Question

Show that the relation R in the set N × N defined by (a, b) R (c, d) if a² + d² = b² + c² $\forall\;a,\;b,\;c,\;d\in N,$ is an equivalence relation.

Let $\stackrel{\rightarrow}{a}\$ = $\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$, $\stackrel{\rightarrow}{b}\$ = $4\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$ , and $\stackrel{\rightarrow}{c}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ Find a vector of magnitude 6 units,which is parallel to the vector $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}+3\stackrel{\rightarrow}{c}$.

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+1)}}\,.$

Show that the equation of normal at any point t on the curve x = 3 cos t – cos³t and y = 3 sin t – sin³t is 4(y cos2t – x sin³t) = 3 sin 4t

Evaluate : $\large\textstyle\int\!{\frac{d x}{x(x^{3}+8)}}\,.$

Using properties of determinants, solve for x : $\begin{vmatrix}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{vmatrix}$ = 0

Let f : N → N be defined as $f(n)={\left\{\begin{array}{l l}{\displaystyle{\frac{n+1}{2}},{\mathrm{when~}}n{\mathrm{~is~odd}}}\\ {\displaystyle{\frac{n}{2}},{\mathrm{when~}}n{\mathrm{iseven}}}\end{array}\right.}$ for all n ∈ N. State whether the function f is bijective. Justify your answer.