# NCERT Solutions for Class 8 Maths

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# Algebraic Expressions and Identities

Exercise 1

1.    Identify the terms, their coefficients for each of the following expressions:

(i) 5xyz2 – 3zy                                          (ii) 1 + x + x2

(iii) 4x2y– 4x2y2z2 + z2                           (iv) 3 – pq + qr – rp

(v)                                            (vi) 0.3a – 0.6ab + 0.5b

2.    Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories:

x + y, 1000, x + x2 + x3+ x4, 7 + y + 5x, 2y – 3y2, 2y – 3y2 + 4y3, 5x – 4y + 3xy, 4z – 15z2, ab + bc + cd + da, pqr, p2q, pg2, 2p + 2q

(i) ab – bc, bc – ca, ca – ab

(ii) a – b + ab, b – c +bc, c – a + ac

(iii) 2p2q2 – 3pq + 4,5 + 7pq 3p2q2

(iv) l2 + m2, m2 + n2, n+ l2+ 2lm + 2mn + 2nl

4.    (a) Subtract 4a – 7 ab + 3b + 12 from 12a – 9ab + 5b – 3.

(b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz.

(c) 4p2q2 – 3pq + 5p2q2 – 8p + 7 – 10 from 18 – 13q – 11q + 5pq – 2pq2 + 5p2q

Exercise 2

5.    Find the product of the following pairs of monomials:

(i) 4, 7p                        (ii) -4p, 7 p                            (iii) -4p, 7 pq

(iv) 4p3 , -3p               (iv) 4p, 0

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

(p, q); (10m, 5n); (20x2, 5y2); (4x, 3x2); (3mn, 4np)

7.    Complete the table of products:

8.   Obtain the volume of rectangular boxes with the following length, breadth and height respectively:

(i) 5a, 3a27a4                                             (ii) 2p, 4q, 8r

(iii) xy, 2x2y, 2xy2                                    (iv) a, 2b, 3c

9.   Obtain the product of:

(i) xy, yz, zx                                                              (ii) a, -a2, a3

(iii) 2, 4y, 8y2 ,16y3                                                (iv) a, 2b, 3c, 6abc

(v) m, -mn, mnp

Exercise 3

10.  Carry out the multiplication of the expressions in each of the following pairs:

(i) 4p, q + r                                                                 (ii) ab, a – b

(iii) a + b, 7 a2 b2                                                                 (iv) a2 – 9, 4a

(v) pq + qr + rp, 0

11.  Complete the table:

12.  Find the product:

13.  (a) Simplify: 3x (4x – 5) + 3 and find values for (i) x = 3 (ii) x

(b) Simplify: a (a2 + a + 1) + 5 and find its value for (i) a = 0 (ii) a = 1 (iii) a = -1.

14.  (a) Add: p(p – q), q(q – r) and r(r – p).

(b) Add: 2x (z – x – y) and 2y(z – y – zx).

(c) Subtract: 3l (l – 4m + 5n) from 4l(10n – 3m + 21).

(d) Subtract: 3a(a + b + c) -2b(a – b + c) from 4c(-a + b + c).

Exercise 4

15.  Multiply the binomials:

(i) (2x + 5) and (4x – 3)

(ii) (y – 8) and (3y – 4)

(iii) (2.51 – 0.5m) and (2.51 + 0.5m)

(iv) (a + 3b) and (x + 5)

(v) (2pq + 3q2) and (3pq – 2q2)

(vi)

16.  Find the products:

(i) (5 – 2x)(3 + x)                                     (ii) (x + 7y) (7x – y)

(iii) (a2 + b)(a + b2)                               (iv) (p2 – q2)(2p + q)

17.  Simplify:

(i) (x2 – 5)(x + 5) + 25

(ii) (a2 + 5)(b2 + 3) + 5

(iii) (t + s2)(t2 – s)

(iv) (a + b)(c – d) + (a – b)(c + d) +2 (ac + bd)

(v) (x + y)(2x + y) + (x + 2y)(x – y)

(vi) (x + y)(x2 – xy + y2)

(vii) (1.5x – 4y)(1.5x + 4y + 3) -4.5x + 12y

(viii) (a + b + c)(a + b – c).

Exercise 5

18.  Use a suitable identity to get each of the following products:

(i) (x + 3) (x + 3)                         (ii) (2y + 5)(2y + 5)

(iii) (2a -7)(2a – 7)                       (iv)

(v) (1.1m – 0.4)(1.1m + 0.4)       (vi) (a+ b2)(-a+ b2)

(vii) (6x – 7)(6x + 7)                    (viii) (-a + c)(-a + c)

(ix)                      (x) (7 a – 9b) (7 a – 9b)

19.  Use the identity (x + a)(x + b) = x2 + (a + b) x + ab to find the following products:

(i)  (x + 3)(x + 7)                                  (ii) (4x + 5)(4x + 1)

(iii) (4x – 5)(4x – 1)                              (iv) (4x + 5)(4x – 1)

(v) (2x + 5y)(2x + 3y)                         (vi) (2a2 + 9)(2a2 + 5)

(vii) (xyz – 4)(xyz – 2)

20.  Find the following squares by using identities:

21.  Simplify:

(i) (a2 – b2 )2

(ii) (2x + 5)2 -(2x – 5)2

(iii) (7m – 8n)2 + (7m + 8n)2

(iv) (4m +5n)2 + (5m + 4n)2

(v) (2.5 p – 5q)2 – (1.5p – 2.5q)2

(vi) (ab + bc)2 – 2ab2c

(vii) (m2 – n2m)2 + 2m5n3

22.  Show that:

(i) (3x + 7)2 – 84x = (3x – 7)2

(ii) (9p – 502 + 180pq = (9p + 50)2

(iii)

(iv) (4pq + 3q)2 – (4pq – 3q)2 = 48pq2

(v) (a – b)(a + b) + (b – c)(b + c) + (c – a)(c + a) = 0

23.  Using identities, evaluate:

(i) 712                     (ii) 992              (iii) 1022

(iv) 9982                (v) 5.22             (vi) 297 × 303

(vii) 78 × 82          (viii) 8.92         (ix) 1.05 × 9.5

24.  Using a2 – b2 = (a + b)(a – b), find

(i) 512 – 492                    (ii) (1.02)2 – (0.98)2

(iii) 1532 – 147             (iv) 12.12 – 7.92

25.  Using (x + a)(x + b)= x2 + (a + b)x + ab, find

(i)  103 × 104

(ii)  5.1 × 5.2

(iii)  103 × 98

(iv)  9.7 × 9.8.

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