Question 41 of 43advanced🔧 ApplyLong Answer5 marks

If a+b+c=5a + b + c = 5 and ab+bc+ca=10ab + bc + ca = 10, then prove that a3+b3+c3−3abc=−25a^3 + b^3 + c^3 - 3abc = -25.

Correct Answer

We are given that a+b+c=5a + b + c = 5 and ab+bc+ca=10ab + bc + ca = 10. To prove the required equation, we will use the algebraic identities from the chapter.

First, we need to find the value of a2+b2+c2a^2 + b^2 + c^2. Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca, we substitute the given values:

(5)2=a2+b2+c2+2(10)(5)^2 = a^2 + b^2 + c^2 + 2(10) 25=a2+b2+c2+2025 = a^2 + b^2 + c^2 + 20 a2+b2+c2=5a^2 + b^2 + c^2 = 5

Next, we apply the identity (a+b+c)(a2+b2+c2−ab−bc−ca)=a3+b3+c3−3abc(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = a^3 + b^3 + c^3 - 3abc. Substituting the known values into the left-hand side of this identity:

LHS=5×(5−10)\text{LHS} = 5 \times (5 - 10) LHS=5×(−5)\text{LHS} = 5 \times (-5) LHS=−25\text{LHS} = -25

Since the left-hand side equals the right-hand side of the identity, we conclude that a3+b3+c3−3abc=−25a^3 + b^3 + c^3 - 3abc = -25. Hence proved.

Exercise: END-OF-CHAPTER EXERCISES | Q: 11 | (Page 23)
For More Understanding

Explanation

The textbook introduces the identity (x+y+z)(x2+y2+z2−xy−xz−yz)=x3+y3+z3−3xyz(x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) = x^3 + y^3 + z^3 - 3xyz. To use this, we need the value of a2+b2+c2a^2 + b^2 + c^2, which is found using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.

Substituting a+b+c=5a + b + c = 5 and ab+bc+ca=10ab + bc + ca = 10 gives a2+b2+c2=5a^2 + b^2 + c^2 = 5. Plugging all values into the main identity yields the result −25-25.

Solution Steps

  1. Step 1: Find a2+b2+c2a^2 + b^2 + c^2 using (a+b+c)2=a2+b2+c2+2(ab+bc+ca)(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)

  2. Step 2: Substitute values to get 25=a2+b2+c2+2025 = a^2+b^2+c^2+20, so a2+b2+c2=5a^2+b^2+c^2=5

  3. Step 3: Use identity (a+b+c)(a2+b2+c2−ab−bc−ca)=a3+b3+c3−3abc(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc

  4. Step 4: Substitute values: 5×(5−10)=−255 \times (5 - 10) = -25

  5. Step 5: Conclude a3+b3+c3−3abc=−25a^3+b^3+c^3-3abc = -25

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