Assertion and Reason on (a − b)³ + (b − c)³ + (c − a)³ Assertion and Reason Question on Algebraic Identities Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. (a) Statement-1 is true, […]

Factor of (x + y)³ − (x³ + y³) Factor of (x + y)³ − (x³ + y³) Which of the following is a factor of \[ (x+y)^3-(x^3+y^3)? \] (a) \(x^2+y^2+2xy\) (b) \(x^2+y^2-xy\) (c) \(xy^2\) (d) \(3xy\) Solution \[ (x+y)^3=x^3+y^3+3xy(x+y) \] Therefore, \[ (x+y)^3-(x^3+y^3)=3xy(x+y) \] Hence, \[ \boxed{(d)\ 3xy} \] Next Question / Full Exercise

If x/y + y/x = −1, Find the Value of x³ − y³ If x/y + y/x = −1, Find the Value of x³ − y³ If \[ \frac{x}{y}+\frac{y}{x}=-1 \quad (x,y\ne0), \] then the value of \[ x^3-y^3 \] is (a) \(1\) (b) \(-1\) (c) \(0\) (d) \(\frac{1}{2}\) Solution \[ \frac{x}{y}+\frac{y}{x}=-1 \] Multiplying by \(xy\):

If x³ − 3x² + 3x + 7 = (x + 1)(ax² + bx + c), Find a + b + c If x³ − 3x² + 3x + 7 = (x + 1)(ax² + bx + c), Find a + b + c If \[ x^3-3x^2+3x+7=(x+1)(ax^2+bx+c), \] then \[ a+b+c= \] (a) \(4\) (b)

If (x + y)³ − (x − y)³ − 6y(x² − y²) = ky³, Find k If (x + y)³ − (x − y)³ − 6y(x² − y²) = ky³, Find k If \[ (x+y)^3-(x-y)^3-6y(x^2-y^2)=ky^3, \] then \[ k= \] (a) \(1\) (b) \(2\) (c) \(4\) (d) \(8\) Solution \[ (x+y)^3=x^3+3x^2y+3xy^2+y^3 \] \[ (x-y)^3=x^3-3x^2y+3xy^2-y^3 \]

If 3x = a + b + c, Find the Value of (x − a)³ + (x − b)³ + (x − c)³ − 3(x − a)(x − b)(x − c) If 3x = a + b + c, Find the Value of (x − a)³ + (x − b)³ + (x − c)³ −

Factorization of x⁴ + 4 Factorization of x⁴ + 4 The expression \[ x^4+4 \] can be factorized as (a) \((x^2+2x+2)(x^2-2x+2)\) (b) \((x^2+2x+2)(x^2+2x-2)\) (c) \((x^2-2x-2)(x^2-2x+2)\) (d) \((x^2+2)(x^2-2)\) Solution \[ x^4+4 \] \[ =x^4+4x^2+4-4x^2 \] \[ =(x^2+2)^2-(2x)^2 \] \[ =(x^2+2x+2)(x^2-2x+2) \] Therefore, \[ \boxed{(a)\ (x^2+2x+2)(x^2-2x+2)} \] Next Question / Full Exercise

Factors of x³ − 7x + 6 Factors of x³ − 7x + 6 The factors of \[ x^3-7x+6 \] are (a) \(x(x-6)(x-1)\) (b) \((x^2-6)(x-1)\) (c) \((x+1)(x+2)(x-3)\) (d) \((x-1)(x+3)(x-2)\) Solution \[ x^3-7x+6 \] \[ =x^3-x^2+x^2-7x+6 \] \[ =x^2(x-1)+x(x-1)-6(x-1) \] \[ =(x-1)(x^2+x-6) \] \[ =(x-1)(x+3)(x-2) \] Therefore, \[ \boxed{(d)\ (x-1)(x+3)(x-2)} \] Next Question / Full Exercise

Factors of x² + 4y² + 4y − 4xy − 2x − 8 Factors of x² + 4y² + 4y − 4xy − 2x − 8 The factors of \[ x^2+4y^2+4y-4xy-2x-8 \] are (a) \((x-2y-4)(x-2y+2)\) (b) \((x-y+2)(x-4y-4)\) (c) \((x+2y-4)(x+2y+2)\) (d) none of these Solution \[ x^2+4y^2+4y-4xy-2x-8 \] \[ =x^2-4xy+4y^2-2x+4y-8 \] \[ =(x-2y)^2-2(x-2y)-8 \] Let \[

Factors of x⁴ + x² + 25 Factors of x⁴ + x² + 25 The factors of \[ x^4+x^2+25 \] are (a) \((x^2+3x+5)(x^2-3x+5)\) (b) \((x^2+3x+5)(x^2+3x-5)\) (c) \((x^2+x+5)(x^2-x+5)\) (d) none of these Solution \[ x^4+x^2+25 \] \[ =x^4+10x^2+25-9x^2 \] \[ =(x^2+5)^2-(3x)^2 \] \[ =(x^2+5+3x)(x^2+5-3x) \] \[ =(x^2+3x+5)(x^2-3x+5) \] Therefore, \[ \boxed{(a)\ (x^2+3x+5)(x^2-3x+5)} \] Next Question /