1.Find and correct the errors in the following mathematical statements.
1. \(4 (x – 5) = 4x – 5\)

2. \(x(3x + 2) = 3x^2 + 2\)

3. \(2x + 3y = 5xy\)

4.\( x + 2x + 3x = 5xy\)

5. \(5y + 2y + y – 7y = 0\)

6. \(3x + 2x = 5x^2\)

7. \((2x)^2 + 4(2x) + 7 = 2x^2 + 8x + 7\)

8. \((2x)^2 + 5x = 4x + 5x = 9x\)

9. \((3x + 2)^2 = 3x^2 + 6x + 4\)

10. Substituting\(x = -3 in\)

(a) \(x^2 + 5x + 4\) gives \((-3)2 + 5(-3) + 4 = 9 + 2 + 4 = 15\)

(b)\( x^2 – 5x + 4\) gives \((-3)2 -5(-3) + 4 = 9 – 15 + 4 = -2\)

(c)\( x^2 + 5x\) gives \((-3)2 + 5(-3) = -9 – 15 = -24\)

11.\( (y – 3)^2 = y^2 – 9\)

12.\( (z + 5)^2 = z^2 + 25\)

13.\( (2a + 3b)(a – b) = 2a^2 – 3b^2\)

14.\( (a + 4)(a + 2) = a^2 + 8\)

15.\( (a – 4)(a – 2) = a^2 – 8\)

16. \(\frac { 3{ x }^{ 2 } }{ 3{ x }^{ 2 } } =0\)

17. \(\frac { 3{ x }^{ 2 }+1 }{ 3{ x }^{ 2 } } =1+1=2\)

18.\( \frac { 3x }{ 3x+2 } =\frac { 1 }{ 2 } \)

19.\( \frac { 3 }{ 4x+3 } =\frac { 1 }{ 4x } \)

20. \(\frac { 4x+5 }{ 4x } =5\)

21. \(\frac { 7x+5 }{ 5 } =7x\)

1. We have LHS=\(4(x-5)=4\times x -4 \times 5 =4x-20\)

We should have RHS=LHS

Therefore,\(4(x-5)=4x-20\) is the correct statement.

2. We have LHS=\(x(3x + 2)=x \times 3x + x \times 2 =3x^2+2x\)

We should have RHS=LHS

Therefore,\(x(3x + 2)=3x^2+2x\) is the correct statement.

3.The given statement is incorrect only liked terms can be grouped together.Therefore, 2x+3y=2x+3y may be the correct statement

4.Using distributive property of multiplivation over addition:

\(LHS=x+2x+3x=6x\neq\) given RHS

So x+2x+3x=6x is the correct statement

5.Using distributive property of multiplivation over addition:

\(LHS=5y+2y+y-7y=(8-7)y=7y\neq\) given RHS

So 5y+2y+y-7y=7y is the correct statement

6.Using distributive property of multiplivation over addition:

\(LHS=3x + 2x=5x\neq\) given RHS

So 3x + 2x=5x is the correct statement

7.We have LHS:\((2x)^2 + 4(2x) + 7\)

\(=(2)^2\times (x)^2 + 4(2x) + 7= 4x^2 + 8x + 7\neq\) given RHS

So \((2x)^2 + 4(2x) + 7= 4x^2 + 8x + 7\) is the correct statement

8.We have LHS:\((2x)^2 + 5x\)

\(=(2)^2\times (x)^2 +5x= 4x^2 + 5x\neq\) given every value in RHS

So \((2x)^2 + 5x= 4x^2 +5x \) is the correct statement

9.We have LHS:\((3x + 2)^2\)

\(=(3x)^2+2\times (3x)\times(2)+2^2)=9x^2 +12x + 4\neq\) given RHS

So \((3x + 2)^2= 9x^2 +12x + 4\) is the correct statement

10.(a)Substituting x=-3 in \(x^2 +5x + 4\)

\(=(-3)^2+5(-3)+4=9-15+4=-2 \neq \) given RHS

So for x=-3, the value of \(x^2 +12x + 4 =-2\)

(b)Substituting x=-3 in \(x^2 -5x + 4\)

\(=(-3)^2-5(-3)+4=9+15+4=28 \neq \) given RHS

So for x=-3, the value of \(x^2 -5x + 4 =28\)

(c)Substituting x=-3 in \(x^2 +5x\)

\(=(-3)^2+5(-3)=9-15=-6 \neq \) given RHS

So for x=-3, the value of \(9x^2 +5x =62\)

11.We have, We have LHS:\((y-3)^2\)

\(=(y)^2-2\times (y)\times(3)+3^2)=y^2 -6y + 9\neq\) given RHS

So \((y-3)^2= y^2 -6y + 9\) is the correct statement

12.We have, We have LHS:\((z+5)^2\)

\(=(z)^2+2\times (z)\times(5)+5^2)=z^2 +10z + 25\neq\) given RHS

So \((z+5)^2= z^2 +10z + 25\) is the correct statement

13. Here we have LHS=\((2a+3b)(a-b)\)

\(=2a(a-b)+3b(a-b)=2a \times a - 2a times b+3b \times a - 3a \times b\)

\(=2a^2+ab-3b^2\neq \) given RHS

So \((2a+3b)(a-b)=2a^2+ab-3b^2\) is the correct statement

14.Here we have LHS=\((a+4)(a+2)\)

\(=a(a+2)+4(a+2)=a \times a + a times 2+4 \times a + 4 \times 2\)

\(=a^2+6a+8\neq \) given RHS

So \((a+4)(a+2)=a^2+6a+8\) is the correct statement

15.Here we have LHS=\((a-4)(a-2)\)

\(=a(a-2)-4(a-2)=a \times a - a \times 2-4 \times a + 4 \times 2\)

\(=a^2-6a+8\neq \) given RHS

So \((a-4)(a-2)=a^2-6a+8\) is the correct statement

16.We have LHS=\(\frac{3x^2}{3x^2}=1 \neq\) given RHS

So we have \(\frac{3x^2}{3x^2}=1\) as the correct statement

17.We have LHS=\(\frac{3x^2+1}{3x^2}=\frac{3x^2}{3x^2}+\frac{1}{3x^2}=1+\frac{1}{3x^2} \neq\) given RHS

So we have \(\frac{3x^2+1}{3x^2}=1+\frac{1}{3x^2}\) as the correct statement

18.We have LHS=\(\frac{3x}{3x+2}\neq \frac{1}2 \)

So we have \(\frac{3x^2}{3x^2}=\frac{3x}{3x+2}\) as the correct statement

19.Clearly in LHS we can see that, \(\frac{3}{4x+3} \neq \frac{1}{4x}\)

So we have \(\frac{3}{4x+3}=\frac{3}{4x+3}\) as the correct statement

20.Clearly in LHS we can see that, \(\frac{4x+5}{4x}=\frac{4x}{4x}+\frac{5}{4x}=1+\frac{5}{4x} \neq 5\)

So we have \(\frac{4x+5}{4x}=1+\frac{5}{4x}\) as the correct statement

21.Clearly in LHS we can see that, \(\frac{7x+5}{5}=\frac{7x}{5}+\frac{5}{5}=1+\frac{7x}{5} \neq 7x\)

So we have \(\frac{7x+5}{5}=1+\frac{7x}{5}\) as the correct statement