Bsc First Semester Maths Paper 2023

      

Bsc First Semester Model Paper 2023

      Bsc First-year Mathematics Model Paper 2023

        Allahabad state university Bsc Paper 2023

4143

Bsc ( first semester) Examination 2023

नई शिक्षा नीति के अनुसार 

Mathematics

Differential Calculus & Integral Calculus

[Time : Two Hours]             [Maximum Marks: 75]

Note : Question Paper is divided into three sections

'A"B'and'C'. Attempt questions from each section as directed.

नोट: प्रश्न पत्र तीन खंडों में खंडों में' अ'' ब' और ' स' में

विभाजित है सभी खंडों से प्रश्नो का उत्तर निर्देशानुसार

दीजिए।

                      Section -A

                         खण्ड - अ

    Very Short Answer Type Question

                   अति लघुउत्तरीय   प्रश्न

1. Answer any three out of 5 sub questions.Each sub question carries 3 marks.

                                                  (3 x3 = 9)

(a) Definition of Convergent and Divergent Series.

(b) Define Evolutes and Involutes

(c) Find the intrinsic equation of curve 

        r = (1 - cos θ)

(d) Find the pedal equation of curve

      r =  aeθcot α

(e) Define stoke's theorem.

                             Section -B

                                 खण्ड- ब

                  Short Answer Type Question

                             लघु उत्तरीय प्रश्न

 2. Answer any four out of 7 sub questions. Each

     sub question carries 9 marks. (4 x9 =36)

(a) State and prove the Euler theorem.

 (b) Find the length of astroid  

                    x²ᐟ³ + y²ᐟ³ = a²ᐟ³

(c) Prove that every convergent sequence is a

    cauchy sequence. 

(d) Show that Γ1/2 = √π

(e) State and prove Green's theorem in plane.

(f) Show that Continuity is a necessary but not

      a sufficient condition for differentiability.

(g) Find the maximum and minimum of

      following function sin x + sin y + sin(x + y ).

                            Section- C

                              खण्ड- स

Long Answer Type Question

3. Answer any two sub question out of four

questions. Each sub question carries 15

marks. (15x 2 = 30 )

 (a) State and Prove Fundamentals Theorem of

       Integral Calculus.

(b) If u = log( x³+ y³ +z³ -3xyz) show that

       (i) ∂u/∂x + ∂u/∂y + ∂u/∂z = 3/ x+y+z

       (ii) (∂/∂x + ∂/∂y+∂z)² u= -9/ (x +y + z)²

(c) Prove that Γn (n- 1) = π/sin nπ 0<n<1

(d) Prove that   

     grad (u.v) = u x curl v +v x curl u + (u.▽ )T+

 (V.▽ )ū