If b = 0, c < 0, is it true that the roots of x^{2} + bx+ c = 0 are numerically equal and opposite in sign? Justify.
True
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form, ax^{2} + bx + c = 0 Where a, b and c are real numbers with a 0
Roots : If ax^{2} + bx + c = 0 …..(1)
Is a quadratic equation then the values of x which satisfy equation 1 are the roots of the equation.
Here the given equation is x^{2} + bx+ c = 0 …..(2)
It is also given that b = 0, c < 0.
Let c=-y
Put b = 0, c = – y in (2)
Hence both the roots are equal and opposite in sign. Hence the given statement is true.
View Full Answer(1)Is 0.2 a root of the equation x^{2} – 0.4 = 0? Justify
False
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form, ax^{2} + bx + c = 0 Where a, b and c are real numbers with a 0
Roots : If ax^{2} + bx + c = 0 …..(1)
Is a quadratic equation then the values of x which satisfy equation 1 are the roots of the equation.
Here the given equation is x^{2} – 0.4 = 0 …..(2)
If 0.2 is a root of equation 2 then it should satisfy its equation
Put x = 0.2 in (2)
Here 0.2 is not satisfying the equation (2)
Hence 0.2 is not a root of the equation
View Full Answer(1)Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rational? Why?
True
Let us consider an equation with distinct irrational numbers
compare with where
let us find the roots of the equation
The roots are rational. Hence the given statement is true
View Full Answer(1)
Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.
True
Solution
Rational number – A number that can be expressed in the form of where
(p, q are integers)
Irrational number – A number that can not be expressed in the form of ratio of two integers.
Let us suppose a quadratic equation with rational coefficient.
compare with where
Let us find the roots of the equation
Both of its roots are irrational.
Hence the given statement is true.
View Full Answer(2)A quadratic equation with integral coefficient has integral roots. Justify your answer
False
Solution
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is a quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
Let us take a quadratic equation with integral coefficient
compare with where
(a=2,b=-3,c=-5)
let us find the roots of the equation
Here we found that 5/2 is not on integral
Hence the given statement is false
View Full Answer(1)Write whether the following statements are true or false. Justify your answers.
(i) Every quadratic equation has exactly one root.
(ii) Every quadratic equation has at least one real root.
(iii) Every quadratic equation has at least two roots.
(iv) Every quadratic equations has at most two roots.
(v) If the coefficient of x^{2} and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
(vi)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.
(i) False
Solution
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is an quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
Let us consider a quadratic equation
Here – 2, 2 are the two roots of the equation.
Hence it is false that every quadratic equation has exactly one root.
(ii) False
Solution
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is an quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
Let us consider a quadratic equation
compare with where
Hence both the roots of the equation are imaginary.
Hence the statement every quadratic equation has at least one real root is False.
(iii) False
Solution
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is an quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
Let us consider a quadratic equation
The equation has only one root which is x = 2
Hence the given statement is False
(iv) True
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is an quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
As we know that the standard form of a quadratic equation is
It is a polynomial of degree 2.
As per the power of x is 2. There is at most 2 values of x exist that satisfy equation.
Hence the given statement every quadratic equations has at most two roots is true.
(v) True
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is an quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
Let us suppose a quadratic equation
is a quadratic equation in which coefficient of x^{2} and constant term have opposite signs.
compare with where
Here
Hence these types of equations have real roots.
(vi) True
Quadratic Equation – A quadratic equation in x is an equation that can be written in the standard form,
Where a, b and c are real numbers with
Root: If …..(1)
Is an quadratic equation then the values of x which satisfy equation (1) are the roots of the equation.
Let us suppose a quadratic equation
In , the coefficient of x^{2} and constant term has same sign and coefficient of x is 0.
Compare with where
Here
Hence these types of equations have no real roots.
View Full Answer(2)State whether the following quadratic equations have two distinct real roots. Justify your answer.
(i) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here a=1,b=-3,c=4
Hence the equation has no real roots.
(ii) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here a=2,b=-1,c=-1
Here for the equation ,
Hence it is a quadratic equation with two distinct real roots
(iii) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here
here
Hence the equation have two real and equal roots.
(iv) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here a=3,b=-4,c=1
Here
Hence the equation has distinct and real roots.
(v) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here
Hence the equation has no real roots.
(vi) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Compare with where
Here
Hence the equation has two distinct real roots.
(vii) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here
Here
Hence the equation has distinct real roots.
(viii) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
compare with where
Here
Hence the equation has no real roots.
(ix) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Here
Hence the equation has two distinct real roots.
(x) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form Where a, b and c are real numbers with
The given equation is
Compare with where
Here
Hence the equation has two distinct real roots.
View Full Answer(1)Write whether the following statement is True or False. Justify your answer.
The graph of every linear equation in two variables need not be a line.
Answer:
False
Solution:
We know that the standard form of any linear equation is ax + by + c = 0
If we put different value of x we get different values of y corresponding to the values of x.
So we have x proportional to y.
Thus we can say = constant.
Now, the general equation of a line is , which also gives a direct proportionality between x and y.
So if we plot a graph with the help of such points, it will always be a line
Therefore the given statement is false.
View Full Answer(1)Write whether the following statement is True or False. Justify your answer.
Every point on the graph of a linear equation in two variables does not represent a solution of the linear equation.
Answer:
False
Solution:
The graph of a linear equation in two variables is a line which can be plotted by finding two solutions (a_{1}, b_{1}) and (a_{2}, b_{2}).
Hence the graph of a linear equation is constructed by joining these points. So any point on the graph must be a solution of this linear equation.
Hence the given statement is False.
View Full Answer(1)Write whether the following statement is True or False. Justify your answer.
The coordinates of points in the table:
x | 0 | 1 | 2 | 3 | 4 |
y | 2 | 3 | 4 | -5 | 6 |
represent some of the solutions of the equation x – y + 2 = 0.
Answer:
False
Solution:
The given equation is x – y + 2 = 0
Put x = 0 in the given equation
0 – y + 2 = 0
y = 2, i.e., the point is (0, 2)
Put x = 1 in the given equation
1 – y + 2 = 0
y = 3, i.e., the point is (1, 3)
Put x = 2 in the given equation
2 – y + 2 = 0
4 = y, i.e., the point is (2, 4)
Put x = 3 in the given equation
3 – y + 2 = 0
5 = y, i.e., the point is (3, 5)
Put x = 4 in the given equation
4 – y + 2 = 0
6 = y, i.e., the point is (4, 6)
The coordinates of points in the table:
x | 0 | 1 | 2 | 3 | 4 |
y | 2 | 3 | 4 | -5 | 6 |
It is given that the above represents some of the solutions of the equation x – y + 2 = 0.
Here all the points satisfy the given equation except (3, -5).
Therefore the given statement is false because one of the table entry is an incorrect solution for the given equation.
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