CLASS – XII WORKSHEET (MACRO-ECONOMICS) · CLASS – XII . WORKSHEET (MACRO-ECONOMICS) (MULTIPLE...

CLASS – XII

WORKSHEET (MACRO-ECONOMICS)

(MULTIPLE CHOICE QUESTIONS)

Q1The study of cotton textiles comes under-------------

a) Micro economics b) Macro economics c) Modern economics d) Indian economy

Q2Under which branch of economics you will study the problem of unemployment in india

a) Micro economics b) Macro economics c) Modern economics d) Indian economy

Q3 Net investment is equal to

a) Gross investment-depreciation b) Gross investment + depreciation c) Gross investment – capital investments d) Gross investment – inventory

Q4 It is a quantity measured at a particular point of time.

a) Flow b) Depreciation c) Stock d) Wealth

Q5Which of the following is the consumption sector?

a) Household b) Firms c) Government d) Foreign sector

Q6Which of the following is a transfer income?

a) Rent b) Wages c) Interest d) Old age pension

Q7Which of the following a factor income?

a) Royalty b) Corporate tax c) Pension d) Unemployment allowance

Q8 Which of the following is not a method to estimate national income of a country ?

a) Value added method b) Income method c) Expenditure method d) Total addition method

Q9Which of the following is included in the national income?

a) Domestic services of mother b) Change in stock c) Both (a) and (b) d) Neither (a) nor (b)

Q10 Which of the following is not a part of profit ?

a) Corporate tax b) Dividends c) Retained earning d) Royalty

Q11Which of the following is a part of the expenditure method?

a) Rent and royalty b) Mixed income c) Net exports d) Sales

Q12 Net exports is calculated as:

a) Exports + imports b) Exports – imports c) Imports-exports d) None of these

Q13 Out of the following which aggregate represents “national income”?

a) NNP MP b) GNP FC c) NNP FC d) GNP MP

Q14 Domestic factor income is the another name for:

a) NDP FC b) NNP MP c) GDP FC d) NNP FC

Q15 private income includes:

a) Transfer income b) Factor income c) Both (a) and (b) d) Neither (a) nor (b)

Q16 It is a system in which goods are exchanged for goods.

a) Barter system b) Trade c) Money supply d) Balance of trade

Q17 -------------- refers to demand deposits created by the commercial banks.

a) Money supply b) Demand deposits c) Bank money d) Bank deposits

Q18The rate at which central bank lends to commercial banks.

a) SLR b) CRR c) LRR d) Bank rate

Q19Which of the following is the apex bank of india?

a) RBI b) World bank c) SBI d) PNB

Q20 Banks create credit:

a) On the basis of their fixed deposits b) On the basis of their securities c) On the basis of their total deposits d) On the basis of deposits

Q21What are the components of aggregate supply?

a) Consumption (c) b) Saving (s) c) Both (a) and (b) d) None of these

Q22 ------------ refers to the functional relationship between saving (s) and income (y), S = f (Y).

a) Consumption function b) Saving function c) Production function d) None of these

Q23 ---------- is the ratio of aggregate saving to aggregate income.

a) APC b) MPC c) APS d) Consumption function

Q24What is the value of MPC when MPS is zero?

a) MPC =0 b) MPC = 1 c) MPC > I d) None of these

Q25 What is the relationship between APS and APC ?

a) APS + APC = 1 b) APC – APS = 1 c) APS – APC = 1 d) APC + APS = 0

Q26 At equilibrium level:

a) C = I b) AD = S c) S = I d) C =S

Q27 What will be the value of multiplier when MPC = 0 ?

a) 1 b) 0 c) Symbol infinity d) – 1

Q28 What can be the minimum value of K ?

a) 1 b) 0 c) Symbol infinity d) – 1

Q29The excess demand leads to -------------

a) Increase in output b) Increase in employment c) Increase in general price level d) Increase in output and employment both

Q30 Deficit demand adversely effects.

a) Output b) Employment c) Price level d) All

Q31 Which of the following is a capital receipt ?

a) Loan from the world bank b) Recovery of loans c) Both are capital receipts d) Only (a) is capital receipts

Q32Which of the following is a source of revenue receipt?

a) Direct tax b) Indirect tax c) Both (a) and (b) d) Only (a)

Q33 Which of the following is not an indirect tax :

a) Corporate tax b) Value added tax c) Service tax d) Excise duty

Q34 Balance of payment is a…………concept.

a) Stock b) Flow c) Both (a) and (b) d) Neither (a) nor (b)

Q35 Which of the following is a visibleitem ?

a) Jute b) Banking c) Cotton d) Both (a) and (c)

Q36.. Define full employment in an economy . Discuss the situation when aggregate demand is more than aggregate supply at full employment income level.

Or

What are two alternative ways of determining equilibrium level of income ?How are these related ?

Q37. Define autonomous consumption. Difference between Induced investment and autonomous investment.

Q38.What is government budget ?Explain its major components.

Q39.Explain the components of current account and capital account.

40. Explain the process of credit creation by commercial bank.

HOLIDAYS HOMEWORK SUBJECT-POLITICAL SCIENCE CLASS-XII

Read the chapters from 1 to 9 (Book-1) in detail and answer the following questions.

Q1 NAM was considered as the third option by the third world countries.How did this option benefit their growth during the peak of the cold war?

Q2 Mention three ways in which US dominance is different from its position as super power during the cold war.

Q3 Draw and describe in detail about the ASEAN flag.

Q4 The emerging economies of India and China have great potential to challenge the unipolar world.Do you agree with the statement?

Q5 The peace and prosperity of the country lay in the establishment and strengthening of regional economic organisation .Justify.

Q6 What are some of the commonalities and differences between the Bangladesh and Pakistan in their democratic experience?

Q7 How are external powers influencing bilateral relations in Southasia?

Q8 Draw and describe in detail about UN logo.

Q9 What is terrorism? Is it a traditional or non traditional threat? How can we overcome terrorism?

Q10 Write short notes on the following

A) Balance of power B) Disarmament C) Confidence building measures.(CBM’s) D) Alliance formation E) Arma race F) Arms control G) Non traditional security and its components

Q11 Most serious challenge before the states today is to pursue economic development without causing further damage to the natural environment. How can we achieve it ? Explain with the help of examples.

Q12 Read all the cartoons from the textbook and describe briefly about them.

Q13 Locate and mark all the states on the Indian political map along with capital cities and union territories. Q14 Locate and mark all important countries of the world which are part of our syllabus(book-1) on World map.

QUESTIONS REGARDING GENERAL STUDIES

Q 1Describe briefly about the massive victory of BJP government in 2019 parliamentary elections. Q2 Write and memorise the names of all the cabinet ministers in power. Q3 Write and memorise the names of chief ministers of all the states of India. Q4 Describe in brief about the history of J&K state.( You can refer book-2) Q5 Listen to the news daily and read news paper.

ASSIGNMENT FOR MATHEMATICS CLASS XII

RAJESH KUMAR, PGT MATHEMATICS 14-Jun-19

COMPILED BY: - RAJESH KUMAR, PGT MATHEMATICS, APS AKHNOOR 2019-20

RELATION AND FUNCTION

VERY SHORT ANSWER QUESTIONS

Q1. Let R be a relation on A defined as R = {(a, b) ∈ A × A : a is a husband of b} can we say R is symmetric? Explain your answer. Q2. Let A = {a, b, c} and R is a relation on A given by R = {(a, a), (a, b), (a, c), (b, a), (c, c)}. Is R symmetric? Give reasons. Q3. Let R = {(a, b), (c, d), (e, f )}, write R–1. Q4. Let L be the set of are straight lines in a given plane and R = {(x, y) : x ⊥ y ∀ x, y ∈ L}. Can we say that R is transitive? Give reasons. Q5. The relation R in a set A = {x : x ∈ z and 0 ≤ x ≤ 12} is given by R = {(a, b) : |a – b| is a multiple if 4} is an equivalence relation. Find the equivalence class related to {3}. Q6. Let R1 be the relation on R defined as R = {(a, b) : a ≤ b2}. Can we say that R is reflexive? Give reasons. Q7. Let R {(a, b) : a, b ∈ Z (Integers) and |a – b| ≤ 5}. Can we say that R is transitive? Give reason. Q8. If A = {2, 3, 4, 5}, then write the relation R on A, where R = {(a, b) : a + b = 6}. Q9. If A {1, 2}, and B = {a, b, c}, then what is the number of relations on A × B? Q10. State reason for the relation R in the set {1, 2, 3} given by R {(1, 2), (2, 1)} not to be transitive. Short Answer Type Questions (4 Marks) Q11.If 𝑅1 𝑎𝑛𝑑 𝑅2 are equivalence relations in a set A , show that 𝑅1 ∩ 𝑅2 is also an equivalence relation.

Q12.Let R be the relation on set A of ordered pairs of positive integers defined by ( x , y ) R (( u , v ) if and only if xv = yu. Show that R is an equivalence relation.

Q13. Let A = { 1 , 2 , 3 }. Then show that the number of relations containing ( 1 , 2 ) and ( 2 , 3 ) which are reflexive and transitive but not symmetric is four.

Q14. Consider a function f : [ 0 , 𝜋2 ] → 𝑅 given by f(x) = sinx and g : [ 0 , 𝜋

2 ] → 𝑅 given

by g(x) = cosx. Show that f and g are one one but f o g is not one one.

Q15. Show that the relation R defined by : (a , b) R (c , d) ⇒ a + d = b + c on the set N is an equivalence relation.

Q16 If the function f : R → R is given by f(x) = 𝑥 + 32

and g : R → R is given by g(x) = 2x – 3 , find fog and gof. Is f -1 = g .

Q17. Let Y = { 𝑛2 ∶ 𝑛 ∈ 𝑁 } ⊂ 𝑁. Consider f : N → Y given by f(n) = 𝑛2. Show that f is invertible. Find the inverse of f. ( Ans. 𝑓−1(𝑛) = √𝑛 ) Q18. Given the functions f(x) = sin x and g(x) = cos x are one-one in [ 0 , 𝜋

2 ]. Prove that

f + g is not one-one in [ 0 , 𝜋2 ].


Q19. Let f , g : R → R be two functions defined as f (x) = │x│+ x . g(x) = │x│ - x ∀ 𝑥 ∈𝑋. Then , find fog and gof.( Ans. (gof)(x) = 0 , ∀ 𝑥 ∈ 𝑥. (fog)(x) = 0 , x ≥ 0 - 4x , x < 0 Q20. If N denote the set of all natural numbers and R be the relation on N X N defined by ( a , b ) R ( c , d ) , if ad(b + c) = bc(a + d). Then show that R is an equivalence relation.


CHAPTER 2

INVERSE TRIGONOMETRIC FUNCTION

1 mark question:

1. Write the principal value of tan−1 1 + cos−1 (−12)

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

3. Write into simplest form : 𝑠𝑖𝑛−1 [ √𝑥√1 − 𝑥2 − 𝑥 √1 − 𝑥 ]. 4. Prove that 𝑠𝑒𝑐2(tan−1 2) + 𝑐𝑜𝑠𝑒𝑐2(cot−1 3) = 15. 5. Simlify cot−1 1

√𝑥2−1 𝑓𝑜𝑟 𝑥 < −1

6. Write the principal value of tan−1 √3 − sec−1(−2).

7. Evaluate: tan−1 {sin {cos−1√2

3}} .

8. Find the value of cos ((2 cos−1 𝑥) + sin−1 𝑥) at x = 1

5 .

9. Simlify : tan−1𝑥

√𝑎2−𝑥2

10. Find the principal value of cos−1 (cos7𝜋

6)

2 marks questions:

11. Find the value of 𝑡𝑎𝑛−1 ( 𝑥𝑦 ) − 𝑡𝑎𝑛−1 (

𝑥−𝑦

𝑥+𝑦 ).

12. If 𝑎1 , 𝑎2 , 𝑎3 , ……… . , 𝑎𝑛 be an arithmetic progression with common difference d, then evaluate the following expression

13. Evaluate: tan[𝑡𝑎𝑛−1 ( 𝑑

1+ 𝑎1.𝑎2 ) + 𝑡𝑎𝑛−1 (

𝑑

1+ 𝑎2.𝑎3 ) +

𝑡𝑎𝑛−1 ( 𝑑

1+ 𝑎3.𝑎4 )……… . .+ 𝑡𝑎𝑛−1 (

𝑑

1+ 𝑎𝑛−1.𝑎𝑛 ) ].

14. Solve cos ( tan -1 x ) = sin ( cot -1 3

4 )

15. Prove that tan−1 5 + tan−1 3 − cot−1 47=

𝜋

2

16. Prove the following: cot−1 ( 𝑥𝑦+1𝑥−𝑦

) + cot−1 ( 𝑦𝑧+1

𝑦−𝑧 ) + cot−1 (

𝑧𝑥+1

𝑧−𝑥 ) = 0 ,

( 0 < 𝑥𝑦 , 𝑦𝑧 , 𝑧𝑥 < 1 )

17. Solve sin−1 2𝑎

1+𝑎2+ cos−1

1−𝑏2

1+𝑏2= 2 tan−1 𝑥

18. Show that 𝑡𝑎𝑛 (12sin−1

3

4) =

4−√7

3

19. Solve : tan−11−𝑥

1+𝑥=

1

2tan−1 𝑥 ; 𝑥 > 0.

20. Evaluate: tan {2 tan−1(1

5) +

𝜋

4}.

21. Simplify : tan−1 (𝑎+𝑏𝑥

𝑏−𝑎𝑥) , 𝑥 <

𝑏

𝑎


4 marks questions:

22. Solve for x , if 𝑡𝑎𝑛−1 2𝑥 + 𝑡𝑎𝑛−1 3𝑥 = 𝜋4.

23. Solve for x , 𝑡𝑎𝑛−1 ( 𝑥 + 1 ) + 𝑡𝑎𝑛−1 ( 𝑥 − 1 ) = 𝑡𝑎𝑛−1 8

31

24. Prove that 𝑠𝑖𝑛−1 45+ 𝑠𝑖𝑛−1

5

13+ 𝑠𝑖𝑛−1

16

65=

𝜋

2

25. Prove that 𝑐𝑜𝑠−1 1213+ 𝑠𝑖𝑛−1

3

5= 𝑠𝑖𝑛−1

56

65

26. Solve for x: tan−1( 𝑥 − 1 ) + tan−1 𝑥 + tan−1( 𝑥 + 1 ) = tan−1 3𝑥

27. Prove that tan−1 ( 6𝑥−8𝑥3

1−12 𝑥2 ) − tan−1 (

4𝑥

1 − 4𝑥2 ) = tan−1 2𝑥 ; │2𝑥│ <

1

√3

28. Prove that 2tan−1 15+ sec−1

5√2

7+ 2 tan−1

1

8=

𝜋

4

29. Show that cos ( 2 tan−1 17 ) = sin ( 4 tan−1

1

3 )

30. Find the value of x satisfying the equation: cos−1 ( 𝑥2− 1

𝑥2+ 1 ) + tan−1 (

2𝑥

1− 𝑥2 ) =

2𝜋

3 , 𝑥 >

0

31. Solve the equation tan−1( 2𝑥1− 𝑥2

) + cot−1 ( 1− 𝑥2

2𝑥 ) =

𝜋

3 , 𝑥 > 0

32. Find the value of x , if sin−1 6𝑥 + sin−1 6√3𝑥 = − 𝜋2

33. Prove that 2sin−1 35− tan−1

17

31=

𝜋

4

34. Show that 𝑡𝑎𝑛−11 + 𝑡𝑎𝑛−12 + 𝑡𝑎𝑛−13 = 2 ( 𝑡𝑎𝑛−11 + 𝑡𝑎𝑛−11

2+ 𝑡𝑎𝑛−1

1

3 )

35. Show that cot−1 1 + cot−1 2 + cot−1 3 = 𝜋2

36. If sin[ cot−1( 𝑥 + 1 )] = cos( tan−1 𝑥 ) , then find the value of x.

37. If ( tan−1 𝑥 )2 + ( cot−1 𝑥 )2 = 5𝜋2

8 , then find the value of x.

38. Prove that sin -1( 45 ) + sin -1( 5

13 ) + sin-1 ( 16

65 ) = 𝜋

2

39. Solve sin -1 5𝑥 + sin -1

12

𝑥 = 𝜋

2

40. Solve sin -1 ( 1 – x ) – 2 sin -1 x = 𝜋2

41. Solve the equation sin [ 2 𝑐𝑜𝑠−1 (cot( 2 𝑡𝑎𝑛−1𝑥))] = 0.

42. Prove that :cot−1 (√1+sin 𝑥+√1−sin 𝑥√1+sin 𝑥−√1−sin 𝑥

) =𝑥

2 , 𝑥 ∈ (0,

𝜋

4)

43. Prove that :tan−1 (√1+𝑥−√1−𝑥√1+𝑥+√1−𝑥

) =𝜋

4−

1

2cos−1 𝑥,

−1

√2≤ 𝑥 ≤ 1

44. If tan−1 𝑥 + tan−1 𝑦 + tan−1 𝑧 = 𝜋

2, 𝑥, 𝑦, 𝑧 > 0 then prove that xy+yz+zx=1

45. Prove that tan−1 ( 𝑐𝑜𝑠𝑥

1+𝑠𝑖𝑛𝑥) =

𝜋

4−

𝑥

2 , 𝑥𝜖 (

−𝜋

2 ,𝜋

2)

CHAPTER 3 AND 4


MATRICES AND DETERMINANT

1. The sale figure of three car dealers during march 2015 showed that dealer A sold 6 deluxe, 4 premium and 5 standard cars, dealer B sold 8 deluxe, 3 premium and 4 standard cars, dealer C sold 4 deluxe, 2 premium and 3 standard cars. Write 3 x 3 order matrices summarizing sales data for March.

2. If a matrix has 14 elements, then what are the possible orders it can have? What if it has 17 elements?

3. If 𝐴 = [2 4 15 −6 22 1 5

], then find trace of A.

4. Construct a 2 × 3 matrix A=[𝑎𝑖𝑗] whose elements are given by

(i)aij=1

2(𝑖 − 2𝑗)2 (ii) aij=

1

2|4𝑖 + 𝑗| (iii) aij={

𝑖 − 𝑗, 𝑖𝑓 𝑖 ≥ 𝑗𝑖 + 𝑗, 𝑖𝑓 𝑖 < 𝑗

(iv)𝑎𝑖𝑗 = 𝑖−𝑗

𝑖+𝑗

5. If A is a square matrix such that 𝐴2 = 𝐼, then find the simplest value of (𝐴 − 𝐼)3 + (𝐴 + 𝐼)3 −7𝐴.

6. Write the number of all possible matrices of order 2 x 2 with each entry 1,2 or 3.

7. If [2 1 3](−1 0 −1−1 1 00 1 1

)(10−1) = 𝐴, the write the order of matrix A.

8. If 2 [3 45 𝑥

] + [1 𝑦0 1

] = [7 010 5

], then find (x-y).

9. Solve the following matrix equation for x. (𝑥 1) [1 0−2 0

]=O.

10. If matrix A=[ 1 −1−1 1

] 𝑎𝑛𝑑 𝐴2 = 𝑘𝐴, 𝑡ℎ𝑒𝑛 𝑤𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑘.

11. In the interval 𝜋2< 𝑥 < 𝜋, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑓𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 [

2 sin𝑥 31 2 sin 𝑥

] is

singular matrix.

12. If |𝑥 + 1 𝑥 − 1𝑥 − 3 𝑥 + 2

| = |4 −11 3

|, then find the value of x.

13. If ∆= |5 3 82 0 11 5 −7

|, the write the value of 𝑎32. 𝐴32

14. For what value of x, A=[2(𝑥 + 1) 2𝑥𝑥 𝑥 − 2

] 𝑖𝑠 𝑎 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑎𝑡𝑟𝑖𝑥?

15. What is the value of |4 𝑎 𝑏 + 𝑐4 𝑏 𝑐 + 𝑎4 𝑐 𝑎 + 𝑏

| ?

16. Write the value of |𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎𝑏 − 𝑐 𝑐 − 𝑎 𝑎 − 𝑏𝑐 − 𝑎 𝑎 − 𝑏 𝑏 − 𝑐

|.

17. For what value of k, the system of linear equations 𝑥 + 𝑦 + 𝑧 = 2; 2𝑥 + 𝑦 − 𝑧 = 3; 3𝑥 + 2𝑦 + 𝑘𝑧 = 4. Has a unique solution?

18. If A is a square matrix of order 3 such that |𝑎𝑑𝑗𝐴| = 64, find the |𝐴|. Short Answer (2 marks)

19. If |𝑥 sin 𝜃 cos 𝜃

− sin𝜃 −𝑥 1cos 𝜃 1 𝑥

| = 8, 𝑡ℎ𝑒𝑛 write the value of x.


20. From the following matrix equation, find the value of x: [1 34 5

] [𝑥2] = [

56]

21. Write the value of (x-y+z) from the following matrix [𝑥 + 𝑦 + 𝑧𝑥 + 𝑧𝑦 + 𝑧

] = [957].

22. If 2𝑋 + 3𝑌 = [2 34 0

] 𝑎𝑛𝑑 3𝑋 + 2𝑌 = [−2 21 −5

], find X and Y.

23. If 𝐴 = [3 5], 𝐵 = [7 3] 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑚𝑎𝑡𝑟𝑖𝑥 𝐶 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐴𝐶 = 𝐵𝐶.

24. If A=[0 05 0

] 𝑓𝑖𝑛𝑑 𝐴16 .

25. If A=[ 𝑐𝑜𝑠𝛼 𝑆𝑖𝑛𝛼−𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼

], find 𝛼 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 0 < 𝛼 < 𝜋

2 𝑤ℎ𝑒𝑛 𝐴 + 𝐴𝑇 = √2𝐼2; where 𝐴𝑇 is

transpose of A. 26. If A and B are square matrix of same order and A is skew-symmetric, prove that 𝐵′AB is also

skew-symmetric.

27. if A=[1 24 2

], then find the value of k, if |2𝐴| = 𝑘|𝐴|

28. If A=[𝟏 𝟎 𝟐𝟎 𝟐 𝟏𝟐 𝟎 𝟑

] find A(adj A).

29. Show that [2 −1 3−5 3 1−3 2 3

] 𝑖𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 [−7 −9 10−12 −15 171 1 −1

].

Long Answer type type I (4 marks)

30. If 𝐴 = [1 −3 10 2 −12 1 3

] , 𝐵 = [−1 0 15 4 62 1 1

] 𝑎𝑛𝑑 𝐶 = [−1 −2 11 −2 22 −1 1

] ; 𝑣𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡

(i) A(𝐵𝐶) = (𝐴𝐵)𝐶 (ii) (A+B)C=AC+BC

31. If A=[ 0 3−7 5

] , 𝐼 = [1 00 1

] , 𝑓𝑖𝑛𝑑 𝑘 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑘𝐴2 = 5𝐴 − 21𝐼.

32. Find the matrix A such that [2 −11 0−3 4

]𝐴 = [−1 −8 −101 −2 −59 22 15

].

33. Let A=[1 −2 30 4 7

] 𝑎𝑛𝑑 𝐵 = [0 4 22 −2 1

] verify that (i)(𝐴 + 𝐵)𝑡 = 𝐴𝑡 +𝐵𝑡(ii)(2A)t=2At.

34. The co-operative store of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 45, 40 and 30 each respectively. Find the total amount the store will receive by selling all books.

35. If A=[𝑎 𝑏0 1

] and 𝑎 ≠ 1prove by P.M.I that 𝐴𝑛 = [𝑎𝑛 𝑏(𝑎𝑛−1)

𝑎−1

0 1] for all natural number.

36. If A=[352] 𝑎𝑛𝑑 𝐵 = [1 0 4], 𝑣𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 (𝐴𝐵)′ = 𝐵′𝐴′.

37. Express [2 4 −13 5 81 −2 1

] as the sum of a symmetric and a skew-symmetric matrix.

38. If A=[1 2 22 1 22 2 1

] 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝐴2 − 4𝐴 − 5𝐼 = 𝑂. ℎ𝑒𝑛𝑐𝑒 𝑓𝑖𝑛𝑑 𝐴−1.


39. Using elementary row transformation, find the inverse of each of the following matrices :

(i)[1 23 7

](ii) [2 17 4

]

40. Using properties of determinant prove that |𝑥 − 𝑦 − 𝑧 2𝑥 2𝑥

2𝑦 𝑦 − 𝑧 − 𝑥 2𝑦2𝑧 2𝑧 𝑧 − 𝑥 − 𝑦

| = (𝑥 + 𝑦 + 𝑧)3

41. Using properties of determinant prove that |𝑥 𝑥 + 𝑦 𝑥 + 2𝑦

𝑥 + 2𝑦 𝑥 𝑥 + 𝑦𝑥 + 𝑦 𝑥 + 2𝑦 𝑥

| = 9𝑦2(𝑥 + 𝑦).

42. If 𝑓(𝑥) = |𝑎 −1 0𝑎𝑥 𝑎 −1𝑎𝑥2 𝑎𝑥 𝑎

| , 𝑢𝑠𝑖𝑛𝑔 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠 𝑜𝑓 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡𝑠, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑓(2𝑥) −

𝑓(𝑥).

43. Let f(t)=|𝑐𝑜𝑠𝑡 𝑡 12𝑠𝑖𝑛𝑡 𝑡 2𝑡𝑠𝑖𝑛𝑡 𝑡 𝑡

| 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 lim𝑡→0

𝑓(𝑡)

𝑡2.

44. Find the area of triangle whose vertices are 𝐴(𝑎𝑡12 , 2𝑎𝑡1), 𝐵(𝑎𝑡22 , 2𝑎𝑡2) 𝑎𝑛𝑑 𝐶(𝑎𝑡32 , 2𝑎𝑡3). 45. Find the equation of line containing points (1,2),(3,8). 46. Show that points (a+5,a-4),(a-2,a+3) and (a,a) do not lie on a straight line for any value of a.

47. If 𝐴 = [2 31 4

] 𝑎𝑛𝑑𝐵 = [1 23 4

] , 𝑣𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 𝑎𝑑𝑗(𝐴𝐵) = (𝑎𝑑𝑗𝐵)(𝑎𝑑𝑗 𝐴).

48. Find the adjoint of matrix 𝐴 = [4 −6 1−1 −1 1−4 11 −1

] 𝑎𝑛𝑑 𝑣𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 𝐴(𝑎𝑑𝑗𝐴) = (𝑎𝑑𝑗𝐴)𝐴 = |𝐴|𝐼3.

49. Let 𝐴 = [3 72 5

] 𝑎𝑛𝑑𝐵 = [6 87 9

] . 𝑣𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 (𝐴𝐵)−1 = 𝐵−1𝐴−1.

50. Find the inverse of A=[2 1 34 −1 0−7 2 1

] 𝑎𝑛𝑑 𝑣𝑒𝑟𝑖𝑓𝑦 𝑡ℎ𝑎𝑡 𝐴−1𝐴 = 𝐴𝐴−1 = 𝐼3.

51. For the matrix A=[3 −3 42 −3 40 −1 1

] 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝐴3 = 𝐴−1.

52. Find the minor and cofactor of each entry of the first column of the matrix A and hence find the value of determinant in each case :

(i)A=[5 200 1

] (ii)A=[1 −3 24 −1 23 5 2

]

53. Find the matrix X satisfying the matrix equation [1 22 3

] 𝑋 [4 73 5

] = [1 00 1

]

54. Find the value of k, if area of a triangle is 4 sq. unit when its vertices are (k ,0), (4, 0),(0,2).

55. Solve the equation |𝑥 − 2 2𝑥 − 3 3𝑥 − 4𝑥 − 4 2𝑥 − 9 3𝑥 − 16𝑥 − 8 2𝑥 − 27 3𝑥 − 64

| = 0

56. Prove that the product of matrix (in any order) [ 𝑐𝑜𝑠2𝜃 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃 𝑠𝑖𝑛2𝜃

] and

[ 𝑐𝑜𝑠2∅ 𝑐𝑜𝑠∅𝑠𝑖𝑛∅𝑐𝑜𝑠∅𝑠𝑖𝑛∅ 𝑠𝑖𝑛2∅

]

is a null matrix, where 𝜃 𝑎𝑛𝑑 ∅ 𝑑𝑖𝑓𝑓𝑒𝑟 𝑏𝑦 𝑎𝑛 𝑜𝑑𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝜋2.


57. If A=[1 0 20 2 12 0 3

] , 𝑎𝑛𝑑 𝐴3 − 6𝐴2 + 7𝐴 + 𝐾𝐼3 = 𝑂, 𝑓𝑖𝑛𝑑 𝑘.

58. A coaching institute of mathematics (subjects) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs.9000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs. 26000. Using matrix method, find monthly fees paid by each child of two types. Long Answer type II (6 marks)

59. A total amount of Rs.7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and 8 1

2% respectively. The total annual interest from these three accounts

is Rs.550. Equal amounts have been deposited in the 5% and 8% savings accounts. Find the amount deposited in each of the three accounts, with the helps of matrices.

60. Given that 𝐴 = [−4 4 4−7 1 35 −3 −1

] 𝑎𝑛𝑑 𝐵 = [1 −1 11 −2 −22 1 3

] find AB. Use this result to solve the

following system of linear equations :x-2y+z=4; x-2y-2z=9; 2x+y+3z=1.

61. Find A-1 if 𝐴 = [−1 2 52 −3 1−1 1 1

] and hence, solve the system of linear equations :

-x+2y+5z=2; 2x-3y+z=15; -x +y +z=-3 62. Using elementary row transformation, find the inverse of each of the following matrices :

(i)[2 3 −3−1 −2 21 1 −1

](ii) [2 0 −15 1 00 1 3

]

63. Using properties of determinants, prove that|(𝑏 + 𝑐)2 𝑏𝑎 𝑐𝑎

𝑎𝑏 (𝑐 + 𝑎)2 𝑐𝑏

𝑎𝑐 𝑏𝑐 (𝑎 + 𝑏)2| = 2𝑎𝑏𝑐(𝑎 + 𝑏 +

𝑐)3

64. In a ∆𝐴𝐵𝐶, if |1 1 1

1 + 𝑠𝑖𝑛 𝐴 1 + 𝑠𝑖𝑛 𝐵 1 + 𝑠𝑖𝑛 𝑐𝑠𝑖𝑛 𝐴 + sin2 𝐴 𝑠𝑖𝑛 𝐵 + sin2𝐵 𝑠𝑖𝑛 𝐵 + sin2𝐵

|=0, then prove that ∆𝐴𝐵𝐶 is an

isosceles triangle.

65. If |𝑝 𝑏 𝑐𝑎 𝑞 𝑐𝑎 𝑏 𝑟

| = 0, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝

𝑝−𝑎+

𝑞

𝑞−𝑏+

𝑟

𝑟−𝑐, 𝑝 ≠ 𝑎, 𝑞 ≠ 𝑏, 𝑟 ≠ 𝑐.

66. Solve the following system of equations using matrix method : 𝟏

𝒙−𝟏

𝒚+𝟏

𝒛= 𝟓 ,

𝟐

𝒙+𝟑

𝒚−𝟒

𝒛= −𝟕 ;

𝟐

𝒙+𝟏

𝒚−𝟐

𝒛= −𝟑

67. In a survey of 20 richest persons of three residential society A, B, C it is found that in society A,5 believe in honesty, 10 in hard work, 5 in unfair means while in B, 5 in honesty, 8 in hard work, 7 in unfair means and in C, 6 in honesty, 8 in hard work, 6 in unfair means. If the per day income of 20 richest persons of society A ,B, C are Rs.32500, 30500,Rs.31000 respectively, then find the per day income of each type of people by matrix method.

a. Which type of people has more per day income?


68. Using properties of determinants, prove that|(2𝑥 + 2−𝑥)2 (2𝑥 − 2−𝑥)2 1

(3𝑥 + 3−𝑥)2 (3𝑥 − 3−𝑥)2 1(4𝑥 + 4−𝑥)2 (4𝑥 − 4−𝑥)2 1

| = 0

69. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others using matrix method, find the number of awardees of each category. Apart from these values, namely honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

70. Prove that |𝑦𝑧 − 𝑥2 𝑧𝑥 − 𝑦2 𝑥𝑦 − 𝑧2

𝑥𝑧 − 𝑦2 𝑦𝑥 − 𝑧2 𝑦𝑧 − 𝑥2

𝑥𝑦 − 𝑧2 𝑦𝑧 − 𝑥2 𝑥𝑧 − 𝑦2|is divisible by (x+y+z) and hence find the quotient.

71. Using properties of determinants, prove that |(𝑎 + 1)(𝑎 + 2) 𝑎 + 2 1(𝑎 + 3)(𝑎 + 2) 𝑎 + 3 1(𝑎 + 3)(𝑎 + 4) 𝑎 + 4 1

| = −2.

72. If A=[1 0 20 2 12 0 3

] , 𝑎𝑛𝑑 𝐴3 − 6𝐴2 + 7𝐴 + 𝐾𝐼3 = 𝑂, 𝑓𝑖𝑛𝑑 𝑘.


CHAPTER 5 & 6

LIMITS, CONTINUITY AND DIFFERENTATION & APPLICATION OF DERIVATIVE

Each question carry 1 marks

1. Find the differentiation of tan(2𝑥 + 𝑏) w.r.t. 𝑥. 2. If 𝑥 = 𝑠𝑖𝑛𝑡 and 𝑦 = 𝑐𝑜𝑠𝑡, then find 𝑑𝑦

𝑑𝑥 at 𝑡 = 𝜋

4.

3. Find 𝑑𝑦𝑑𝑥

if 𝑥 + 𝑦 = 𝑥𝑦. 4. Find 𝑓′(1), if 𝑓(𝑥) = tan−1 𝑥. 5. Differentiate log (1−2𝑥). 6. Differentiate tan 𝑥 w.r.t. 𝑥2.

7. Find the value of k for which 𝑓(𝑥) = {𝑠𝑖𝑛2𝑥

5𝑥, 𝑤ℎ𝑒𝑛 𝑥 ≠ 0

𝑘, 𝑤ℎ𝑒𝑛 𝑥 = 0 is continuous at 𝑥 = 0.

8. Find the value of 𝑓(0) so that the function 𝑓(𝑥) = 𝑒𝑎𝑥−𝑒𝑏𝑥

𝑥 is continuous at 𝑥 = 0.

9. Show that the function 𝑓(𝑥) = |𝑥| + |𝑥 − 1| is continuous for all 𝑥 ∈ 𝑅. 10. Give an example of a function which is continuous but not differentiable at at least

two points. 11. Differentiate 𝑠𝑖𝑛2( 𝑥2 ) w.r.t x2 12. Differentiate ( 3𝑥−7 )

√𝑥3 w.r.t x

13. Determine the value of k for which the function

( 𝑥+3 )2 −3

𝑥 − 3 , x≠3

F(x) = K , x = 3 at x = 3

14. Determine the value of p for which the function 𝑝 𝑥

𝐼𝑥𝐼 , x <0

F(x) = 3 x ≥0 Continuous at x = 0

15. The amount of pollution content added in air in a city due to x diesel Vehicles is given by p(x) = 0.005𝑥3 + 0.02 𝑥2 + 30 x . Find the marginal Increase in pollution content when 3 diesel vehicles are added .

16. Find the slope of the tangent to the curve y = 3x2 – 6 at the point on it Whose x coordinate is 2.

17. Find the slope of normal to the curve y = 3x4 – 4x at x = 1 18. Write the derivative of f(x) = [x] at x = 3/2. 19. Is the function f(x) = sec x derivable at x = π/2 ?

20. Differentiate √𝑒√𝑥 w.r.t x


21. Differentiate 𝑥𝑎 𝑎𝑥 w.r.t x 22. Find the maximum values of f(x) = |𝑥 − 2| + 5 23. Find the minimum values of f(x) = |𝑥| + 3 24. If f(x) = 𝑡𝑎𝑛−1(𝑐𝑜𝑡 𝑥 ) , show that f`(x) = -1 25. If y = 3sinx – 4 𝑠𝑖𝑛3𝑥 , find 𝑑𝑦

𝑑𝑥 at x =π/6

26. Is the function f(x) = x – |𝑥| derivable at x = o ? 27. If the radius of the circle is increasing at the rate of 0.7 mm/sec, at what rate is its

circumference changing ? 28. Determine the point on the curve y = 3x2 -1 at which the slope of the curve is 3. 29. If f(x) = log(log x ) , find f`(e)

30. If y = √𝑎𝑐𝑜𝑠−1𝑥 find 𝑑𝑦𝑑𝑥

SEC-B

Every Questions carry 2 marks:

31. Prove that the function f given by 𝑓(𝑥) = |𝑥 − 5|, 𝑥 ∈ 𝑅 is not differentiable at 𝑥 = 5

32. Differentiate tan−1 (1 + cos𝑥sin𝑥

) with respect to 𝑥.

33. If (𝑥2 + 𝑦2)2 = 𝑥𝑦, find 𝑑𝑦𝑑𝑥

.

34. If 𝑥 = 𝑎 (2𝜃 − sin 2𝜃) and 𝑦 = 𝑎 (1 − cos 2𝜃), find 𝑑𝑦𝑑𝑥

when 𝜃 = 𝜋

3.

35. Determine the value of ‘k’ for which the following function is continuous

at 𝑥 = 3: 𝑓(𝑥) = {(𝑥+3)2−36

𝑥−3, 𝑥 ≠ 3

𝑘 , 𝑥 = 3.

36. Find the value of ‘c’ in Rolle’s theorem for the function 𝑓(𝑥) = 𝑥3 − 3𝑥 in [−√3, 0]

37. Find: 𝑑𝑑𝑥cos−1 (

𝑥 − 𝑥−1

𝑥 + 𝑥−1).

38. Find the relationship between ‘a’ and ‘b’ so that the function ‘𝑓’ defined by

𝑓(𝑥) = {𝑎𝑥 + 1, 𝑖𝑓 𝑥 ≤ 3

𝑏𝑥 + 3, 𝑖𝑓 𝑥 > 3 is continuous at 𝑥 = 3

39. Find the points of discontinuity of 2x + 3 if x ≤ 2 F(x) = 2x – 3 if x > 2 40. Find f `(x) at x = 1 , y = 𝜋

4 if 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 𝑥𝑦 = 𝑘

41. If 𝑠𝑖𝑛−1( 6𝑥√1 − 9𝑥2 ) , −13√2

< x < 1

3√2


42. Show that the function f(x)=𝑥3 − 3𝑥2 + 6x – 100 is increasing on R 43. Show that the function f defined by f(x) = Ix -2I is continuous at x =2 but not derivable at x =2. 44. If y = 𝑡𝑎𝑛−1 5𝑎𝑥

𝑎2− 6𝑥2 , prove that 𝑑𝑦

𝑑𝑥=

3𝑎

𝑎2+ 9𝑥2 + 2𝑎

𝑎2+ 4𝑥2

45. Use implicit differentiation to verify that 𝑑𝑦𝑑𝑥 .𝑑𝑥

𝑑𝑦 = 1 when

𝑥3 + 𝑦3 = 3𝑎𝑥𝑦 46. Find the second order derivatives of 𝑒𝑠𝑖𝑛𝑥2 w.r.t x 47. If y = cot x , find y`` at x = π/4 48. Verify Rolle`s theorem for the function and the point(s) in the interval

Where derivatives vanishes f(x) = x2 + 2 on [ -2 , 2 ] 49. Differentiate 𝑠𝑖𝑛−1 ( 3𝑥 − 4𝑥3 ) 50. Find the point of local maxima and minima for f(x) = sin x + cos x , o < x < π/2 . 51. Show that the function f(x) = (log x )/x has maximum value at x = e. 52. Find the least value of a so that the function f(x) = x2 + ax + 1 is strictly increasing

on [ 1, 2]. 53. Prove that the function f(x) = log sinx is strictly decreasing on (π/2 , π). 54. Find the point on the curve y =x2 – 2x +3 at which tangent is parallel to

X axis . 55. If the gradient of the curve 2y2 = ax2 + b at 1 and -1 is -1 , find a and b 56. Find the slope of the normal to the curve x = a 𝑐𝑜𝑠3θ , y =𝑠𝑖𝑛3θ at θ=π/4 57. If x = at2 , y = 2at , find y2 58. Differentiate log( 1 + θ ) w. r. t 𝑠𝑖𝑛−1θ

SEC-C

Each question carry 4marks

59. If y = 3𝑒2𝑥 + 2𝑒3𝑥, prove that 𝑑2𝑦

𝑑𝑥2− 5

𝑑𝑦

𝑑𝑥+ 6𝑦 = 0.

60. Find the value of p and q for which 𝑓(𝑥) =

{

1−sin3 𝑥

3 cos2 𝑥𝑖𝑓 𝑥 <

𝜋

2

𝑝 𝑖𝑓 𝑥 =𝜋

2𝑞(1−sin 𝑥)

(𝜋−2𝑥)2𝑖𝑓 𝑥 >

𝜋

2

if f(x) is continuous

at x=𝜋2

61. Find the value of k so that 𝑓(𝑥) = {1−cos 4𝑥

8𝑥2𝑖𝑓 𝑥 ≠ 0

𝑘 𝑖𝑓 𝑥 = 0, if f(x) is continuous at x=𝜋

2


62. Find all the point of discontinuity, where f is defined as 𝑓(𝑥) =

{|𝑥| + 3 𝑖𝑓 𝑥 ≤ −3−2𝑥 𝑖𝑓 − 3 < 𝑥 < 36𝑥 + 2 𝑥 ≥ 3

.

63. If 𝑥𝑦 + 𝑦𝑥 = 𝑎𝑏, find 𝑑𝑦𝑑𝑥

.

64. If 𝑒𝑦(𝑥 + 𝑦) = 1, 𝑡ℎ𝑒𝑛 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑑2𝑦

𝑑𝑥2= (

𝑑𝑦

𝑑𝑥)2

.

65. Differentiate to tan−1 (√1+𝑥2−1

𝑥) w. r. t sin−1 ( 2𝑥

1+𝑥2), when x≠ 0.

66. If 𝑥𝑐𝑜𝑠 (𝑎 + 𝑦) = cos 𝑦, 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑑𝑦𝑑𝑥=

cos2(𝑎+𝑦)

sin 𝑎. Hence, show that

sin 𝑎𝑑2𝑦

𝑑𝑥2+ sin 2(𝑎 + 𝑦)

𝑑𝑦

𝑑𝑥= 0.

67. If 𝑥 = 𝑠𝑖𝑛𝑡 𝑎𝑛𝑑 𝑦 = sin 𝑝𝑡, 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 (1 − 𝑥2)𝑑2𝑦

𝑑𝑥2− 𝑥

𝑑𝑦

𝑑𝑥+ 𝑝2 = 0.

68. Find the value of a and b, if the function defined by 𝑓(𝑥) = 𝑓(𝑥) =

{𝑥2 + 3𝑥 + 𝑎, 𝑥 ≤ 1

𝑏𝑥 + 2, 𝑥 ≥ 1is differentiable at x=1.

69. The length x of a rectangle is decreasing at the rate of 5 cm/min and the width y is increasing at the rate of 4 cm /min, when x=8 cm and y=6 cm, find the rate of change of : a. Perimeter b. Area of rectangle.

70. The side of an equilateral triangle is increasing at the rate of 2 cm/sec. At what rate is its area is increasing, when the side of triangle is 20 cm?

71. Find the intervals in which 𝑦 = [𝑥(𝑥 − 2)]2 is an increasing function and decreasing function also.

72. Find the interval in which 𝑦 = 𝑥4 − 8𝑥3 + 22𝑥2 − 24𝑥 + 21 is an increasing function and decreasing function also.

73. Find the interval in which 𝑦 = sin4 𝑥 + cos4 𝑥 is an increasing function and

decreasing function in [0, 𝜋2].

74. Find the approximate value of 𝑓(3.02),𝑢𝑝𝑡𝑜 𝑡𝑤𝑜 𝑝𝑙𝑎𝑐𝑒𝑠 𝑜𝑓 𝑑𝑒𝑖𝑚𝑎𝑙,𝑤ℎ𝑒𝑟𝑒 𝑓(𝑥) =3𝑥2 + 15𝑥 + 3.

75. Using differential find the approximate value of √49.5. 76. Find the equation of tangent to the curve 𝑦 = 𝑥3 + 2𝑥 − 4, which are perpendicular

to the line x+14y-3=0. 77. Find the point on the curve 9𝑦2 =

𝑥3, 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑚𝑎𝑘𝑒𝑠 𝑒𝑞𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑎𝑥𝑒𝑠. 78. Find the points on the curve 𝑦 = 𝑥3 − 11𝑥 + 5, at which equation of tangent is y=x-

11. 79. Find the equation of tangent to the curve 𝑥 = sin 3𝑡, 𝑦 = cos2𝑡 𝑎𝑡 𝑡 = 𝜋

4.


80. Find the equation of tangent to the curve y= 𝑥−7

𝑥2−5𝑥−6 at the point, where it cuts the X-

axis.

81. If 𝑦 = sin(sin 𝑥), prove that 𝑑2𝑦

𝑑𝑥2+ tan 𝑥

𝑑𝑦

𝑑𝑥+ 𝑦 𝑐𝑜𝑠2𝑥 = 0.

82. Differentiate 𝑥sin𝑥 + (sin 𝑥)cos𝑥 with respect to 𝑥. 83. If y = 2 cos (log 𝑥) + 3 sin (log 𝑥), prove that 𝑥2 𝑑

2𝑦

𝑑𝑥2+ 𝑥

𝑑𝑦

𝑑𝑥+ 𝑦 = 0.

84. If 𝑥 = a sin 2t (1+cos2t) and y = bcos2t (1–cos2t), find 𝑑𝑦𝑑𝑥

at t = 𝜋4.

85. If 𝑥 = 𝛼 sin 2t (1+cos2t) and y = 𝛽cos2t (1–cos2t), show that 𝑑𝑦𝑑𝑥

= 𝛽𝛼

tan t. 86. Find the derivative of the following function 𝑓(𝑥) w.r.t. 𝑥, at 𝑥 = 1:

cos−1 [sin√1+𝑥

2] + 𝑥𝑥.

87. Find the value of 𝑑𝑦𝑑𝑥

at 𝜃 = 𝜋

4, if 𝑥=a𝑒𝜃(sin 𝜃 − cos 𝜃) and ya𝑒𝜃(sin 𝜃 +

cos 𝜃). 88. If y = P𝑒𝑎𝑥 + Q𝑒𝑏𝑥, show that 𝑑

2𝑦

𝑑𝑥2− (𝑎 + 𝑏)

𝑑𝑦

𝑑𝑥+ 𝑎𝑏𝑦 = 0.

89. If 𝑥 = a sin 2t (1+cos2t) and y = bcos2t (1–cos2t), show that at t = 𝜋4,

𝑑𝑦

𝑑𝑥 = 𝑏

𝑎.

90. If 𝑥 = cos t (3– 2 𝑐𝑜𝑠2𝑡)and y = sin t (3−2 𝑠𝑖𝑛2𝑡), find the value of 𝑑𝑦𝑑𝑥

at t = 𝜋4.

91. If 𝑦𝑥 = 𝑒𝑦−𝑥, prove that 𝑑𝑦𝑑𝑥

= (1+log𝑦)2

log𝑦.

92. Differentiate the following with respect to 𝑥: sin−1 (2𝑥+1.3𝑥

1+(36)𝑥).

93. Find the value of k, for which𝑓(𝑥) = {√1+𝑘𝑥−√1−𝑘𝑥

𝑥, 𝑖𝑓 − 1 ≤ 𝑥 < 0

2𝑥+1

𝑥−1, 𝑖𝑓 0 ≤ 𝑥 < 1

is continuous at 𝑥 = 0.

94. If 𝑥 = a𝑐𝑜𝑠3𝜃 and y = a𝑠𝑖𝑛3𝜃, then find the value of 𝑑2𝑦

𝑑𝑥2 at 𝜃 = 𝜋

6.

95. If 𝑥𝑦 = 𝑒𝑥−𝑦, prove that 𝑑𝑦𝑑𝑥

= log𝑥

(1+log𝑥)2.

96. If 𝑥 sin (a + y) + sin a cos (a + y) = 0, prove that 𝑑𝑦𝑑𝑥

= 𝑠𝑖𝑛2(𝑎+𝑦)

sin𝑎.

97. If 𝑥 = √𝑎sin−1 𝑡, y = √𝑎cos−1 𝑡, show that 𝑑𝑦𝑑𝑥

= − 𝑦

𝑥.


98. Differentiate tan−1 [√1 + 𝑥2 − 1

𝑥]with respect to 𝑥.

99. If 𝑥 = a (cos t + t sin t) and y = a (sin t – t cos t), 0 < t <𝜋2, find 𝑑

2𝑥

𝑑𝑡2,

𝑑2𝑦

𝑑𝑡2and 𝑑

2𝑦

𝑑𝑥2.

100. If y = (tan−1 𝑥)2, show that(𝑥2 + 1)2 𝑑2𝑦

𝑑𝑥2+ 2𝑥(𝑥2 + 1)

𝑑𝑦

𝑑𝑥= 2.

101. If 𝑥 = a (cos t + log tan 𝑡2), y = a sin t, find 𝑑

2𝑦

𝑑𝑡2 and 𝑑

2𝑦

𝑑𝑥2.

102. If 𝑥𝑦 = 𝑒𝑥−𝑦, show that 𝑑𝑦𝑑𝑥

= log 𝑥

[log(𝑥𝑒)]2

103. If 𝑥 = tan (1𝑎log 𝑦), show that (1 + 𝑥2) 𝑑

2𝑦

𝑑𝑥2+ (2𝑥 − 𝑎)

𝑑𝑦

𝑑𝑥= 0.

104. If y = 𝑒𝑎 sin−1 𝑥 , −1 ≤ 𝑥 ≤ 1, then show that (1 − 𝑥2) 𝑑2𝑦

𝑑𝑥2− 𝑥

𝑑𝑦

𝑑𝑥−

𝑎2𝑦 = 0

105. If y = cos−1 (3𝑥+4√1−𝑥2

5), find 𝑑𝑦

𝑑𝑥.

106. If y = cosec−1 𝑥 , 𝑥 > 1, then show that 107. 𝑥(𝑥2 − 1) 𝑑

2𝑦

𝑑𝑥2+ (2𝑥2 − 1)

𝑑𝑦

𝑑𝑥= 0.

108. If sin y = 𝑥sin (a + y), prove that 𝑑𝑦𝑑𝑥

= 𝑠𝑖𝑛2(𝑎+𝑦)

sin𝑎.

109. If (cos 𝑥)𝑦 = (sin 𝑦)𝑥, find 𝑑𝑦𝑑𝑥

.

110. If y = 𝑒𝑥(sin 𝑥 + cos 𝑥), then show that 𝑑2𝑦

𝑑𝑥2− 2

𝑑𝑦

𝑑𝑥+ 2𝑦 = 0.

If 𝑓(𝑥) defined by the following, is continuous at 𝑥 = 0, find the values of ‘a’, ‘b’

and ‘c’. 𝑓(𝑥) =

{

sin(𝑎+1)𝑥+sin𝑥

𝑥 , 𝑖𝑓𝑥 < 0

𝑐 , 𝑖𝑓𝑥 = 0√𝑥+𝑏𝑥2−√𝑥

𝑏𝑥3 2⁄ , 𝑖𝑓𝑥 > 0.

111. If y = sin−1 [5𝑥+12√1−𝑥2

13], find 𝑑𝑦

𝑑𝑥.

112. If 𝑥 = a (cos 𝜃 + log tan 𝜃2) and y = a sin 𝜃, find the value of 𝑑𝑦

𝑑𝑥 at

𝜃 =𝜋

4.

SEC-D

*Every Question carry 6 Marks.


113. If =sin−1 𝑥

√1−𝑥2, 𝑡ℎ𝑒 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 (1 − 𝑥2)

𝑑2𝑦

𝑑𝑥2− 3𝑥

𝑑𝑦

𝑑𝑥− 𝑦 = 0.

114. If 𝑦 = (cot−1 𝑥)2, 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 (𝑥2 + 1)2𝑑2𝑦

𝑑𝑥2+ 2𝑥(𝑥2 + 1)

𝑑𝑦

𝑑𝑥= 2.

115. If log𝑒(√1 + 𝑥2 − 𝑥) = 𝑦√1 + 𝑥2, then show that (1 + 𝑥2)𝑑𝑦

𝑑𝑥+ 𝑥𝑦 + 1 = 0.

116. Find the equations of tangent to the curve 3𝑥2 − 𝑦2 =

8,𝑤ℎ𝑖𝑐ℎ 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (4

3, 0).

117. Prove that all normal to the curve 𝑥 = acos 𝑡 + 𝑎𝑡𝑠𝑖𝑛 𝑡 𝑎𝑛𝑑 𝑦 = asin 𝑡 −

𝑎𝑡𝑐𝑜𝑠𝑡 𝑎𝑟𝑒 at constant distance “a” from the origin.

118. Find the equation of tangent and normal to the curve 𝑥 = 1 − cos𝜃 , 𝑦 = 𝜃 −

sin 𝜃 𝑎𝑡 𝜃 =𝜋

4.

119. Prove that the function x defined by f(x)=𝑥2 − 𝑥 +

1 𝑖𝑠 𝑛𝑒𝑖𝑡ℎ𝑒𝑟 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑛𝑜𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑖𝑛 (−1, 1). Hence, find the interval in

which function is :

a) Strictly increasing

b) Strictly decreasing

120. Prove that: 𝑦 =4 sin 𝜃

2+cos𝜃− 𝜃 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 (0,

𝜋

2).

121. Find the interval in which the function f given by 𝑓(𝑥) = sin 𝑥 − 𝑐𝑜𝑠𝑥 , 0 ≤ 𝑥 ≤

2𝜋, is strictly increasing and strictly decreasing.

122. Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

123. AB is the diameter of a circle and C is any point on the circle. Show that the area of triangle ABC is maximum, when it is an isosceles triangle.

124. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

125. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6√3r.

126. If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is 𝜋

3 .

127. If the function 𝑓(𝑥) = 2𝑥3 – 9m𝑥2 + 12m2𝑥 + 1, where m > 0 attains its maximum and minimum at p and q respectively such that p2 = q, then find the value of m.

128. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3. Also show that the maximum volume of the cone is 8/27 of the volume of the sphere.

129. Prove 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.

130. If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 600.


131. Prove that the radius of the right circular cylinder of greatest curved surface which can be inscribed in a given cone is half of that of the cone.

132. An open box with a square base is to be made out of a given quantity of cardboard of area c2 square units. Show that the maximum volume of the box is c3/6√3 cubic units.

133. Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 times the radius of the base.

134. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.

135. If the length of three sides of a trapezium other than the base is 10 cm each, find the area of the trapezium, when it is maximum.

136. Find the intervals in which the following function 𝑓 (𝑥) = 20 – 9 𝑥 + 6 𝑥2 − 𝑥3 is (a) strictly increasing (b) strictly decreasing.

137. Find the equations of the tangent and the normal to the curve 𝑥 = 1 – cos𝜃; y = 𝜃 − sin 𝜃 at 𝜃 = 𝜋

4.

138. If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 𝜋

3.

139. A manufacturer can sell 𝑥 items at a price of `(5 − 𝑥

100) each. The cost price of 𝑥

items is (

𝑥

5+ 500). Find the number of items he should sell to earn maximum profit.

140. Show that the semi-vertical angle of the cone of maximum volume and of given slant height is tan−1 √2.Show that the volume of the greatest cylinder that can be inscribed in a cone of height h and semi-vertical angle 𝛼 is (4/27) πh3𝑡𝑎𝑛2𝛼.

***************************BEST OF LUCK*******************************

ARMY PUBLIC SCHOOL AKHNOOR UNIT 1 ELECTROSTATICS

IMPORTANT TOPICS FOR SLOW BLOOMERS (MLL) Coulombs law Electric dipole-electric field on axial and equatorial line, torque acting on the dipole. Statement of Gauss Theorem and its application. Electric field due to infinite plane sheet of charge Electric field due to spherical shell Electric field due to infinite uniformly charged line charge Electric potential due to dipole and point charge. Electrostatic Potential energy and equipotential surfaces Capacity of a parallel plate capacitor with (i) air (ii) dielectric (iii) conducting medium between the plates Numerical on series and parallel combination of capacitor. Energy stored in a capacitor.

ONE MARK QUESTIONS

1. Define dipole moment of an electric dipole. Is it a scalar or a vector? 2. In which orientation a dipole placed in a uniform electric field is in a) Stable, b) Unstable

Equilibrium? 3. What is the electric potential due to electric dipole at an equatorial point? 4. What is the shape of equipotential surface due to a single isolated charge? 5. Name a physical quantity whose SI unit is J/C. Is it a scalar or a vector quantity? 6. A hollow metal sphere of radius 5 cm is charged such that the potential on its surface is 10V. What is

the potential at the centre of the sphere? TWO MARKS QUESTION

1. What is the work done to move a test charge q through a distance of 1 cm along the equatorial axis of dipole?

2. A 500μC charge is at the centre of square of side 10cm. Find work done in moving a charge of 10 μC between two diagonally opposite points on the square.

3. Can two equipotential surfaces intersect each other? Give reasons. 4. The given graph shows the variation of charge, q versus potential difference V

for capacitors C1 and C2 . The two capacitors have same plate area of C2 is double than that C1. Which of the lines in the graph correspond to C1 and C2 and why?

5. Depict the equipotential surfaces for a system of two identical positive point charges placed at a

distance ‘d’ apart.

(3 MARKS & 5 MARKS QUESTIONS)

1. Derive expression for electric field at a point on the axial line of the dipole. Give the direction of electric field at the point.

2. Derive expression for electric field at a point on the equatorial line of dipole. 3. An electric dipole is held in uniform electric field

(i) Show that no net force acts on it. (ii) Derive an expression for the torque acting on it

4. State Gauss Theorem. A thin charged wire of infinite length has line charge density ‘λ’. Derive expression for electric field at a distance ‘r’.

5. Charge q is distributed uniformly on a spherical shell of radius R. Using gauss law derive expression of electric field at a distance r from the centre when (i)r>R (ii) r=R (iii) r<R

6. Derive expression for capacitance of parallel plate capacitor. 7.Derive expression for capacitance of parallel plate capacitor with dielectric as medium between the plates. 8. Derive expression for energy stored in a capacitor. 9. Derive expression for capacitance of parallel plate capacitor with conductor as medium between the plates. 10. Using Gauss theorem show that electric field due to infinite plane sheet of charge is independent of distance of the point from the sheet.

Prepared by: ANKUR SINGH JAMWAL (PGT PHYSICS) APS AKHNOOR

ANSWERS

One mark question solution

Ans 1 Electric dipole moment of an electric dipole is equal to the product of either charge or distance between the two charges.

Where p is dipole moment. It is a scalar quantity.

Ans 2. (a) For stable equilibrium the angle between p and E must be 00 (b) For unstable equilibrium the angle between p and E must be 1800 Ans 3. Potential at a point on equatorial line is 0. Ans 4. For an isolated charge equipotential surface are concentric spherical shells and distance between them increases with the decrease in field.

Ans 5. J/C is unit of electric potential. It is a scalar quantity. Ans6. 10V

2 marks question solution Ans 1. Potential at any point on the equatorial line is 0. Hence work done W = qΔV =0 as ΔV=0

Ans2. Two diagonally opposite points are equidistant from the centre of square hence potential at these points due to given charge will be equal. W= QΔV=0 as ΔV=0. Ans 3. No, two equipotential surfaces cannot intersect each other because two normals can be drawn at intersecting point on the two surfaces which gives two directions of E at the same point which is not possible. Ans 4.

The slope of graph represents capacity of capacitor A has greater slope than that of B So capacitance of A is greater than that of B

Ans 5 Equipotential surfaces for two equal and opposite charges

Long question solution

Ans 1. Electric field at a point on an axial point of electric dipole.

The axial line of a dipole is the line passing through the positive and negative charges of the electric dipole. Consider a system of charges (-q and +q) separated by a distance 2a. Let 'P' be any point on an axis where the field intensity is to be determined. Electric field at P (EB) due to +q

Electric field at P due to -q (EA)

Net field at P is given by

Simplifying, we get

As a special case :

Ans 2. An equatorial line of a dipole is the line perpendicular to the axial line and passing through a point mid way between the charges.

Consider a dipole consisting of -q and +q separated by a distance 2a. Let P be a point Consider a point P on the equatorial line.

The resultant intensity is the vector sum of the intensities along PA and PB. EA and EB can be resolved into vertical and horizontal components. The vertical components of EA Sinθ and EB Sinθ cancel each other as they are equal and oppositely directed. It is the horizontal components which add up to give the resultant field.

E = 2EA cos

As 2qa = p

As a special case,

Ans 3.

Force on +q charge=qE along direction of E Force on –q charge =qE opposite to E Fnet=qE-qE =0 The forces are equal in magnitude, opposite in direction acting at different points, therefore they form a couple which rotates the dipole.

Ans 4. Gauss’s Law: ‘Electric flux over a closed surface is 1/ε0 times the charge enclosed by it.’

To calculate the field at P we consider a Gaussian surface with wire as axis, radius r and length l as shown in the figure. The electric lines of force are parallel to the end faces of the cylinder and hence the component of the field along the normal to the end faces is zero. The field is radial everywhere and hence the electric flux crosses only through the curved surface of the cylinder.

If E is the electric field intensity at P, then the electric flux through the Gaussian surface is

According to gauss theorem electric flux is

Ans 5. Consider a hollow conducting sphere of radius R with its centre at O. let σ be its surface density. The field at any point P, outside or inside depends upon the distance from the centre of the spherical shell. Let the distance between the centre of the spherical shell and the point be r.

Case (i) r>R

At points outside the sphere the electric field is radial every where because of spherical symmetry.

Ans 6. Let the surface charge density on the plates be σ such that

Electric field between the plates is given by

Ans 7. Let the surface charge density on the plates be σ

Such that

where E0 is electric field in air and Ei is electric field in dielectric. Potential difference between the plates is given by

Ans 8. Consider a parallel plate capacitor of capacity C. Let at any instant the charge on the capacitor be Q’. Then potential difference between the plates will be Suppose the charge on the plates increases by d Q’. The work done will be

1

Subject – History Class- xii Session 2019-20 practice of Important questions for during Summer Vacations

Ch-1

1. What do you know about the seals and sealings? (2) 2. Write about the script used by the harappans (4) 3. Write about the weight system used by the Harappans (2) 4. Mention the causes of the end of the civilisation (2) 5. Mention the domestic architecture of Harappans.(2) 6. List the raw materials required for craft production in the Harappan civilisation and discuss how

these might have been obtained. (2) 7. Two methods of procuring materials for craft production:- (2) 8. “Our knowledge about the Indus Valley Civilization is poorer than that of the other

Civilizations”. Explain it by your arguments? (2) 9. what were the confusions in the mind of Cunningham while studying Harappan

civilization? (2) 10. what were the differences in the techniques adopted by Marshall and Wheeler in studying

Harappan civilization? (2) 11. “Burials is a better source to trace social differences prevalent in the Harappan civilization”.

Discuss. (2) 12. Write a note on the Drainage system of the Harappans. (2) 13. discuss the functions that may have been performed by rulers in Harappan Society. (4) 14. How can you say that the Harappan culture was an urban one. (4) 15. Write a note on the agricultural technology of Harappans. (4) 16. Discuss how archaeologist reconstruct the past. (8) Ch-2

1. Mention the important features of Magadha Empire 2. What are megaliths? 3. Define Dhamma Mahamatta? 4. Write any two sources of Mauryan history? 5. Who was a Gahapati? 2 6. Who were Kushanas? 2 7. In which languages and script, Ashokan script was written? 2 8. Discuss factors responsible for the rise of Magadha – 4 9. Describe five features of Mahajanapadas? 4

10.Explain main features of Ashoka’s Dhamma? 4 11.Important changes in agriculture during the period between 600 BCE to 600 CE . 4

12.How do inscription help in reconstruction of history? 4 13.Main features of Mauryan administration? 8 Ch-3

Q1Critically examine the duties as laid down inManusmriti for the chandalas. Q2 In what ways was the Buddhist theory of a social contract different from the Brahmanical view of society derived from the Purusha sukta. Q3. Why Mahabharata is considered a colossal epic?

Q.4 What were three strategies adopted by the Brahmins for enforcing Social norms? Q5.How new jatis were grouped?

Q6.Critically examine the duties as laid down inManusmriti for the chandalas.

2

Q7.The Mahabharata is a good source to study the social value of ancient times. Prove it. Ch-4

Q1. Mention four places associated with the life of the Buddha. 2 Q2. What do you mean by Tri -ratna? 2 Q3. Into how many categories the religious sects that originated during the 6th century B.C. can be divided? 2

Q4. What do you mean by “Dharma Chakra Pravartana”? 2 Q5. Mention the various incarnatins of Vishnu according to Vaishnavism. 2 Q6. Mention the teachings of Mahatma Buddha? 4 Q7. What was the Budha Sangha? Discuss its characteristics. 4 Q8. How Buddhist text were prepared and preserved? 4

Q9. Discuss how and why Stupas were built? (4+4)=10 Ch-5 Q.1 Name any two travellers who came India during the medieval period (11th to 17th Q.2 What was the Al-Biruni’s objective to came India? 2 Q.3Do you think Al-Biruni depended only on Sanskrit texts for his information and understanding of Indian society? 2 Q.4 Name the Plants found in India which amazed Ibn-Battuta. 2 Q5.What was the more complex social reality which Bernier’s notice in the Mughal Empire? 2 Q.6.What were the “barriers” discussed by Al-Biruni that obstructed him in understanding India?2 Q7. According to Bernier, What were the evils-effects of the crown ownership of land? 4 Q8. What did Bernier write about the Sati system? 5 Q9. Who wrote ‘Kitab-ul-Hind’? Throw light in its main features? 4 Q10. Analyze the evidence for slavery provided Ibn-Battuta. 8 Ch-6

1. What does Bhakti movement mean? 2 2. Who were Alavars? 2 3. What do you mean by Sufism? 2

4. What is the importance of Murshid in Sufi ideology? 5. Describe the causes of the rise of Bhakti movement? 4

6. Describe the main Principal of Bhakti movement?4 7 Discuss the influences &importance of Bhakti movement? 4

8.What were the main Principle of Sufism? 4 9. What were Attitude of the Nayanars & Alavars sects towards the cast? 4

10. What were the similarities and differences between the be-Shari’ a and ba - Shari’a Sufi traditions?4

11. Write a note on the relationship between Sufism and orthodox Islam. 2 12. Write a short note on Amir Khusrau as a great artist and scholar. 2 Q.-9 Describe the teaching of Kabir? How does he describe the ultimate reality through the poems?

CLASS – XII WORKSHEET (MACRO-ECONOMICS) · CLASS – XII . WORKSHEET (MACRO-ECONOMICS) (MULTIPLE...

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