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False

Expression with two unlike terms is called a ‘Binomial’.

(i) Like terms – 4xy, -3yx and Numerical coefficients – 4, -3

(ii) Like terms (-7a²bc, −3ca²b) and (−4/3cba²) and their Numerical coefficients – 7, -3,

(-4/3)

Like terms are (−5/2abc²) and (3/2 abc²) and numerical coefficients are (−5/2) and (3/2).

(i) 5 – 3t²

Term other than constant = -3/2

numerical co-efficient = -3

(ii) 1 + t + t² + t³

Term other than constant = t, t², t³

Numerical co-efficients = 1, 1, 1.

(iii) x + 2xy + 3y

Terms other than constants = x, 2xy + 3y

Numerical co-efficients = 1, 2, 3

(iv) 100m + 1000n

Terms other than constants = 100m, 1000n

Numerical co-efficients = 100, 1000.

(v) -p²q² + 7pq

Terms other than constants = -p² q², 7 pq

Numerical co-efficients = -1, 7

(vi) 1.2a + 0.8b

Terms other than constants = 1.2a, 0.8b

Numerical co-efficients = 1.2, 0.8

(vii) 3.14 r²

Terms other than constants = 3.14r²

Numerical co-efficients = 3.14

(viii) 2 (l + b)

Terms other than constants = 2l, 2b

(∵ 2(l + b) = 2l + 2b)

Numerical co-efficients = 2, 2

(ix) 0.1 y + 0.01 y²

Terms other than constants = 0.1 y, 0.01 y²

Numerical co-efficients = 0.1, 0.01

False.

We know that, total number of planets are 8 is constant. We are not going to denote planets in variables.

False.

Sum of 2 and p is 2 + p.

False.

The terms having the same algebraic factors are called like terms. Numerical of the coefficient can be vary.

True

Sum of x and y is x+y.

False.

Expression with three unlike terms is called a ‘Trinomial’.

True.

In general, an expression with one or more than one term (with nonnegative integral exponents of the variables) is called a ‘Polynomial’.

False.

If we add a monomial and binomial, then answer can be a monomial.

For example: sum of y² and –y² + x²

= y² + (-y² + x²)

= y² – y² + x²

= x²

False.

If we subtract a monomial from a binomial, then answer is at least a monomial.

For example: difference of y² and y² + x²

= y² – (y² + x²)

= y² – y² – x²

= – x²

The product of xy, yz and zx is 2xyz.

Coefficient of x⁵ in 8/5 x⁵ is 8/5.

False

4y – 7z is a trinomial.

1. 2/8

2. 3xyz

3. 8x² + 6x + 2

4. \(\frac{16x^3}{3}\)

5. (x + b)

2n, – 3n are like terms.

Like terms similar to 7xy² are – 7xy², 3xy², – 4xy²

Like term of expression 7x²y is – 10x²y.

(3x + 8) (5 – 2x)

= 3x (5 – 2x) + 8(5 – 2x)

= 3x × 5 – 3x × 2x + 8 × 5 – 8 × 2x

= 15x – 6x² + 40 – 16x

= – 6x² + 15x – 16x + 40

= – 6x² – x + 40

(x + 3y) (3x – y)

= x (3x – y) + 3y (3x – y)

= x × 3x – x × y + 3y × 3x – 3y × y

= 3x² – xy + 9yx – 3y²

= 3x² – xy + 9xy – 3y²

= 3x² + 8xy – 3y²

The terms and the respective coefficients of the given expressions are as follows.

(i) 21b – 32 + 7b – 20b

(re-arranging the terms)

= 21b + 7b – 20b – 32

= 28b – 20b – 32

= 8b – 32

(ii) z² + 13z² – 5z + 7Z² – 15z

(re-arranging the terms)

= -z² + 13z² – 5z – 15z + 7z³

= 12z² – 20z + 7z³

(iii) p – (p – q) – q – (q – p)

= p – p + q – q – q + p

(re-arranging the terms)

= p + p - p + q - q - q

= p – q

(iv) 3a – 2b – ab – (a – b + ab) + 3ab + b – a

= 3a – 2b – ab – a + b – ab + 3ab + b – a

= 3a – a – a – 2b + b + b – ab – ab + 3ab 

= 3a - 2a - 2b + 2b - 2ab + 3ab

= a + ab

(v) 5x²y – 5x² + 3yx² – 3y² + x² – y² + 8xy² – 3y²

(re-arranging the terms)

= 5x²y + 3yx² – 5x² + x² – 3y² – y² – 3y² + y²

= 8x²y – 4x² – 7y² + 8xy²

(vi) (3y² + 5y – 4) – (8y – y² – 4) .

3y² + 5y – 4 – 8y – y² – 4

(re-arranging the terms)

3y² + y² + 5y – 8y – 4 + 4

= 14y² – 3y

(a² + b) (a + b²)

= a² (a + b²) + b (a + b²)

= a² × a + a² × b² + b × a + b × b²

= a³ + a²b² + ba + b³

= a³ + a²b² + ab + b³

The product of pq + qr + 2p and 0 is 0.

(p² – q²) (2p + q)

= p² (2p + q) – q² (2p + q)

= p² x 2p + p² x q – q² x 2p – q² x q

= 2p³ + p²q – 2q²p – q³

(x + 5) (x – 7) + 35

= x (x – 7) + 5 (x – 7) + 35

= x² – 7x + 5x – 35 + 35

= x² – 2x

(a² – 3) (b² + 3) + 5

= a² (b² + 3) – 3 (b² + 3) + 5

= a²b² + 3a² – 3b² – 9 + 5

= a²b² + 3a² – 3b² – 4

(a + b + c) (a + b – c)

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

= a² + ab – ac + ab + b² – bc + ac + bc – c²

= a² + b² – c² + 2ab

(t + s²) (t² – s)

= t (t² – s) + s² (t² – s)

= t³ – ts + s²t² – s³

= t³ – s³ + s²t² – ts

(a + b) (a² – ab + b²)

= a (a² – ab + b²) + b (a² – ab + b²)

= a³ – a²b + ab² + ba² – ab² + b³

= a³ + b³ – a²b + a²b + ab² – ab²

= a³ + b³

(i) (a² − b2)² 

= (a²)² − 2(a²) (b²) + (b²)²    [(a − b)² = a² − 2ab + b² ]

= a⁴ − 2a²b² + b⁴

(ii) (2x +5)² − (2x − 5)² 

= (2x)² + 2(2x) (5) + (5)² − [(2x)² − 2(2x) (5) + (5)²]    

 [(a − b)2 = a² − 2ab + b²]  [(a + b)² = a² + 2ab + b²]

= 4x² + 20x + 25 − [4x² − 20x + 25]

= 4x² + 20x + 25 − 4x² + 20x − 25 

= 40x

(iii) (7m − 8n)² + (7m + 8n)²

= (7m)² − 2(7m) (8n) + (8n)² + (7m)2 + 2(7m) (8n) + (8n)²

[(a − b)² = a² − 2ab + b² and (a + b)² = a² + 2ab + b²]

= 49m² − 112mn + 64n² + 49m² + 112mn + 64n²

= 98m² + 128n²

(iv) (4m + 5n)² + (5m + 4n)²

= (4m)² + 2(4m) (5n) + (5n)² + (5m)² + 2(5m) (4n) + (4n)²    [ (a + b)² = a² + 2ab + b²]

= 16m² + 40mn + 25n² + 25m² + 40mn + 16n²

= 41m² + 80mn + 41n²

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

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

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

= ac – ad + bc – bd + ac + ad – bc – bd + 2 ac + 2 bd

= ac + ac + 2ac – ad + ad + be – bc – bd – bd – 2 bd

= 4ac

(i) (a + b) (c − d) + (a − b) (c + d) + 2 (ac + bd)

= a (c − d) + b (c − d) + a (c + d) − b (c + d) + 2 (ac + bd)

= ac − ad + bc − bd + ac + ad − bc − bd + 2ac + 2bd

= (ac + ac + 2ac) + (ad − ad) + (bc − bc) + (2bd − bd − bd)

= 4ac

(ii) (x + y) (2x + y) + (x + 2y) (x − y)

= x (2x + y) + y (2x + y) + x (x − y) + 2y (x − y)

= 2x² + xy + 2xy + y² + x² − xy + 2xy − 2y²

= (2x² + x²) + (y² − 2y²) + (xy + 2xy − xy + 2xy)

= 3x² − y² + 4xy

(i) (x + y) (x² − xy + y²)

= x (x² − xy + y²) + y (x² − xy + y²)

= x³ − x²y + xy² + x²y − xy² + y³

= x³ + y³ + (xy² − xy²) + (x²y − x²y)

= x³ + y³

(ii) (1.5x − 4y) (1.5x + 4y + 3) − 4.5x + 12y

= 1.5x (1.5x + 4y + 3) − 4y (1.5x + 4y + 3) − 4.5x + 12y

= 2.25 x² + 6xy + 4.5x − 6xy − 16y² − 12y − 4.5x + 12y

= 2.25 x² + (6xy − 6xy) + (4.5x − 4.5x) − 16y² + (12y − 12y)

= 2.25x² − 16y²

(iii) (a + b + c) (a + b − c)

= a (a + b − c) + b (a + b − c) + c (a + b − c)

= a² + ab − ac + ab + b² − bc + ca + bc − c²

= a² + b² − c² + (ab + ab) + (bc − bc) + (ca − ca)

= a² + b² − c² + 2ab

(i) (x + 3) (x + 7) 

= x² + (3 + 7) x + (3) (7) 

= x² + 10x + 21

(ii) (4x + 5) (4x + 1) 

= (4x)² + (5 + 1) (4x) + (5) (1) 

= 16x² + 24x + 5

(iii) (4x - 5) (4x - 1)

= (4x)² + [ (-5) + (-1) ] (4x) + (5) (-1)

= 16x² + 16x -5

(iv) (4x + 5) (4x - 1) 

= (4x)² + [ (5) + (-1) ] (4x) + (5) (-1)

=16x² + 16x − 5

(i) (2x +5y) (2x + 3y) 

= (2x)² + (5y + 3y) (2x) + (5y) (3y)

= 4x² + 16xy + 15y²

(ii) (2a² +9) (2a² + 5) 

= (2a²)² + (9 + 5) (2a²) + (9) (5)

= 4a⁴ + 28a² + 45

(vii) (xyz − 4) (xyz − 2)

+ (xyz)² + [ (-4) + (-2) ] (xyz) + (-4) (-2)

= x²y²z² - 6xyz + 8

(i) (x² − 5) (x + 5) + 25

= x² (x + 5) − 5 (x + 5) + 25

= x³ + 5x² − 5x − 25 + 25

= x³ + 5x² − 5x

(ii) (a² + 5) (b³ + 3) + 5

= a² (b³ + 3) + 5 (b³ + 3) + 5

= a²b³ + 3a² + 5b³ + 15 + 5

= a²b³ + 3a² + 5b³ + 20

(iii) (t + s²) (t² − s)

= t (t² − s) + s² (t² − s)

= t³ − st + s²t² − s³

The products will be as follows.

(i) (x + 3) (x + 3)  = (x + 3)²

= (x)² + 2(x) (3) + (3)² [(a + b)²  = a² + 2ab + b²]

= x² + 6x + 9

(ii) (2y + 5) (2y + 5)  = (2y + 5)²

= (2y)² + 2(2y) (5) + (5)² [(a + b)² = a² + 2ab + b²]

= 4y² + 20y + 25

(iii) (2a − 7) (2a − 7)  = (2a − 7)²

= (2a)² − 2(2a) (7) + (7)² [(a − b)² = a² − 2ab + b²]

= 4a² − 28a + 49

(i) (a² + b) (a + b²) 

= a² (a + b²) + b (a + b²)

= a³ + a²b² + ab + b³

(ii) (p² − q²) (2p + q) 

= p² (2p + q) − q² (2p + q)

= 2p³ + p²q − 2pq² − q³

(i) (5 − 2x) (3 + x) 

= 5 (3 + x) − 2x (3 + x)

= 15 + 5x − 6x − 2x²

= 15 − x − 2x²

(ii) (x + 7y) (7x − y) 

= x (7x − y) + 7y (7x − y)

= 7x² − xy + 49xy − 7y²

= 7x² + 48xy − 7y²

(i) (2x + 5) × (4x − 3) 

= 2x × (4x − 3) + 5 × (4x − 3)

= 8x² − 6x + 20x − 15

= 8x² + 14x −15 (By adding like terms)

(ii) (y − 8) × (3y − 4) 

= y × (3y − 4) − 8 × (3y − 4)

= 3y² − 4y − 24y + 32

= 3y² − 28y + 32 (By adding like terms)

a(a – b) + b(b – c) + c(c – a)

= a × a – a × b + b × b – b × c + c × c – c × a

= a² – ab + b² – bc + c² – ca

= a² + b² + c² – ab – bc – ca

x(x + y – r) + y(x – y + r) + z(x – y – z)

= x² + xy – xr + xy – y² + yr + zx – yz – z²

= x² – y² – z² + 2xy – xr + yr + zx – yz

3x(x + 2y) – 2x(5x – y)

= (3x × x + 3x × 2y) - (2x × 5x – 2x × y)

= 3x² + 6xy – (10x² – 2xy)

= 3x² + 6xy- 10x² + 2xy

= 8xy – 7x²

1.5x (10x²y – 100xy²)

= 1.5x × 10x²y – 1.5x × 100xy²

= 15 × x³ × y – 150 × x² × y²

= 15x³y – 150x²y²

Given 2x² – 3x + 1, 3x² – 2x and 3x + 7

sum of 3x² – 2x and 3x + 7

= (3x² – 2x) + (3x +7)

=3x² – 2x + 3x + 7

= (3x² + x + 7)

Now, required expression = 2x² – 3x + 1+ (3x² + x + 7)

= 2x² + 3x² – 3x + x + 1 + 7

= 5x² – 2x + 8

Correct option is  B) 6391

Correct option is (B) 6391

83 \(\times\) 77 = (80+3) (80-3)

= \(80^2-3^2\) = 6400 - 9 = 6391.

D) a² + b² + c² – ab – bc – ca

Correct option is  A) a³ – b²a