NCERT Solutions for Class 7 Maths Chapter 12 – Algebraic Expressions
Page No 234:
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 a and b subtracted from their product.
Answer:
(i) y − z
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1603/Chapter%2012_html_13f9a6de.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1603/Chapter%2012_html_13f9a6de.gif)
(iii) z2
(iv) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1603/Chapter%2012_html_63ad2cd1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1603/Chapter%2012_html_63ad2cd1.gif)
(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) y − y3
(d)
(e) − ab + 2b2 − 3a2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_m21fb93b1.gif)
(ii) Identify terms and factors in the expressions given below:
(a) − 4x + 5 (b) − 4x + 5y (c) 5y + 3y2
(d)
(e) pq + q
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_m7435dba3.gif)
(f) 1.2 ab − 2.4 b + 3.6 a (g) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_3a9d4463.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_3a9d4463.gif)
(h) 0.1p2 + 0.2 q2
Answer:
(i)
(a)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_1ca7c6ce.jpg)
(b)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_63e8838a.jpg)
(c)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_481845b6.jpg)
(d)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_m1f751a2c.jpg)
(e)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1606/Chapter%2012_html_m35760108.jpg)
(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
|
Page No 235:
Question 3:
Identify the numerical coefficients of terms (other than constants) in the following expressions:
(i) 5 − 3t2 (ii) 1 + t + t2 + t3 (iii) x + 2xy+ 3y
(iv) 100m + 1000n (v) − p2q2 + 7pq (vi) 1.2a + 0.8b
(vii) 3.14 r2 (viii) 2 (l + b) (ix) 0.1y + 0.01 y2
Answer:
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 + z + 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
Answer:
(a)
- RowExpressionTerms with xCoefficient of x(i)y2x + yy2xy2(ii)13y2 − 8yx− 8yx−8y(iii)x + y + 2x1(iv)5 + z + zxzxz(v)1 + x + xyxxy1y(vi)12xy2 + 2512xy212y2(vii)7x+ xy27xxy27y2
(b)
- RowExpressionTerms with y2Coefficient of y2(i)8 − xy2−xy2− x(ii)5y2 + 7x5y25(iii)2x2y + 7y2−15xy27y2−15xy27−15x
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) 4p2q − 4pq2 (viii) 7mn (ix) z2 − 3z + 8
(x) a2 + b2 (xi) z2 + z (xii) 1 + x + x2
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1616/Chapter%2012_html_m7fe2bf40.gif)
(iv) 14xy, 42yx (v) 4m2p, 4mp2 (vi) 12xz, 12 x2z2
Answer:
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, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1616/Chapter%2012_html_48a08fe9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1616/Chapter%2012_html_48a08fe9.gif)
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, −6x2, y, 2xy,3x
(b) 10pq, 7p, 8q, − p2q2, − 7qp, − 100q, − 23, 12q2p2, − 5p2, 41, 2405p, 78qp, 13p2q, qp2, 701p2
Answer:
(a) −xy2, 2xy2
−4yx2, 20x2y
8x2, −11x2, −6x2
7y, y
−100x, 3x
−11xy, 2xy
(b) 10pq, −7qp, 78qp
7p, 2405p
8q, −100q
−p2q2, 12p2q2
−23, 41
−5p2, 701p2
13p2q, qp2
Page No 239:
Question 1:
Simplify combining like terms:
(i) 21b − 32 + 7b − 20b
(ii) − z2 + 13z2 − 5z + 7z3 − 15z
(iii) p − (p − q) − q − (q − p)
(iv) 3a − 2b − ab − (a − b + ab) + 3ab + b − a
(v) 5x2y − 5x2 + 3y x2 − 3y2 + x2 − y2 + 8xy2 −3y2
(vi) (3 y2 + 5y − 4) − (8y − y2 − 4)
Answer:
(i) 21b − 32 + 7b − 20b = 21b + 7b − 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
= p − q
(iv) 3a − 2b − ab − (a − b + ab) + 3ba + b − a
= 3a − 2b − ab − a + b − ab + 3ab + b − a
= 3a − a − a − 2b + b + 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 + 3yx2 − 5x2 + x2 − 3y2 − y2 − 3y2 + 8xy2
= x2y (5 + 3) + x2 (−5 + 1) + y2(−3 − 1 − 3) + 8xy2
= 8x2y − 4x2 − 7y2 + 8xy2
(vi) (3y2 + 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:
Add:
(i) 3mn, − 5mn, 8mn, −4mn
(ii) t − 8tz, 3tz − z, z − 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 − 10y + 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) x2 − y2 − 1 , y2 − 1 − x2, 1− x2 − y2
Answer:
(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 − 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 + b − 3) + (b − a + 3) + (a − b + 3)
= a + b − 3 + b − a + 3 + a − b + 3
= a − a + a + b + b − b − 3 + 3 + 3
= a (1 − 1 + 1) + b (1 + 1 − 1) + 3 (− 1 + 1 + 1)
= a + b + 3
(v) (14x + 10y − 12xy − 13) + (18 − 7x − 10y + 8yx) + 4xy
= 14x + 10y − 12xy − 13 + 18 − 7x − 10y + 8yx + 4xy
= 14x − 7x + 10y − 10y − 12xy + 8yx + 4xy − 13 + 18
= x (14 − 7) + y (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 y − 3xy2 − 5xy2 + 5x2y = 4x2 y + 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 p2q2 + 7p2q2 − 4pq + 9pq + 5 + 15
= p2q2 (3 − 10 + 7) + pq (− 4 + 9) + 5 + 15
= 5pq + 20
(ix) (ab − 4a) + (4b − ab) + (4a − 4b)
= ab − 4a + 4b − ab + 4a − 4b
= ab − ab − 4a + 4a + 4b − 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 − x2 − x2 − y2 + y2 − y2 − 1 − 1 + 1
= x2(1 − 1 − 1) + y2 (−1 + 1 − 1) + (− 1 − 1 + 1)
= − x2 − y2 − 1
Page No 240:
Question 3:
Subtract:
(i) − 5y2 from y2
(ii) 6xy from − 12xy
(iii) (a − b) from (a + 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 + 3q2 − pq
Answer:
(i) y2 − (−5y2) = y2 + 5y2 = 6y2
(ii) − 12xy − (6xy) = −18xy
(iii) (a + b) − (a − b) = a + b − a + b = 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) = 5x − 10 + x2 − 10x + 5
= x2 + 5x − 10x − 10 + 5
= x2 − 5x − 5
(vii) (3ab − 2a2 − 2b2) − (5a2− 7ab + 5b2)
= 3ab − 2a2 − 2b2 − 5a2 + 7ab − 5 b2
= 3ab + 7ab − 2a2 − 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 2a + 8b + 10 to get − 3a + 7b + 16?
Answer:
(a) Let a be the required term.
a + (x2 + y2 + xy) = 2x2 + 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) − p = − 3a + 7b + 16
p = 2a + 8b + 10 − (− 3a + 7b + 16)
= 2a + 8b + 10 + 3a − 7b − 16
= 2a + 3a + 8b − 7b + 10− 16
= 5a + b − 6
Question 5:
What should be taken away from 3x2 − 4y2 + 5xy + 20 to obtain
− x2 − y2 + 6xy + 20?
Answer:
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 + x2 − 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.
Answer:
(a) (3x − y + 11) + (− y − 11)
= 3x − y + 11 − y − 11
= 3x − y − y + 11 − 11
= 3x − 2y
(3x − 2y) − (3x − y − 11)
= 3x − 2y − 3x + y + 11
= 3x − 3x − 2y + y + 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 − x2 − 5x + 2x + 5
= 2x2 − 3x + 5
(− x + 2x2 + 9) − (2x2 − 3x + 5)
= − x + 2x2 + 9 − 2x2 + 3x − 5
= − x + 3x + 2x2 − 2x2 + 9 − 5
= 2x + 4
Page No 242:
Question 1:
If m = 2, find the value of:
(i) m − 2 (ii) 3m − 5 (iii) 9 − 5m
(iv) 3m2 − 2m − 7 (v) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1641/Chapter%2012_html_m2ab48de3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1641/Chapter%2012_html_m2ab48de3.gif)
Answer:
(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) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1641/Chapter%2012_html_22b29e85.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1641/Chapter%2012_html_22b29e85.gif)
Question 2:
If p = −2, find the value of:
(i) 4p + 7
(ii) −3p2 + 4p + 7
(iii) −2p3 − 3p2 + 4p + 7
Answer:
(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
Answer:
(i) 2x − 7
= 2 × (−1) − 7 = −9
(ii) − x + 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) a2 + b2 (ii) a2 + ab + b2 (iii) a2 − b2
Answer:
(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) 2a2 + b2 + 1
(iii) 2a2 b + 2ab2 + ab (iv) a2 + ab + 2
Answer:
(i) 2a + 2b = 2 × (0) + 2 × (−1) = 0 − 2 = −2
(ii) 2a2 + b2 + 1
= 2 × (0)2 + (−1) × (−1) + 1
= 0 + 1 + 1 = 2
(iii) 2a2b + 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) + 5x − 7
(iii) 6x + 5 (x − 2) (iv) 4 (2x −1) + 3x + 11
Answer:
(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
Answer:
(i) 3x − 5 − x + 9 = 3x − x − 5 + 9
= 2x + 4 = (2 × 3) + 4 = 10
(ii) 2 − 8x + 4x + 4 = 2 + 4 − 8x + 4x
= 6 − 4x = 6 − (4 × 3) = 6 − 12 = −6
(iii) 3a + 5 − 8a + 1 = 3a − 8a + 5 + 1
= − 5a + 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 + a − 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
Answer:
(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?
Answer:
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 (a2 + ab) + 3 − ab
Answer:
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
Page No 246:
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)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_2e6e3335.jpg)
(b)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_6e863c99.jpg)
(c)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_m7deb739b.jpg)
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 −
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_72b1af8c.jpg)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_m6666eacf.jpg)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_m2fce33ee.jpg)
Answer:
(a) It is given that the number of segments required to form n digits of the kind
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_72b1af8c.jpg)
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).
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_m6666eacf.jpg)
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).
![](https://img-nm.mnimgs.com/img/study_content/curr/1/7/3/42/1676/Chapter%2012_html_m2fce33ee.jpg)
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
Page No 247:
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
|
–
|
Answer:
The given table can be completed as follows.
S.No.
|
Expression
|
Terms
| |||||||||
1st
|
2nd
|
3rd
|
4th
|
5th
|
…
|
10th
|
…
|
100th
|
…
| ||
(i)
|
2n − 1
|
1
|
3
|
5
|
7
|
9
|
–
|
19
|
–
|
199
|
–
|
(ii)
|
3n + 2
|
2
|
5
|
8
|
11
|
17
|
–
|
32
|
–
|
302
|
–
|
(iii)
|
4n + 1
|
5
|
9
|
13
|
17
|
21
|
–
|
41
|
–
|
401
|
–
|
(iv)
|
7n + 20
|
27
|
34
|
41
|
48
|
55
|
–
|
90
|
–
|
720
|
–
|
(v)
|
n2 + 1
|
2
|
5
|
10
|
17
|
26
|
–
|
101
|
–
|
10,001-
|
–
|