y+20

Subject to constraints: x-y≥0 , 2y-x≥2 ; x≥3 , y≤4 , x,y≥0

31. Prove that: {�� -

{� sin-1 � ={ �sin-1�√� .

32. Find the value of a and b such that the following function f(x) is a continous function:

0�� = � 5, � ≤ 2�� + K, 2 < � < 10 21, � ≥ 10

OR

�� =_ab ��fgc�66��√6��r=√6��

S0, � < 0S0, � = 0 S0, � > 0

SECTION –D

Questions numbers from 33 to 36 carry 6 marks each.

33. The monthly incomes of Aryan and Babban are in the ratio 3:4 and their monthly

expenditures are in ratio 5:7. If each saves 15,000 per month ,find their monthly incomes

using matrix method.

34. If y= xx cosx+67=�67�� , find

CDC6.

35. Using properties of determinants, prove the following :

;1 + a 1 11 1 + b 11 1 1 + c;=ab+bc+ca+abc.

36. A dealer wishes to purchase a number of fans and sewing machines. He has only

5,760 Rs to invest and has space for at the most 20 items. A fan costs him 360Rs and a

sewing machine 240Rs. He expects to sell a fan at a profit of 22Rs and a sewing machine

for a profit of 18Rs. Assuming that he can sell all the items that he buys, how should he

invest his money to maximize his profit? Solve it graphically.

OR An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400/- is made on each first class ticket and a profit of Rs.600 is made on each economy class ticket. The airline reserves at least 20 seats of first class. Determine how many each type of tickets must be sold in order to maximise the profit for the airline. What is the maximum profit?

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