Write a description of each shaded area. Use symbols U, A, B, C, U, ∩,  and as necessary.

Draw venn diagram of three sets A, B and C which illustrates the following:
A and B disjoint but both are subsets of C.

If R is the set of all real numbers, what do the cartesian products R \(\times\) Rand R \(\times\)R \(\times\)R represent?

Solve the inequation x \(\ge\) 2 graphically.

Solve the quadratic equation 5^2x- 5^{x + 3}+ 125 = 5^x.

Resolve into partial fractions: \({{x^3+1}\over{x(x+1)^2}}\)

Prove that \(cos\left( \frac { \pi }{ 4 } -A \right) cos\left( \frac { \pi }{ 4 } -B \right) -sin\left( \frac { \pi }{ 4 } -A \right) sin\left( \frac { \pi }{ 4 } -B \right) \)

Prove that \([1+cot\alpha-sec(\alpha+\frac{\pi}{2})][1+cot\alpha+sec(\alpha+\frac{\pi}{2})]=2cot\alpha\)

Prove that \(\frac { sin(A-B) }{ sin(A+B) } =\frac { { a }^{ 2 }-{ b }^{ 2 } }{ { c }^{ 2 } } \)

In any triangle ABC, prove that \(\frac { { a }^{ 2 }sin(B-C) }{ sinA } +\frac { { b }^{ 2 }sin(C-A) }{ sinB } +\frac { { c }^{ 2 }sin(A-B) }{ sinC } =0\)

If (n+2)! = 60(n-1)! find n.

In how many ways can 9 examination papers be arranged so that the best and the worst papers are never together?

A question paper has two parts A and B, each containing 10 questions. If a student has to choose 8 from part A, 5 from Part B, in how many ways can he choose the questions?

Let p(n) be the statement "3ⁿ>n". If p(n) is true, prove that p(n+1) is true.

Find n if ^{n - 1}P₃ : ⁿP₄ = 1 : 9

Find the number of 4-digt numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

The first three terms in the expansion of (1 + ax)ⁿ are 1 + 12x + 64x². Find n and a

Show that the sequence where log a,\(log\frac { { a }^{ 2 } }{ b^{ 1 } } log\frac { { a }^{ 2 } }{ { b }^{ 2 } } \)  ..is an A.P

Find the A.P in which the sum of any number of terms is always there times the square of the number of these terms

If a, b, c are in A.P b, c, d are in G.P, c, d, e are in H.P then show that a, c, e in G.P

If p^th term of an AP is q and q^th term is p, find (p + q)^th term.

Find the coefficient of the term involving x³² and x^-17 in the expansion of \((x^{4}-\frac{1}{x^{3}})^{15}\).

If the p^th, q^th and r^th terms of an A.P. are a, b, c respectively, prove that a (q - r) + b (r - p) + c (p - q) = 0.

The sum of first three terms of a G.P. is to the sum of the first six terms as 125: 152. Find the common ratio of the G.P.

Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, -1).