If x, y, z are real and distinct then f( x,y ) = x^2 + 4y^2 + 9z^2 - 6yz - 3zx - 2xy is always

f( x,y ) = 12[ ( x - 2y )^2 + ( 2y - 3z )^2 + ( 3z - x )^2 ] 0.

if x y z are real and distinct then fleft xy x 2 4y 2 9z 2

If $$x, y, z$$ are real and distinct then $$f\left( {x,y} \right) = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$ is always

A. non-negative

B. non-positive

C. zero

D. None of these

Answer : non-negative

Solution :
$$f\left( {x,y} \right) = \frac{1}{2}\left[ {{{\left( {x - 2y} \right)}^2} + {{\left( {2y - 3z} \right)}^2} + {{\left( {3z - x} \right)}^2}} \right] \geqslant 0.$$

Releted MCQ Question on
Algebra >> Quadratic Equation

Releted Question 1

If $$\ell ,m,n$$ are real, $$\ell \ne m,$$ then the roots by the equation: $$\left( {\ell - m} \right){x^2} - 5\left( {\ell + m} \right)x - 2\left( {\ell - m} \right) = 0$$ are

A. Real and equal

B. Complex

C. Real and unequal

D. None of these

View Answer

Releted Question 2

The equation $$x + 2y + 2z = 1{\text{ and }}2x + 4y + 4z = 9{\text{ have}}$$

A. Only one solution

B. Only two solutions

C. Infinite number of solutions

D. None of these

View Answer

Releted Question 3

Let $$a > 0, b > 0$$ and $$c > 0$$ . Then the roots of the equation $$a{x^2} + bx + c = 0$$

A. are real and negative

B. have negative real parts

C. both (A) and (B)

D. none of these

View Answer

Releted Question 4

Both the roots of the equation $$\left( {x - b} \right)\left( {x - c} \right) + \left( {x - a} \right)\left( {x - c} \right) + \left( {x - a} \right)\left( {x - b} \right) = 0$$ are always

A. positive

B. real

C. negative

D. none of these.

View Answer

If $$x, y, z$$ are real and distinct then $$f\left( {x,y} \right) = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$ is always

Releted MCQ Question on
Algebra >> Quadratic Equation

If $$\ell ,m,n$$ are real, $$\ell \ne m,$$ then the roots by the equation: $$\left( {\ell - m} \right){x^2} - 5\left( {\ell + m} \right)x - 2\left( {\ell - m} \right) = 0$$ are

The equation $$x + 2y + 2z = 1{\text{ and }}2x + 4y + 4z = 9{\text{ have}}$$

Let $$a > 0, b > 0$$ and $$c > 0$$ . Then the roots of the equation $$a{x^2} + bx + c = 0$$

Both the roots of the equation $$\left( {x - b} \right)\left( {x - c} \right) + \left( {x - a} \right)\left( {x - c} \right) + \left( {x - a} \right)\left( {x - b} \right) = 0$$ are always

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Quadratic Equation

Practice More MCQ Question on Maths Section

If $$x, y, z$$ are real and distinct then $$f\left( {x,y} \right) = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$ is always

Releted MCQ Question on Algebra >> Quadratic Equation

If $$\ell ,m,n$$ are real, $$\ell \ne m,$$ then the roots by the equation: $$\left( {\ell - m} \right){x^2} - 5\left( {\ell + m} \right)x - 2\left( {\ell - m} \right) = 0$$ are

The equation $$x + 2y + 2z = 1{\text{ and }}2x + 4y + 4z = 9{\text{ have}}$$

Let $$a > 0, b > 0$$ and $$c > 0$$ . Then the roots of the equation $$a{x^2} + bx + c = 0$$

Both the roots of the equation $$\left( {x - b} \right)\left( {x - c} \right) + \left( {x - a} \right)\left( {x - c} \right) + \left( {x - a} \right)\left( {x - b} \right) = 0$$ are always

Practice More Releted MCQ Question on Quadratic Equation

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Algebra >> Quadratic Equation

Practice More Releted MCQ Question on
Quadratic Equation