To do:We have to draw a perpendicular to $\overline{XY}$ at Y.Solution: Steps of construction: (i) Let us draw a line 'l' and point a point X on it.(ii) Now, let us draw an arc intersecting the line 'l' at two points by placing the pointer of the compasses on point ... Read More

To do:We have to draw a perpendicular to $\overline{PQ}$.Solution: Steps of construction:(i) Let us draw a line segment $\overline{PQ}$ and mark a point R outside it.(ii) Now, place a set square in such a way that one of its right angles aligns with $\overline{PQ}$ (iii) Now, by placing a ruler along ... Read More

Given:A line segment $\overline{AB}$.To do:We have to construct $\overline{PQ}$ such that the length of $\overline{PQ}$ is twice that of $\overline{AB}$.Solution: Steps of construction: (i) To draw the line segment $\overline{PQ}$.(ii) Let us place the pointer of the compasses on point A of the given line segment AB and adjust the ... Read More

To do:We have to draw a perpendicular to $\overline{AB}$.Solution:Steps of construction: (i) Let us draw a line segment $\overline{AB}$ and mark a point M on it.(ii) Now, by taking M as a centre let's draw an arc intersecting the line segment $\overline{AB}$ and name the intersecting points as C and ... Read More

Given: $(i)$. $5-3t^2$$(ii)$. $ 1+t+t^{2}+t^3$$(iii)$ . $x+2xy+3y$$(iv)$. $100m+1000n$$(v)$. $-p^2q^{2}+7pq$$(vi)$. $1.2a+0.8b$$(viii)$. $3.14r^2$$(viii)$. $2(l+b)$$(ix)$. $0.1y+0.01y^2$To do: To identify the numerical coefficients of terms $(other\ than\ constants)$ in the given expressions.Solution: $(i)$. $-3$ is the coefficient of $t^2$.$(ii)$. $1$, $1$ and $1$ are $t$he coefficients of $t$, $t^2$ and $t^3$.$(iii)$. $1$, $2$ and $3$ are ... Read More

Given: $(a)$ $(i)$. $y^2x+y$            $(ii)$. $13y^2-8yx$                    $(iii)$ . $x+y+2$$(iv)$ . $5+z+zx$          $(v)$ . $1+x+xy$                 $(vi)$. $12xy^2+25$$(viii)$. $7x+xy^2$ $(b)$.  $(i)$. $8-xy^2$      $(ii)$. $5y^2+7x$     $(iii)$. $2x^2y-15xy^2+7y^2$To ... Read More

Given: $(i)$. $4y-7z$$(ii)$. $y^2$$(iii)$. $x+y-xy$$(iv)$. $100$$(v)$. $ab-a-b$$(vi)$. $5-3t$$(viii)$. $4p^2q-4pq^2$ $(viii)$. $7mn$$(ix)$. $z^2-3z+8$$(x)$. $a^2+b^2$$(xi)$. $z^2+z$$(xii)$. $1+x+x^2$To do: To classify the given terms into monomials, binomials, and trinomials. Solution: The expressions having a single term in the most simplified form are called monomials. The expressions having exactly two terms in the most simplified form are ... Read More

Given: Pairs:$(i)$. $1, \ 100$$(ii)$. $-7x, \ \frac{5}{2}x$$(iii)$. $-29x, \ -29y$$(iv)$. $14xy, \ 42yx$$(v)$. $4m^2p, \ 4mp^2$ $(vi)$. $12xz, \ 12x^2y^2$To do: To state whether the given pair of terms is of like or unlike terms:Solution:We know that like terms are those terms whose variables and their exponent power are the same. ... Read More

To do: $(a)$. To subtract $3x-y-11$ from the sum of $3x-y+11$ and $-y-11$.$(b)$. To subtract the sum of $3x^2-5x$ and $-x^2+2x+5$ from the sum of $4+3x$ and $5-4x+2x^2$. Solution:a). The sum of $3x-y+11$ and $-y-11$.$=(3x-y+11)+(-y-11)$$=3x-y+11-y-11$$=3x-2y$Now, we subtract $3x-y-11$ from $3x-2y$$(3x-2y)-(3x-y-11)$$=3x-2y-3x+y+11$$=-y+11$b). The sum of $4+3x$ and $5-4x+2x^2$.  $=(4+3x)+(5-4x+2x^2)$$=4+3x+5-4x+2x^2$$=-x+2x^2+9$Sum of $3x^2-5x$ and $-x^2+2x+5$$=(3x^2-5x)+(-x^2+2x+5)$$=3x^2-5x-x^2+2x+5$$=2x^2-3x+5$Now, we subtract ... Read More

Given: $(i)$. $21b-32+7b-20b$$(ii)$. $-z^2+13z^2-5z+7z^3-15z$$(iii)$. $p-(p-q)-q-(q-p)$$iv)$. $3a-2b-ab-(a-b+ab)+3ab+b-a$$(v)$. $5x^2y-5x^2+3yx^2-3y^2+x^2-y^2+8xy^2-3y^2$$(vi)$. $(3y^2+5y-4)-(8y-y^2-4)$ To do: To simplify combining like terms.Solution:$(i)$ $21b-32+7b-20b$$=21b+7b-20b-32$             [On arranging like terms]$=b(21+7-20)-32$$=8b-32$$(ii)$ $-z^2+13z^2-5z+7z^3-15z$$=7z^{3}+z^2+13z^2-5z+5z-15z$      [On arranging like terms]$=7z^3+(-1+13)z^2+(-5-15)z$$= 7z^3 + 12z^2 – 20z$$(iii)$ $p-(p-q)-q-(q-p)$$=p-p+q-q-q+p$                 [On arranging like terms]$=p-q$$(iv)$ $3a-2b-ab-(a-b+ab)+3ab+b-a$$=3a-2b-ab-a+b-ab+3ab+b-a$        ... Read More