UNIT-I: SETS
1. Define the
following terms:
a. Subset
b. Universal set
c. Finite set
d. Infinite set
2. Given sets A =
{1, 2, 3, 4} and B = {3, 4, 5, 6}, find:
a. A union B
b. A
intersection B
c. The
complement of A with respect to the universal set U = {1, 2, 3, 4, 5, 6}
3. Find the
cardinality of set A = {x | x is a prime number less than 10}. Explain your
reasoning.
4. Solve for x:
If A = {x | x is a positive even number less than 10} and B = {x | x is a
multiple of 3 less than 10}, find A ∪ B.
UNIT-II: RELATIONS AND FUNCTIONS
5. Define what an equivalence relation is and provide an
example.
Determine whether the following functions are onto
(surjective), into (injective), or one-to-one (bijective):
a. f(x) = 2x, where x ∈ R
b. g(x) = x^2, where x ∈ R
6. 1Given
functions f(x) = 2x and g(x) = x + 3, find:
a. The composite function (f ∘ g)(x)
b. The inverse function of f(x)
UNIT-III: PARTIAL ORDER RELATIONS AND LATTICES
8. Draw the Hasse diagram for the partial order set defined
by the relation "divides" on the set {1, 2, 3, 4, 5, 6}.
9. Define the
terms "glb" and "lub" in the context of lattices and
provide an example.
10 Find a
sublattice of the power set of {a, b, c} and explain why it is a sublattice.
UNIT-IV: FUNCTIONS OF SEVERAL VARIABLES
11. Calculate the partial derivatives of the function f(x, y)
= x^2 + 2y^2 - 3xy with respect to x and y.
Use the concept of partial differentiation to find the
extrema of the function f(x, y) = 2x^2 + 3y^2 - 4xy.
UNIT-V: 3D COORDINATE GEOMETRY
13. Calculate the direction cosines of a line passing through
the points A(1, 2, 3) and B(4, -1, 5).
Determine the equation of the plane passing through the
points P(1, 2, 3), Q(2, -1, 4), and R(3, 0, 2).
Find the shortest distance between the two lines:
Line 1: r = (1, 2, 3) + t(2, -1, 1)
Line 2: r = (3, -1, 4) + s(-1, 3, -2)
UNIT-VI: MULTIPLE INTEGRATION
16. Calculate the double integral ∬R (x^2 +
y^2) dA, where R is the region bounded by the curves y = x and y = x^2.
Evaluate the triple integral ∭V (x^2 + y^2 + z^2) dV,
where V is the region enclosed by the sphere x^2 + y^2 + z^2 = 4.
Find the volume of the region bounded by the planes x = 0, y
= 0, z = 0, x + y = 1, and z = 2.
0 Comments