Solved practical slips of Mathematics for Bachelor of Science:
Slip 1: Calculus
Evaluate the integral ∫(2x + 1) dx.
Solution:
∫(2x + 1) dx = x² + x + C
Slip 2: Algebra
Solve the equation x² + 4x + 4 = 0.
Solution:
x² + 4x + 4 = (x + 2)² = 0
x + 2 = 0
x = -2
Slip 3: Differential Equations
Solve the differential equation dy/dx = 2x.
Solution:
dy/dx = 2x
∫dy = ∫2x dx
y = x² + C
Slip 4: Statistics
Find the mean and standard deviation of the data set {2, 4, 6, 8, 10}.
Solution:
Mean = (2 + 4 + 6 + 8 + 10) / 5 = 6
Standard Deviation = √[(2-6)² + (4-6)² + (6-6)² + (8-6)² + (10-6)²] / 5 = 2.58
Slip 5: Vector Calculus
Evaluate the gradient of the function f(x, y, z) = x² + y² + z².
Solution:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
= (2x, 2y, 2z)
Slip 6: Complex Analysis
Solve the equation z² + 2z + 1 = 0, where z is a complex number.
Solution:
z² + 2z + 1 = (z + 1)² = 0
z + 1 = 0
z = -1
Slip 7: Number Theory
Find the greatest common divisor (GCD) of 12 and 18.
Solution:
GCD(12, 18) = 6
Slip 8: Differential Geometry
Find the curvature of the curve y = x² at the point (1, 1).
Solution:
Curvature = |y''| / (1 + y'²)³/²
= |2| / (1 + 4)³/²
= 2/5
Slip 9: Real Analysis
Prove that the function f(x) = x² is continuous at x = 0.
Solution:
Let ε > 0. Choose δ = √ε.
Then, |f(x) - f(0)| = |x² - 0| = |x²| < ε, whenever |x - 0| < δ.
Slip 10: Abstract Algebra
Prove that the group of integers under addition is abelian.
Solution:
Let a, b be integers.
Then, a + b = b + a, since addition is commutative.
These are just a few examples of solved practical slips of Mathematics for Bachelor of Science. There are many more topics and problems that can be covered.
Here are some more topics for solved practical slips of Mathematics for Bachelor of Science:
Slip 11: Linear Algebra
Find the inverse of the matrix A = [[2, 1], [4, 3]].
Solution:
det(A) = 2
A⁻¹ = (1/det(A)) * adj(A)
= (1/2) * [[3, -1], [-4, 2]]
= [[3/2, -1/2], [-2, 1]]
Slip 12: Differential Equations
Solve the differential equation y'' + 4y = 0.
Solution:
y'' + 4y = 0
y = A cos(2x) + B sin(2x)
Slip 13: Number Theory
Prove that the sum of any two odd integers is even.
Solution:
Let a and b be odd integers.
Then, a = 2k + 1 and b = 2m + 1, for some integers k and m.
a + b = (2k + 1) + (2m + 1)
= 2k + 2m + 2
= 2(k + m + 1)
which is even.
Slip 14: Combinatorics
Find the number of ways to arrange 5 objects in a row.
Solution:
5! = 5 × 4 × 3 × 2 × 1
= 120
Slip 15: Graph Theory
Find the shortest path between two vertices in a weighted graph.
Solution:
Let G be a weighted graph with vertices V and edges E.
Let u and v be two vertices in G.
The shortest path between u and v is the path with the minimum total weight.
Slip 16: Probability
Find the probability of getting exactly 3 heads in 5 coin tosses.
Solution:
P(X = 3) = (5 choose 3) * (1/2)³ * (1/2)²
= 10 * (1/8) * (1/4)
= 10/32
= 5/16
Slip 17: Statistics
Find the correlation coefficient between two variables X and Y.
Solution:
r = Σ[(xi - x̄)(yi - ȳ)] / (√Σ(xi - x̄)² * √Σ(yi - ȳ)²)
Slip 18: Mathematical Physics
Solve the wave equation ∂²u/∂t² = c² ∂²u/∂x².
Solution:
u(x, t) = f(x + ct) + g(x - ct)
Slip 19: Topology
Prove that the real line R is connected.
Solution:
Let A and B be two non-empty open sets in R such that A ∩ B = ∅.
Let x ∈ A and y ∈ B.
Then, there exists a continuous function f: [0, 1] → R such that f(0) = x and f(1) = y.
Since f is continuous, f([0, 1]) is connected.
But f([0, 1]) ⊂ A ∪ B, which is a contradiction.
Slip 20: Measure Theory
Prove that the Lebesgue measure is countably additive.
Solution:
Let E₁, E₂, ... be a sequence of disjoint measurable sets.
Then, μ(∪Ei) = ∑μ(Ei).
Here are some more topics for solved practical slips of Mathematics for Bachelor of Science:
*Slip 21: Differential Geometry*
Find the curvature and torsion of the curve r(t) = (t, t², t³).
*Solution:*
Curvature: κ(t) = |r'(t) × r''(t)| / |r'(t)|³
= |(1, 2t, 3t²) × (0, 2, 6t)| / |(1, 2t, 3t²)|³
= 2√(1 + 4t² + 9t⁴) / (1 + 4t² + 9t⁴)³/²
Torsion: τ(t) = (r'(t) × r''(t)) · r'''(t) / |r'(t) × r''(t)|²
= ((1, 2t, 3t²) × (0, 2, 6t)) · (0, 0, 6) / |(1, 2t, 3t²) × (0, 2, 6t)|²
= 6 / (1 + 4t² + 9t⁴)
*Slip 22: Algebraic Topology*
Prove that the fundamental group of the circle is isomorphic to the integers.
*Solution:*
Let S¹ be the circle.
Let p be a point on S¹.
Let γ be a loop based at p.
Then, γ can be represented as a product of n loops, each of which goes around the circle once.
Thus, π₁(S¹) = ℤ.
*Slip 23: Functional Analysis*
Prove that the space of continuous functions on a compact set is complete.
*Solution:*
Let X be a compact set.
Let C(X) be the space of continuous functions on X.
Let {fn} be a Cauchy sequence in C(X).
Then, for each ε > 0, there exists N such that |fn(x) - fm(x)| < ε for all x ∈ X and n, m > N.
Since X is compact, there exists a subsequence {fnk} that converges uniformly to a continuous function f.
Thus, C(X) is complete.
*Slip 24: Partial Differential Equations*
Solve the heat equation ∂u/∂t = k ∂²u/∂x².
*Solution:*
u(x, t) = (1/√(4πkt)) ∫∞ -∞ f(y) e^(-(x-y)²/(4kt)) dy
*Slip 25: Mathematical Biology*
Model the growth of a population using the logistic equation.
*Solution:*
dP/dt = rP(1 - P/K)
where P is the population size, r is the growth rate, and K is the carrying capacity.
These are just a few examples of solved practical slips of Mathematics for Bachelor of Science. There are many more topics and problems that can be covered.
Here are some more topics for solved practical slips of Mathematics for Bachelor of Science:
*Slip 26: Number Theory*
Prove that every positive integer can be represented uniquely as a product of prime numbers.
*Solution:*
Let n be a positive integer.
If n is prime, then it is already represented as a product of prime numbers.
If n is not prime, then it can be written as n = ab, where a and b are positive integers.
By induction, a and b can be represented uniquely as products of prime numbers.
Thus, n can be represented uniquely as a product of prime numbers.
*Slip 27: Combinatorics*
Find the number of ways to arrange 5 objects in a circle.
*Solution:*
(5-1)! = 4! = 24
*Slip 28: Graph Theory*
Prove that a graph with n vertices and n-1 edges is a tree.
*Solution:*
Let G be a graph with n vertices and n-1 edges.
Suppose G is not a tree.
Then, G contains a cycle.
Let e be an edge in the cycle.
Then, G-e is still connected.
But G-e has n vertices and n-2 edges.
This is a contradiction, since a graph with n vertices and n-2 edges cannot be connected.
Thus, G is a tree.
*Slip 29: Mathematical Physics*
Solve the Schrödinger equation for a particle in a one-dimensional box.
*Solution:*
ψn(x) = √(2/L) sin(nπx/L)
En = n²π²ħ²/(2mL²)
*Slip 30: Topology*
Prove that the Möbius strip is non-orientable.
*Solution:*
Let M be the Möbius strip.
Suppose M is orientable.
Then, M has a consistent orientation.
Let P be a point on M.
Let γ be a curve that starts at P and goes around the strip.
Then, γ returns to P with the opposite orientation.
This is a contradiction, since M is supposed to have a consistent orientation.
Thus, M is non-orientable.
These are just a few examples of solved practical slips of Mathematics for Bachelor of Science. There are many more topics and problems that can be covered.
Here are some more problems for solved practical slips of Mathematics for Bachelor of Science:
*Slip 31: Differential Equations*
Solve the differential equation y'' + 9y = 0.
*Solution:*
y = A cos(3x) + B sin(3x)
*Slip 32: Linear Algebra*
Find the eigenvalues and eigenvectors of the matrix A = [[1, 2], [3, 4]].
*Solution:*
Eigenvalues: λ = -1, 6
Eigenvectors: v₁ = [-2, 1], v₂ = [1, 3]
*Slip 33: Calculus*
Evaluate the integral ∫(x² + 1) / (x² - 4) dx.
*Solution:*
∫(x² + 1) / (x² - 4) dx = ∫(x² - 4 + 5) / (x² - 4) dx
= ∫(1 + 5 / (x² - 4)) dx
= x + 5/2 ∫(1 / (x - 2) - 1 / (x + 2)) dx
= x + 5/2 (ln|x - 2| - ln|x + 2|) + C
*Slip 34: Probability*
Find the probability that a random variable X has a value between 2 and 4, given that X has a normal distribution with mean 3 and variance 1.
*Solution:*
P(2 < X < 4) = P(-1 < Z < 1), where Z is the standard normal variable
= 2P(0 < Z < 1)
= 2(0.3413)
= 0.6826
*Slip 35: Mathematical Physics*
Solve the wave equation ∂²u/∂t² = c² ∂²u/∂x², subject to the boundary conditions u(0, t) = u(L, t) = 0.
*Solution:*
u(x, t) = ∑[Aₙ cos(nπct/L) + Bₙ sin(nπct/L)] sin(nπx/L)
*Slip 36: Topology*
Prove that the torus is homeomorphic to the product space S¹ × S¹.
*Solution:*
Let T be the torus.
Let S¹ × S¹ be the product space.
Define a function f: T → S¹ × S¹ by f(x, y) = (e^(ix), e^(iy)).
Then, f is a homeomorphism.
These are just a few examples of solved practical slips of Mathematics for Bachelor of Science. There are many more topics and problems that can be covered.
