True False[1 Marks ]

Question 11 Mark

Write whether the following statements are true or false. Justify your answers.
Every quadratic equations has at most two roots.

Answer

Consider the quadratic equation $x^2-5 x+6=0$. put 2 and 3 in $x$ and the quadratic expression $x^2-5 x+6$ becomes equal to 0 . So, 2 and 3 are the roots of the quadratic equation $x^2-5 x+6=0$. So, any quadratic equation can have atmost two roots i.e., one or two roots, but not more than two.

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Question 21 Mark

Write whether the following statements are true or false. Justify your answers.
Every quadratic equation has at least one real root.

Answer

False:
Consider a quadratic equation $x^2+1=0$ which has no real root. So, the given statement is false.

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Question 31 Mark

Write whether the following statements are true or false. Justify your answers.
If the coefficient of $x^2$ and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.

Answer

True:
In quadratic equation $a x^2+b x+c=0$, if $a$ and $c$ have same sign and $b=0$, then $b^2-4 a c=(0)^2-4 a c=-4 a c<0$.So, the quadratic equation has no real roots.
Hence, the given statement is true.

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Question 41 Mark

Write whether the following statements are true or false. Justify your answers.
If the coefficient of $x^2$ and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.

Answer

True:
In quadratic equation $a x^2+b x+c=0$, if a and $c$ have opposite signs, then $a c<0$. Therefore, $b ^2-4 ac >0$. So, the quadratic equation has real roots. Hence, the given statement is true.

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Question 51 Mark

Write whether the following statements are true or false. Justify your answers.
Every quadratic equation has at least two roots.

Answer

False:
Consider the quadratic equation $x^2-4 x+4=0$ which has only 2 as root. So, the given statement is false.

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Question 61 Mark

Write whether the following statements are true or false. Justify your answers.
Every quadratic equation has exactly one root.

Answer

False:
Consider a quadratic equation $x^2-4=0$ which has two distinct roots -2 and 2 . So, the given statement is false.

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MATHS True False[1 Marks ]

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