A simple cipher takes a number and codes it, using the function f(x) = 3x - 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x(by drawing the lines)

Graph the function f(x) = x³ and \(g(x)=\sqrt[3]x\) on the same co-ordinate plane. Find f o g and graph it on the plane as well. Explain your results.

Write the values of f at -3, 5, 2, -1, 0 if
\(f(x)=\begin{cases} x^2+x-5\quad if\ x \in(-\infty, 0) \\x^2+3x-2\quad if\ x\in(3,\infty) \\x^2\quad \quad \quad \quad \quad if\ x\ \in(0,2) \\x^2-3 \quad \quad \quad otherwise \end{cases}\)

If a² = by + cz, b² = cz + ax and c² ax + by, prove that \({{x}\over{a+x}}+{{y}\over{b+y}}+{{z}\over{c+z}}=1.\)

Determine the region in the plane determined by the inequalities.
\(2x+3y\le 6,\ x+4y\le 4,\ x\ge 0,\ y\ge 0.\)

Resolve the following rational expressions into partial fractions.
\({{2x^2+5x-11}\over{x^2+2x-3}}\)

If the sides of a \(\triangle\)ABC are a = 4, b = 6, and c = 8, show that \(4\cos { B } +3\cos { C } =2\)

Using the Mathematical induction, show that for any natural number n; with the assumption i² = -1, (r(cos ፀ + i sin ፀ))ⁿ = rⁿ (cos nፀ+i sin nፀ)

Find the fourth root of 623 correct to seven places of decimal.

The product of three increasing numbers in GP is 5832. if we add 6 to the second number and 9 to the third number, then resulting number form an AP. Find the numbers in GP

Show that the straight lines joining the origin to the points of intersection of 3x - 2y + 2 = 0 and 3x² + 5xy - 2y² + 4x + 5y = 0 are at right angles.

If \(A=\left[ \begin{matrix} 1 & 8 \\ 4 & 3 \end{matrix} \right] \quad B=\left[ \begin{matrix} 1 & 3 \\ 7 & 4 \end{matrix} \right] \quad C=\left[ \begin{matrix} -4 & 6 \\ 3 & -5 \end{matrix} \right] \) 
Prove that 
(l) AB ≠ BA
(it)A(BC) = (AB)C
(iii) A(B + C) = AB + AC
(iv) AI = IA = A

 If \(A=\left[ \begin{matrix} 1 & 2 \\ 2 & 0 \end{matrix} \right] ,B=\left[ \begin{matrix} 3 & -1 \\ 1 & 0 \end{matrix} \right] \) verify the following:

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

Find the projection of:
(i) \(\hat { i } -\hat { j } \) on Z-axis
(ii) \(\hat { i } +2\hat { j } -2\hat { k } \)  on \(2\hat { i } -\hat { \quad j } -2\hat { k } \) 
(iii) \(3\hat { i } +\hat { j } -\hat { k } \) on \(4\hat { i } -\hat { j } +2\hat { k } \) 

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(sin x\over |x|\)

Find y' if x⁴ + y⁴ = 16.

Integrate the following functions with respect to x : \(x+1\over (x+2)(x+3)\)

The probability that a new railway bridge will get an award for its design is 0.48, the probability that it will get an award for the efficient use of materials is 0.36, and that it will get both awards is 0.2. What is the probability, that (i) it will get at least one of the two awards (ii) it will get only one of the awards.

A purse contains 3 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled out at random from one of the two purses, what is the probability that it is a silver coin?