MB Subedi, Lecturer of Mathematics

For B.C.Sc.IT First Semester

November 18, 2011 — MB Subedi

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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August 31, 2011 — MB Subedi

$y = mx + c$

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Best of Luck to All the examinees 2068 of Grade XI

May 3, 2011 — MB Subedi

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.

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Operation on Mathematics for Grade XI

May 2, 2011 — MB Subedi