📚 FYUG (NEP) previous year question papers solution

ASSAM UNIVERSITY, SILCHAR

FYUG 1st semester Physics DSC 101 Previous Year Question Papers Solutions

UNIT 3

Vector Differentiation: Directional derivatives and normal derivatives. Gradient of a scalar field and its geometrical interpretation. Divergence and curl of a vector field. Laplacian operator. Vector identities.
Vector Integration: Ordinary Integrals of Vectors. Line, surface and volume integrals of Vector fields. Gauss's divergence theorem and Stokes Theorem.

2019

(FYUG PHY-DSC101 Same as CBCS PHҮНСС–101T )

1. If A + B + C = 0, show that
(A × B) = (B × C) = (C × A)

2. Find the value of a such that the vectors
2i − j + k, i + 2j − 3k and 3i + aj + 5k
are coplanar.

3. Show that ∇φ is a vector normal to the surface
φ(x, y, z) = C

4. For any three vectors A, B and C, show that
A · (B × C) = C · (A × B) = B · (C × A)

5. Show that
∇ × (∇ × A) = ∇(∇ · A) − ∇²A

6. A vector field is defined as
A = r r²

where r = xi + yj + zk
Evaluate ∇ · A and ∇ × A and hence state whether the field is solenoidal or irrotational.

7. Show that
A × (B × C) + C × (A × B) + B × (C × A) = 0

8. Calculate the volume integral of
(∇ · r)
over the volume enclosed by a sphere of radius a, where
r = xi + yj + zk.

(b) If

F = ∇φ

everywhere in a region R and φ is single-valued and has continuous derivatives in R, then show that

∫_A^B F · dr

is independent of the path joining the points A and B.

(c) Calculate the work done when a force

F = 3xyi − y²j

moves a particle in xy-plane from (0, 0) to (1, 2) along the parabola

y = 2x².

6. Answer (a) or (b) :

(a) Verify Gauss divergence theorem for the vector

A = x²i + y²j + z²k

over the surface of a cube bounded by

0 ≤ x, y, z ≤ 1.

(b) State and prove Stokes’ theorem.

2020

(FYUG PHY-DSC101 Same as CBCS PHҮНСС–101T )

(e) Prove that

A × (B × C) + B × (C × A) + C × (A × B) = 0

(f) Find the value of m for which the vectors A, B and C are coplanar :

A = 2i − j + k

B = i + 2j − 3k

C = 3i + mj + 5k

(g) For vector

R = xi + yj + zk

find the divergence.

(h) Find the value of b for which the vector

A = (2x + 3y)i + (6y − 3z)j + (6x − 12z)k

is solenoidal.

4. What is gradient of a scalar function? Give its physical interpretation. Show that

∇rⁿ = nrⁿ⁻²r

where r = xi + yj + zk

5. (a) Give the physical significance of ‘divergence’ and ‘curl’.

(b) Prove that

curl (grad φ) = 0

div (curl A) = 0

where φ is a scalar and A is a vector.

(i) Evaluate :

∫_x=0¹ ∫_y=0² (x² + 3xy²) dxdy

(j) Find the value of

∫₀¹ ∫₀¹ ∫₀¹ (x² + y² + z²) dxdydz

(k) For a given force

F = 4xyi − 8yj − 2k,

find the work done along straight line from

(0, 0, 0) to (3, 1, 2).

(l) Using Gauss divergence theorem, express the Gauss law in electrostatics in differential form.

6. (a) Evaluate

∬_S ( yzi + zxj + xyk ) · dS

where S is the surface of the sphere

x² + y² + z² = 4

in the first octant.

6. (b) Evaluate

∭_V (x² + y² + z²) dx dy dz

where V is sphere having centre at origin and radius r.

7. State and prove Gauss’ divergence theorem.

2021

(FYUG PHY-DSC101 Same as CBCS PHҮНСС–101T )

4. Find the area of the parallelogram whose adjacent sides are

i − 2j + 3k and 2i + j − 4k.

5. Find the volume of the parallelepiped if

a = −3i + 7j + 5k,

b = −3i + 7j − 3k

and

c = 7i − 5j − 3k

are the three coterminous edges of the parallelepiped.

6. At any point of the curve

x = 3 cos t, y = 3 sin t, z = 4t,

find the tangent vector.

18. (a) Find m so that the vectors

2i − 4j + 5k,

i − mj + k

and

3i + 2j − 5k

are coplanar.

(b) Let

a = i + j − k,

b = i − j + k,

c = i − j − k.

Find the vector

a × (b × c)

19. (a) If

da dt = u × a

and

db dt = u × b,

then prove that

d(a × b) dt = u × (a × b)

(b) If

φ = 3x²y − y³z²,

then find grad φ at the point (1, −2, −1).

2023

(FYUG PHY-DSC101 )

23. Define order and degree of a differential equation.

25. What do you mean by ordinary differential equation (ODE)? Give an example of ODE.

26.hen is a differential equation of the form Mdx+ Ndy = 0 said to be exact or inexact

18. (a) Solve the differential equation

(x² + y²)dx + 2xy dy = 0

(b) Solve the differential equation

dy dx + 2xy = 2e^−x²

19. (a) What is auxiliary equation? If m₁ and m₂ are the two roots of the auxiliary equation, then write the expression of complementary function for the cases

m₁ = m₂

and

m₁ ≠ m₂

(b) Find the solution of the differential equation

d²y dx² + 4 dy dx + 4y = x²

when y(0) = 0 and y′(0) = ¹/₂.

4. State Green's theorem.

7. If a force

F = 2x²yi + 3xyj

displaces a particle in the xy-plane from (0, 0) to (1, 4) along a curve

y = 4x²,

then find the work done.

8. Evaluate by Stokes’ theorem

∮_C ( yz dx + zx dy + xy dz )

where C is the curve

x² + y² = 1, z = y².

20. (a) If

F = 2yi − zj + xk,

then evaluate

∫_C F · dr

along the curve

x = cos t, y = sin t, z = 2 cos t

from t = 0 to t = π/2.

20. (b) A vector field is given by

F = (sin y)i + x(1 + cos y)j.

Evaluate the line integral over a circular path

x² + y² = a², z = 0.

21. (a) Using Green’s theorem, evaluate

∮_C ( x²y dx + x² dy )

where C is the boundary described counter-clockwise of the triangle with vertices

(0, 0), (1, 0), (1, 1).

21. (b) Using Stokes’ theorem, evaluate

∮_C [ (2x − y)dx − yz²dy − y²z dz ]

where C is the circle

x² + y² = 1

corresponding to the surface of the sphere of unit radius.

2023

(FYUG PHY-DSC101 )

7. State and explain Stokes’ theorem.

8. Define divergence of a vector. Whether divergence of a vector is scalar or vector?

9. Check and predict whether the vector

r = xi + yj + zk

is irrotational vector or not.

20. (a) Show that

∇ · (∇ × A) = 0

and

∇ × (∇φ) = 0

(b) Show that

∇ × (rⁿr) = 0

(a) State and prove Gauss’ divergence theorem.

(b) Find the value of n for which the vector

rⁿr

is solenoidal, where

r = xi + yj + zk.

2024

(FYUG PHY-DSC101 )

Define solenoidal and irrotational fields.

State and explain Gauss divergence theorem.

(c) Find grad φ at the point

(−1, −2, 1)

where

φ = x²y + xz.

8. (a) Define directional derivative. Find its expression.

(b) Define gradient of a scalar function. Explain the geometrical interpretation of gradient.

OR

9. (a) State and prove Stokes’ theorem.

(b) Show that

∇rⁿ = nrⁿ⁻²r

where r is the position vector.

CBCS PYQ SAME AS FYUG PHYSICS DSC 101

CBCS PHYHCC-101 Question Paper 2019	Click Here
CBCS PHYHCC-101 Question Paper 2020	Click Here
CBCS PHSHCC-101 Question Paper 2021	Click Here

FYUG PYQ PHYSICS DSC 201

FYUG PHYSICS DSC 101 Question Paper 2023	Click Here
FYUG PHYSICS DSC 101 Question Paper 2024	Click Here

See more content

FYUG Syllabus

FYUG SYLLABUS SEMSTER 1	Click Here
FYUG SYLLABUS SEMSTER 2	Click Here
FYUG SYLLABUS SEMSTER 3	Click Here
FYUG SYLLABUS SEMSTER 4	Click Here
FYUG SYLLABUS SEMSTER 5	Click Here
FYUG SYLLABUS SEMSTER 6	Click Here
FYUG SYLLABUS SEMSTER 7	Click Here
FYUG SYLLABUS SEMSTER 8	Click Here

ASSAM UNIVERSITY, SILCHAR

UNIT 3

2019

(FYUG PHY-DSC101 Same as CBCS PHҮНСС–101T )

1. If A + B + C = 0, show that (A × B) = (B × C) = (C × A)

2. Find the value of a such that the vectors 2i − j + k, i + 2j − 3k and 3i + aj + 5k are coplanar.

3. Show that ∇φ is a vector normal to the surface φ(x, y, z) = C

4. For any three vectors A, B and C, show that A · (B × C) = C · (A × B) = B · (C × A)

5. Show that ∇ × (∇ × A) = ∇(∇ · A) − ∇2A

6. A vector field is defined as A = r r2 where r = xi + yj + zk Evaluate ∇ · A and ∇ × A and hence state whether the field is solenoidal or irrotational.

7. Show that A × (B × C) + C × (A × B) + B × (C × A) = 0

8. Calculate the volume integral of (∇ · r) over the volume enclosed by a sphere of radius a, where r = xi + yj + zk.

(b) If

everywhere in a region R and φ is single-valued and has continuous derivatives in R, then show that

is independent of the path joining the points A and B.

(c) Calculate the work done when a force

moves a particle in xy-plane from (0, 0) to (1, 2) along the parabola

6. Answer (a) or (b) :

(a) Verify Gauss divergence theorem for the vector

over the surface of a cube bounded by

(b) State and prove Stokes’ theorem.

2020

(FYUG PHY-DSC101 Same as CBCS PHҮНСС–101T )

(e) Prove that

(f) Find the value of m for which the vectors A, B and C are coplanar :

(g) For vector

find the divergence.

(h) Find the value of b for which the vector

is solenoidal.

4. What is gradient of a scalar function? Give its physical interpretation. Show that

5. (a) Give the physical significance of ‘divergence’ and ‘curl’.

(b) Prove that

where φ is a scalar and A is a vector.

(i) Evaluate :

(j) Find the value of

(k) For a given force

find the work done along straight line from

(l) Using Gauss divergence theorem, express the Gauss law in electrostatics in differential form.

6. (a) Evaluate

where S is the surface of the sphere

in the first octant.

6. (b) Evaluate

where V is sphere having centre at origin and radius r.

7. State and prove Gauss’ divergence theorem.

2021

(FYUG PHY-DSC101 Same as CBCS PHҮНСС–101T )

4. Find the area of the parallelogram whose adjacent sides are

5. Find the volume of the parallelepiped if

are the three coterminous edges of the parallelepiped.

6. At any point of the curve

find the tangent vector.

18. (a) Find m so that the vectors

are coplanar.

(b) Let

Find the vector

19. (a) If

then prove that

(b) If

then find grad φ at the point (1, −2, −1).

2023

(FYUG PHY-DSC101 )

23. Define order and degree of a differential equation.

25. What do you mean by ordinary differential equation (ODE)? Give an example of ODE.

26.hen is a differential equation of the form Mdx+ Ndy = 0 said to be exact or inexact

18. (a) Solve the differential equation

(b) Solve the differential equation

19. (a) What is auxiliary equation? If m1 and m2 are the two roots of the auxiliary equation, then write the expression of complementary function for the cases

(b) Find the solution of the differential equation

when y(0) = 0 and y′(0) = 1/2.

4. State Green's theorem.

7. If a force

displaces a particle in the xy-plane from (0, 0) to (1, 4) along a curve

then find the work done.

8. Evaluate by Stokes’ theorem

where C is the curve

20. (a) If

then evaluate

along the curve

from t = 0 to t = π/2.

20. (b) A vector field is given by

1. If A + B + C = 0, show that
(A × B) = (B × C) = (C × A)

2. Find the value of a such that the vectors
2i − j + k, i + 2j − 3k and 3i + aj + 5k
are coplanar.

3. Show that ∇φ is a vector normal to the surface
φ(x, y, z) = C

4. For any three vectors A, B and C, show that
A · (B × C) = C · (A × B) = B · (C × A)

5. Show that
∇ × (∇ × A) = ∇(∇ · A) − ∇²A

6. A vector field is defined as
A = r r²

where r = xi + yj + zk
Evaluate ∇ · A and ∇ × A and hence state whether the field is solenoidal or irrotational.

7. Show that
A × (B × C) + C × (A × B) + B × (C × A) = 0

8. Calculate the volume integral of
(∇ · r)
over the volume enclosed by a sphere of radius a, where
r = xi + yj + zk.

19. (a) What is auxiliary equation? If m₁ and m₂ are the two roots of the auxiliary equation, then write the expression of complementary function for the cases

when y(0) = 0 and y′(0) = ¹/₂.