Fundamental Concepts of Algebra — Mental Maths

Fill in the blanks:

(i) $(2a^2 + 3b^2) - (3a^2 + 2b^2)$ = ...............

(ii) $(5x - 6y) + (6x - 5y)$ = ...............

(iii) $(a^2 - 2ab + b^2) - (a^2 + 2ab - b^2)$ = ...............

(iv) How much is $(a + 2b)$ greater than $(b - a)$? ...............

(v) How much is $(a - 2b + 3c)$ less than $(3c + 5 - a)$? ...............

(vi) What must be added to $(3x^2 + 5x - 2)$ to get $(3x + 1)?$ ...............

Answer

(i) $(2a^2 + 3b^2) - (3a^2 + 2b^2)$

$\Rightarrow 2a^2 + 3b^2 - 3a^2 - 2b^2\\[1em] \Rightarrow (2a^2 - 3a^2) + (3b^2 - 2b^2)\\[1em] \Rightarrow {-a^2 + b^2}$

Hence, $\bm{(2a^2 + 3b^2) - (3a^2 + 2b^2) = -a^2 + b^2}$

(ii) $(5x - 6y) + (6x - 5y)$

$\Rightarrow (5x + 6x) + (-6y - 5y)\\[1em] \Rightarrow {11x - 11y}$

Hence, $\bm{(5x - 6y) + (6x - 5y) = 11x - 11y}$

(iii) $(a^2 - 2ab + b^2) - (a^2 + 2ab - b^2)$

$\Rightarrow a^2 - 2ab + b^2 - a^2 - 2ab + b^2\\[1em] \Rightarrow (a^2 - a^2) + (-2ab - 2ab) + (b^2 + b^2)\\[1em] \Rightarrow {-4ab + 2b^2}$

Hence, $\bm{(a^2 - 2ab + b^2) - (a^2 + 2ab - b^2) = -4ab + 2b^2}$

(iv) Required expression $= (a + 2b) - (b - a)$

$\Rightarrow a + 2b - b + a\\[1em] \Rightarrow {2a + b}$

Hence, $\bm{(a + 2b) - (b - a) = 2a + b}$

(v) Required expression $= (3c + 5 - a) - (a - 2b + 3c)$

$\Rightarrow 3c + 5 - a - a + 2b - 3c\\[1em] \Rightarrow (3c - 3c) + (-a - a) + 2b + 5\\[1em] \Rightarrow {-2a + 2b + 5}$

Hence, $\bm{(3c + 5 - a) - (a - 2b + 3c) = -2a + 2b + 5}$

(vi) Required expression $= (3x + 1) - (3x^2 + 5x - 2)$

$\Rightarrow 3x + 1 - 3x^2 - 5x + 2\\[1em] \Rightarrow -3x^2 + (3x - 5x) + (1 + 2)\\[1em] \Rightarrow {-3x^2 - 2x + 3}$

Hence, $\bm{(3x + 1) - (3x^2 + 5x - 2) = -3x^2 - 2x + 3}$

Write (T) for true and (F) for false for each of the following statements:

(i) 3 subtracted from −6 is −3.

(ii) $(a + b)$ subtracted from $(a - b)$ is $2b$.

(iii) (−3) multiplied by (−6) is −18.

(iv) $(x^2 + 5)$ is a polynomial of degree 2.

(v) $(x^2y^2 + 8)$ is a polynomial of degree 2.

(vi) A variable has always a fixed value.

Answer

(i) False (F).

Reason :

3 subtracted from −6 is −6 − 3 = −9, not −3.

(ii) False (F).

Reason :

$(a - b) - (a + b) = a - b - a - b = -2b$, not $2b$.

(iii) False (F).

Reason :

(−3) × (−6) = 18, since the product of two negative numbers is positive.

(iv) True (T).

Reason :

The highest power of $x$ in $x^2 + 5$ is 2, so its degree is 2.

(v) False (F).

Reason :

The degree of $x^2y^2$ is (2 + 2) = 4, so the polynomial is of degree 4.

(vi) False (F).

Reason :

A variable can take various values; it is a constant that has a fixed value.