Mathematics_Gen_DSC1A

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B.Sc. General Examinations 2020

(Under CBCS Pattern)

Semester - I

Subject: MATHEMATICS

Paper: DSC 1A/2A/3A-T (Differential Calculus)

Candidates are required to give their answers in their own words as far as practicable.

The figures in the margin indicate full marks.

Question Paper

VIDYASAGAR UNIVERSITY

Answer any three from the following questions : 3×20

1. (a) If

 

cos , if

2 2

7 if

2

a x

x x f x

x



 

  

 



for what values of ‘a’

 

2

lim ?

2

x

f x f





   

 

(b) If y x ⁿ^¹logx, then show that



^{1 !}



n

y n

x

  .

(c) If a and b are distinct real numbers, show that there exists a real number c between a and b such that _a²__{ab b}_ ² _₃_c².

Full Marks : 60

Time : 3 Hours

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(d) Find the radius of curvarture of the curvey y xe ^^x at the point where ^y is maximum.

(e) If ^{f x y}



^



^ ^{f x f y}

     

^. ^, ^f^ ⁰ ^^3, ^f

 

⁵ ^^2,^find^f^

 

⁵ ^. ^4×5=20

2. (a) If ^{f x}

 

^^tan ^x, then show that

 

⁰ ₂ ²

 

⁰ ₄ ⁴

 

⁰ ^{... sin}

2

n n n

C C

f n f _ n f _ n

   

(b) If ^u^ ^{f x y}

 

^, ^andx r cos , y r sin , then prove that

2 2 2 2

2

1

u u u u

x y r r 

 

   

        

      

        10+10

3. (a) If lx my 1 is a normal of the parabola y² 4ax, then show that _al³_₂_alm² __m². (b) Find the value of a, so that 0 3

sin sin 2 lim^x tan

a x x

x



 exists finitely and find the limit. 10+10

4. (a) A function ^f ^{: 0,1}

 

is continuous on

 

0,1 . Prove that there exists a point c in

 

^{0,1 such}

that ^{f c}

 

^^c.

(b) Using mean value theorem show that 1 1

0 log 1

ex

x x

  

  

  , for x0. 10+10

5. (a) If p p₁, ₂ be the radii of curvarture at the extremities of any chord of the cardioid



^{1 cos}



r a   , which passes through the pole, then prove that ₁² ₂² 16 ² 9 p  p  a .

(b) Find the infinite series expansion of the function sin ,x x. 10+10

6. (a) Show that

     

   

2 2, , 0,0

,

0 , , 0,0

xy x y

x y f x y

x y

 

  

 



has partical derivatives but is not

continuous at (0, 0).

(b) If the normal to the curve

2 2 2

3 3 3

x y a makes an angel  with the axis of x, show that its

equation is ycosxsin acos 2. 10+10

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