Question

If p = −2, q = −1 and r = 3, find the value of
(i) p² + q² − r²
(ii) 2p² − q² + 3r²
(iii) p − q − r
(iv) p³ + q³ + r³ + 3pqr
(v) 3p²q + 5pq² + 2pqr
(vi) p⁴ + q⁴ − r⁴

Answer 1

(i) ${\mathrm{p}}^2+{\mathrm{q}}^2-{\mathrm{r}}^2$
Substituting p = -2, q = -1 and r = 3 in the given expression:
$$\begin{gathered} (-2{)}^2+(-1{)}^2-(3{)}^2=(-2\times -2)+(-1\times -1)-(3\times 3) \\ \Rightarrow 4+1-9=-4 \end{gathered}$$

(ii) $2{\mathrm{p}}^2-{\mathrm{q}}^2+3{\mathrm{r}}^2$
Substituting p = -2, q = -1 and r = 3 in the given expression:

$$\begin{gathered} 2\times (-2{)}^2-(-1{)}^2+3\times (3{)}^2=2\times (-2\times -2)-(-1\times -1)+3\times (3\times 3) \\ \Rightarrow 8-1+27=\mathbf{34} \end{gathered}$$

(iii) $\mathrm{p}-\mathrm{q}-\mathrm{r}$
Substituting p = -2, q = -1 and r = 3 in the given expression:
$$\begin{gathered} (-2)-(-1)-\left(3\right)=-2+1-3 \\ =-4 \end{gathered}$$

(iv) ${\mathrm{p}}^3+{\mathrm{q}}^3+{\mathrm{r}}^3+3\mathrm{pqr}$
Substituting p = -2, q = -1 and r = 3 in the given expression:

$$\begin{gathered} (-2{)}^3+(-1{)}^3+(3{)}^3+3\times (-2\times -1\times 3) \\ =(-2\times -2\times -2)+(-1\times -1\times -1)+(3\times 3\times 3)+3\times (6) \\ =(-8)+(-1)+(27)+18 \\ =36 \end{gathered}$$

(v) $3{\mathrm{p}}^2\mathrm{q}+5{\mathrm{pq}}^2+2\mathrm{pqr}$
Substituting p = -2, q = -1 and r = 3 in the given expression:

$$\begin{gathered} 3\times (-2{)}^2\times (-1)+5\times (-2)\times (-1{)}^2+2\times (-2\times -1\times 3) \\ =3\times (-2\times -2)\times (-1)+5\times (-2)\times (-1\times -1)+2\times (-2\times -1\times 3) \\ =-12-10+12 \\ =-10 \end{gathered}$$

(vi) ${\mathrm{p}}^4+{\mathrm{q}}^4-{\mathrm{r}}^4$
Substituting p = -2, q = -1 and r = 3 in the given expression:
$$\begin{gathered} (-2{)}^4+(-1{)}^4-(3{)}^4 \\ =(-2\times -2\times -2\times -2)+(-1\times -1\times -1\times -1)-(3\times 3\times 3\times 3) \\ =16+1-81 \\ =-64 \end{gathered}$$