M l aggarwal understanding icse mathematics class 8 Solutions for Chapter 10 algebraic expressions and identities

Identify the terms, their numerical as well as literal coefficients in each of the following

expressions:

(i) 12x2yz – 4xy2

(ii) 8 + mn + nl – lm

(iii) x2/3 + y/6 – xy2

(iv) -4p + 2.3q + 1.7r

Identify the terms, their numerical as well as literal coefficients in each of the following

expressions:

(i) 12x2yz – 4xy2

(ii) 8 + mn + nl – lm

(iii) x2/3 + y/6 – xy2

(iv) -4p + 2.3q + 1.7r

Identify monomials, binomials, and trinomials from the following algebraic expressions :

(i) 5p × q × r2

(ii) 3x2 + y ÷ 2z

(iii) -3 + 7x2

(iv) (5a2 – 3b2 + c)/2

(v) 7x5 – 3x/y

(vi) 5p ÷ 3q – 3p2 × q2

Identify monomials, binomials, and trinomials from the following algebraic expressions :

(i) 5p × q × r2

(ii) 3x2 + y ÷ 2z

(iii) -3 + 7x2

(iv) (5a2 – 3b2 + c)/2

(v) 7x5 – 3x/y

(vi) 5p ÷ 3q – 3p2 × q2

Identify which of the following expressions are polynomials. If so, write their degrees.

(i) 2/5x4 – √3x2 + 5x – 1

(ii) 7x3 – 3/x2 + √5

(iii) 4a3b2 – 3ab4 + 5ab + 2/3

(iv) 2x2y – 3/xy + 5y3 + √3

Identify which of the following expressions are polynomials. If so, write their degrees.

(i) 2/5x4 – √3x2 + 5x – 1

(ii) 7x3 – 3/x2 + √5

(iii) 4a3b2 – 3ab4 + 5ab + 2/3

(iv) 2x2y – 3/xy + 5y3 + √3

Identify which of the following expressions are polynomials. If so, write their degrees.

(i) 2/5x4 – √3x2 + 5x – 1

(ii) 7x3 – 3/x2 + √5

(iii) 4a3b2 – 3ab4 + 5ab + 2/3

(iv) 2x2y – 3/xy + 5y3 + √3

Add the following expressions:

(i) ab – bv, bv – ca, ca – ab

(ii) 5p2q2 + 4pq + 7, 3 + 9pq – 2p2q

(iii) l2 + m2 + n2, lm + mn, mn + nl, nl + lm

(iv) 4x3 – 7x2 + 9, 3x2 – 5x + 4, 7x3 – 11x + 1, 6x2 – 13x

Add the following expressions:

(i) ab – bv, bv – ca, ca – ab

(ii) 5p2q2 + 4pq + 7, 3 + 9pq – 2p2q

(iii) l2 + m2 + n2, lm + mn, mn + nl, nl + lm

(iv) 4x3– 7x2 + 9, 3x2– 5x + 4, 7x3– 11x + 1, 6x2– 13x

Subtract:

(i) 8a + 3ab – 2b + 7 from 14a – 5ab + 7b – 5

(ii) 8xy + 4yz + 5zx from 12xy – 3yz – 4zx + 5xyz

(iii) 4p2q – 3pq + 5pq2 – 8p + 7q -10 from 18 – 3p – 11q + 5pq – 2pq2 + 5p2q

Subtract:

(i) 8a + 3ab – 2b + 7 from 14a – 5ab + 7b – 5

(ii) 8xy + 4yz + 5zx from 12xy – 3yz – 4zx + 5xyz

(iii) 4p2q – 3pq + 5pq2 – 8p + 7q -10 from 18 – 3p – 11q + 5pq – 2pq2 + 5p2q

Subtract the sum of

3x2 + 5xy + 7y2 + 3 and 2x2 – 4xy – 3y2 + 7 from 9x2 – 8xy + 11y2

Subtract the sum of 3x2 + 5xy + 7y2 + 3 and 2x2 – 4xy – 3y2 + 7 from 9x2 – 8xy + 11y2

What must be subtracted from 3a2 – 5ab – 2b2 – 3 to get 5a2 – 7ab – 3b2 + 3a?

What must be subtracted from 3a2 – 5ab – 2b2 – 3 to get 5a2 – 7ab – 3b2 + 3a?

Find the product of:

(i) 4x3 and -3xy

(ii) 2xyz and 0

(iii) –(2/3)p 2q, (3/4)pq2 and 5pqr

(iv) -7ab, -3a3 and –(2/7)ab2

(v) –½x 2– (3/5)xy, (2/3)yz and (5/7)xyz

Find the product of:

(i) 4x3 and -3xy

(ii) 2xyz and 0

(iii) –(2/3)p2q, (3/4)pq2 and 5pqr

(iv) -7ab, -3a3 and –(2/7)ab2

(v) –½x2 – (3/5)xy, (2/3)yz and (5/7)xyz

Multiply:

(i) (3x – 5y + 7z) by – 3xyz

(ii) (2p2 – 3pq + 5q2 + 5) by – 2pq

(iii) (2/3a2b – 4/5ab2 + 2/7ab + 3) by 35ab

(iv) (4x2 – 10xy + 7y2 – 8x + 4y + 3) by 3xy

Multiply:

(i) (3x – 5y + 7z) by – 3xyz

(ii) (2p2– 3pq + 5q2 + 5) by – 2pq

(iii) (2/3a2b – 4/5ab2 + 2/7ab + 3) by 35ab

(iv) (4x2– 10xy + 7y2– 8x + 4y + 3) by 3xy

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:

(i) (p2q, pq2)

(ii) (5xy, 7xy2)

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths

respectively:

(i)\left(p^{2} q, p q^{2}\right) (ii) \left(5 x y, 7 x y^{2}\right)

Find the volume of rectangular boxes with the following length, breadth, and height respectively:

(i) 5ab, 3a2b, 7a4b2

(ii) 2pq, 4q2, 8rp

Find the volume of rectangular boxes with the following length, breadth and height

respectively:(i) 5 a b, 3 a^{2} b, 7 a^{4} b^{2} (ii) 2 \mathrm{pq}, 4 \mathrm{q}^{2}, 8 \mathrm{rp},

Simplify the following expressions and evaluate them as directed(i) x^{2}\left(3-2 x+x^{2}\right) for x=1 ; x=-1 ; x=2 / 3 and x=-1 / 2 (ii) 5 x y(3 x+4 y-7)-3 y\left(x y-x^{2}+9\right)-8 for x=2, y=-1

Simplify the following expressions and evaluate them as directed:

(i) x2(3 – 2x + x2) for x = 1; x = -1; x = 2/3 and x = –1/2

(ii) 5xy(3x + 4y – 7) – 3y(xy – x2 + 9) – 8 for x = 2, y = -1

Add the following:

(i) 4p(2 – p2) and 8p3– 3p

(ii) 7xy(8x + 2y – 3) and 4xy2(3y – 7x + 8)

Add the following:

(i) 4p(2 – p2) and 8p3 – 3p

(ii) 7xy(8x + 2y – 3) and 4xy2(3y – 7x + 8)

Subtract:

(i) 6x(x – y + z)- 3y(x + y – z) from 2z(-x + y + z)

(ii) 7xy(x^2-2xy+3y^2)-8x(x^2y-4xy+7xy^2)\ from\ 3y(4x^2y-5xy+8xy^2)

Subtract:(i) 6 x(x-y+z)-3 y(x+y-z) from 2 z(-x+y+z) (ii) 7 \mathrm{xy}\left(\mathrm{x}^{2}-2 \mathrm{xy}+3 \mathrm{y}^{2}\right)-8 \mathrm{s}\left(\mathrm{x}^{2} \mathrm{y}-4 \mathrm{xy}+7 \mathrm{xy}^{2}\right) from 3 \mathrm{y}\left(4 \mathrm{r}^{2} \mathrm{y}-5 \mathrm{xy}+8 \mathrm{ry}^{2}\right)

Multiply:

(i) (5x – 2) by (3x + 4)

(ii) (ax + b) by (cx + d)

(iii) (4p – 7) by (2 – 3p)

(iv) (2x2 + 3) by (3x – 5)

(v) (1.5a – 2.5b) by (1.5a + 2.56)

\begin{aligned} &(\mathbf{v})\\ &\left(\frac{3}{7} p^{2}+4 q^{2}\right) \text { by } 7\left(p^{2}-\frac{3}{4} q^{2}\right) \end{aligned}

Multiply:

(i) (5x – 2) by (3x + 4)

(ii) (ax + b) by (cx + d)

(iii) (4p – 7) by (2 – 3p)

(iv) (2x^2 + 3) by (3x – 5)

(v) (1.5a – 2.5b) by (1.5a + 2.56)

(vi) (3/7p^2+4q^2)\ by\ 7(p^2-3/4q^2)

Multiply:

(i) (x – 2y + 3) by (x + 2y)

(ii) (3-5x+2x^2)\ by\ (4x-5)

Multiply:

(i) (x – 2y + 3) by (x + 2y)

(ii) (3 – 5x + 2x2) by (4x – 5)

Multiply:\left(3 x^{2}-2 x-1\right) by \left(2 x^{2}+x-5\right) (ii) \left(2-3 y-5 y^{2}\right) by \left(2 y-1+3 y^{2}\right)

Multiply:

(i) (3x2– 2x – 1) by (2x2 + x – 5)

(ii) (2 – 3y – 5y2) by (2y – 1 + 3y2)

Simplify:

(i) (x2 + 3) (x – 3) + 9

(ii) (x + 3) (x – 3) (x + 4) (x – 4)

(iii) (x + 5) (x + 6) (x + 7)

(iv) (p + q – 2r) (2p – q + r) – 4qr

(v) (p + q) (r + s) + (p – q)(r – s) – 2(pr + qs)

(vi) (x + y + z) (x – y + z) + (x + y – z) (-x + y + z) – 4zx

Simplify:

(i) (x2 + 3) (x – 3) + 9

(ii) (x + 3) (x – 3) (x + 4) (x – 4)

(iii) (x + 5) (x + 6) (x + 7)

(iv) (p + q – 2r) (2p – q + r) – 4qr

(v) (p + q) (r + s) + (p – q)(r – s) – 2(pr + qs)

(vi) (x + y + z) (x – y + z) + (x + y – z) (-x + y + z) – 4zx

\text { If two adjacent sides of a rectangle are } 5 x^{2}+25 x y+4 y^{2} \text { and } 2 x^{2}-2 x y+3 y^{2}, \text { find its area }

\text { If two adjacent sides of a rectangle are } 5 x^{2}+25 x y+4 y^{2} \text { and } 2 x^{2}-2 x y+3 y^{2}, \text { find its area }

\text { If two adjacent sides of a rectangle are } 5 x^{2}+25 x y+4 y^{2} \text { and } 2 x^{2}-2 x y+3 y^{2}, \text { find its area }

Divi

(i) -39 \mathrm{pq}^{2} \mathrm{r}^{5} by -24 \mathrm{p}^{3} \mathrm{q}^{3} \mathrm{r} (ii) -a^{2} b^{3} by a^{3} b^{2}

Divide:

(i) -39pq^2r^5\ by\ -24p^3q^3r

(ii) -a^2b^3\ by\ a^3b^2

Divide:(i) 9 x^{4}-8 x^{3}-12 x+3 by 3 x (ii) 14 p^{2} q^{3}-32 p^{3} q^{2}+15 p q^{2}-22 p+18 q b y-2 p^{2} q

Divide:

(i) 9x4– 8x3– 12x + 3 by 3x

(ii) 14p2q3– 32p3q2 + 15pq2– 22p + 18q by – 2p2q.

Divide:

(i) 6x^2+13x+5\ by\ 2x+1

(ii) 1+y^3\ by\ 1+y

(iii) 5+x-2x^2\ by\ x+1

(iv) x^3-6x^2+12x-8\ by\ x-2

Divide:

(i) 6x2 + 13x + 5 by 2x + 1

(ii) 1 + y3 by 1 + y

(iii) 5 + x – 2x2 by x + 1

(iv) x3 – 6x2 + 12x – 8 by x – 2

Divide:

(i) 6 x^{3}+x^{2}-26 x-25 by 3 x-7 (ii) \mathrm{m}^{3}-6 \mathrm{m}^{2}+7 \mathrm{bv} \mathrm{m}-1

Divide:

(i) 6x^3+x^2-26x-25\ by\ 3x-7

(ii) m^3-6m^2+7\ by\ m-1

Divide:

(i) a^{3}+2 a^{2}+2 a+1 by a^{2}+a+1 (ii) 12 x^{3}-17 x^{2}+26 x-18 by 3 x^{2}-2 x+5

Divide:

(i) a^3+2a^2+2a+1\ by\ a^2+a+1

(ii) 12x^3-17x^2+26x-18\ by\ 3x^2-2x+5

If the area of a rectangle is 8x^2– 45y^2+18xy and one of its sides is 4x + 15y, find the length of

adjacent side.

If the area of a rectangle is 8x2 – 45y2 + 18xy and one of its sides is 4x + 15y, find the length of

adjacent side

Using suitable identities, find the following products

(i) (3x + 5) (3x + 5)

(ii) (9y – 5) (9y – 5)

(iii) (4x + 11y) (4x – 11y)

(iv) (3m/2 + 2n/3) (3m/2 – 2n/3)

(v) (2/a + 5/b) (2a + 5/b)

(vi) (p2/2 + 2/q2) (p2/2 – 2/q2)

Using suitable identities, find the following products:

(i) (3x + 5) (3x + 5)

(ii) (9y – 5) (9y – 5)

(iii) (4x + 11y) (4x – 11y)

(iv) (3m/2 + 2n/3) (3m/2 – 2n/3)

(v) (2/a + 5/b) (2a + 5/b)

(vi) (p^2/2+2/q^2)(p^2/2-2/q^2)

Using the identities, evaluate the following:

(i) 81^2

(ii) 97^2

(iii) 105^2

(iv) 997^2

(v) 6.1^2

(vi) 496 × 504

(vii) 20.5 × 19.5

(viii) 9.62

Using the identities, evaluate the following:

(i) 812

(ii) 972

(iii) 1052

(iv) 9972

(v) 6.12

(vi) 496 × 504

(vii) 20.5 × 19.5

(viii) 9.62

Find the following squares, using the identities

(i)(p q+5 r)^{2} (ii) (5 a / 2-3 b / 5)^{2} ( iii )(\sqrt{2} a+\sqrt{3 b})^{2} (i v)(2 x / 3 y-3 v / 2 x)^{2}

Find the following squares, using the identities:

(i)\ (pq+5r)^2\ \\(ii)\ (5a/2-3b/5)^2\\(iii)(\sqrt{2}a+\sqrt{3}b)^2\\ (iv)\ (2x/3y-3y/2x)^2

Using the identity, (x + a) (x + b) = x2 + (a + b)x + ab, find the following products:

(i) (x + 7) (x + 3)

(ii) (3x + 4) (3x – 5)

(iii) (p2 + 2q) (p2 – 3q)

(iv) (abc + 3) (abc – 5)

Using the identity, (x+a)(x+b)=x^2+(a+b)x+ab , find the following products:

(i) (x + 7) (x + 3)

(ii) (3x + 4) (3x – 5)

(iii) (p^2+2q)(p^2-3q)

(iv) (abc + 3) (abc – 5)

Using the identity, (x + a) (x + b) = x2 + (a + b)x + ab, evaluate the following:

(i) 203 × 204

(ii) 8.2 × 8.7

(iii) 107 × 93

Using the identity, (x+a)(x+b)=x^2+(a+b)x+ab , evaluate the following:

(i) 203 × 204

(ii) 8.2 × 8.7

(iii) 107 × 93

Using the identity a^2-b^2=(a+b)(a-b) , find

(i) 53^2-47^2

(ii) (2.05)^2-(0.95)^2

(iii) (14.3)^2-(5.7)^2

\text { Using the identity } a^{2}-b^{2}=(a+b)(a-b), \text { find }

(i) 53^{2}-47^{2} (ii) (2.05)^{2}-(0.95)^{2} (iii) (14.3)^{2}-(5.7)^{2}

Simplify the following:

(i) (2x + 5y)^2 + (2x – 5y)^2

(ii) (7a/2 – 5b/2)^2 – (5a/2 – 7b/2)^2

(iii) (p^2-q^2r)^2+2p^2q^2r

Simplify the following:

(i) (2 x+5 y)^{2}+(2 x-5 y)^{2} (ii) (7 a / 2-5 b / 2)^{2}-(5 a / 2-7 b / 2)^{2} (iii) \left(p^{2}-q^{2} r\right)^{2}+2 p^{2} q^{2} r

Show that:

(i) (4x+7y)^2-(4x-7y)^2=112xy

(ii) (3p/7-7q/6)^2+pq=9p^2/49+49q^2/36

(iii) (p – q)(p + q) + (q – r)(q + r) + (r – p) (r + p) = 0

Show that:

(i) (4x + 7y)2 – (4x – 7y)2 = 112xy

(ii) (3p/7 – 7q/6)2 + pq = 9p2/49 + 49q2/36

(iii) (p – q)(p + q) + (q – r)(q + r) + (r – p) (r + p) = 0

Divide 10x^4 – 19x^3 + 17x^2 + 15x – 42 \ by \ 2x^2 – 3x + 5.

If x+1 / s=2, evaluate: (i) x^{2}+1 / x^{2} (ii)x^{4}+1 / x^{4}

If x – 1/x = 7, evaluate:

(i) x^{2}+1 / x^{2} (ii) x^{4}+1 / x^{4}

If x – 1/x = 7, evaluate:

(i) x^{2}+1 / x^{2} (ii) x^{4}+1 / x^{4}

\text { If } x^{2}+1 / x^{2}=23, \text { evaluate: }

(i) x+1 / x (ii) x-1 / x

If x^2 + 1/x^2 = 23 , evaluate:

(i) x + 1/x

(ii) x – 1/x

Prove that following:

(i) (a + b)^2 – (a – b)^2 + 4ab

(ii) (2a + 3b)^2 + (2a – 3b)^2 = 8a^2 + 18b^2

If a + b = 9 and ab = 10, find the value of\mid a^{2}+b^{2}

\text { If } a-b=6 \text { and } a^{2}+b^{2}=42, \text { find the value of }

If x + 1/x = 5, evaluate

(i) x^2 + 1/x^2

(ii) x^4 + 1/x^4

\text { If } a^{2}+b^{2}=41 \text { and } a b=4, \text { find the values of }

(i) a + b

(ii) a – b

If a^2 + b^2 = 41 and ab = 4, find the values of

(i) a + b

(ii) a – b

Add the following expressions:

(i) -5 x^{2} y+3 x y^{2}-7 x y+8,12 x^{2} y-5 x y^{2}+3 x y-2 (ii) 9 x y+3 y z-5 z x, 4 y z+9 z x-5 y,-5 x z+2 x-5 x y

Add the following expressions:

(i) -5 x^{2} y+3 x y^{2}-7 x y+8,12 x^{2} y-5 x y^{2}+3 x y-2 (ii) 9 x y+3 y z-5 z x, 4 y z+9 z x-5 y,-5 x z+2 x-5 x y

Subtract:

(i) 5a + 3b + 11c – 2 from 3a + 5b – 9c + 3

(ii) 10x2 – 8y2 + 5y – 3 from 8x2 – 5xy + 2y2 + 5x – 3y

Subtract:

(i) 5a + 3b + 11c – 2 from 3a + 5b – 9c + 3

(ii) 10x2 – 8y2 + 5y – 3 from 8x2 – 5xy + 2y2 + 5x – 3y

\text { What must be added to } 5 x^{2}-3 x+1 \text { to get } 3 x^{3}-7 x^{2}+8 ?

\text { What must be added to } 5 x^{2}-3 x+1 \text { to get } 3 x^{3}-7 x^{2}+8 ?

Find the product of

\text { (i) } 3 x^{2} y \text { and }-4 x y^{2}

(ii) –(4/5)xy, (5/7)yz and –(14/9)zx

Find the product of

\text { (i) } 3 x^{2} y \text { and }-4 x y^{2}

(ii) –(4/5)xy, (5/7)yz and –(14/9)zx

Multiply:

(i) \left(3 p q-4 p^{2}+5 q^{2}+7\right) by 7 p q (ii) \left(3 / 4 x^{2} y-4 / 5 x y+5 / 6 x y^{2}\right) by -15 x y z

Multiply:

(i) \left(3 p q-4 p^{2}+5 q^{2}+7\right) by 7 p q (ii) \left(3 / 4 x^{2} y-4 / 5 x y+5 / 6 x y^{2}\right) by -15 x y z

Multiply:

(i) \left(5 x^{2}+4 x-2\right) by \left(3-x-4 x^{2}\right) (ii) \left(7 x^{2}+12 x y-9 y^{2}\right) by \left(3 x^{2}-5 x y+3 y^{2}\right)

Multiply:

(i) \left(5 x^{2}+4 x-2\right) by \left(3-x-4 x^{2}\right) (ii) \left(7 x^{2}+12 x y-9 y^{2}\right) by \left(3 x^{2}-5 x y+3 y^{2}\right)

Simplify the following expressions and evaluate them as directed

\text { (i) }\left(3 a b-2 a^{2}+5 b^{2}\right) x\left(2 b^{2}-5 a b+3 a^{2}\right)+8 a^{3} b-7 b^{4} \text { for } a=1, b=-1

\text { (ii) }(1.7 x-2.5 y)(2 y+3 x+4)-7.8 x^{2}-10 y \text { for } x=0, y=1

Simplify the following expressions and evaluate them as directed

\text { (i) }\left(3 a b-2 a^{2}+5 b^{2}\right) x\left(2 b^{2}-5 a b+3 a^{2}\right)+8 a^{3} b-7 b^{4} \text { for } a=1, b=-1

\text { (ii) }(1.7 x-2.5 y)(2 y+3 x+4)-7.8 x^{2}-10 y \text { for } x=0, y=1

Carry out the following divisions:

(i) 66 \mathrm{pq}^{2} \mathrm{r}^{3} \div \mathrm{l} \mathrm{lqr}^{2} (ii) \left(x^{3}+2 x^{2}+3 x\right) \div 2 x

Carry out the following divisions:

(i) 66 \mathrm{pq}^{2} \mathrm{r}^{3} \div \mathrm{l} \mathrm{lqr}^{2} (ii) \left(x^{3}+2 x^{2}+3 x\right) \div 2 x

\text { Divide } 10 x^{4}-19 x^{3}+17 x^{2}+15 x-42 \text { by } 2 x^{2}-3 x+5

\text { Divide } 10 x^{4}-19 x^{3}+17 x^{2}+15 x-42 \text { by } 2 x^{2}-3 x+5

Using identities, find the following products:

(i) (3x + 4y) (3x + 4y)

(ii) (5a/2 - b) (5a/2 – b)

(iii) (3.5m – 1.5n) (3.5m + 1.5n)

(iv) (7xy – 2) (7xy + 7)

Using identities, find the following products:

(i) (3x + 4y) (3x + 4y)

(ii) (5a/2 - b) (5a/2 – b)

(iii) (3.5m – 1.5n) (3.5m + 1.5n)

(iv) (7xy – 2) (7xy + 7)

Using suitable identities, evaluate the following

(i) 105*2

(ii) 972

(iii) 201 × 199

(iv) 872 – 132

(v) 105 × 107