## NCERT Solutions for Class 7 Maths Chapter 12 – Algebraic Expressions

#### Question 1:

Get the algebraicexpressions in the following cases using variables, constants and arithmetic operations.
(i) Subtraction of z from y.
(ii) One-half of the sum of numbers x and y.
(iii) The number z multiplied by itself.
(iv) One-fourth of the product of numbers p and q.
(v) Numbers x and y both squared and added.
(vi) Number 5 added to three times the product of number m and n.
(vii) Product of numbers y and z subtracted from 10.
(viii)Sum of numbers and b subtracted from their product.

(i) y − z
(ii)
(iii) z2
(iv)
(v) x2 + y2
(vi) 5 + 3 (mn)
(vii) 10 − yz
(viii) ab − (a + b)

#### Question 2:

(i) Identify the terms and their factors in the following expressions
Show the terms and factors by tree diagrams.
(a) x − 3 (b) 1 + x + x2 (c) − y3
(d)  (e) − ab + 2b2 − 3a2
(ii) Identify terms and factors in the expressions given below:
(a) − 4x + 5 (b) − 4x + 5y (c) 5+ 3y2
(d)  (e) pq + q
(f) 1.2 ab − 2.4 b + 3.6 (g)
(h) 0.1p2 + 0.2 q2

(i)
(a)
(b)
(c)
(d)
(e)
(ii)
 Row Expression Terms Factors (a) − 4x + 5 − 4x 5 − 4, x 5 (b) − 4x + 5y − 4x 5y − 4, x 5, y (c) 5y + 3y2 5y 3y2 5, y 3, y, y (d) xy + 2x2y2 xy 2x2y2 x, y 2, x, x, y, y (e) pq + q pq q p, q q (f) 1.2ab − 2.4b + 3.6a 1.2ab − 2.4b 3.6a 1.2, a, b − 2.4, b 3.6, a (g) (h) 0.1p2 + 0.2q2 0.1p2 0.2q2 0.1, p, p 0.2, q, q

#### Question 3:

Identify the numerical coefficients of terms (other than constants) in the following expressions:
(i) 5 − 3t2 (ii) 1 + t2 + t3 (iii) x + 2xy+ 3y
(iv) 100m + 1000n (v) − p2q2 + 7pq (vi) 1.2a + 0.8b
(vii) 3.14 r2 (viii) 2 (b) (ix) 0.1y + 0.01 y2

 Row Expression Terms Coefficients (i) 5 − 3t2 − 3t2 − 3 (ii) 1 + t + t2 + t3 t t2 t3 1 1 1 (iii) x + 2xy + 3y x 2xy 3y 1 2 3 (iv) 100m + 1000n 100m 1000n 100 1000 (v) − p2q2 + 7pq − p2q2 7pq − 1 7 (vi) 1.2a +0.8b 1.2a 0.8b 1.2 0.8 (vii) 3.14 r2 3.14 r2 3.14 (viii) 2(l + b) 2l 2b 2 2 (ix) 0.1y + 0.01y2 0.1y 0.01y2 0.1 0.01

#### Question 4:

(a) Identify terms which contain x and give the coefficient of x.
(i) y2x + y (ii) 13y2− 8yx (iii) x + y + 2
(iv) 5 + zx (v) 1 + x+ xy (vi) 12xy2 + 25
(vii) 7x + xy2
(b) Identify terms which contain y2 and give the coefficient of y2.
(i) 8 − xy2 (ii) 5y2 + 7x (iii) 2x2y −15xy2 + 7y2

(a)
 Row Expression Terms with x Coefficient of x (i) y2x + y y2x y2 (ii) 13y2 − 8yx − 8yx −8y (iii) x + y + 2 x 1 (iv) 5 + z + zx zx z (v) 1 + x + xy x xy 1 y (vi) 12xy2 + 25 12xy2 12y2 (vii) 7x+ xy2 7x xy2 7 y2
(b)
 Row Expression Terms with y2 Coefficient of y2 (i) 8 − xy2 −xy2 − x (ii) 5y2 + 7x 5y2 5 (iii) 2x2y + 7y2 −15xy2 7y2 −15xy2 7 −15x

#### Question 5:

Classify into monomials, binomials and trinomials.
(i) 4y − 7z (ii) y2 (iii) x + y − xy
(iv) 100 (v) ab − a − b (vi) 5 − 3t
(vii) 4p2− 4pq2 (viii) 7mn (ix) z2 − 3z + 8
(x) a2 + b2 (xi) z2 + z (xii) 1 + x + x2

The monomials, binomials, and trinomials have 1, 2, and 3 unlike terms in it respectively.
(i) 4y − 7z
Binomial
(ii) y2
Monomial
(iii) x + y − xy
Trinomial
(iv) 100
Monomial
(v) ab − a − b
Trinomial
(vi) 5 − 3t
Binomial
(vii) 4p2q − 4pq2
Binomial
(viii) 7mn
Monomial
(ix) z2 − 3z + 8
Trinomial
(x) a2 + b2
Binomial
(xi) z2 + z
Binomial
(xii) 1 + x + x2
Trinomial

#### Question 6:

State whether a given pair of terms is of like or unlike terms.
(i) 1, 100 (ii)  (iii) − 29x, − 29y
(iv) 14xy, 42yx (v) 4m2p, 4mp2 (vi) 12xz, 12 x2z2

The terms which have the same algebraic factors are called like terms. However, when the terms have different algebraic factors, these are called unlike terms.
(i) 1, 100
Like
(ii) − 7x
Like
(iii) −29x, −29y
Unlike
(iv) 14xy, 42yx
Like
(v) 4m2p, 4mp2
Unlike
(vi) 12xz, 12x2z2
Unlike

#### Question 7:

Identify like terms in the following:
(a) −xy2, − 4yx2, 8x2, 2xy2, 7y, − 11x2, − 100x, −11yx, 20x2y, −6x2y, 2xy,3x
(b) 10pq, 7p, 8q, − p2q2, − 7qp, − 100q, − 23, 12q2p2, − 5p2, 41, 2405p, 78qp, 13p2qqp2, 701p2

(a) −xy2, 2xy2
−4yx2, 20x2y
8x2, −11x2, −6x2
7yy
−100x, 3x
−11xy, 2xy
(b) 10pq, −7qp, 78qp
7p, 2405p
8q, −100q
p2q2, 12p2q2
−23, 41
−5p2, 701p2
13p2qqp2

#### Question 1:

Simplify combining like terms:
(i) 21b − 32 + 7b − 20b
(ii) − z2 + 13z2 − 5z + 7z3 − 15z
(iii) p − (p − q) − q − (− p)
(iv) 3a − 2b − ab − (a − b + ab) + 3ab + b − a
(v) 5x2y − 5x2 + 3y x2 − 3y2 + x− y+ 8xy2 −3y2
(vi) (3 y+ 5y − 4) − (8y − y2 − 4)

(i) 21b − 32 + 7− 20b = 21b + 7− 20b − 32
b (21 + 7 − 20) −32
= 8b − 32
(ii) − z2 + 13z2 − 5z + 7z3 − 15z = 7z3 − z2 + 13z2 − 5z − 15z
= 7z3 + z2 (−1 + 13) + z (−5 − 15)
= 7z3 + 12z2 − 20z
(iii) p − (p − q) − q − (q − p) = p − p + q − q − q + p
− q
(iv) 3a − 2b − ab − (a − b + ab) + 3ba + − a
= 3a − 2b − ab − a + b − ab + 3ab + − a
= 3a − a − a − 2b + b − ab − ab + 3ab
a (3 − 1 − 1) + b (− 2 + 1 + 1) + ab (−1 −1 + 3)
a + ab
(v) 5x2y − 5x2 + 3yx2 − 3y2 + x2 − y2 + 8xy2 − 3y2
= 5x2y + 3yx− 5x2 + x2 − 3y2 − y2 − 3y+ 8xy2
x2(5 + 3) + x2 (−5 + 1) + y2(−3 − 1 − 3) + 8xy2
= 8x2y − 4x2 − 7y2 + 8xy2
(vi) (3y+ 5y − 4) − (8y − y2 − 4)
= 3y2 + 5y − 4 − 8y + y2 + 4
= 3y2 + y2 + 5y − 8y − 4 + 4
y2 (3 + 1) + y (5 − 8) + 4 (1 − 1)
= 4y2 − 3y

#### Question 2:

(i) 3mn, − 5mn, 8mn, −4mn
(ii) − 8tz, 3tz − zz − t
(iii) − 7mn + 5, 12mn + 2, 9mn − 8, − 2mn − 3
(iv) a + b − 3, b − a + 3, a − b + 3
(v) 14x + 10y − 12xy − 13, 18 − 7x − 10+ 8xy, 4xy
(vi) 5m − 7n, 3n − 4m + 2, 2m − 3mn − 5
(vii) 4x2y, − 3xy2, − 5xy2, 5x2y
(viii) 3p2q2 − 4pq + 5, − 10p2q2, 15 + 9pq + 7p2q2
(ix) ab − 4a, 4b − ab, 4a − 4b
(x) x− y2 − 1 , y2 − 1 − x2, 1− x2 − y2

(i) 3mn + (−5mn) + 8mn + (−4mn) = mn (3 − 5 + 8 − 4)
= 2mn
(ii) (t − 8tz) + (3tz − z) + (z − t) = t − 8tz + 3tz − z + z − t
− t − 8tz + 3tz − z + z
t (1 − 1) + tz (− 8 + 3) + z (− 1 + 1)
= −5tz
(iii) (− 7mn + 5) + (12mn + 2) + (9mn − 8) + (− 2mn − 3)
= − 7mn + 5 + 12mn + 2 + 9mn − 8 − 2mn − 3
= − 7mn + 12mn + 9mn − 2mn + 5 + 2 − 8 − 3
= mn (− 7 + 12 + 9 − 2) + (5 + 2 − 8 − 3)
= 12mn − 4
(iv) (a + − 3) + (b − a + 3) + (a − b + 3)
a + − 3 + b − a + 3 + a − b + 3
a − a + a + − − 3 + 3 + 3
a (1 − 1 + 1) + b (1 + 1 − 1) + 3 (− 1 + 1 + 1)
a + b + 3
(v) (14+ 10y − 12xy − 13) + (18 − 7x − 10y + 8yx) + 4xy
= 14+ 10y − 12xy − 13 + 18 − 7x − 10y + 8yx + 4xy
= 14− 7x + 10y − 10y − 12xy + 8yx + 4xy − 13 + 18
= x (14 − 7) + (10 − 10) + xy (− 12 + 8 + 4) − 13 + 18
= 7x + 5
(vi) (5m − 7n) + (3n − 4m + 2) + (2m − 3mn − 5)
= 5m − 7n + 3n − 4m + 2 + 2m − 3mn − 5
= 5m − 4m + 2m − 7n + 3n − 3mn + 2 − 5
m (5 − 4 + 2) + n (− 7 + 3) −3mn + 2 − 5
= 3m − 4n − 3mn − 3
(vii) 4x2 − 3xy2 − 5xy2 + 5x2y = 4x2 + 5x2y − 3xy2 − 5xy2
x2 y (4 + 5) + xy2 (− 3 − 5)
= 9x2y − 8xy2
(viii) (3p2q2 − 4pq + 5) + (−10 p2q2) + (15 + 9pq + 7p2q2)
= 3p2q2 − 4pq + 5 − 10 p2q2 + 15 + 9pq + 7p2q2
= 3p2q2 − 10 p2q+ 7p2q− 4pq + 9pq + 5 + 15
p2q2 (3 − 10 + 7) + pq (− 4 + 9) + 5 + 15
= 5pq + 20
(ix) (ab − 4a) + (4− ab) + (4a − 4b)
ab − 4a + 4− ab + 4a − 4b
ab − ab − 4+ 4a + 4− 4b
ab (1 − 1) + a (− 4 + 4) + b(4 − 4)
= 0
(x) (x2 − y2 − 1) + (y2 − 1 − x2) + (1 − x2 − y2)
x2 − y2 − 1 + y2 − 1 − x2 + 1 − x2 − y2
x2 − x− x− y2 + y− y− 1 − 1 + 1
x2(1 − 1 − 1) + y2 (−1 + 1 − 1) + (− 1 − 1 + 1)
= − x2 − y2 − 1

#### Question 3:

Subtract:
(i) − 5yfrom y2
(ii) 6xy from − 12xy
(iii) (a − b) from (b)
(iv) a (b − 5) from b (5 − a)
(v) − m2 + 5mn from 4m2 − 3mn + 8
(vi) − x2 + 10x − 5 from 5x − 10
(vii) 5a2 − 7ab + 5b2 from 3ab − 2a2 −2b2
(viii) 4pq − 5q2 − 3p2 from 5p2 + 3q− pq

(i) y2 − (−5y2) = y2 + 5y2 = 6y2
(ii) − 12xy − (6xy) = −18xy
(iii) (a + b) − (a − b) = a + − a + = 2b
(iv) b (5 − a) − a (b − 5) = 5b − ab − ab + 5a
= 5a + 5b − 2ab
(v) (4m2 − 3mn + 8) − (− m2 + 5mn) = 4m2 − 3mn + 8 + m2 − 5 mn
= 4m2 + m2 − 3mn − 5 mn + 8
= 5m2 − 8mn + 8
(vi) (5x − 10) − (− x2 + 10x − 5) = 5− 10 + x2 − 10x + 5
x2 + 5− 10x − 10 + 5
x2 − 5− 5
(vii) (3ab − 2a2 − 2b2) − (5a2− 7ab + 5b2)
= 3ab − 2a2 − 2b2 − 5a2 + 7ab − 5 b2
= 3ab + 7ab − 2a− 5a2 − 2b2 − 5 b2
= 10ab − 7a2 − 7b2
(viii) 4pq − 5q2 − 3p2 from 5p2 + 3q2 − pq
(5p2 + 3q2 − pq) − (4pq − 5q2− 3p2)
= 5p2 + 3q2 − pq − 4pq + 5q2 + 3p2
= 5p2 + 3p2 + 3q2 + 5q2 − pq − 4pq
= 8p2 + 8q2 − 5pq

#### Question 4:

(a) What should be added to x2 + xy + y2 to obtain 2x2 + 3xy?
(b) What should be subtracted from 2+ 8b + 10 to get − 3a + 7b + 16?

(a) Let a be the required term.
a + (x2 + y2 + xy) = 2x+ 3xy a = 2x2 + 3xy − (x2 + y2 + xy)
a = 2x2 + 3xy − x2 − y2 − xy
a = 2x2 − x2 − y2 + 3xy − xy
x2 − y2 + 2xy
(b) Let p be the required term.
(2a + 8b + 10) − = − 3a + 7b + 16
p = 2a + 8b + 10 − (− 3a + 7b + 16)
= 2a + 8b + 10 + 3a − 7− 16
= 2a + 3a + 8b − 7b + 10− 16
= 5− 6

#### Question 5:

What should be taken away from 3x2 − 4y2 + 5xy + 20 to obtain
− x2 − y+ 6xy + 20?

Let p be the required term.
(3x2 − 4y2 + 5xy + 20) − p = − x2 − y2 + 6xy + 20
p = (3x2 − 4y2 + 5xy + 20) − (− x2 − y2 + 6xy + 20)
= 3x2 − 4y2 + 5xy + 20 + x2 + y2 − 6xy − 20
= 3x2 + x− 4y2 + y2 + 5xy − 6xy + 20 − 20
= 4x2 − 3y2 − xy

#### Question 6:

(a) From the sum of 3x − y + 11 and − y − 11, subtract 3x − y − 11.
(b) From the sum of 4 + 3x and 5 − 4x + 2x2, subtract the sum of 3x2 − 5x and − x2 + 2x + 5.

(a) (3x − y + 11) + (− y − 11)
= 3x − + 11 − y − 11
= 3x − − y + 11 − 11
= 3x − 2y
(3x − 2y) − (3x − − 11)
= 3x − 2y − 3x + + 11
= 3x − 3x − 2y + + 11
= − y + 11
(b) (4 + 3x) + (5 − 4x + 2x2) = 4 + 3x + 5 − 4x + 2x2
= 3x − 4x + 2x2 + 4 + 5
= − x + 2x2 + 9
(3x2 − 5x) + (− x2 + 2x + 5) = 3x2 − 5x − x2 + 2x + 5
= 3x2 − x− 5x + 2x + 5
= 2x2 − 3x + 5
(− x + 2x2 + 9) − (2x2 − 3x + 5)
= − x + 2x2 + 9 − 2x2 + 3x − 5
= − x + 3x + 2x2 − 2x+ 9 − 5
= 2x + 4

#### Question 1:

If m = 2, find the value of:
(i) m − 2 (ii) 3m − 5 (iii) 9 − 5m
(iv) 3m2 − 2m − 7 (v)

(i) m − 2 = 2 − 2 = 0
(ii) 3m − 5 = (3 × 2) − 5 = 6 − 5 = 1
(iii) 9 − 5m = 9 − (5 × 2) = 9 −10 = −1
(iv) 3m2 − 2m − 7 = 3 × (2 × 2) − (2 × 2) − 7
= 12 − 4 − 7 = 1
(v)

#### Question 2:

If p = −2, find the value of:
(i) 4p + 7
(ii) −3p2 + 4p + 7
(iii) −2p3 − 3p2 + 4p + 7

(i) 4p + 7 = 4 × (−2) + 7 = − 8 + 7 = −1
(ii) − 3p2 + 4p + 7 = −3 (−2) × (−2) + 4 × (−2) + 7
= − 12 − 8 + 7 = −13
(iii) −2p3 − 3p2 + 4p + 7
= −2 (−2) × (−2) × (−2) − 3 (−2) × (−2) + 4 × (−2) + 7
= 16 − 12 − 8 + 7 = 3

#### Question 3:

Find the value of the following expressions, when x = − 1:
(i) 2x − 7 (ii) − x + 2 (iii) x2 + 2x + 1
(iv) 2x2 − x − 2

(i) 2− 7
= 2 × (−1) − 7 = −9
(ii) − + 2 = − (−1) + 2 = 1 + 2 = 3
(iii) x2 + 2x + 1 = (−1) × (−1) + 2 × (−1) + 1
= 1 − 2 + 1 = 0
(iv) 2x2 − x − 2 = 2 (−1) × (−1) − (−1) − 2
= 2 + 1 − 2 = 1

#### Question 4:

If a = 2, b = − 2, find the value of:
(i) ab2 (ii) a2 + ab + b2 (iii) a2 − b2

(i) a2 + b2
= (2)2 + (−2)2 = 4 + 4 = 8
(ii) a2 + ab + b2
= (2 × 2) + 2 × (−2) + (−2) × (−2)
= 4 − 4 + 4 = 4
(iii) a2 − b2
= (2)2 − (−2)2 = 4 − 4 = 0

#### Question 5:

When a = 0, b = − 1, find the value of the given expressions:
(i) 2a + 2b (ii) 2ab2 + 1
(iii) 2ab + 2ab2 + ab (iv) a2 + ab + 2

(i) 2a + 2= 2 × (0) + 2 × (−1) = 0 − 2 = −2
(ii) 2a2 + b2 + 1
= 2 × (0)2 + (−1) × (−1) + 1
= 0 + 1 + 1 = 2
(iii) 2a2+ 2ab2 + ab
= 2 × (0)2 × (−1) + 2 × (0) × (−1) × (−1) + 0 × (−1)
= 0 + 0 + 0 = 0
(iv) a2 + ab + 2
= (0)2 + 0 × (−1) + 2
= 0 + 0 + 2 = 2

#### Question 6:

Simplify the expressions and find the value if x is equal to 2
(i) x + 7 + 4 (x − 5) (ii) 3 (x + 2) + 5− 7
(iii) 6x + 5 (− 2) (iv) 4 (2x −1) + 3+ 11

(i) x + 7 + 4 (x − 5) = x + 7 + 4x − 20
x + 4x + 7 − 20
= 5x − 13
= (5 × 2) − 13
= 10 − 13 = −3
(ii) 3 (x + 2) + 5x − 7 = 3x + 6 + 5x − 7
= 3x + 5x + 6 − 7 = 8x − 1
= (8 × 2) − 1 = 16 − 1 =15
(iii) 6x + 5 (x − 2) = 6x + 5x − 10
= 11x − 10
= (11 × 2) − 10 = 22 − 10 = 12
(iv) 4 (2x − 1) + 3x + 11 = 8x − 4 + 3x + 11
= 11x + 7
= (11 × 2) + 7
= 22 + 7 = 29

#### Question 7:

Simplify these expressions and find their values if x = 3, a = − 1, b = − 2.
(i) 3x − 5 − x + 9 (ii) 2 − 8x + 4x + 4
(iii) 3a + 5 − 8a + 1 (iv) 10 − 3b − 4 − 5b
(v) 2a − 2b − 4 − 5 + a

(i) 3x − 5 − x + 9 = 3x − x − 5 + 9
= 2x + 4 = (2 × 3) + 4 = 10
(ii) 2 − 8+ 4x + 4 = 2 + 4 − 8+ 4x
= 6 − 4= 6 − (4 × 3) = 6 − 12 = −6
(iii) 3a + 5 − 8+ 1 = 3a − 8a + 5 + 1
= − 5+ 6 = −5 × (−1) + 6
= 5 + 6 = 11
(iv) 10 − 3b − 4 − 5b = 10 − 4− 3b − 5b
= 6 − 8b = 6 − 8 × (−2)
= 6 + 16 = 22
(v) 2a − 2b − 4 − 5 + a = 2a + − 2b − 4 − 5
= 3a − 2b − 9s
= 3 × (−1) − 2 (−2) − 9
= − 3 + 4 − 9 = −8

#### Question 8:

(i) If z = 10, find the value of z3 − 3 (z − 10).
(ii) If p = − 10, find the value of p2 − 2p − 100

(i) z3 − 3 (z − 10) = z3 − 3z + 30
= (10 × 10 × 10) − (3 × 10) + 30
= 1000 − 30 + 30 = 1000
(ii) p2 − 2p − 100
= (−10) × (−10) − 2 (−10) − 100
= 100 + 20 − 100 = 20

#### Question 9:

What should be the value of a if the value of 2x2 + x − a equals to 5, when x = 0?

2x2 + x − a = 5, when x = 0
(2 × 0) + 0 − a = 5
0 − a = 5
a = −5

#### Question 10:

Simplify the expression and find its value when a = 5 and b = −3.
2 (aab) + 3 − ab

2 (a2 + ab) + 3 − ab = 2a2 + 2ab + 3 − ab
= 2a2 + 2ab − ab + 3
= 2a2 + ab + 3
= 2 × (5 × 5) + 5 × (−3) + 3
= 50 − 15 + 3 = 38

#### Question 1:

Observe the patterns of digits made from line segments of equal length. You will find such segmented digits on the display of electronic watches or calculators.
(a)
(b)
(c)
If the number of digits formed is taken to be n, the number of segments required to form n digits is given by the algebraic expression appearing on the right of each pattern.
How many segments are required to form 5, 10, 100 digits of the kind −
,  , .

(a) It is given that the number of segments required to form n digits of the kind
is (5n + 1).
Number of segments required to form 5 digits = (5 × 5 + 1)
= 25 + 1 = 26
Number of segments required to form 10 digits = (5 × 10 + 1)
= 50 + 1 = 51
Number of segments required to form 100 digits = (5 × 100 + 1)
= 500 + 1 = 501
(b) It is given that the number of segments required to form n digits of the kind  is (3n + 1).
Number of segments required to form 5 digits = (3 × 5 + 1)
= 15 + 1 = 16
Number of segments required to form 10 digits = (3 × 10 + 1)
= 30 + 1 = 31
Number of segments required to form 100 digits = (3 × 100 + 1)
= 300 + 1 = 301
(c)It is given that the number of segments required to form n digits of the kind  is (5n + 2).
Number of segments required to form 5 digits = (5 × 5 + 2)
= 25 + 2 = 27
Number of segments required to form 10 digits = (5 × 10 + 2)
= 50 + 2 = 52
Number of segments required to form 100 digits = (5 × 100 + 2)
= 500 + 2 = 502

#### Question 2:

Use the given algebraic expression to complete the table of number patterns.
 S. No Expression Terms 1st 2nd 3rd 4th 5th … 10th … 100th … (i) 2n − 1 1 3 5 7 9 – 19 – – – (ii) 3n + 2 2 5 8 11 – – – – – – (iii) 4n + 1 5 9 13 17 – – – – – – (iv) 7n + 20 27 34 41 48 – – – – – – (v) n2 + 1 2 5 10 17 – – – – 10, 001 –