KENDRIYA VIDYALAYA SANGATHAN · 2018-08-29 · chapter but taught in bits that supplement the...

1

KENDRIYA VIDYALAYA SANGATHAN HYDERABAD REGION

Workshop for Mathematics PGTs

Venue: Kendriya Vidyalaya No:II, SVN, Visakhapatnam

(17th

August &18th

August 2018)

2

Deputy Commissioner

Sri D MANIVANNAN RO HYDERABAD, KVS

Assistant commissioners:

Shri. K. SASEENDRAN

Dr. D. MANJUNATH Smt. JSV LAKSHMI

Course Director

Sri MVRSSVLN SASTRI

Principal

KV Steel Plant

Venue Principal

Sri Ch. BABU RAO

Resource Persons

Sri. R.S.N. Acharyulu (PGT Math, KV2 GOLCONDA)

Sri. V. Simhadri (PGT Math, KV2, SVN, VISAKHAPATNAM)

3

PARTICIPANTS

S.No Name of the Teacher Name of the KV 1 Ms Rekha Raj BEGUMPET AFS

2 Mr S Srinivas BOLARUM

3 Mr K S R Gopalakrishna BOWENPALLY

4 Mrs M chandrika CUDDAPAH

5 Mr J Jagan Mohan ELURU

6 Mr Kishore G GACHIBOWLI

7 Mr Ajay Kumar Naik GOLCONDA No.1

8 Mr Y Srinivasa Reddy GOOTY

9 Mrs Ch Hanumayamma GUNTUR

10 Mr P Ranga Swamy HAKIMPET AFS

11 Mr K Dasharatha Ram HYDERABAD CRPF

12 Mr G Sudha Reddy HYDERABAD No.2 AFA

13 Mr T Trinadha Rao KALINGA INS

14 Mr P Srinivas KANCHANBAGH

15 Mr Bhukya Chitti Babu KHAMMAM

16 Mr R Vamana Rao KHARIM NAGAR

17 Mr Shaik Sha Vali KURNOOL

18 Mr J Satyanarayana MACHILIPATNAM

19 Mr G Veera Bhadra Rao MAHABUBNAGAR

20 Mr A Ravi Kumar NALGONDA

21 Mr G Raghu Vara Prasada Rao NAUSENABAGH No.1

22 Mr K V Govardhana Rao NAUSENABAGH No.2

23 Mr K Balaji NELLORE

24 Mr Rajendra Prasad NFC NAGAR

25 Mr M Srinivasa Rao ONGOLE

26 Mr T Veera Babu PICKET

27 Mr Y S R Prasad RAJAHMUNDRY ONGC

28 Mr N Brahmaiah RAMAGUNDAM NTPC

29 Mrs L Surya Chandra SHIVARAMPALLY NPA

30 Mr B L Narasimham SRIKAKULAM

31 Mrs G Venkata Lakshmi SRIVIJAYANAGAR No.1

32 Mrs G Sabitha STEEL PLANT

33 Mr P Rangaiah SURYALANKA AFS

34 Mr P S Sundara Rao TIRUMALAGIRI

35 Mr E V L N Vamsi Krishna TIRUPATI No.1

36 Mr V Savari Muthu UPPAL No.1

37 Mr S Sunil UPPAL No.2

38 Dr T Ravi Kishore VIJAYAWADA No.1

39 Mrs Kalavathi Gupta VISAKHAPATNAM NAD

40 Mr K Rama Krishan VIZIANAGARAM

41 Mr N V Neelachalam WALTAIR

42 Mrs Josephina Bala Raju WARANGAL

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From Director’s pen…

It is the basic duty of a Mathematics teacher is to remove the fear complex

amongst the students and make it student friendly. This needs to begin from

primary classes itself. In spite of various constraints when the child crossover the

primary/middle stage the PGT’s/being the people to show the Gateway of

Mathematics world at least introspect/identify and remove the fear complex in

class IX to the possible extent if not at least at class XI.

Now the question of achieving 100% pass and good P.I. amongst late

bloomers/mediocre one has to address the difficult topics of the student

individually and see that these difficult topics need to be kept aside whose weight

age is nearly 10/15/20 marks at board exams so that the target may be for

preparation of 80 marks with achievable target of 60/40 (having known very

well that the energies of difficult topics need to be utilized for easy topics to score

more here). Once the module is in the hands of all teachers they are advised to

split up and see that small work sheets need to be got prepared at their level and

have thorough drilling on these.

Word of caution is: These are in addition to thorough drilling of worked out

examples as notified at NCERT book of class XII.

The Board Exam Question Papers are also used for practice with marking

scheme (2015, 2016, 2017 & 2018). A well mix of new set of two Question

Papers are made with marking scheme/Blue Print for practice of these before 1st

Pre Board Examination.

Last but not least make the mathematics a joy full subject with easy success

rate but not make it difficult one in the life.

With all good wishes

MVRSSVLN SASTRY

PRINCIPAL

KV STEEL PLANT

5

KENDRIYA VIDYALAYA SANGATHAN

HYDERABAD REGION

FOR

STRATEGY FOR ASSURED PERFORMANCE AT BOARD EXAMS

2 DAY WORKSHOPFOR PGT (MATHS) : AUGUST-2018

KENDRIYA VIDYALAYA NO:2 SRI VIJAYANAGAR

TARGET

100% Pass & Better P.I

MODULE

6

FORE WORD

In this project effort is made to see the task of passing as a job simple for the student. The

areas of focus are identified with reference to the latest board Q.P pattern. The content is divided into

modules of 4 to 6 marks weightage at board exam. The questions are chosen from the previous board

exams to make the child acquainted with the expectation in the exams.

Teacher needs to identify the necessary basics and formulae/methods necessary to solve the

set of questions in each module instead of listing all the formulae/methods at once. Integration is area

of concern for them. My idea is to make the child learn only those concepts of integration which are

helpful in solving the questions of other chapters. For example: simplifying 𝑎2 − 𝑥2 𝑑𝑥 , which

is helpful in finding area under curves, integration by-parts which is helpful in solving a LDE,

𝑙𝑖𝑛𝑒𝑎𝑟

𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 dx that is helpful in solving a HDE. If child selects to learn these methods along with

finding integral as limit of sum he/she can for go with the chapter. So teaching Integration not as a

chapter but taught in bits that supplement the acquiring of necessary skills for solving the problems of

DE and Application of Integrals would be a great help for the child in overcoming the fear. The topics

such as scalar triple product, limit of sum (in summation format) help child score marks if properly

practiced. While dealing the modules chapter-wise focus may be given on the 1 and 2 mark questions

related to the topics that are being dealt.

Presenting the task in the form of small modules, making them practice the question banks

followed by testing and error analysis forms a proper diagnosis for the issue. A small formulae test is

included to make children focus on this key skill. The question modules and the tests are given in word

form for easy editing. A grand test is included to assess the overall performance.

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KENDRIYA VIDYALAYA SANGATHAN: HYDERABAD REGION

MODULE FORASSURED PERFORMANCE IN BOARD EXAMS: 2018-19

SCORING AREAS - STRATEGIC ACTION PLAN FOR SLOW LEARNERS

CLASS XII (MATHEMATICS)

S No Topic Areas identified Marks

1

Functions

Inverse Trigonometric

functions

Finding inverse of a function, Equivalence Relation

tan−1 𝑥 ± tan−1 𝑦 application

6

4/2 m

2

Matrices & Determinants

Finding inverse of matrix by elementary operations

method

Solving system of linear equations by using matrix

method

6/4

3

Differentiation

Application of Derivatives

Parametric differentiation & logarithmic

Differentiation

Mean Value Theorem, Rolle’s Theorem

Increasing and Decreasing functions.

Finding f’(x) and critical points

4

4

Integration

Integration as Limit of Sum,

Integration by partial fractions

𝐋𝐢𝐧𝐞𝐚𝐫

𝐐𝐮𝐚𝐝𝐫𝐚𝐭𝐢𝐜 dx , 𝐋𝐢𝐧𝐞𝐚𝐫 𝐪𝐮𝐚𝐝𝐫𝐚𝐭𝐢𝐜 dx

Properties of definite integrals

6/4 m

5

Application of Integrals

Area between two curves /

Line and curve, Area of a triangle 6/4 m

6

Differential Equations

Linear/Homogeneous differential Eqn.

General / Particular solution

6

7

Linear Programming

Diet problem

Manufacturing Problem

6

8

8

Vectors

Projection of a Vector, Unit vector

Dot product - Vector Product,

Scalar Triple Product

4

9

3–D Geometry

Shortest distance between two lines

Plane passing through intersection of planes,

Plane perpendicular to two planes, Image

4

10

Probability

Baye’s Theorem

Probability Distribution

4/6

Total >50M

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MODULE FORASSURED PERFORMANCE IN BOARD EXAMS 2018-19

Topics: 1. Relations &Functions2. Inverse Trigonometric functions

1. Let f be a function defined by f(x) = 9x2 +6x -5 Prove that f is not invertible. Modify,

only the co domain of ‗f‘ to make f invertible and then find its inverse.

2. Let f: N→N be a function defined by f(x) = 4x2 +12x +15 Prove that f from

N→.Range of f be invertible. Find inverse of f and also find f-1

(31) and f-1

(87)

3. Show that f : R- {-4/3} to R- { 4/3} given by f(x) = 4𝑥+3

3𝑥+4 . Show that f is bijective.

Find the inverse of f and hence find f-1

(0) and x such that f-1

(x) = 2.

4. If the function f : R→R be defined by f (x) = 2x + 3and g : R→R be defined by g(x) = x3+ 5,

then find f o g and show that f o g is invertible. Hence find ( f o g) -1

(9).

5. Let A= {x ∈ Z:0≤ x≤ 12}. Show that R = { (a, b) ∈ A, │a-b│is divisible by 4} is an

equivalence relation.

6. Show that tan-1(

1+𝑥 − 1−𝑥

1+𝑥 + 1−𝑥) =

𝝅

𝟒 -

1

2 cos

-1x

7. Simplify cot-1 1

x2−1 for x< -1

8. Find the real solutions of the equation tan–1

1−𝑥

1+𝑥 =

1

2 tan

–1(x )

9. If tan-1

(𝑥−3

𝑥−4) + tan

-1 (𝑥+3

𝑥+4) =

𝜋

4 find the value of x.

10. What is the principal value of tan–1

( tan 2π

3 )

11. Does the following trigonometric equation have any solutions? If Yes, obtain the solution(s):

tan-1

(x+1

x−1) + tan

-1 (

x−1

x) = - tan

-1 (7)

12. If 4sin-1

(x) + cos-1

(x) = 𝜋 then find the value of x.

13. If sin-1

(x) + sin-1

(1-x) = cos-1

(x) find x.

14. Prove that 2 sin-1

( 3

5 ) – tan

-1 (

17

31) =

𝜋

4

10



Matrices and Determinants

1. of l A l

2.

3. Find the value of x ,y, and z if 𝑥 + 𝑦 25 + 𝑧 𝑥𝑦

= 6 25 8

.

4. Evaluate 𝑥 𝑥 + 1

𝑥 − 1 𝑥

5. Find the value of x if

2 45 1

= 2𝑥 46 𝑥

6.

7. Given A= 2 −3

−4 7 compute A

-1 and show that 2A

-1 =9I—A

8. Find the value of x and y if 2 1 30 𝑥

+ 𝑦 01 2

= 5 61 8

9. If 𝐴 = 3 −24 −2

and 𝐼 = 1 00 1

find k so that 𝐴2 = 𝑘𝐴 − 2𝐼.

10. Using matrices find the area of a triangle whose vertices are (3, 8), (-4, 2) and (5, 1).

11.

12.

13.

11

14. Using properties of determinants prove that

𝑥 𝑥2 𝑦𝑧

𝑦 𝑦2 𝑧𝑥

𝑧 𝑧2 𝑥𝑦

=(x-y) (y-z) (z-x) (xy +yz +zx)

15. Find the equations of the line joining (1, 2) and (3, 6) using determinants.

16. Using elementary row transformations find the inverse of the matrix

A= 1 3 −2

−3 0 −52 5 0

17. Using properties of determinants prove that

𝑥 𝑥2 𝑦𝑧

𝑦 𝑦2 𝑧𝑥

𝑧 𝑧2 𝑥𝑦

=(x-y) (y-z) (z-x) (xy +yz +zx)

18. Prove that 1 𝑎 𝑎2

1 𝑏 𝑏2

1 𝑐 𝑐2

= (a-b) (b-c) (c-a)

19. Find A2-5A+6I if A=

2 0 12 1 31 −1 0

20. Find the matrix X so that 𝑋 1 2 34 5 6

= −7 −8 −92 4 6

6 Mark Questions

21.

22.

12

23.

24. If A== 3 3 24 2 0

𝑎𝑛𝑑 𝐵 = 2 −1 21 2 4

verify that

(I) (A-1

)1 =A (ii) (A+B)

1= A

I+B

1 (iii) (KB)

I= KB

I where K is any constant

25. Solve system of linear equations using Matrix method x – y + z = 4, 2x + y- 3z=0

and x + y + z = 2

13



Topic: Continuity - Differentiation

1.) Discuss the continuity of the function f defined by f( x) = 𝑥 + 2, 𝑖𝑓𝑥 ≤ 1𝑥 − 2, 𝑖𝑓𝑥 > 1.

2. ). Find the values of k 𝑓 𝑥 =

𝑘 cos 𝑥

𝜋−2𝑥 , 𝑖𝑓𝑥 ≠

𝜋

2

3, 𝑖𝑓𝑥 =𝜋

2

is continuous at x = 𝜋

2

3 .)Find the values of a and b such that the function defined by

f ( x) = 𝑥2 + 𝑎𝑥 + 𝑏, 𝑖𝑓 0 ≤ 𝑥 < 2

3𝑥 + 2, 𝑖𝑓 2 ≤ 𝑥 ≤ 42𝑎𝑥 + 5𝑏, 𝑖𝑓 4 < 𝑥 ≤ 8 .

is a continuous function.

4.) Find all points of discontinuity of f, where 𝑓 𝑥 = sin 𝑥

𝑥 𝑖𝑓 𝑥 < 0

𝑥 + 1 𝑖𝑓 ≥ 0

5.) Find all points of discontinuity off, where 𝑓 𝑥 =

𝑥 + 3 , 𝑖𝑓 𝑥 ≤ −3 −2𝑥 , 𝑖𝑓 − 3 < 𝑥 < 3

6𝑥 + 2 , 𝑖𝑓 𝑥 ≥ 3.

6.) Find all points of discontinuity of f, where 𝑓 𝑥 =

𝑥

𝑥 , 𝑖𝑓 𝑥 < 0

−1 , 𝑖𝑓 𝑥 ≥ 0

7) Find the values of a and b such that the function defined by

5, 2

, 2 10

21, 10

x

f x ax b x

x

8) Differentiate (usinglogarithms)

(i) xx +(sinx)x

(ii) cos( )x xx +

2 1

1

x

x

(iii) ( cos x)

y = ( cos y )

x

(iv) yx = e x – y

, Show that

2

log

log( )

dy x

dx xe (v)

1/(cos ) (sin )x xx x

9.) Show that the function defined by g (x) = x – [x] is discontinuous at all integral

points. Here [x] denotes the greatest integer less than or equal to x.

10) Find 𝑑𝑦

𝑑𝑥 of , x = 2at

2, y = at

4.

14

11) Find 𝑑𝑦

𝑑𝑥 of x = cos𝜃– cos 2𝜃, y = sin 𝜃– sin 2𝜃.

12) Find 𝑑𝑦

𝑑𝑥 ofx = a (– sin ), y = a (1 + cos)

13) If y = (tan–1 x)2, show that (x

2 + 1)

2y2 + 2x (x

2 + 1) y1 = 2.

14) Verify Mean Value Theorem for the function f (x) = x2 in the interval [2, 4].

15) Verify Rolle‘s theorem for the function f (x) = x2 + 2x – 8, x ∈[– 4, 2].

15



Topic: Application of Derivatives

1. Show that the function f(x) = x3-3x

2+6x-100 is increasing on R.

2. The volume of a sphere is increasing at the rate of 8 cm3/s. How fast is the surface area increasing

when the length of an edge is 12 cm? Ans: 8

3 cc/sec

3. A balloon, which always remains spherical has a variable diameter 3

2 (2x+1). Find the rate of change

of its volume with respect to x. Ans: 27

8π (2x +1)

2

4. The total cost C(x) in Rupees associated with the product of x units of an item is given by

C(x) = 0.007x3 – 0.003x

2 + 15x + 4000.

Find the marginal cost when 17 units are produced? Ans: Rs. 20,967.

5. Find the intervals in which the function f given by f(x) = sin x + cos x, 0 ≤ x ≤ 2π is strictly

increasing or strictly decreasing.

Ans: [0,𝜋

4) and (

5𝜋

4 , 2π] is strictly increasing, (

𝜋

4,

5𝜋

4) strictly decreasing.

6. Show that y = log (1+x) - 2𝑥

2+𝑥 , x > -1 is an increasing function of x throughout its domain.

7. Find the point at which the tangent to the curve y = 4𝑥 − 3 -1 has its slope 2

3 . Ans: (3,2)

8. Find the equation of the tangent and normal to the curve 𝑥2/3 + 𝑦2/3 = 2 at (1, 1).

Ans:Equation of the tangent is y+x -2 = 0 and Equation of the normal is y –x =0.

9. Use differential to approximate 36.6 Ans: 6.05

10. Find the approximate value of f(3.02), where f(x) = 3x2 + 5x + 3. Ans: 45.46

11. Find the absolute maximum and minimum values of a function f given by

f(x) = 2x3 – 15x

2 + 36x +1 on the interval (1, 5).

Ans: absolute maxima 56 at x =5 and absolute minima 24 at x=1.

12. A rectangular sheet of tin 45 cm by 24 cm is to be made into the box without top by cutting of

square from each corner and folding of the flaps. What should be the side of the square to be

cutoff so that the volume of the box is maximum? Ans: 3cm

16



Topics: Integration

1. 𝑠𝑖𝑛5𝜋

2−𝜋

2

𝑥 𝑑𝑥

2. 𝑑𝑥

1+𝑥2

1

0

3. 𝑑𝑥

𝑥+𝑥𝑙𝑜𝑔 𝑥

4. 3𝑥3

2 𝑑𝑥

5.

6.

7. 𝐼𝑓 1

4+𝑥2

𝑎

0 𝑑𝑥=𝜋/2 then find a

8. 𝑥+𝐶𝑂𝑆 6𝑋

3𝑋2+sin 6𝑥 𝑑𝑥

9. If (3𝑥2 + 5𝑥 + 𝑘)1

0 𝑑𝑥 = 0 𝑡𝑕𝑒𝑛 𝑓𝑖𝑛𝑑 𝑘

10. 𝑒 cos 𝑥

𝑒 cos 𝑥 +𝑒− cos 𝑥

𝜋

0 𝑑𝑥

11. 𝑑𝑥

𝑠𝑖𝑛2𝑥𝑐𝑜𝑠 2𝑥

12. 𝑥 + 2 5

−5 𝑑𝑥

13. Find 𝑥𝑐𝑜𝑠𝑥 𝑑𝑥

14. Find 𝑥2𝑒𝑥 𝑑𝑥

15.

17

16.

17. Evaluate as limit of sum

18.

19.

20.

21. (3 sin 𝜃−2)𝑐𝑜𝑠𝜃

5−𝑐𝑜𝑠 2𝜃−4𝑠𝑖𝑛𝜃d𝜃

22. (𝑥−4)𝑒𝑥𝑑𝑥

(𝑥−2)3

23. 𝑥 sin 𝑥

1+𝑐𝑜𝑠 2𝑥

𝜋

0 𝑑𝑥

24. 𝑥

𝑎2𝑐𝑜𝑠 2𝑥+𝑏2𝑠𝑖𝑛2𝑥

𝜋

0 𝑑𝑥

25. log 1 + 𝑡𝑎𝑛𝜃 𝑑𝜃𝜋

40

18



Topics: Area under Curves

1.

2.

3.

4.

5.

6. Find the area bounded by the curves (x-1)2 + y

2 = 1 and x

2 + y

2 = 1

7.

8.

9.

10. Find the area bounded by the curves {(x, y): y ≥ x2 and y = | x|.

11. Prove that the curves y2

= 4x and x2 = 4y divide the area of the square bounded by x=0, x=4,

y=4, and y=0 into three equal parts.

12. Using integration find the area of the region in the first quadrant enclosed by the x axis , the

line y = x and the circle x2 + y

2 = 32.

19



Topics: Differential Equations

1. Obtain the differential equation of the family circles

1) passing through the points (a,0) and (-a,0) 2) in the and touching the coordinate axis

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. Find the particular solution of the differential equation 𝑑𝑦

𝑑𝑥 + 2y tanx = sinx given that y=0

when x= 𝜋

3

13. Find the particular solution of the differential equation 𝑑𝑦

𝑑𝑥 = 1+ x + y + xy given that y=0

when x=1

14. Solve the differential equation (1+x2 )

𝑑𝑦

𝑑𝑥 + y = 𝑒𝑡𝑎𝑛 −1𝑥

15. Find the particular solution of the DE ex

tany dx + (2- ex) sec

2ydy = 0 given that y =

𝜋

4 when x=0

20



Topics: Vectors and 3-D

1. Write the vector equation of the plane: 3𝑥 − 4𝑦 + 2𝑧 − 8 = 0

2. Find the equation of the line passing through the points (-3, 1, 2) and (1, -2, -2).

3. Find the value of x and y so that the vectors 2𝑖 + 3𝑗 − 𝑦𝑘 and 𝑥𝑖 + 3𝑗 + 5𝑘 are equal.

4. Find 𝑎 𝑋𝑏 if 𝑎 = 𝑖 + 3𝑗 + 5𝑘 and 𝑏 = 𝑖 + 𝑗 + 𝑘

5. Scalar product of vectors 𝑎 and 𝑏 is 1

2 and angle between them is 60 degrees.

Find magnitude of the vectors 𝑎 and 𝑏 when they are equal.

6. Find the vector and Cartesian equation of the line passing through two points −1,0,2 and 3,4,6

7. If 𝑎 + 𝑏 + 𝑐 = 0 find 𝑎 . 𝑏 + 𝑏 . 𝑐 + 𝑐 . 𝑎 when 𝑎 = 1, 𝑏 = 4, 𝑐 = 2

8. Find 𝑎 𝑎𝑛𝑑 𝑏 if 𝑎 + 𝑏 . 𝑎 − 𝑏 = 8 and 𝑎 = 8 𝑏

9. Find the direction cosine of the line joining the points (2, -1, 4) and (-1, 3, 3).

10.

11.

12.

13.

14.

15. Find the angle between two lines 𝑟 = 3𝑖 + 2𝑗 − 4𝑘 + 𝜗(𝑖 + 2𝑗 + 2𝑘 ) and

𝑟 = 5𝑖 − 2𝑗 + 𝜇(3𝑖 + 2𝑗 + 6𝑘 ).

21

16. Find the angle between two lines 𝑥−2

2=

𝑦−1

7=

𝑧+3

−3 and

𝑥+2

−1=

𝑦−4

2=

𝑧−5

4

17. Find the equation of the plane passing through three points:

2, 5, −3 ; −2, −3, 5 𝑎𝑛𝑑 5, 3, −3

18.

19.

20.

21.

22. Find the equation of the plane through the intersection of the planes 𝑟. 𝑖 + 𝑗 + 𝑘 = 6 and

𝑟. 2𝑖 + 3𝑗 + 4𝑘 = −5 and the point 1,1,1

23.

24.

25.

22

26.

27.

28.

29.

30.

23



Topic: Linear Programming Problem

1.

2.

3.

4.

5.

24

6. In a midday meal programme, an NGO wants to provide vitamin rich diet to the students of

an MCD school. The dietician of NGO wishes to mix two types of food in such a way that

vitamin contents of the mixture at least contains 8 units of vitamin A and 10 units of vitamin C.

Food 1 contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C.Food 2 contains

1unit per kg of vitamin A and 2 unit per kg of vitamin C. It costs Rs 50 per kg to purchase food

1 and costs Rs 70 per kg to purchase food 2 .Formulate the problem as LPP and solve it

graphically for the minimum cost of such a mixture.

7. A manufacturer produces two products A and B .Both the products are processed on two

machines. The available capacity of first machine is 12 hours and that of second machine is 9

hours per day. Each unit of product A requires 3 hours on both machines and each unit of

product B requires 2 hours on first machine and 1 hour on second machine. Each unit of

product A is sold at profit Rs 7 and that of B at the profit of rs 4.Find the production level per

day for maximum profit graphically.

8.

9. A dietitian wishes to mix two types of foods in such a way that the vitamin contents of mixture

contains at least 8 units of vitamin A and 10 units of vitamin C. Food 1 contains 2 units/ kg. of

vitamin A and 1 unit/kg. of vitamin C while food II contains 1 unit/kg. of vitamin A and 2 units /kg.

of vitamin C. It cost Rs. 5 per kg. to purchase food I and Rs. 7/- per kg. to purchase food II find the

minimum cost of such a mixture. Formulate above as LPP and solve graphically.

10.

25



Topic: Probability

1. Two cards are drawn successively with replacement, from a well shuffled deck of 52 cards.

Find the probability distribution of number of aces.

2. In answering a question on a multiple choice test, a student either knows the answer or

guesses. Let 3

5 be the probability that he knows the answer and

2

5 be the probability that he

guesses. If a student who guesses the answer will be correct with probability1

3, what is the

probability that the student knows the answer, given that he answered it correctly?

3. Of the students in a college, it is known that 60% reside in hostel and 40% do not reside in

hostel. Previous year result report that 30% of students residing in hostel attain A grade and 20

% of one‘s not residing in hostel attain A grade in their annual examination. At the end of the

year, one student is chosen at random from the college and he has an A grade. What is the

probability that selected student is a hosteller.

4. Three boxes contain 6 red, 4 black; 4 red, 6 black and 5 red, 5 black balls respectively. One of

the boxes is selected at random and a ball is drawn from it. If the ball drawn is red, find the

probability that it has come from first box.

5. A factory has two machines A and B. Past record shows that machine A produced 60% of

items of output and machine B produced 40% of items. Further, 2% of items, produced by

machine A and 1% produced by machine B were defective. One item is chosen at random from

total output and this is found to be defective. What is the probability that it was produced by

machine B?

6. Find the mean number of heads in three tosses of a coin.

7. A random variable X has following probability distributions.

X 0 1 2 3 4 5 6 7

P(X) 0 k 2k 2k 3k k2 2k

2 7k

2 + k

Find (i) k (ii) P(X<3) (iii) P(X> 6) (iv) P(0<X<3)

26



Questions for Assessment

Topics: 1. Relations and Functions 2. Inverse trigonometric functions

1. Let f : N → N be a function defined by f(x) = 4x2 +12x +15 Prove that f from N N →.Range of

f be invertible. Find its inverse of f and also find f-1

(31) and f-1

(87)

2. If the function f : R → R be defined by f (x) = 2x + 3and g : R → R be defined by

g(x) =x3+ 5, then find f ogand show that f ogis invertible.

3. If tan-1

(𝑥−3

𝑥−4) + tan

-1 (𝑥+3

𝑥+4) =

𝜋


4. If 4sin-1

(x) + cos-1

(x) =𝜋 then find the value of x.

5. Find the value of cos-1( 3

2) + sin-1(−

1

2) +cot-1( 3




Topics : Continuity -Differentiation

1. Find the relation between a and b so that 1, 3

3, 3

ax xf x

bx x

is continues at x = 3.

2. Find the values of m𝑓 𝑥 =

𝑚 cos 𝑥

𝜋−2𝑥 , 𝑖𝑓 𝑥 ≠

𝜋

2

3, 𝑖𝑓 𝑥 =𝜋

2

is continuous at x = 𝜋

2.

3. Differentiate: (x)cosx +(sinx)tanx.

1. Verify Mean Value Theorem, if f (x) = x2 – 4x – 3 in the interval [a, b], where a = 1 and b = 4.

2. Verify Rolle‘s theorem for the function y = x2 + 2, a = – 2 and b = 2.

27


MODULE FOR ASSURED PERFORMANCE IN BOARD EXAMS 2018-19


Topics: MATRICES & DETERMINANTS

1.

ofdetA. (1m)

2. Find the value of x if

2 45 1

= 2𝑥 46 𝑥

(1m)

3. Given A= 2 −3

−4 7 compute A

-1 and show that 2A

-1 =9I—A (2m)

4.

(2m)

5. Using properties of determinants prove that (4m)

𝑥 𝑥2 𝑦𝑧

𝑦 𝑦2 𝑧𝑥

𝑧 𝑧2 𝑥𝑦

=(x-y) (y-z) (z-x) (xy +yz +zx)

6. Find the matrix X so that 𝑋 1 2 34 5 6

= −7 −8 −92 4 6

(4m)

7. Solve system of linear equation using Matrix method x – y + z=4, 2x + y- 3z=0

and x+y+ z=2 (6m)

28




Topic : Application of derivatives

1. The total cost C(x) in Rupees associated with the product of x units of an item is given by

C(x) = 0.9x3 – 0.03x2- 11x + 2000.

Find the marginal cost when 17 units are produced?

2. Use differential to approximate 49.5

3. Find the intervals in which the function f given by f(x) = sin x -cos x, 0 ≤ x ≤ 2π

is strictly increasing or strictly decreasing. .

4. Show that y = log (1+x) - 2𝑥

2+𝑥 , x > -1 is an increasing function of x throughout its domain.

5. Find the point at which the tangent to the curve y = 3𝑥 − 4-1 has its slope 2

3

6. Find the equation of the tangent and normal to the curve 𝑥2/3 + 𝑦2/3 = 2 at (-1, 2).




Topics: Area under Curves

1.

2. Find the area bounded by the curves {(x, y): y ≥ x2 and y = | x|.

3. Using integration find the area of the region in the first quadrant enclosed by the x axis , the

line y = x and the circle x2 + y

2 = 32.

4.

29




Topic : Integration

Instructions: Questions 1 to 4 of 1 mark each, questions 5to 8 of 2 marks each

questions 9 and 10 of 6 marks each

1. 𝑠𝑒𝑐2 𝑑𝑥 =

2. (2𝑥)2 − 𝑎2 dx

3 𝑥2 + 42dx

4 𝑥2 + 𝑥 +1

𝑥𝑑𝑥

5 𝑥 + 2 5

−5 𝑑𝑥

6. Find 𝑥𝑐𝑜𝑠𝑥 𝑑𝑥

7 If (3𝑥2 + 5𝑥 + 𝑘)1

0 𝑑𝑥 = 0 𝑡𝑕𝑒𝑛 𝑓𝑖𝑛𝑑 𝑘

8 𝑒 cos 𝑥

𝑒 cos 𝑥 +𝑒− cos 𝑥

𝜋

0 𝑑𝑥

9.

10.

30



Questions for Assessment:

Topic: Differential Equations

1.

2.

3.

4.

5.




Subject Mathematics Topics: 3-D & Vectors

Max. Marks: 20 Time : 45 min

1. Find the equation of the line passing through the points (-3, 1, 2) and (1, -2, -2). (1M)

2. Find 𝑎 𝑋𝑏 if 𝑎 = 𝑖 + 3𝑗 + 5𝑘 and 𝑏 = 𝑖 + 𝑗 + 𝑘 (1M)

3. Find the vector and Cartesian equation of the line passing through two points −1,0,2 and

3,4,6 (2M)

4. If 𝑎 + 𝑏 + 𝑐 = 0 find 𝑎 . 𝑏 + 𝑏 . 𝑐 + 𝑐 . 𝑎 when 𝑎 = 1, 𝑏 = 4, 𝑐 = 2 (2M)

5.

4M

6. Find the angle between two lines 𝑟 = 3𝑖 + 2𝑗 − 4𝑘 + 𝜗(𝑖 + 2𝑗 + 2𝑘 ) and

𝑟 = 5𝑖 − 2𝑗 + 𝜇(3𝑖 + 2𝑗 + 6𝑘 ). 4M

7. Find the equation of the plane through the intersection of the planes 𝑟. 𝑖 + 𝑗 + 𝑘 = 6 and

𝑟. 2𝑖 + 3𝑗 + 4𝑘 = −5 and the point 1,1,1 6M

31




Topic: Linear Programming

1. Solve the following linear programming problem graphically (2 marks)

Maximise Z=3x+4y , Subject to the constraints x+y≤4 x≥0 . y≥0

2. A manufacturer company makes two models A and B . Each piece of model A requires 9 labour

hours for fabricating and 1 labour hour for finishing. . Each piece of model B requires 12

labour hours for fabricating and 3 labour hour for finishing. For fabricating and finishing

maximum hours available are 180 and 30 respectively. The company makes profit of Rs 8000

on each piece of model A and Rs 12000 on each piece of model B. How many pieces of

model A and model B should be manufactured to get maximum profit. (6 marks)

3. A dietitian wishes to mix two types of foods in such a way that the vitamin contents of mixture

contains at least 8 units of vitamin A and 11 units of vitamin C. Food 1 contains 3units/ kg. of

vitamin A and 5 unit/kg. of vitamin C while food II contains 4 unit/kg. of vitamin A and 2 units /kg.

of vitamin C. It cost Rs. 60 per kg. to purchase food I and Rs.80- per kg. to purchase food II find the

minimum cost of such a mixture. Formulate above as LPP and solve graphically (6 marks)




Topics: Probability

1. An insurance company insured 2000 scooter drivers, 4000 car drivers, 6000 truck drivers. The probabilities of their meeting with an accident are 0.01, 0.03 and 0.15, respectively. One of the insured people meets with an accident. What is the probability that he is a scooter driver?

2. From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of number of defective bulbs

3. There are three coins. One is a two headed coin another is a biased coin that comes up heads 75% of the time and third is unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is a probability that it was two headed coin?

4. A random variable X has following probability distributions.

X 0 1 2 3 4 5 6 7

P(X) 0 k 2k 2k 3k k2 2k2 7k2+ k

32

Find (i) k (ii) P(X<3) (iii) P(X> 6) (iv) P(0<X<3) .

33



Additional Assessment on Integration Formulae

1. 𝑠𝑒𝑐2 𝑑𝑥 = 11. d

dx (sec x) =

2. 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 𝑑𝑥 12. 1

𝑎2 −𝑥2 dx =

3. 𝑑

𝑑𝑥 (u. v) 13. (2𝑥)2 − 𝑎2 dx

4. 𝑑𝑥

𝑎2−𝑥2 = 14.

𝑑𝑥

𝑥2−𝑎2

5. 1

𝑥2 −𝑎2 dx 15.

𝑑𝑥

𝑥 𝑥2−1 =

6. 1

𝑥2 + 𝑎2 dx 16. 𝑒𝑥 [ 𝑓 𝑥 + 𝑓′ 𝑥 ] =

7. 𝑓 𝑥 . 𝑔(𝑥) 𝑑𝑥 17. 𝑡𝑎𝑛𝑥 𝑑𝑥

8. 𝑓 ′(𝑥)

𝑓(𝑥)𝑑𝑥 18.

1

𝑥𝑑𝑥

9. 𝑥2 + 42 dx 19. 𝑥2 + 𝑥 +1

𝑥𝑑𝑥

10. 32 + 𝑥2 dx 20. 𝑓 ′(𝑥)

𝑓(𝑥)𝑑𝑥

34



Questions for Assessment of improvement in the performance

Subject: Mathematics Topics: Grand Test

Solve the following Questions:- Questions 1 to 5 carries 4m; Questions 6 to 10 carries 6m

1.

2. If tan-1

(𝑥−3

𝑥−4) + tan

-1 (𝑥+3

𝑥+4) =

𝜋


3.Find the intervals in which the function f(x) = 3

2x

4-4x

3-45x

2+51 is increasing anddecreasing

4. Evaluate 𝑥 + 𝑒2𝑥 𝑑𝑥4

0as limit of a sum

5. Let f : R- {-4/3} to R given by f(x) = 4𝑥+3

3𝑥+4 . Show that f is one -one. Check whether f is

an on-to function. Hence find the inverse of f in Range of f to R- {-4/3}.

6. A Manufacturing company makes two models A and B of a product. Each piece of model A

requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of model B

requires 12 labour hours for fabricating and 3 labour hour for finishing. For Fabricating and

finishing, the maximum labour hours available are 180 and 30 respectively. The company

makes a profit of Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B.

How many pieces of model A and Model B should be manufactured per week to realise a

maximum profit ? What is the maximum profit per week?

7. Evaluate 5𝑥+3 𝑑𝑥

𝑥2+4𝑥+10

8.

9. Find the mean and variance of number of heads in three tosses of a coin.

10.

35

ALL THE BEST

HOTS

Chapterwise Higher Order Thinking

Questions for top scorers.

-By Group 1

36

HOTS questions for Class XII MATHEMATICS

1. Relations and Functions

1. Let A = {–1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g: A→ B be functions defined by

𝑓 𝑥 = 𝑥2– 𝑥, 𝑥 ∈ 𝐴and 𝑔(𝑥) = 2 |𝑥 − 1

2 | – 1, 𝑥 ∈ 𝐴. Find 𝑓𝑜𝑔(𝑥), 𝑔𝑜𝑓 (𝑥) and hence

show that 𝑓𝑜𝑔 = 𝑔𝑜𝑓.

2. If the function f : R → R be defined by f(x) = 2𝑥 – 3 and g : R → R by g(x) = 𝑥3 + 5, then

find the value of 𝑓𝑜𝑔 –1(𝑥).

3. Let A= N×N. Let * be a binary operation on A defined by (𝑎, 𝑏) ∗ (𝑐, 𝑑) = (𝑎𝑑 +

𝑏𝑐, 𝑏𝑑)a,b,c,dϵN. Show that (i)* is commutative (ii)* is associative and (iii) identity element

w.r.t.* does not exist.

4. Draw the graph of the function 𝑓 𝑥 = 𝑥2on R and Show that it is not invertible. Restrict its

domain suitably so that 𝑓−1may exist. Also find 𝑓−1and draw its graph.

5. Show that the relation ― congruence modulo 2‖ on the set Z is an equivalence relation. Also

find the equivalence class of 1. ( Ans: Equivalence class of 1 is ={……-3,-1,1,3,5,7,9,……}

2. Inverse Trigonometric Functions

1. If a1,a2,a3,....,an is in an arithmetic progression with common difference d, then evaluate the

following expression

𝑡𝑎𝑛 tan−1 𝑑

1+𝑎1𝑎2 + tan−1

𝑑

1+𝑎2𝑎3 + tan−1

𝑑

1+𝑎3𝑎4 + ⋯ + tan−1

𝑑

1+𝑎𝑛−1𝑎𝑛 .

(𝐻𝑖𝑛𝑡: write d as 𝑎2 − 𝑎1 = 𝑎3 − 𝑎2 = 𝑎4 − 𝑎3 = ⋯… . . )

2. Show that 2 tan−1 𝑡𝑎𝑛𝛼

2. 𝑡𝑎𝑛

𝜋

4−

𝛽

2 = tan−1

𝑠𝑖𝑛𝛼 .𝑐𝑜𝑠𝛽

𝑐𝑜𝑠𝛼 +𝑠𝑖𝑛𝛽

3. Prove that 2 tan−1 𝑎−𝑏

𝑎+𝑏𝑡𝑎𝑛

𝜃

2 = cos−1

𝑎𝑐𝑜𝑠𝜃 +𝑏

𝑎+𝑏𝑐𝑜𝑠𝜃 .

4. If sin−1 𝑥 + sin−1 𝑦 + sin−1 𝑧 = 𝜋, prove that 𝑥. 1 − 𝑥2 + 𝑦. 1 − 𝑦2 + 𝑧. 1 − 𝑧2 = 2𝑥𝑦𝑧.

5. If tan−1 𝑥 2 + cot−1 𝑥 2 =5𝜋2

8, then find 𝑥. ( Ans: -1)

6. If tan−1 𝑥 + tan−1 𝑦 + tan−1 𝑧 =𝜋

2, prove that 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥 = 1.

7. Prove that cos−1 𝑥 + cos−1 𝑥

2+

3−𝑥2

2 =

𝜋

3.

8. If cos−1 𝑥

2 + cos−1

𝑦

3 = 𝜃, then show that 9𝑥2 − 12𝑥𝑦𝑐𝑜𝑠𝜃 + 4𝑦2 = 36 sin2 𝜃.

37

3&4.Matrices& Determinants

1) If the matrix 0 𝑎 32 𝑏 −1𝑐 1 0

is a skew symmetric matrix, then find the values of a,b and c.

(Answer : 𝑎 = −2, 𝑏 = 0 𝑎𝑛𝑑 𝑐 = −3)

2) If 𝐴 = 0 −𝑥𝑥 0

, 𝐵 = 0 11 0

and 𝑥2= −1,then show that (A + B)2 = A

2 + B

2

3) To promote the making toilets for women an organization tried to generate awareness through

I) house calls II) letters and III) announcements. The cost for each mode per attempt is given

below I) Rs 50,II) Rs.20 III) Rs. 40.The number of attempts made in three villages X,Y and Z

are given below.

I II III

X 400 300 100

Y 300 250 75

Z 500 400 150

Find the total cost incurred by the organization for the three villages separately usingmatrices.

(Answer : X = 30000,Y = 23000,Z = 39000)

4) Find the maximum value of ∆ = 1 1 11 1 + 𝑠𝑖𝑛𝜃 1

1 + 𝑐𝑜𝑠𝜃 1 1 for all 𝜃𝜖𝑅

(Answer :Max. ∆ = ½)

5) If 𝑥 , 𝑦 and 𝑧 are not all zero such that 𝑎𝑥 + 𝑦 + 𝑧 = 0, 𝑥 + 𝑏𝑦 + 𝑧 = 0, 𝑥 + 𝑦 + 𝑐𝑧 = 0

then find the value of 1 – 𝑎 −1

+ 1 − 𝑏 −1 + 1 − 𝑐 −1. (Answer : 1)

6) If 𝑓 𝑥 = 𝑎 −1 0𝑎𝑥 𝑎 −1𝑎𝑥2 𝑎𝑥 𝑎

, using properties of determinants find the value of 𝑓(2𝑥) – 𝑓(𝑥).

(Answer :𝑎𝑥 (3𝑥 + 2𝑎).

7) The sum of three numbers is 6. If we multiply third number by 3 and add second number to it

we get 11.By adding first and third numbers we get double of the second number. Represent it

algebraically and find the numbers using matrix method. (Answer : 1,2,3)

8) A sum of saving amount Rs 10000 is put into three investments at the rate of 10%,12% and

15% per annum.The combined income of first and second investments is Rs. 190 less of the

income of third investment.If the combined income is Rs.1310 then find the investment in each

using matrices. (Answer ;2000,3000,5000)

38

9) If p, q ,r are not in G.P. and

1𝑞

𝑝𝛼 +

𝑞

𝑝

1𝑟

𝑝𝛼 +

𝑟

𝑝

𝑝𝛼 + 𝑞 𝑞𝛼 + 𝑟 0

= 0 ,

show that𝑝𝛼2 + 2𝑞𝛼 + 𝑟 = 0

5. Continuity and Differentiability

1. If f(x) = 2𝑐𝑜𝑠𝑥 − 1

𝑐𝑜𝑡𝑥 −1 , 𝑥 ≠

𝜋

4 , find the value of 𝑓

𝜋

4 so that 𝑓(𝑥) becomes continuous at 𝑥 =

𝜋

4.

2. Show that the function f given by f 𝑥 = e

1𝑥 − 1

e1

𝑥 + 1, 𝑥 ≠ 0

0, 𝑥 = 0

is discontinuous at 𝑥= 0.

3. Prove that the function f defined by 𝑓 𝑥 =

𝑥

𝑥 + 2𝑥2 , 𝑥 ≠ 0

𝑘, 𝑥 = 0 remains discontinuous at x = 0,

regardless the choice of ‗k‘.

4. If 𝑦 = 𝑥 + 𝑥2 + 𝑎2 𝑛

, then prove that 𝑑𝑦

𝑑𝑥=

𝑛𝑦

𝑥2+𝑎2

5. If 𝑦 = sin−1 𝑥2 1 − 𝑥2 + 𝑥 1 − 𝑥4 , then prove that 𝑑𝑦

𝑑𝑥=

2𝑥

1−𝑥4+

1

1−𝑥2.

6. If 𝑦 1 − 𝑥2 + 𝑥 1 − 𝑦2 = 1 , prove that 𝑑𝑦

𝑑𝑥= −

1−𝑦2

1−𝑥2.

7. If 𝑦 𝑥2 + 1 = log 𝑥2 + 1 − 𝑥 , show that 𝑥2 + 1 𝑑𝑦

𝑑𝑥+ 𝑥𝑦 + 1 = 0

8. Differentiate w.r.to 𝑥: 𝑦 = sin−1 𝑥 + sin−1 1 − 𝑥2, 0 < x < 1

9. Differentiate w.r.to 𝑥: 𝑦 = sec−1 𝑥+1

𝑥−1 + sin−1

𝑥 − 1

𝑥 + 1

10. Differentiate w.r.to 𝑥: y = 𝑥 +1

𝑥

𝑥

+ 𝑥 1+1

𝑥

11. Find 𝑑𝑦

𝑑𝑥 if 𝑥 =

1+𝑙𝑜𝑔𝑡

𝑡2 & 𝑦 =3+2𝑙𝑜𝑔𝑡

𝑡

12. If 𝑥 = 𝑒𝑐𝑜𝑠2𝑡 and 𝑦 = 𝑒𝑠𝑖𝑛2𝑡 , prove that 𝑑𝑦

𝑑𝑥=

−𝑦𝑙𝑜𝑔𝑥

𝑥𝑙𝑜𝑔𝑦

6. Applications of Derivatives

1. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and

semi vertical angle 30° is4

81𝜋𝑕3

2. Find the angle between the parabolas 𝑦2 = 4𝑎𝑥 and 𝑥2 = 4𝑏𝑦 at their point of intersection other

than the origin. (Ans 𝜃 = tan−1 3𝑎

13 𝑏

13

2 𝑎23+ 𝑏

23

)

39

3. Show that the volume of greatest cylinder which can be inscribed in a cone of height ‗h‘ and semi

vertical angle 𝛼 is 4

27𝜋𝑕3 tan2 𝛼

4. Show that the right circular cone of least curved surface area and given volume is 2 times the

radius of the base.

5. A point on the hypotenuse of a right angled triangle is at distances 𝑎 and 𝑏 from the sides, show

that the length of the hypotenuse is at least 𝑎2

3 + 𝑏2

3

3

2.

6. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex

lowermost. Its semi-vertical angle is tan−1 0.5 . Water is poured into it at a constant rate of 5

cubic m per hr. Find the rate at which the level of the water is rising at the instant when the depth

of water in the tank is 4 m.

7. A car starts from a point P at time t = 0 seconds and stops at point Q. The distance x, in meters,

covered by it, in t seconds is given by = 𝑡2 2 −𝑡

3 . Find the time taken by it to reach Q and also

find distance between P and Q.

8. Find the co-ordinates of the point on the curve 𝑥 + 𝑦 = 4 at which tangent is equally inclined

to the axes.

9. Sand is pouring from a pipe at the rate of 12cm3

/sec.The falling sand forms a cone on the ground

in such a way that the height of the cone is always one-sixth of the radius of the base.How fast is

the height of the sand cone increasing when the height is 4 cm?

(Ans. 1

48𝜋 cm/s.)

10. An isosceles triangle of vertical angle 2is inscribed in a circle of radius a. Show that the area of

triangle is maximum when = 𝜋

6

7. Integrals

1. Evaluate: cot−1(1 − 𝑥 + 𝑥2)1

0𝑑𝑥. (Ans:

𝜋

2− log 2 )

2. Evaluate: sin−1 𝑥

𝑎+𝑥

𝑎

0𝑑𝑥. ( Hint: let 𝑥 = 𝑎𝑡𝑎𝑛2𝜃 ) (Ans:

𝑎

2 𝜋 − 2 )

3. Evaluate 1+𝑥2

𝑥4𝑑𝑥. (Ans:

−1

3 1 +

1

𝑥2

3/2

+ 𝑐)

4. Evaluate 𝑐𝑜𝑠 2𝑥

1+𝑎𝑥𝑑𝑥, 𝑎 > 0.

𝜋

−𝜋(Ans:

𝜋

2)

5. Evaluate 𝑥 + log 1+𝑥

1−𝑥

1

2−1

2

𝑑𝑥. (Ans: −1

2)

6. Evaluate 𝑥3 + 3𝑥2 + 3𝑥 + 3 + 𝑥 + 1 cos(𝑥 + 1) 𝑑𝑥0

−2.(Ans: 4)

7. Evaluate

dxx

x

1

0

21

1log8. Ans: 𝜋𝑙𝑜𝑔2

8. Evaluate

4/342 1xx

dx

.

(Ans: 1

1+1

𝑥4 1/4 + 𝑐)

40

9. Evaluate

335

912

1

52

xx

xx dx (Ans:

3

1+1

𝑥2+1

𝑥5 4 + 𝑐 )

10. Show that 𝑠𝑖𝑛2𝑥 .𝑐𝑜𝑠 2𝑥

𝑠𝑖𝑛 5𝑥+𝑐𝑜𝑠 3𝑥𝑠𝑖𝑛2𝑥+𝑠𝑖𝑛3𝑥𝑠𝑖𝑛2𝑥+𝑐𝑜𝑠 5𝑥 2 𝑑𝑥=−1

3 1+𝑡𝑎𝑛 3𝑥 + 𝑐.

11. Evaluate

dx

xx

2/1

04

162

11

31 (Ans :2)

8. Application of Integrals 1. Sketch the region bounded by the curves y = 5 − 𝑥2 and 𝑦 = | 𝑥 − 1| and find its area.

(Ans :5

2 { 𝑠𝑖𝑛−1 2

5 + 𝑠𝑖𝑛−1 1

5} −

1

2)

2. Find the area of the region lying above the X axis and included between the curves

𝑥2 + 𝑦2 = 2𝑎𝑥 𝑎𝑛𝑑 𝑦2 = 𝑎𝑥. (Ans :𝑎2 { 𝜋

4 −

2

3 })

3. Sketch the region bounded by the curves 𝑦 = 𝑥2 and y =2

1+𝑥2 and find its area (Ans : 𝜋 − 2

3)

4. Find the area included between the curves 𝑥 − 1 2 + 𝑦2 = 1 and x2 + y

2 = 1

(Ans : ( 2𝛱

3 -

3

2 )

5. Find the area of the region given by { (x,y) : x2≤ 𝑦 ≤ |𝑥| } (Ans: 1/3)

6. Sketch the graph of y = | x -1| and evaluate | x − 1|4

−2 dx (Ans : 9)

7. Draw the rough sketch of y = sin2x and determine the area enclosed by the lines x = Π /4 and

x = 3 Π /4 (Ans : 1)

8. Compute area bounded by the lines x +2y = 2 , y –x = 1 and 2x +y = 7. (Ans : 6)

9. Diferential Equations

1. Solve : (1 + 𝑦2) tan−1 𝑥. 𝑑𝑥 + 2𝑦(1 + 𝑥2)𝑑𝑦 = 0

2. Solve : y + 𝑑

𝑑𝑥 𝑥𝑦 = 𝑥(𝑠𝑖𝑛𝑥 + 𝑙𝑜𝑔𝑥)

3. Find the general solution of (1 + tany) (𝑑𝑥 – 𝑑𝑦) + 2𝑥𝑑𝑦 = 0

4. Solve 𝑑𝑦

𝑑𝑥= cos 𝑥 + 𝑦 + sin(𝑥 + 𝑦)

5. Solve the differential equation: 2𝑥2 + 3𝑦2 + 1 𝑥𝑑𝑥 + 4𝑥2 + 6𝑦2 + 3 𝑦𝑑𝑦 = 0

6. Solve the differential equation 𝑥𝑑𝑦

𝑥2+𝑦2 = 𝑦

𝑥2+𝑦2 − 1 𝑑𝑥

7. Solve 2(y + 3) – 𝑥𝑦𝑑𝑦

𝑑𝑥 = 0, given that y(1) = − 2

8. Form the differential equation of circles represented by 𝑥 − 𝛼 2 + 𝑦 − 𝛽 2 = 𝑟2 by eliminating

𝛼 𝑎𝑛𝑑 𝛽(Ans: 1 + 𝑑𝑦

𝑑𝑥

2

3

= 𝑟2 𝑑2𝑦

𝑑𝑥2

2

)

9. Show that 𝐴𝑥2 + 𝐵𝑦2 = 1 is a solution of the differential equation

𝑥 𝑦 𝑑2𝑦

𝑑𝑥2 + 𝑑𝑦

𝑑𝑥

2

= 𝑦𝑑𝑦

𝑑𝑥

10. Solve the differential equation: 𝑑𝑦

𝑑𝑥+

2

𝑥𝑦 = 3𝑥2𝑦

4

3.

41

10. Vectors

1. If 𝑎 = 3𝑖 –𝑗 and 𝑏 = 2𝑖 + 𝑗 – 3𝑘 , express 𝑏 as 𝑏1

+ 𝑏2 where 𝑏1

is parallel to 𝑎 and 𝑏2 is

perpendicular to 𝑎 ? ( Ans: 𝑏1 =

3

2𝑖 -

1

2𝑗 , 𝑏2

= 1

2𝑖 +

3

2𝑗 – 3𝑘 )

2. Find 𝑥 such that the four points with position vectors −6𝑖 + 3𝑗 + 2𝑘 , 3𝑖 + 𝑥𝑗 + 4𝑘 ,

5𝑖 + 7𝑗 + 3𝑘 and−13𝑖 + 17𝑗 – 𝑘 are coplanar ?

( Hint : Let us consider OA , OB , OC , OD are position vectors then find AB , BC and CD vectors

then [ AB , BC , CD ] = 0 then find x.( Ans : 𝑥 = -2 )

3. If 𝑎 =𝑖 + 𝑗 + 𝑘 , 𝑏 = 𝑗 – 𝑘 then find a vector c such that 𝑎 × 𝑐 = 𝑏 and 𝑎 . 𝑐 = 3.

( Ans : 5

3𝑖 +

2

3𝑗 +

2

3𝑘 )

4. For three vectors 𝑎 , 𝑏 , 𝑐 if 𝑎 × 𝑏 = 𝑐 , 𝑎 × 𝑐 = 𝑏 then prove that 𝑎 , 𝑏 , 𝑐 are mutually

perpendicular and also prove that 𝑎 = 𝑏 = 𝑐 .

5. Show that the vectors𝑎 , 𝑏 ,𝑐 are coplanar iff 𝑎 + 𝑏 , 𝑏 + 𝑐 and 𝑐 + 𝑎 are coplanar ?

6. Prove that for any three vectors 𝑎 , 𝑏 and 𝑐 then prove that

[𝑎 + 𝑏 , 𝑏 + 𝑐 , 𝑐 + 𝑎 ] = 2 [𝑎 , 𝑏 , 𝑐 ]

7. If 𝑎 = 𝑖 + 4𝑗 + 2𝑘 , 𝑏 = 3𝑖 – 2𝑗 + 7𝑘 and 𝑐 = 2𝑖 –𝑗 + 4𝑘 then find a vector 𝑑 which is perpendicular

to both 𝑎 , 𝑏 and 𝑐 .𝑑 . = 15 ? (Ans : 5

3 ( 32 𝑖 – 𝑗 -14 𝑘 ))

8. The scalar product of the vector 𝑎 = 𝑖 + 𝑗 + 𝑘 with a unit vector along the sum of vectors

𝑏 =𝑖 + 4𝑗 – 5𝑘 and 𝑐 = x𝑖 + 2𝑗 + 3𝑘 is equal to one then find ? (Ans : x = 1 )

11. Three Dimensional Geometry

1. Find the distance of the point (1, −2, 3) from the plane 𝑥 − 𝑦 + 𝑧 = 5 measured parallel to the

line 𝑥−1

2=

𝑦−3

3=

𝑧+2

−6. (Ans: 1)

2. If 4x + 4y – λz =0 is the equation of the plane through the origin that contains the line

𝑥−1

2=

𝑦+1

3=

𝑧

4, find the value of ‗λ’. (Ans: 5)

3. Find the image of the point (1, 6, 3) in the line𝑥

1=

𝑦−1

2=

𝑧−2

3. (Ans: (1, 0, 7)

4. Find the equations of the line of shortest distance between the lines :𝑥−8

3=

𝑦+9

16=

𝑧−10

7 and

𝑥−15

3=

𝑦−29

8=

5−𝑧

5also find the shortest distance between the lines.

(Ans: 14 units, 𝑥−5

2=

𝑦−7

3=

𝑧−3

6)

5. Find the length and foot of perpendicular from the point (1,3

2, 2) to the plane

2𝑥 − 2𝑦 + 4𝑧 + 5 = 0 . (Ans: 1

3 𝑢𝑛𝑖𝑡𝑠 and

11

9,

19

9,

34

9 )

6. A plane meets the coordinate axes in A, B, C such that the centroid of the triangle ABC is the

point (𝛼, 𝛽, 𝛾). Show that the equation of the plane is 𝑥

𝛼+

𝑦

𝛽+

𝑧

𝛾= 3.

42

7. Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and

l, m, n respectively, from the origin, then prove that 1

𝑎2 +1

𝑏2 +1

𝑐2 =1

𝑙2 +1

𝑚2 +1

𝑛2 .

8. Find the coordinates of the point where the line through (3, −4, 5) and (2, −3, 1) crosses the

plane passing through three points (2, 2, 1) (3, 0, 1) (4, −1, 0).𝐴𝑛𝑠: (1, −2,7)

9. A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes

at A, B , C. Show that the locus of the centroid of ΔABC is 1

𝑥2 +1

𝑦2 +1

𝑧2 =1

𝑝2.

10. Find the distance of the point (1,-2,3) from the plane x - y + z = 5 measured parallel to the

line𝑥+1

2=

𝑦+3

3=

𝑧+1

−6 . (Ans: 1 Unit)

12. Linear Programming

1) Kelloggs is a new cereal formed of a mixture of bran and rice that contains at least 88 grams

of protein and at least 36 milligrams of iron.Knowing that bran contains 80 grams of protein

and 40 milligrams of iron per kilogram,and that rice contains 100 grams of protein and 30

milligrams of iron per kilogram,find the minimum cost of producing this new cereal if bran

costs Rs.5 per kilogram and rice costs Rs.4 per kilogram.

(Answer:Rs.4.6 when 0.6grams of bran and 0.4 grams of rice are mixed).

2) Amanufacturer of electronic circuits has a stock 200 resistors,120 transistors and 150

capacitors and is required to produce two types of circuits A and B.Type A requires 20

resistors,10 transistors and 10 capacitors.If the profit on type A circuit is Rs.50 and that on

type B circuit is Rs.60,formulate and solve this problem as an LPP,so that the manufacturer

can maximize his profit.

(Answer: Maximum at (28/3,4/3) and the maximum profit isRs. 546.67).

3) Maximise Z = x + 2y subject to the constraints 𝑥 − 𝑦 ≥ 0, 2𝑦 ≤ 𝑥 + 2, 𝑥 , 𝑦 ≥ 0

(Answer : z = 6 at (2,2)).

4) A company makes two kinds of leather belts A and B.Belt A is high quality and belt B is of

low quality.The respective profits are Rs.4 and Rs. 3 per belt.Each belt of type A requires twice

as much time as a belt of type B and if all belts were of type B the company could make 1000

belts per day.The supply of leather is sufficient for only 800 belts per day.(both A and B

combined). Belt A requires a fancy buckle and only 400 buckles per day are available.Thereare

only 700 buckles available for belt B .What should be the daily production of each type of

belt? Formulate the problem as an LPP.

(Answer : P = 4x + 3y, 2x + y ≤ 1000, x + y ≤ 800 , x ≤ 400, y ≤ 700, x,y ≥ 0)

43

5) A manufacturing company makes two types of television sets one is black and white and other

is colour.The company has resource to make at most 300sets a week.It take Rs.1800 to make a

black and white set and Rs.2700 to make coloured set. The company can spend not more than

648000 a week to make television sets.If it makes a profit of Rs.510 per black and white set

and Rs.675 per coloured set. Howmany sets of each type should be produced so that the

company has maximum profit?Formulate this problem as an LPP given that the objective is to

maximize the profit.

Answer : Z = 172800 at (180,120)

13. Probability

1. Urn A contains 1 white, 2 black and 3 red balls. Urn B contains 2 white , 1 black ,1 red ball.Urn C contains 4 white ,5 black and 3 red balls. Two balls are drawn from one of the urnsand found to be one white and one red. Find probabilities that they come from urns A,B or C.

(Ans : 16/39) 2. A die is thrown 120 times and getting 1 or 5 is considered as success. Find the mean and

variance of number of success. (Ans : Mean = 40 Variance = 26.7) 3. Three cards are drawn with replacement from well shuffled pack of cards. Find the probability

that the cards are king ,queen and a jack. (Ans : 6/2197) 4. A man is known to speak truth 3 times out of 4. He throws a die and reports that is a six. Find

the probability that is actually six.(Ans :3/4) 5. Find the probability that the sum of numbers appearing and showing on the two dice is 8

given that atleast one of the dice does not show 5. (Ans : 1/9) 6. Six dice are thrown 729 times. How many times do you expect atleast three dice to show 5

or 6. (Ans : 233) 7. Two dice are thrown.Find the probability that the number appeared have a sum 8 if it is

known that the second dice always exihibits 4. (Ans :1/6) 8. Two cards are drawn from a pack and kept out.Then one card is drawn from remaining 50

cards. Find the probability that it is an Ace. (Ans : 1/13).

44

Class XII

Sample Paper

-By Group 3

45

MATHEMATICS

CLASS XII

Set : 1

Time allowed:3 hours Maximum Marks : 100

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 29 questions divided into four sections A, B, C and D. Section A

comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each,

Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of

six marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact

requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in 3 questions of four

marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in

all such questions.

(v) Use of calculators is not permitted. You may ask for logarithmic tables, if require.

SECTION A Question numbers 1 to 4 carry 1 mark each.

1. If 𝑎 𝑎𝑛𝑑 𝑏 are unit vectors, then what is the angle between 𝑎 𝑎𝑛𝑑 𝑏 for 𝑎 − 2𝑏 to be a unit

vector?

2. Find the value of 𝑡𝑎𝑛−1 3 - 𝑐𝑜𝑡−1(− 3).

3. If the matrix A = 0 𝑎 −32 0 −1𝑏 1 0

is skew symmetric, find the value of a and b.

4. If a *b denotes the larger of a and b and if a°b = (a*b) +3, then write the value of

5 ° 10, where * and ° are binary operations.

SECTION B

Question numbers 5 to 12 carry 2 marks each.

5. Given A = 2 −3

−4 7 , compute A-1 and show that 2A-1 = 9I-A.

6. Differentiate 𝑡𝑎𝑛−1 1+cos 𝑥

sin 𝑥 with respect to x.

7. The total cost C(x) associated with the production of x units of an item is given by C(x)

= 0.005 x3 – 0.02 x2+30x + 5000. Find the, marginal cost when 3 units are produced,

46

where by marginal cost we mean the instantaneous rate of change of total cost at

any level of output.

8. Evaluate: cos 2𝑥+ 2 𝑠𝑖𝑛2𝑥

𝑐𝑜𝑠 2𝑥 dx.

9. Show that 𝑡𝑎𝑛−1 2

3 =

1

2𝑡𝑎𝑛−1 12

5.

10. Find the value of p, where f(x) = 𝑝, 𝑥 = 01−𝑐𝑜𝑠4𝑥

𝑥2 , 𝑥 ≠ 0 is continuous at x = 0.

11. Find the angle between the lines

r = (2i-3j+k)+𝛼(i+j+3k); r = (i-j-k) + 𝛽(2i-3j+k).

12. A couple has 2 children. Find the probability that both are boys, if it is known that (i)

one of them is a boy (ii) the older child is a boy.

SECTION C Question numbers 13 to 23 carry 4 marks each.

13. Find k, f(x) = 𝑘 𝑠𝑖𝑛

𝜋

2 𝑥 + 1 , 𝑥 ≤ 0

tan 𝑥−sin 𝑥

𝑥3 , 𝑥 > 0 is continuous at x = 0.

14. Find equation of normal to the curve ay2 = x3 at the point whose x coordinate is am2.

15. There are two bags A and B. Bag A contains 3 white and 4 red balls whereas bag B contains 4

white and 3 red balls. There balls are drawn at random(without replacement) from one of the

bags and are found to be two white and one red. Find the probability that these were drawn

from bag B.

16. Find the equation of the tangent and the normal, to the curve 16x2 + 9 y2 = 145 at the point

(𝑥1 , 𝑦1), where 𝑥1=2 and 𝑦1 >0.

OR

Find the intervals in which the function f(x) = 𝑥4

4 –x3 – 5 x2 + 24x +12 is

(a) strictly increasing, (b) strictly decreasing.

17. An open tank with a square base and vertical sides is to be constructed from a metal sheet so

as to hold a given quantity of water. Show that the cost of material will be least when depth of

the tank is half of its width. If the cost is to be borne by nearby settled lower income families,

for whom water will be provided, what kind of value is hidden in this question?

18. Find: 2 cos 𝑥

(1−sin 𝑥)(1+𝑠𝑖𝑛 2𝑥)𝑑𝑥.

OR

Find: 𝑥+2

𝑥2+2𝑥+3𝑑𝑥.

19. Let 𝑎 = 4i +5j –k; 𝑏 = i -4j +5 k and 𝑐 = 3i+j-k. Find a vector 𝑑 which is perpendicular to both

𝑐 and 𝑏 and 𝑑 . 𝑎 = 21.

20. Two numbers are selected at random(without replacement) from the first five positive

integers. Let X denote the larger of the two numbers obtained. Find mean and variance of X.

47

21.

Using properties of determinants prove that

𝑏 + 𝑐 𝑎 − 𝑏 𝑎𝑐 + 𝑎 𝑏 − 𝑐 𝑏𝑎 + 𝑏 𝑐 − 𝑎 𝑐

= 3𝑎𝑏𝑐 − 𝑎3 − 𝑏3 − 𝑐3 .

OR

Using properties of determinants prove that

𝑥2 + 1 𝑥𝑦 𝑥𝑧

𝑥𝑦 𝑦2 + 1 𝑦𝑧

𝑥𝑧 𝑦𝑧 𝑧2 + 1

= 1 + 𝑥2 + 𝑦2 + 𝑧2.

22. Find the particular solution of the differential equation

𝑥2 𝑑𝑦

𝑑𝑥− 𝑥𝑦 = 1+cos(

𝑦𝑥 ), x ≠0, when x = 1, y = 𝜋 2 .

23. Find the particular solution of the differential equation

(1+x2)𝑑𝑦

𝑑𝑥 = (𝑒𝑚𝑡𝑎𝑛 −1𝑥 - y), give that y = 1 when x = 0.

SECTION D Question numbers 24 to 29 carry 6 marks each.

24. If A = 2 −3 53 2 −41 1 −2

, find A-1. Use it to solve the system of equations

2x-3y+5z = 11

3x+2y -4z = -5

x+ y – 2z = -3.

OR

Using elementary row transformations, find the inverse of the matrix

1 2 32 5 7

−2 −4 −5 .

25. Evaluate: sin 𝑥+cos 𝑥

16+9 sin 2𝑥𝑑𝑥.

𝜋

40

OR

Evaluate: (𝑥2 + 3𝑥 + 𝑒𝑥)3

1dx.by limit as a sum.

26. Find the distance of the point (-1, -5, -10) from the point of intersection of the line

𝑟 =(2i –j +2k) + 𝜇(3𝑖 + 4𝑗 + 2𝑘) and the plane 𝑟 . (𝑖 − 𝑗 + 𝑘) = 5.

27. Let f: N→N be a function defined as f(x) = 4x2 + 12x +15 show that f: N→ 𝑆 is invertible (where

S is range of f). Find the inverse of f, and find f-1(31) and f-1(87).

28. Using integration , find the area of the region {(x,y): y2≤6ax and x2 +y2≤16a2}.

48

29. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and

F2 are available costing Rs. 5 per unit and Rs. 6 per unit respectively. One unit of food F1

contains 4 units of vitamin A and 3 units of minerals. Where as one unit of food F2 contains 3

units of vitamin A and 6 units of minerals. Formulate this as a LPP. Find the minimum cost of

diet that consist of mixture of these two foods and also meets minimum nutritional

requirement.

MATHEMATICS

CLASS XII

BLUE PRINT

SNO CHAPTER 1 2 4 6 TOTAL

1 Relations and functions 1(1) 1(6) 2(7)

2 Inverse trigonometric functions 1(1) 1(2) 2(3)

3 Matrices 1(1) 1(6) 2(7)

4 Determinants 1(2) 1(4) 2(6)

5 Continuity and Differentiability 2(4) 1(4) 3(8)

6 Application of derivatives 1(2) 3(12) 4(14)

7 Integration 1(2) 1(4) 1(6) 3(12)

8 Application of integrals 1(6) 1(6)

9 Differential Equations 2(8) 2(8)

10 Vectors 1(1) 1(4) 2(5)

11 3-D Geometry 1(2) 1(6) 2(8)

12 LPP 1(6) 1(6)

13 Probability 1(2) 2(8) 3(10)

Total 4(4) 8(16) 11(44) 6(36) 29(100)

49

MATHEMATICS

CLASS XII

Set : 2

Time allowed:3 hours Maximum Marks : 100

General Instructions :

(vi) All questions are compulsory.

(II) The question paper consists of 29 questions divided into four sections A, B, C and D. Section A

comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each,

Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of

six marks each.

(III) All questions in Section A are to be answered in one word, one sentence or as per the exact

requirement of the question.

(IV) There is no overall choice. However, internal choice has been provided in 3 questions of four

marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in

all such questions.

(V) Use of calculators is not permitted. You may ask for logarithmic tables, if require.

SECTION A

Question numbers 1 to 4 carry 1 mark each.

30. If det A = 4, find det(A-1).

31. Find the product of the order and degree of the following differential equation:

x 𝑑2𝑦

𝑑𝑥 2 2

+ 𝑑𝑦

𝑑𝑥

2+ 𝑦2 = 0.

32. Write the projection of 2i+3j-k along the vector i+j.

33. 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

50

SECTION B


34. If A = 3 57 9

is written as A = P+ Q, where P is a symmetric matrix and Q is Skew- symmetric

matrix, then write the matrix P.

35. Find the value of c in Rolle’s Theorem for the function f(x) = x3- 3 x in [- 3,0].

36. The volume of a cube is increasing at the rate of 9 cm3/s. How fast is its surface area

increasing when the length of an edge is 10 cm?

37. Evaluate: 𝑥−1

𝑥2 𝑒𝑥𝑑𝑥.

38. Evaluate: 𝑑𝑥

𝑥(𝑥5+3) .

39. The x- coordinate of a point on the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its

z - coordinate.

40. Two tailors A and B earn Rs.300 and Rs.400 per day respectively. A can stitch 6 shirts and 4

pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many

days should each of them work and if it is desired to produce atleast 60 shirts and 32 pairs of

trousers at minimum labour cost. Formulate this as an LPP.

41. A die, whose faces are marked 1, 2,3 in red and 4,5,6 in green, is tossed. Let A be the event

“number obtained is even” and B be the event “ number obtained is red”. Find whether A and

B are independent events.

SECTION C


42. Solve for x: 𝑇𝑎𝑛−1(𝑥 + 1) + 𝑇𝑎𝑛−1(𝑥 − 1) = 𝑇𝑎𝑛−1 8

31

𝑂𝑟

Prove the following

𝐶𝑜𝑡−1( 𝑥𝑦 + 1

𝑥 − 𝑦 )+ 𝐶𝑜𝑡−1(

𝑦𝑧 + 1

𝑦 − 𝑧 ) + 𝐶𝑜𝑡−1(

𝑧𝑥 + 1

𝑧 − 𝑥 ) = 0

43. Using properties of determinants ,prove that

51

∆ = 𝑎2 + 2𝑎 2𝑎 + 1 12𝑎 + 1 𝑎 + 2 1

3 3 1

= (𝑎 − 1)3

44. If 𝑥𝑦 + 𝑦𝑥 = 𝑎𝑏 then find 𝑑𝑦

𝑑𝑥 .

45. Find the value of P for which the curves 𝑥2 = 9p (9 – y) and 𝑥2 = p ( y + 1) cut each other at right

angles.

46. Find the particular solution of differential equation 𝑑𝑦

𝑑𝑥 + 2 y tan x = Sin x given that y = 0 when x =

𝜋

3 .

47. Find the shortest distance between the lines 𝑟 = ( 4i – j ) + 𝜆 ( i+ 2j – 3k ) and

𝑟 = ( i – j + 2k ) + 𝜇 ( 2 i +4 j - 5 k ) .

48. The random variable X can be taken only the values 0,1,2,3. Given that P (X = 0) = P (X =1) = p and

P(x=2) = P(X=3) such that 𝛴𝑝𝑖𝑥𝑖2 = 2 𝛴𝑝𝑖𝑥𝑖 , find the value of p.

49. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find

the probability that is actually a six.

50. Solve the following graphically:

Minimise Z= 5x + 10y subject to constraints x + 2y ≤120, x +y ≥ 60, x – y ≥ 0 and x, y ≥ 0.

51. If 𝑎 = i+2j+k , 𝑏 = 2i + j and 𝑐 = 3i – 4j -5k ,

Then find a unit vector perpendicular to both vectors (𝑎 - 𝑏 ) and ( 𝑐 - 𝑑 )

52. Find 𝑑𝑥

5− 8𝑥 − 𝑥2 .

Or

Find 2 cos 𝑥

1−𝑆𝑖𝑛 𝑥 ( 1+𝑆𝑖𝑛 𝑥 ) dx

SECTION D


24. If A = 2 3 11 2 2

−3 1 −1 , find 𝐴−1 and hence solve the system of equations

2x+ y - 3 z = 13 , 3x + 2y + z = 4, x + 2y - z = 8.

25. Consider f : R – { - 4

3 } → R – { −

4

3 } given by f(x) =

4𝑥+3

3𝑥+4 . Show that f is bijective. Find the inverse of

f and hence find 𝑓−1( 0 ) and x such that 𝑓−1 ( x ) = 2

26. A metal box with a square base and vertical sides is to contain 1024 𝑐𝑚3. The material Rs.5 per 𝑐𝑚2

and the material for the sides costs Rs. 2.50 per 𝑐𝑚2 . Find the least cost of the box.

27. Find the coordinates of the points where the line through the points ( 3, -4, -5) and (2, -3, 1 )

52

crosses the plane determined by the points ( 1, 2, 3 ), ( 4, 2, -3 ) and ( 0, 4 , 3).

Or

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes

at A, B, C. Show that the locus of the centroid of ∆ ABC is 1

𝑥2 + 1

𝑦2 + 1

𝑧2 = 1

𝑝2 .

28. Using the integration find the area of the triangle formed by positive X – axis and the tangent and

the normal to the circle 𝑥2 + 𝑦2 = 4 at ( 1, 3).

Or

Find the area of the region bounded by lines y = 5

2x – 5, x + y - 9 = 0 and y =

3

4x –

3

2

29. Evaluate ( 3

1𝑒2−3𝑥 + 𝑥2 + 1 ) dx as a limit of a sum

MATHEMATICS

CLASS XII

BLUE PRINT

SNO CHAPTER 1 2 4 6 TOTAL

1 Relations and functions 1 1(6)

2 Inverse trigonometric functions 1 1(4)

3 Matrices 1 1(2)

4 Determinants 1 1 1 3(11)

5 Continuity and Differentiability 1 1 1 3(7)

6 Application of derivatives 1 1 1 3(12)

7 Integration 2 1 1 4(14)

8 Application of integrals 1 1(6)

9 Differential Equations 1 1 2(5)

10 Vectors 1 1 1 3(7)

53

11 3-D Geometry 1 1 2(10)

12 LPP 1 1 2(6)

13 Probability 1 2 3(10)

Total 4(4) 8(16) 11(44) 6(36) 29(100)

54

Class XI

Sample Paper

-By Group 4

55

KENDRIYA VIDYALAYA SANGATHAN , HEDERABAD REGION

BLUE PRINT FOR HALF YEARLY EXAMINATION

CLASS – XI MATHEMATICS

S.No

.

NAME OF THE CHAPTER VSA

(1 Mark)

SA

(2 Marks)

LA-1

(4 Marks)

LA-2

(6 Marks)

TOTAL

01 SETS 1(1) 1(4) 1(6) 3(11)

02 RELATIONS AND FUNCTIONS 1(2) 1(4) 2(06)

03 TRIGONOMETRIC

FUNCTIONS

1(2) 1(4) 1(6) 3(12)

04 PRINCIPLE OF

MATHEMATICAL INDUCTION

1(4) 1(6) 2(10)

05 COMPLEX NUMBERS AND

QUADRATIC EQUATIONS

1(1) 1(2) 1(4) 3(7)

06 LINEAR INEQUALITIES 1(4) 1(6) 2(10)

07 PERMUTATIONS AND

COMBINATIONS

1(2) 2(8) 3(10)

08 BINOMIAL THEOREM 1(1) 1(2) 1(4) 1(6) 4(13)

09 SEQUENCES AND SERIES 1(1) 2(4) 1(4) 4(09)

10 STRAIGHT LINES 1(2) 1(4) 1(6) 3(12)

TOTAL 4(4) 8(16) 11(44) 6(36) 29(100)

56


HALF YEARLY EXAMINATION - 2018-19

CLASS – XI SUB:MATHEMATICS

Max.Mark:100 Duration : 3 hrs


1. All questions are compulsory.

2. This question paper contains 29 questions.

3. Question 1 – 4 in Section A are very short – answer type questions carrying 1 mark each.

4. Question 5 – 12 in Section B are short – answer type questions carrying 2 marks each.

5. Question 13 – 23 in Section C are long – answer – I type questions carrying 4 marks each.

6. Question 24 – 29 in Section D are long – answer – II type questions 6 marks each.

SECTION - A

1. Write the set

50

7,

37

6,

26

5,

17

4'

10

3,

5

2,

2

1in the set builder form.

2. Find the modulus of (2-3i).

3. The fourth term of a G.P. is 27 and the 7th

term is 729, find the G.P.

4. Write 3rd term of the Expanantion(3𝑥 +2

𝑥 )4

SECTION - B

5. Evaluate cos(-1710 ).

6. Let A={1, 2, 3, 4}, B={1, 4, 9, 16, 25} and R be a relation defined from A to B as R={ 𝑥. 𝑦 : 𝑥 ∈

𝐴, 𝑦 ∈ 𝐵 𝑎𝑛𝑑 𝑦 = 𝑥2}

a. Find domain of R

b. Find range of R

7. Express (1

2+ 2𝑖)3 in the form (a+ib)

8. How many different words (with or without meaning) can be made using all the vowels

without repetition of letters at a time?

9. Find the middle term in (𝑥 −1

𝑥)8

10. Find the 18th

term of the sequence 20,19 ¼,18 ½ ,17 ¾ ,…..

11. Find two numbers whose AM=25 and GM=10.

12. Find the value of K such that the line joining the points (2, K) and (-1, 3) is parallel to the line

joining (0, 1) and (-3, 1).

57

SECTION - C

13. Answer the following questions:

a) What is the interval form of set A= {x : x∈ R, -3 ≤ x < 5}

b) If B = { -1, 0, 1}, what is n(P(B))?

c) What is represented by the adjoining Venn- diagram?

d) For any set A, what does A ∩ A‘ equal to?

14. In triangle ABC with vertices A(2,3) B(4,-1) and C(1,2). Find the equation of the altitude from the

vertex A to the side BC.

15. Let f =

Rx

x

xx :

1,

2

2

be a function from R into R. Determine the range of f.

OR

(i)Define greatest integer function. Draw the graph of the function

(ii)Find the domain of the Function x2-4

16. If sinx= ,2

,4

1

x find the value of

2sin,

2cos,

2tan

xxx

OR

Prove that cos2xcos2

5sin 5sin

2

9 cos 3cos

2

xx

xx

x

17. Prove by mathematical induction P(n) : 1 + 4 + 7 + .... + (3n – 2) = 1

2n(3n-1)

(or)

Given P(n) : 1 + 2 + 3 + ..... + n<1

8 (2n+1)

2

18. Convert the complex number i

i

35

362

into polar form.

(or)

Find the square root of i3612

19. How many numbers greater than 1000000 can be formed by using the digits

0,1,2,3,4,5,6 if the repetition of the digits is not allowed?

OR

In how many of the distinct permutations of the letters in MISSISSIPPI do the four I‘s

Notcome together?

20. From a group of 7 boys and 5 girls, a team consisting of 5 is to be made. Find how many different

ways it can be done

(i) Ifthe team consists of at least 3 girls

(ii) If the team consists of at most 3 girls?

21. Find (x+1)6+(x-1)

6 .Hence evaluate .)12()12( 66

22. Find the sum to n terms of the series: 5+11+19+29+41……………………

OR

58

The sums of n terms of two arithmetic progressions are in the ratio (5n+4) : (9n+6).Find

the ratio of their 18th

terms.

23. Find the equation of circle passing through the points (4,1) and (6,5) and whose centre

is on the line 4x+y=16.

SECTION –D

24. Find the general solution of the equations

i) Sec 2 2x = 1 – tan 2x

ii) Sin x + sin 3x + sin 5x = 0.

25. A college awarded 38 medals in football,15 in basketball and 20 in cricket.If these medals went to

a total of 58 men and only three men got medals in all the three sports, how many received medals

in exactly two of the three sports?

OR In a survey of 60 people,it was found that 25 people read newspaper H,26 read newspaper T,26

read newspaper I,9 read both H and I,11 read both H and T,8 read both T and I,3 read all three

newspapers.Find:

(i) the number of people who read at least one of the newspapers.

(ii) the number of people who read exactly one newspaper.

26. Using principle of mathematical induction, prove that:

983 22 nn is divisible by 8, .Nn

27. A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid

solution must be added to it so that acid content in the resulting mixture will be more

than 15% but less than 18%?

OR

Solve the following system of inequations graphically

2x y 24 ; x y 11 ; 2x 5y 40 ; x 0 ; y 0.

28. The coefficient of the ( r-1 )th

, rth

and ( r + 1 )th

terms in the expansion of ( x + 1 )n are

in the ratio 1 : 3 : 5 . Find n and r ?

29. If p and q are the lengths of perpendiculars from the origin to the lines

,cossec and 2cossincos kecyxkyx respectively,

prove that:222 4 kqp .

59

KENDRIYA VIDYALAYA SANGATHAN --- Half Yearly, 2018-19

CLASS: XI MAX.MARKS: 100

SUBJECT: MATHEMATICS TIME: 3 Hrs

ANSWER KEY

S.NO. ANSWERS Marks

1. {x:x=𝑛

𝑛2+1 , n 𝜖 N and n≤ 7} 1M

2. √13 1M

3. 1,3,9,… 1M

4. 4C2(3x)2(2/x)

2 1M

5. Solving and getting to the solution -1 2M

6. Domain={1,2,3,4}

Range = {1,4,9,16}

1 M

1M

7. Simplifying

-(47/8) – (13/2)i

1M

1M

8. 5! = 120 1 M

+

1M

9. T5 = 8C4

= 70

1M

1 M

10. A=20, d=-(3/4)

A18=23/4

1M

1M

11. Writing formula

A=20, b=5

1M

1M

60

12. For formula of slope

K = 3

1 M

1 M

13. a) [-3,5)

b) 23=8

c) (A-B)U(B-A)

d) ∅

1 M

1 M

1 M

1 M

14. Finding the slope of BC m = -1

Finding the slope of the altitude through A is m=1

Equation of the altitude through A to BC

1 M

1 M

2 M

15. f(x) = 1- (1/1+x2)

as x𝜖R -∝ < 𝑥 < ∝

solving to 0≤f(x)<1

(OR)

(i) For the correct definition

For correct graph

(ii) x2 – 4 ≥ 0

(-∝ , -4] U [4, ∝)

1 M

1 M

2 M

1 M

1 M

1 M

1 M

16. stating x lies in the first quadrant

finding tax(𝑥

2 ), cos(

𝑥

2 ), sin(

𝑥

2 )

1M

(1+1+1)=

3M

17. For proving n= 1 is true

For assuming P(n=k) is true

1 M

1 M

61

For proving P(n=k+1) is also true and for conclusion 2 M

18. For writing a+ib form.

For finding Modulus

For finding amplitude

For writing in polar form

(or)

Let 12 + 6 3𝑖 = x + iy

For getting correct equations

Solving for x=3 and y=√3

Writing correct square root

1 M

1 M

1 M

1 M

1 M

1 M

1 M

1 M

19. 7! -6!

6!(7-1)

6! X 6 = 4320

1 M

1 M

2 M

20. (i) 7C2 5C3 + 7C15C4 + 7C05C4

(ii) 7C2 5C3+ 7C35C2 + 7C45C1 +7C55C0

2 M

2 M

21. Expanding the Binomial (x+1)6+(x-1)

6 and simplifying

Applying the simplified form of (x+1)6+(x-1)

6 in finding

.)12()12( 66

2 M

2 M

22. Sn= 5+11+19+29+….. n terms

(-)Sn= 5+11+19+29+… +an

an = 2n+7

Sn= 2n + 7 simplifying to get n(n+8)

1 M

2 M

1 M

62

23. Equating the centres and framing the equations

Substituting the cenre in the equation 4x+y=16

Solving the two equations

A = 3, b=4

1 M

1 M

2 M

24. (i) 1+tan22x = 1 – tan 2x

tan22x + tan 2x = 0

Solving tan 2x= 0, tan 2x= -1

G S x = n𝜋, n𝜖 Z

(ii) Solving upto sin 3x(2 cos2x+1)=0

Sin x = 0, cos 2x= -1/2

GS x = n𝜋/3, n𝜖 Z (or) x= n𝜋 ± 𝜋/3

1 M

1 M

1 M

1 M

2M

25. n(F)= 38, n(B)=15, n(C)=20

For correct venn diagram

Solving using n(FUBUC)

No.of students who got exactly two of the three medals is 9

(or)

Writing all the given in correct notation

Solving n(HUTUI) = 52

No of people who read exactly two news papers = 19

1M

1M

1 M

3 M

1 M

2 M

3 M

26. Proving P(1) is true

Suppose it is true for P(k)

Proving it is true for P(k+1)

1 M

1 M

3 M

63

By Principle of Mathematical Induction, it is true for every nN 1 M

27. Table for each inequality (three inequalities) (3 x ½ M)

Graph and shading (each one line 3 x 1½ M)

1 ½ M

4 ½ M

28. Solving and simplifying

(nCr-2) : (nCr-1) = 1: 3 and (nCr-1):(nCr) = 3:5

Framing equations n-4r =-5

3n-8r = -3

Finding n=7, r = 3

3 M

2 M

1 M

29. For Using the formula of perpendicular from a point to a line

Simplifying and using the identities to

Prove p2 +4q

2 = k

2

3 M

3 M

64


BLUE PRINT

SESSION ENDING EXAMINATION – 2018-19

CLASS – XI MATHEMATICS

S No Name of the Chapter VSA (1 M)

SA (2 M)

LA – I (4 M)

LA– II (6 M) Total

1 Sets 4(1) 6(1) 10(2)

29(M) 2 Relations and Functions

8(2) 08(2)

3 Trigonometric Functions 1(1)

4(1) 6(1) 11(3)

4 Mathematical Induction 4(1)

4(1)

37(M)

5 Complex Numbers 1(1) 2(1) 4(1) 7(3)

6 Linear Inequalities -- --

6(1) 6(1)

7 Permutations and Combinations

2(1) 4(1) 6(2)

8 Binomial Theorem --

6(1) 6(1)

9 Sequences and Series -- 2(1) -- 6(1) 8(2)

10 Straight Lines

2(1) 4(1)

6(1)

13(M) 11 Conic Sections 1(1) -- 4(1) -- 5(2)

12 Introduction to 3D Geometry -- 2(1)

-- 2(1)

12 Limits and Derivatives -- 2(1) 4(1) -- 6(2) 06(M)

14 Mathematical Reasoning 1(1) 2(1) -- -- 3(2) 03(M)

15 Statistics -- -- -- 6(1) 6(1) 12(M)

16 Probability -- 2(1) 4(1) -- 6(2)

Total 4(4) 16(8) 44(11)) 36(6) 100(29)

65


SESSION ENDING E EXAMINATION - 2018-19

CLASS – XI SUB:MATHEMATICS

Max.Mark:100 Duration: 3hrs


7. All questions are compulsory.

8. This question paper contains 29 questions.

9. Question 1 – 4 in Section A are very short – answer type questions carrying 1 mark each.

10. Question 5 – 12 in Section B are short – answer type questions carrying 2 marks each.

11. Question 13 – 23 in Section C are long – answer – I type questions carrying 4 marks each.

12. Question 24 – 29 in Section D are long – answer – II type questions 6 marks each.

SECTION – A

1. Write the value of tan(11𝜋

3).

2. Write the solutions of quadratic equation: x2 + 1 = 0.

3. Find the slope of the line 2x+3y +4=0.

4. State whether the ‗Or‘ used in the following statement is inclusive or exclusive.

―The school is closed if it is a holiday or a Sunday‖.

SECTION – B

5. Find the least positive integral value of ‗m‘ for which 1+𝑖

1−𝑖 𝑚

= 1

6. Find the sum of all multiples of 3 between the integers 1 to 100

7. If 𝑛𝐶9= 𝑛𝐶8

, find 𝑛𝐶15.

8. 20 cards are numbered from 1 to 20. One card is drawn at random. What is the probability of the

number on the card drawn(i) is a multiple of 5 (ii) an odd number?

9. Equation of the line is 2x – 4 y + 10 = 0. Find its (i) x- intercept (ii) y – intercept

10. The centroid of a triangle ABC is at the point (1, 1, 1). If the co – ordinates of A and B are (3, – 5,

7) and (– 1, 7, – 6), respectively, then find the co – ordinates of the point C.

11. Evaluate: lim𝑥→0 1+𝑥 − 1

𝑥 .

12. Given below are two statements :

p: 35 is a multiple of 5.

q: 35 is a factor of 75.

Write the compound statements connecting these two statements with ―And‖ and ―Or‖. In both

cases check the validity of the compound statement.

66

SECTION – C

13. There are 200 individuals with a skin disorder, 120 had been exposed to the chemical C1, 50 to

chemical C2 and 30 to both the chemicals C1 and C2. Find the number of individuals exposed to

(i) Chemical C2 but not chemical to C1.

(ii) Chemical C1 or Chemical C2.

14. Let f(x) = x2 and g(x) = 2x + 1 be real functions. Find (f + g) (x) and (f.g)(x) hence find (f +

g) (2) and (f.g)(-1)

15. Find the domain and range of the function f(x) = 9 − 𝑥2

16. Solve : 2cos2x + 3sinx = 0.

(OR)

Prove that cos2xcos2

5sin 5sin

2

9 cos 3cos

2

xx

xx

x .

17. Convert the complex number 1

1 + 𝑖 into polar form.

(OR)

Find the square root of : 5 + 12 i.

18. In how many ways can the letters of the word AMARAVATHI be arranged, so that (i)

All the A‘s are together? (ii) all the A‘s are not together.

19. Prove by mathematical induction

1+1

2

..........4321

1......................

4321

1

321

1

21

1

n

n

n

20. Two students A and B appeared in an examination. The probability that A will qualify the

examination is 0.05 and that B will qualify the examination is 0.1. The probability that both will

qualify the examination is 0.02. Find the probability that

(i) Both A and B will not qualify the examination.

(ii) At least one of them will not qualify the examination and

(OR)

In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC

and NSS. If one of these students is selected at random, then find the probability that

(i) The student opted for NCC or NSS

(ii) The student has opted for neither NCC nor NSS

21. Find the foot of the perpendicular of the point (3, 8) on the line x + 3y = 7.

(OR)

Find the equation of the line passing through the intersection of the lines x + y + 3 = 0 an x – y + 2

= 0 and having y – intercept equal to 4.

22. Find the coordinates of the foci, the vertices, the eccentricity and the length of the latusrectum of

the ellipse 𝑥2

16+

𝑦2

36= 1.

67

23. Find the derivative of f(x) = tanx from first principle.

SECTION – D

24. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 3, 5, 7, 8}, B = {1, 5, 9}

Verify that (i) (AB) = A B (ii) (A B) = AB. (iii) A – B ≠ B – A

25. Prove that : (i) cos6x = 32 cos6x – 48 cos

4x + 18 cos

2x – 1.

(ii) cotx.cot2x – cot2x.cot3x – cot3x.cotx = 1.

(OR)

If tanx = 3

4 , 𝜋 < 𝑥 <

3𝜋

2, find the value of sin

𝑥

2 , cos

𝑥

2 and tan

𝑥

2

26. Find the value of k so that the term independent of x if the expansion of

( 𝑥 + 𝑘

𝑥2 )10 is 405.

(OR) The 2nd, 3rd, 4th terms in the expansion of (𝑎 + 𝑏)𝑛are 240,720 and 1080 respectively.

Find the values a,b,n.

27. Solve the following system of in-equations graphically :

x + y ≥ 1, 3x+ 4y −12 < 0, x−2y ≤ 2, x ≥0, y ≥ 0

28. Calculate the mean, variance and standard deviation for the following distribution.

Class 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100

Frequency 3 7 12 15 8 3 2

29. Show that1×22+2×32……+𝑛×(𝑛+1)2

12×2+22×3……+𝑛2×(𝑛+1)=

3𝑛+5

3𝑛+1.

68

KENDRIYA VIDYALAYA SANGATHAN --- SEE, 2018-19

CLASS: XI MAX.MARKS: 100

SUBJECT: MATHEMATICS TIME: 3 Hrs

ANSWER KEY

S.NO. ANSWERS Marks

30. -3 1M

31. I 1M

32. -2

3 1M

33. Exclusive OR 1M

34. Simplifying to im

im

= 1

m = 4

1M

½ M

½ M

35. 3,6,9,…,99

a= 3, d = 3

for solving n = 32

finding Sn = 1632

½ M

½ M

½ M

½ M

36. Given 𝑛𝐶9= 𝑛𝐶8

Using property, n = 17

now 𝑛𝐶15= 17𝐶15

= 136

½ M

½ M

1M

37. (i) 4/20 = 1/5

(ii) 10/20 = ½

1 M

1M

69

38. (i) x – intercept = - 5

(ii) y – intercept = 5/2

1M

1 M

39. Let coordinates of C are (x3, y3, z3)

Centroid = (1, 1, 1) = 3−1+𝑥3

3,−5+7+𝑦3

3,

7−6+𝑧3

3

Solving x3 = 1, y3 = 1, z3 = 2 by equating corresponding co-ordinates.

1M

1M

40. =lim𝑥→0

1+𝑥 − 1

𝑥

1+𝑥 + 1

1+𝑥 + 1

= lim𝑥→0 1+𝑥 − 1

𝑥( 1+𝑥 + 1)

= ½

1M

1M

41. Compound statement with ‗AND‘ : 25 is a multiple of 5 and 8

Compound statement with ‗OR‘ : 25 is a multiple of 5 or 8.

Truthness :

p true and q false, so p and q is false

p true or q false , so p or q is true

½ M

½ M

½ M

½ M

42. Let A: set of individuals exposed to chemical C1

B : set of individuals exposed to chemical C2

n(A) = 120, n(B) = 50, n(AB) = 30

(i) No.of students exposed to Chemical C2 but not chemical to C1

= n(B – A) = n(B) – n(AB) = 50 – 30 = 20

(ii) No.of students exposed to Chemical C1 or Chemical C2

= n(AB) = n(A) + n(B) – n(AB) = 120 + 50 – 30 = 140

Use of wrong chemicals on skin can lead to skin disorders and damaging of

the skin.

1 M

1 M

1 M

1 M

43. (f + g)(x) = f(x) + g(x) = x2 + 2x + 1 1 M

70

(f.g)(x) = f(x).g(x) = x2.(2x + 1) = 2x

3 + x

2

(f+g)(2) = 9

f.g(-1) = -1

1 M

1 M

1 M

44. f(x) = 9 − 𝑥2

9 – x20 => x

2 – 9 0 => x

2 – 3

20 => –3 x 3

Domain = [–3, 3]

Range = [0, 3]

2 M

1 M

1 M

45. 2(1 – sin2x) + 3 sinx = 0

2 – 2sin2x + 3 sinx = 0

2sin2x – 3sinx – 2 = 0

2sin2x – 4sinx + sinx – 2 = 0

(2sinx + 1)(sinx – 2) = 0

2sinx + 1 = 0 or sinx – 2 = 0

Sinx = – ½ or sinx = 2.

But sinx ≠ 2, sinx = – ½ = sin 7𝜋

6

General Solution is given by : x = n + −1 𝑛 7𝜋

6, where nZ

(or)

½{2cos 2xcos2

9 cos 3cos 2

2

xx

x }

½{cos2

3 cos

2

15 cos 3x/2 cos

2

5 xxx

simplifying

½{cos2

15 cos

2

5 xx

= sin 5x sin(5x/2)

1M

2M

1M

(1+1)M

71

1M

46. z = 1

1 + 𝑖 =

1−𝑖

2

= 1

2 −

1

2 𝑖

Compare with a + ib, here a = ½ and b = – ½

r = 1/2

= arg(z) = tan−1 𝑏

𝑎 = tan−1 −1 =

3𝜋

4

so z = r(cos + i sin) = 1

2 𝑐𝑜𝑠

3𝜋

4+ 𝑖 𝑠𝑖𝑛

3𝜋

4

(OR)

Let x + iy = 5 + 12𝑖

(x + iy)2 = 5 + 12i

x2 – y

2 + 2xy i = 5 + 12i

x2 – y

2 = 5 (1)

and 2xy = 12

(x2 + y

2)2 = (x

2 – y

2)2 + (2xy)

2 = (5)

2 +(12)

2 = 25 + 144 = 169

x2 + y

2 = 13 (2)

from (1) and (2) x2 = 9 and y

2 = 4 => x = ±3, y = ±2

since product of xy is positive, so x =3 , y = 2 and x = – 3 and y = – 2

The square root of 5 + 12i are (3 + 2i) and (– 3 – 2i)

1 M

1 M

1 M

1 M

1 M

1 M

1 M

1 M

72

47. No.of letters in the word AMARAVATHI = n = 10

No.of A‘s are 4 .

Total Number of possible words = 10!

4!

(i) If all A‘s are together, then treat these 4 A‘s as one letter then total

number of letters 10 – 4 + 1 = 7 letters .

All these letters can be arranged in 7! Ways.

Total number of words when all A‘s are together is 7!.

(ii) When all A‘s are not together,

Then total number of words = 10!

4! – 7!.

1 M

2 M

1 M

48. Compare with (a – b)n

Here a = 3

2𝑥2 , b =

1

3𝑥, n = 6

General term = (r + 1)th

term = Tr+1 = (– 1)r𝑛𝐶𝑟

an – r

. br

= (– 1)r6𝐶𝑟

3

2𝑥2

6−𝑟

(1

3𝑥)𝑟

= (– 1)r6𝐶𝑟

3

2

6−𝑟

(1

3)𝑟𝑥12−3𝑟

For the term independent of ‗x‘ , 12 – 3r = 0 => 3r = 12 => r = 4.

Term of independent of ‗x‘ = 5th

term = (– 1)46𝐶4

3

2

6−4

(1

3)4

= 15 x 9

4x

1

81 =

5

12

½ M

1 ½ M

1 M

1 M

49. Let E and F be the events that students A and B will qualify the

examination respectively.

Given that P(E) = 0.05, P(F) = 0.10 and P(EF) = 0.02. Then

1 M

73

(i) P(both A and B will not qualify the examination) =

P(EF) = P((EF)) = 1 – P(EF)

= 1 – {P(E) + P(F) – P(EF)}

= 1 – {0.05 + 0.10 – 0.02}

= 1 – 0.13 = 0.87

(ii) P(atleast one of them will not qualify)

= 1 – P(both of them will qualify)

= 1 – 0.02

= 0.98 (OR)

Let A and B denote the students in NCC and NSS respectively.

Given n(A) = 30, n(B) = 32 and n(AB) = 24, n(S) = 60.

P(A) = 30

60; P(A) =

32

60; P(AB) =

24

60

(i) P(student opted for NCC or NSS) = P(AB)

= P(A) + P(B) – P(AB) = 30

60+

32

60−

24

60 =

38

60 =

19

30

(ii) P(student opted neither NCC nor NSS)

= 1 – P(student opted for NCC or NSS) = 1 – 19

30 =

30−19

30 =

11

30

1 ½ M

1 ½ M

1 M

1 ½ M

1 ½ M

50. Given line is x + 3y = 7 (1)

and point is P(3, 8).

Let Q be the foot of the perpendicular of the point Q(3, 8)

Slope of the given line m1 = –1/3.

1M Q

P

74

Slope of the line PQ = m2 = 3.

Equation of PQ is y – 8 = 3(x – 3)

3x – y – 1 = 0 (2).

Solve (1) and (2) to get the foot of the perpendicular.

Which is (1, 2).

(OR)

Find the point of intersection of x + y + 3 = 0 and x – y + 2 = 0

It is (– 5/ 2, –1/2).

Given that y- intercept = c = 4.

The equation of straight line with slope ‗m‘ and y – intercept ‗c‘ is y = mx + c.

Since this line is passing through (–5/2, – ½)

–1/2 = (–5/2) m + 4

Solve it to get m = 9/5

Required line is y = (9/5)x + 4

1 M

2 M

1 M

1 M

1 M

1 M

51. Compare with standard Ellipse, a = 4, b = 6; here a < b

(1) Eccentricity = e = 52

6

(2) Foci = (0, ±be) = (0, ± 52)

(3) Vertices = A(0, ±b) = A(0, ± 6)

(4) Length of Latusrectum = 2𝑎2

𝑏 =

16

3

1 M

1 M

1 M

1M

52.

𝑑

𝑑𝑥 𝑓 𝑥 = lim

𝑕→0

𝑓 𝑥 + 𝑕 − 𝑓(𝑥)

𝑕

Let f(x) = tanx

½ M

75

= lim𝑕→0𝑡𝑎𝑛 𝑥+𝑕 − 𝑡𝑎𝑛 (𝑥)

𝑕

= lim𝑕→01

𝑕

sin (𝑥+𝑕)

cos (𝑥+𝑕)−

𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥

= lim𝑕→01

𝑕

sin 𝑥+𝑕 𝑐𝑜𝑥−cos 𝑥+𝑕 𝑠𝑖𝑛𝑥

cos 𝑥+𝑕 .𝑐𝑜𝑠𝑥

= lim𝑕→01

𝑕

sin (𝑥+𝑕−𝑥)


= lim𝑕→0𝑠𝑖𝑛𝑕

𝑕lim𝑕→0

1


= 1.1

𝑐𝑜𝑠 2𝑥 = sec

2x.

2 M

1 ½ M

53. (i) Finding (AB) = {4, 6}

Finding A B = {4, 6}

(AB) = A B

(ii) Finding (A B) = {1,2,3,4,6,7,8,9}

Finding AB = {1,2,3,4,6,7,8,9}

(A B) = AB.

(iii) Finding A – B = {2, 3, 7,8}

Finding B – A = {1, 9}

A – B ≠ B – A

2M

2M

2M

54. (i) Cos(6x) = cos(2.(3x))

= 2 cos2(3x) – 1

= 2 (4cos3x – 3cosx)

2 – 1

= 2(16cos6x + 9cos

2x – 24cos

4x) – 1

= 32cos6x +18cos

2x – 48cos

4x – 1

(ii) Cot3x = cot (2x + x)

Cot3x = 𝑐𝑜𝑡2𝑥 .𝑐𝑜𝑡𝑥 −1

𝑐𝑜𝑡2𝑥+𝑐𝑜𝑡𝑥

Cot3x(cot2x + cotx) = cot2x.cotx – 1

Cot3x.cot2x + cot3x.cotx = cot2x.cotx – 1

1M

1M

1M

76

Cotx.cot2x – cot2x.cot3x – cot3x.cotx = 1

(OR)

Since 𝜋 < 𝑥 <3𝜋

2, cosx is negative in 3

rd quadrant

Also, 𝜋

2<

𝑥

2<

3𝜋

4, therefore sin

𝑥

2 is positive, cos

𝑥

2 is negative

and tan 𝑥

2 is negative.

Now sec2x = 1 + tan

2x = 1 +

9

16 =

25

16;

cos2x =

16

25; cosx = −

4

5

Now sin 𝑥

2 =

1−𝑐𝑜𝑠𝑥

2 =

3

10

cos 𝑥

2 = −

1 + 𝑐𝑜𝑠𝑥

2 = −

1

10

tan 𝑥

2 =

sin 𝑥

2

cos 𝑥

2 = – 3

1M

2M

1M

1M

1 ½ M

1 ½ M

1M

55. Tr + 1 = 10𝑐 𝑟

kr𝑥 5−

5𝑟

2

Tr + 1 will be independent of x if 5 − 5𝑟

2 = 0 ⇒ r = 2

So term independent of x is T3 = 10𝑐 2k

2

10𝑐 2k

2 = 405 ⇒ k = 3, or -3

2 M

1 M

1 M

2 M

77

(or)

For binomial expansion

Getting equations

Solving a = 2, b= 3, n = 5

1 M

2 M

(1+1+1)=

3M

56. Table for each inequality (three inequalities) (3 x ½ M)

Graph and shading (each one line 3 x 1½ M)

1 ½ M

4 ½ M

57. Preparing the table

Finding the Mean = 62

Finding the variance = 201

Fining Standard Deviation = 14.18 (approximately)

2 M

1 M

2 M

1 M

58. 𝑛×(𝑛+1)2

𝑛2× 𝑛+1 = Expanding and solving

𝑛3+2𝑛2+𝑛

𝑛3+𝑛2 simplifying to 3𝑛+5

3𝑛+1

3 M

3 M

78

The beginning…

For a happy ending…

All the best ***

KENDRIYA VIDYALAYA SANGATHAN · 2018-08-29 · chapter but taught in bits that supplement the...

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Transcript of KENDRIYA VIDYALAYA SANGATHAN · 2018-08-29 · chapter but taught in bits that supplement the...