For B.C.Sc.IT First Semester


Bachelor of Science in Computer Science and Information Technology
Semester:First Course No:MTH-104
Course Title: Calculus and Analytical Geometry

Sub: Maths Full Marks: 80
Time: 3 hrs Pass Marks: 32
Candidades are required to give their answer in their own words as far as practicable

Group – A [10  2 = 20]

1. Verify Rolle’s theorem for the function f(x) = on [ -1,1 ].
2. Find the length of the curve y = from x = 0 to x=2.
3. State integral test.

4. Define normal and curvature of a space curve.
5. Find the eccentricity of the hyperbola .

6. Find the area of the region that lies inside the cardioid r = and outside the
circle r = 1.

7. Express and in terms of r and s if ; x = r – s ; y = r + s.

8. Using partial derivatives , find if .

9. Find the area of the region enclosed by the parabola and the line y = -x.
10. Solve partial differential equation .

Group – B (5X4=20)
11. State and prove Roll’s theorem for a differenciable function.

12. Find the volume of the prism using double integral whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = f(x,y) = 3 – x – y.

OR
Evaluate: ,where R is the semicircular region bounded by the x-axis
and the curve y = .

13. Find curvature and torsion of a space curve: .
14. Test the series:
15. What is meant by directional derivative in the plane? Obtain the derivative of the function f(x,y) = at point (2,0) in the direction of
OR
Define a curvature of a curve.Prove that the curvature of a circle of radius is Ya.

Group-C [ 5X8 = 40]

16. Graph the function y =

17.Find the Taylor series and Taylor polynomials generated by the function f(x) = cosx
at x = 0.

18. Evaluate the double integral by applying the transformation and and integrating over an appropriate region in the uv-plane. OR
Find the volume of the region D enclosed by the surfaces and
.
19.Find the maxima and minima f(x,y) =
Also find the saddle point if it exsists.

20.Find the solution of wave equation.
OR
Find the particular integral of the equation .Where

THE END

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y = mx + c

Best of Luck to All the examinees 2068 of Grade XI


Operation on Mathematics for Grade XI

Five best steps to secure High score on the basis of Basic Mathematics.

First Step: 20 Marks(Q.No.1a,b,c+Q.No.6a,b+Q.No.11)

Title: Sets,Real Number System and Logic : Q.No.1(a) + Q.No.6(a)
(6Marks)

Chapters: Sets,RNS & lOGIC (1SQ+1LQ = 6Marks)

Page9: Eg.2,5,6,7 + Ex.1.1 Q.No.4a,11,13 +
Page19:De’Morgan’s law + Ex.1.2 Q.No.6d +
Page32:absolute Value +Page34: Eg.4 + Page37: Eg.7,8 +
Ex1.3.Q.No.4a + Page49:six definitions.+ Page53: Eg.5.6,7
+ Ex.1.4.Q.No.4all.

Title: Relation,Function & Graphs : Q.No.1(b) + Q.No.11 (8Marks)

Chapters: R,F &G (1SQ+1VLQ = 8Marks)

Definitions: Cartesian Product,Relation ,Domain and Range & Inverse
Relation. Page65:Eg.5,6,7 + Ex.2.1 Q.No.7a,b +
Definitions of Function,Inverse Function & Composite Function +
Page78:Eg.3,4,8,10 + Ex.2.2Q.No.5,9(iii),(v) ,10 + Page91: Eg.3,6 +
Ex 2.3 Q.No.8 all + Page109: Eg.2,5 + Page91: Eg.3,6 + Page113:
Q.No.8,11,12

Title: Curve Sketching : Q.No.1(c) + Q.No.6(b) (6Marks)

Chapters: Curve Sketching (1SQ+1LQ = 6Marks)

Definitions of Even & Odd Function + Increasing &decreasing Function
+ Periodicity and Symmetricity of a Function & Asymptote.
Page122: Eg.1,3,5,6.

Note: (i) Logic and Curve Sketching are two new chapters which carries 6 marks but we should practice the questions from HSEB model , HISHAN exams, and the Send up Exams of reputed Colleges in the Valley.
Like; y = f (x) = (x + 1) (x – 2) (x – 3) .
(ii) From sets or Real number System ,a 4 marks question is asked which is in choice.

Second Step: 20 Marks(Q.No.2a,b,c+Q.No.7a,b+Q.No.12)

Title: General Values : Q.No.2(a) (2Marks)

Chapters: General Values (1SQ = 2Marks)

Page 147: Eg.3,8,910 + Ex.4.2 Q.No.3d,8a,14(i),(ii) + Page155: Eg. 4all

Title: Inverse Circular Function Or Properties & Solution of triangle :
Q.No.7(a) Choice Question(4Marks)

Chapters: Inv Cir Fun Or Prop & Solution of triangle (1LQ = 4Marks)

Page164: Eg.5,6,7,10,12 + Ex.5.1 Q.No.6h,k,7a,b,9(iv),11 +
Page97: Eg.3,4(i)

Or The Cosine Law + Sine Law + Tangent Law +
Page 183: Eg 3,7 + Ex 6.1 Q.No.6,7,8,14 + Page189:6(iii),7(i),8,13
+ Page194: Eg.2,3,4,5 + Ex.7.1:2(i),(ii),(iii)

Title: Sequence and Series & Mathematical Induction
: Q.No.2(b) + Q.No.12 (8Marks)

Chapters: S and S & Mathematical Induction (1SQ+1VLQ = 8Marks)

Page208: Eg.3,4 + Page210:Eg.2,4,6,8 + Ex.8.1 Q.No.1c,2a,b,6 , 8(i)
+ Page225:Eg1,2,4,5 + Page230:3,7b +Ex8.4.Q.No. 2(i),(ii),(iii),4(ii)

Title: Matrix and Determinant : Q.No.2(c) + Q.No.7(b) (6Marks)

Chapters: Matrix and Determinant (1SQ+1LQ = 6Marks)

Page246: Eg.1 + Ex.9.1Q.No.4,10 + Page261:Eg4,5,6 +
Ex9.2Q.No.5(iv),(v) ;6(vi),(viii) + Page278:Eg.3,4,5.

Note: (i) Sequence and Series & Mathematical Induction are two new chapters which carries 8 marks but we should practice the questions from HSEB model , HISHAN exams, and the Send up Exams of reputed Colleges in the Valley.

(ii) From Inverse Circular Function or Properties & Solution of triangle ,a 4 marks question is asked which is in choice.

Third Step: 20 Marks(Q.No.3a,b,c+Q.No.8a,b+Q.No.14)

Title: System of Linear Equation : Q.No.3(a) + Q.No.8(a) (6 Marks)

Chapters: System of Linear Equation (1SQ + 2LQ = 6 Marks)

Page246: Eg.1,2,3 + Ex.10.2.Q.No.2a,j; 3a,j
+Page300:Eg1,2 + Page308:Eg.1,2,3 +
Ex10.3.Q.No.1a,g;2a,f .

Title: Complex Number : Q.No.3(b) + Q.No.14 ( 8Marks )

Chapters: Complex Number (1SQ + 1VLQ = 8Marks)

Page 319: Eg.4 + Ex.11.1.Q.No.4,5 + Page323:absolute
Value + The triangle inequality + Page325:Eg1,2,3,4,5,6,7 +
Page332: Eg.1,2 + De’Moivre’s Theorem + Page342:Eg.7,8 +
Ex.11.4 Q.No.4a,g;5a;7a,b + Page348:Eg.10,13.

Title: Polynomial Equation. : Q.No.3(c) + Q.No.(b) (6Marks)

Chapters: Polynomial Equation (1SQ + 1LQ = 6Marks)

Page353:Theorem.2 + Page355:Eg1,2,3,4 + Ex.12.1 Q.No.2,3 + Page360:Eg.2,4,8 + Ex.12.2Q.No.3a;10a;11b;12 + Page366:Condition
for one root Common (Proof needed) + Page367:Eg2,3 +
Ex.12.3 Q.No.4 + Page370:18,19.

Note: (i) System of linear equations contains different process for solution in which we must be cleared on solving by Row Equivalent Method.
(ii) From Complex Number ,a 6 marks question is asked which is in compulsory to do so give more time and learn Theorems also.

Fourth Step: 14 Marks(Q.No.4a,b + Q.No.9a + Q.No.13)

Title: Co-ordinates(St.+ Pair of Lines):Q.No.4(a) + Q.No.13Choice (8Marks)

Chapters: Co-ordinates(St.+ Pair of Lines) (1SQ + 1VLQ = 8Marks)
Page377:Linear Equation Theorem (Proof) + Page 382 : Eg.4,6,7 +
Ex.13.1 Q.No.9,11 + Page391: Eg. 4,7,8 +
Ex.13.2 Q.No.1a,3c,4,7 +
Page.400:Derivation of Length of Perpendicular +
Bisector of the Angles.
+ Page.405:Eg.3,4,6,8 + Ex13.3 Q.No.2,5b,9,10,11 +
Page416:Condition to represent pair of Lines +
Page420:Eg.3,4,5,7,11 + Ex14.1 Q.No.5a,6a,b;8,10d,11 +
Page430: Q.No.1 to 7 should not leave for
Six Marks question.

Title: Circle : Q.No.4(b) + Q.No.9(a) (6Marks)

Chapters: Circle (1SQ+1LQ = 6Marks)

Page436:Eg2,3,4,6,9 + Ex.15.1 Q.No.1j,2,2e,3c,6 +
Page446:Condition of tangency of a straight line to a circle. +
Page448:Ex 2,5 + Ex15.24a,5c,8,9b

Note: (i) In this Step we give more time to practice derivation types question non related to SLC since a 6 marks question is asked in choice either from St. Lines OR Pair of St. lines.

Fifth Step: 26 Marks(Q.No.4c +Q.No.5a,b,c +Q.No.9b +Q.No.10a,b Q.No.14)
Calculus
Title: Limit & Continuity : Q.No.4(c) + Q.No.9(b)(Choice Question)
(6Marks)

Chapters: Limit & Continuity (1SQ+1LQ = 6Marks)

Page464.Eg.7,9 + Ex.16.1 Q.No.2g,j,k; 4d +
Page468:Theorem + Page 470:Eg.3,4,5 +
Ex.16.2 Q.No.11,17,22,24,27,28 + Page475:Eg.1,2 +
Ex.16.3.Q.No.1d,2b + Page.480.Eg.2,3 +
Ex.16.4Q.No.2(iv),(v) ;4(i) + Page485:11(i) ,(v),(x) +
Page487 :Eg.6,7,8(iii),12,13.

Title: Derivatives : Q.No.5(a) + Q.No.10(a)(First Principle) (6Marks)

Chapters: Derivatives (1SQ + 1LQ = 6Marks)

Page495.Eg.2 + Page504:Eg.3,4,5,9 + Ex.17.1 Q.No.5(xi),(xiii);9(iv) + Page515: Eg.3,4 + Page519:Eg.2 + Ex.17.2 Q.No.1(vi),(viii),(ix),3(iv),(vii);4(vi);5(v);7(x);8(iv);9(i),(v),(vii),10(i) + Page525:Eg.1,2 + Page526:Eg.I;II + Page528:Eg.3,4 + Ex.17.3Q.No.1(iii),(v) ;2(vii);3(i),(vi);,6(iii),(v) + Page533:12(i),(vi) ;13(iv).

Title: Application of derivatives : Q.No.5(b) + Q.No.15 (8Marks)

Chapters: Application of derivatives (1SQ + 1VLQ = 8Marks)

Definitions: Incresing and decreasing of the Curve of a function + Local Max & Min + Absolute Max & Min + Concavity & Convexivity of Curves of a Function + Stationary Points + Point of Inflection + Page 547.Eg.3,5,6 + Ex.18.1.Q.No.5(iv),(vii) ;6(i),(ii) ,10 + Page554:Eg.2,4,7 + Ex.18.2 Q.No.3(a);4(a),7,8,9 + Page561: Eg.8,11

Title: Antiderivative & it’s application : Q.No.5(c) + Q.No.10(b) (6Marks)

Chapters: Antiderivative & it’s application (1SQ + 1LQ = 6Marks)

Page565.Eg.5,7,10,13 + Ex.19.1 Q.No.4(ii),(xi);5all;7(iii) +
Page574: Eg.5,9,10 + Ex.19.2.Q.No.2(iii),(viii),(xv),(xvi),(xvii);(xviii);(xxvii);(xxviii); + Page580:Eg.2,4,6 + Ex.19.3.Q.No.1(i),2(iv),3(i),4(ii) +
Page585:Eg.2,5,6 + Ex.19.4.Q.No.1a,k,o,v,w,x,z +
Page590:Eg.3,5,8 + Ex.19.5.Q.No.9,18,36,38,40 +
Page602:Ex.3,4,5,6 + Ex.19.7.Q.No.1e,f ; 2(ii);4(i),(vi) ; 5(ii) ;6,7,9
+ Page607:6(iii),(iv) ,(vi).

Note: (i) If anyone wants to write to comment and needs questions of different Chapters except the above mentioned questions mail me at the address mbybx@yahoo.com ,I welcome to You.
(ii) The students of Science and Commerce all can practice in the above patern.

Finally Best of Luck to All the examinees 2068.

Operation on Mathematics for Grade XI


Five best steps to secure High score on the basis of Basic Mathematics.

First Step: 20 Marks(Q.No.1a,b,c+Q.No.6a,b+Q.No.11)

Title: Sets,Real Number System and Logic : Q.No.1(a) + Q.No.6(a)
(6Marks)

Chapters: Sets,RNS & lOGIC (1SQ+1LQ = 6Marks)

Page9: Eg.2,5,6,7 + Ex.1.1 Q.No.4a,11,13 +
Page19:De’Morgan’s law + Ex.1.2 Q.No.6d +
Page32:absolute Value +Page34: Eg.4 + Page37: Eg.7,8 +
Ex1.3.Q.No.4a + Page49:six definitions.+ Page53: Eg.5.6,7
+ Ex.1.4.Q.No.4all.

Title: Relation,Function & Graphs : Q.No.1(b) + Q.No.11 (8Marks)

Chapters: R,F &G (1SQ+1VLQ = 8Marks)

Definitions: Cartesian Product,Relation ,Domain and Range & Inverse
Relation. Page65:Eg.5,6,7 + Ex.2.1 Q.No.7a,b +
Definitions of Function,Inverse Function & Composite Function +
Page78:Eg.3,4,8,10 + Ex.2.2Q.No.5,9(iii),(v) ,10 + Page91: Eg.3,6 +
Ex 2.3 Q.No.8 all + Page109: Eg.2,5 + Page91: Eg.3,6 + Page113:
Q.No.8,11,12

Title: Curve Sketching : Q.No.1(c) + Q.No.6(b) (6Marks)

Chapters: Curve Sketching (1SQ+1LQ = 6Marks)

Definitions of Even & Odd Function + Increasing &decreasing Function
+ Periodicity and Symmetricity of a Function & Asymptote.
Page122: Eg.1,3,5,6.

Second Step: 20 Marks(Q.No.2a,b,c+Q.No.7a,b+Q.No.12)

Title: General Values : Q.No.2(a) (2Marks)

Chapters: General Values (1SQ = 2Marks)

Page 147: Eg.3,8,910 + Ex.4.2 Q.No.3d,8a,14(i),(ii) + Page155: Eg. 4all

Title: Inverse Circular Function Or Properties & Solution of triangle :
Q.No.7(a) Choice Question(4Marks)

Chapters: Inv Cir Fun Or Prop & Solution of triangle (1LQ = 4Marks)

Page164: Eg.5,6,7,10,12 + Ex.5.1 Q.No.6h,k,7a,b,9(iv),11 +
Page97: Eg.3,4(i)

Or The Cosine Law + Sine Law + Tangent Law +
Page 183: Eg 3,7 + Ex 6.1 Q.No.6,7,8,14 + Page189:6(iii),7(i),8,13
+ Page194: Eg.2,3,4,5 + Ex.7.1:2(i),(ii),(iii)

Title: Sequence and Series & Mathematical Induction
: Q.No.2(b) + Q.No.12 (8Marks)

Chapters: S and S & Mathematical Induction (1SQ+1VLQ = 8Marks)

Page208: Eg.3,4 + Page210:Eg.2,4,6,8 + Ex.8.1 Q.No.1c,2a,b,6 , 8(i)
+ Page225:Eg1,2,4,5 + Page230:3,7b +Ex8.4.Q.No. 2(i),(ii),(iii),4(ii)

Title: Matrix and Determinant : Q.No.2(c) + Q.No.7(b) (6Marks)

Chapters: Matrix and Determinant (1SQ+1LQ = 6Marks)

Page246: Eg.1 + Ex.9.1Q.No.4,10 + Page261:Eg4,5,6 +
Ex9.2Q.No.5(iv),(v) ;6(vi),(viii) + Page278:Eg.3,4,5.

Third Step: 20 Marks(Q.No.3a,b,c+Q.No.8a,b+Q.No.14)

Title: System of Linear Equation : Q.No.3(a) + Q.No.8(a) (6 Marks)

Chapters: System of Linear Equation (1SQ + 2LQ = 6 Marks)

Page246: Eg.1,2,3 + Ex.10.2.Q.No.2a,j; 3a,j
+Page300:Eg1,2 + Page308:Eg.1,2,3 +
Ex10.3.Q.No.1a,g;2a,f .

Title: Complex Number : Q.No.3(b) + Q.No.14 ( 8Marks )

Chapters: Complex Number (1SQ + 1VLQ = 8Marks)

Page 319: Eg.4 + Ex.11.1.Q.No.4,5 + Page323:absolute
Value + The triangle inequality + Page325:Eg1,2,3,4,5,6,7 +
Page332: Eg.1,2 + De’Moivre’s Theorem + Page342:Eg.7,8 +
Ex.11.4 Q.No.4a,g;5a;7a,b + Page348:Eg.10,13.

Title: Polynomial Equation. : Q.No.3(c) + Q.No.(b) (6Marks)

Chapters: Polynomial Equation (1SQ + 1LQ = 6Marks)

Page353:Theorem.2 + Page355:Eg1,2,3,4 + Ex.12.1 Q.No.2,3 + Page360:Eg.2,4,8 + Ex.12.2Q.No.3a;10a;11b;12 + Page366:Condition
for one root Common (Proof needed) + Page367:Eg2,3 +
Ex.12.3 Q.No.4 + Page370:18,19.

Fourth Step: 14 Marks(Q.No.4a,b + Q.No.9a + Q.No.13)

Title: Co-ordinates(St.+ Pair of Lines):Q.No.4(a) + Q.No.13Choice (8Marks)

Chapters: Co-ordinates(St.+ Pair of Lines) (1SQ + 1VLQ = 8Marks)
Page377:Linear Equation Theorem (Proof) + Page 382 : Eg.4,6,7 +
Ex.13.1 Q.No.9,11 + Page391: Eg. 4,7,8 +
Ex.13.2 Q.No.1a,3c,4,7 +
Page.400:Derivation of Length of Perpendicular +
Bisector of the Angles.
+ Page.405:Eg.3,4,6,8 + Ex13.3 Q.No.2,5b,9,10,11 +
Page416:Condition to represent pair of Lines +
Page420:Eg.3,4,5,7,11 + Ex14.1 Q.No.5a,6a,b;8,10d,11 +
Page430: Q.No.1 to 7 should not leave for
Six Marks question.

Title: Circle : Q.No.4(b) + Q.No.9(a) (6Marks)

Chapters: Circle (1SQ+1LQ = 6Marks)

Page436:Eg2,3,4,6,9 + Ex.15.1 Q.No.1j,2,2e,3c,6 +
Page446:Condition of tangency of a straight line to a circle. +
Page448:Ex 2,5 + Ex15.24a,5c,8,9b

Operation on Mathematics for Grade XI


Five best steps to secure High score on the basis of Basic Mathematics.

First Step: 20 Marks(Q.No.1a,b,c+Q.No.6a,b+Q.No.11)

Title: Sets,Real Number System and Logic : Q.No.1(a) + Q.No.6(a)
(6Marks)

Chapters: Sets,RNS & lOGIC (1SQ+1LQ = 6Marks)

Page9: Eg.2,5,6,7 + Ex.1.1 Q.No.4a,11,13 +
Page19:De’Morgan’s law + Ex.1.2 Q.No.6d +
Page32:absolute Value +Page34: Eg.4 + Page37: Eg.7,8 +
Ex1.3.Q.No.4a + Page49:six definitions.+ Page53: Eg.5.6,7
+ Ex.1.4.Q.No.4all.

Title: Relation,Function & Graphs : Q.No.1(b) + Q.No.11 (8Marks)

Chapters: R,F &G (1SQ+1VLQ = 8Marks)

Definitions: Cartesian Product,Relation ,Domain and Range & Inverse
Relation. Page65:Eg.5,6,7 + Ex.2.1 Q.No.7a,b + Definitions of
Function,Inverse Function & Composite Function + Page78:Eg.3,4,8,10
+ Ex.2.2Q.No.5,9(iii),(v) ,10 + Page91: Eg.3,6 + Ex 2.3 Q.No.8 all +
Page109: Eg.2,5 + Page91: Eg.3,6 + Page113: Q.No.8,11,12

Title: Curve Sketching : Q.No.1(c) + Q.No.6(b) (6Marks)

Chapters: Curve Sketching (1SQ+1LQ = 6Marks)

Definitions of Even & Odd Function + Increasing &decreasing Function
+ Periodicity and Symmetricity of a Function & Asymptote. +
Page122: Eg.1,3,5,6.

Second Step: 20 Marks(Q.No.2a,b,c+Q.No.7a,b+Q.No.12)

Title: General Values : Q.No.2(a) (2Marks)

Chapters: General Values (1SQ = 2Marks)

Page 147: Eg.3,8,910 + Ex.4.2 Q.No.3d,8a,14(i),(ii) + Page155: Eg. 4al
l

Title: Inverse Circular Function Or Properties & Solution of triangle :
Q.No.7(a) Choice Question(4Marks)

Chapters: Inv Cir Fun Or Prop & Solution of triangle (1LQ = 4Marks)

Page164: Eg.5,6,7,10,12 + Ex.5.1 Q.No.6h,k,7a,b,9(iv),11 +
Page97: Eg.3,4(i)

Or The Cosine Law + Sine Law + Tangent Law +
Page 183: Eg 3,7 + Ex 6.1 Q.No.6,7,8,14 + Page189:6(iii),7(i),8,13
+ Page194: Eg.2,3,4,5 + Ex.7.1:2(i),(ii),(iii)

Title: Sequence and Series & Mathematical Induction
: Q.No.2(b) + Q.No.12 (8Marks)

Chapters: S and S & Mathematical Induction (1SQ+1VLQ = 8Marks)

Page208: Eg.3,4 + Page210:Eg.2,4,6,8 + Ex.8.1 Q.No.1c,2a,b,6 , 8(i)
+Page225:Eg1,2,4,5 + Page230:3,7b +Ex8.4.Q.No. 2(i),(ii),(iii),4(ii)

Title: Matrix and Determinant : Q.No.2(c) + Q.No.7(b) (6Marks)

Chapters: Matrix and Determinant (1SQ+1LQ = 6Marks)

Page246: Eg.1 + Ex.9.1Q.No.4,10 + Page261:Eg4,5,6 +
Ex9.2Q.No.5(iv),(v) ;6(vi),(viii) + Page278:Eg.3,4,5.

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