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How many 5 vertices unlabelled trees are there? How would I draw them?
Test the convergence of ∑︁∞
n=1
[︂
n!2n
2
n
]︂
.
Ans: ∑︁un Converges
25. Test the convergence of ∑︁∞
n=1
[︂
n
3 + 1
2
n + 1]︂
.
Ans: ∑︁un Converges
Hint: Use vn as
1
√n
.
26. Test the convergence of
1
2 · 3 · 4
+
2
3 · 4 · 5
+
3
4 · 5 · 6
+
4
5 · 6 · 7
+ · · · .
Ans: ∑︁un Converges
Hint: Use vn as
1
n2
.
27. Test the convergence of
√
2 − 1
3
3 − 1
+
√
3 − 1
4
3 − 1
+
√
4 − 1
5
3 − 1
+ · · · .
Ans: ∑︁un Converges
Hint: Use vn as
1
n
5
2
.
28. Test the convergence of
2
1
p +
3
2
p +
4
3
p + · · · .
Ans: ∑︁un Converges if p > 2 and ∑︁un Diverges if p ≤ 2
Hint: Use vn as
1
n
p−1
.
Find the evolute of
x
2
a2
−
y
2
b
2
= 1 as the envelope of the normals.
9. Find the expression for ∫︁∫︁∫︁∫︁
V
x
l−1y
m−1z
n−1 dx by DZ(Dirichlet’s Integral) in
the form of Gamma integrals, here V is the region x ≥ 0, y ≥ 0, z ≥ 0 and
x + y + z ≤ a.
Hint/Ans: Γ(l)Γ(m)Γ(n)
Γ(l + m + n + 1)
a
l+m+n
10. Evaluate ∫︁
1
1
√
1 − x4
dx
4. Find the evolute of the rectangular hyperbola y
2 = 4ax. Ans: 27ay2 = 4(x −
2a)
3
Hint: Take P (at2
, 2at) be any point on the parabola.
5. Find the envelope of the family of lines of the form y = mx±
√︀
a2m2 − b
2. Ans:
x
2
a2
−
y
2
b
2
= 1
6. Find the envelope of the family of lines of the form
x
a
+
y
b
= 1 subject to the
condition a + b = 1. Ans: √
x +
√
b = 1
7. Find the evolute of
x
2
a2
+
y
2
b
2
= 1 as the envelope of the normals. Ans: (ax)
2/3 +
(by)
2/3 = (a
2 − b
2
)
2/3
1. Find the radius of curvature (ROC) at (a cos3 θ, a sin3
θ) on the curve x
2/3 +
y
2/3 = a
2/3
2. Find the radius of curvature (ROC) for the curve x = a(cos θ + θ sin θ), y =
a(sin θ − θ cos θ).
find the radius of the curvature at (acostheta^3,asintheta^3) on the curve x^2/3+y^2/3=a^2/3
show that ~Q,P—>Q=>~P in mathematical foundations of computer science
Define and what Cryptography in mathematical foundations of computer science
Define and what Cryptography in mathematical foundations of computer science
#339914 on Dec 2023