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If the roots of equations \( \Large ax^{2}+bx+c=0 \) be \( \Large \ \alpha \ and\ \beta \) then the roots of equations \( \Large cx^{2}+bx+a=0 \) are:

A) \( \Large - \alpha ,\ - \beta \)
B) \( \Large \alpha ,\ \frac{1}{ \beta } \)
C) \( \Large \frac{1}{ \alpha },\ \frac{1}{ \beta } \)
D) none of these

Correct Answer:



C) \( \Large \frac{1}{ \alpha },\ \frac{1}{ \beta } \)

Description for Correct answer:

Since \( \Large \alpha \) and \( \Large \beta \) are the roots by \( \Large ax^{2}+bx+c=0 \)

=> \( \Large \alpha + \beta \) = \( \Large -\frac{b}{c} \) and \( \Large \alpha \beta = \frac{c}{a} \)

Let the roots of \( \Large cx^{2}+bx+a=0 \) be \( \Large \alpha' \), \( \Large \beta' \) then

\( \Large \alpha' \) + \( \Large \beta' \) = \( \Large -\frac{b}{c} \) and \( \Large \alpha . \beta = \frac{a}{c} \)

Now, \( \Large \frac{ \alpha + \beta }{ \alpha \beta }=\frac{\frac{-b}{a}}{\frac{c}{a}}=\frac{-b}{c} \)

=> \( \Large \frac{1}{ \alpha }+\frac{1}{ \beta }= \alpha' + \beta' \)

Hence, \( \Large \alpha' \) = \( \Large \frac{1}{ \alpha } \) and \( \Large \beta' \) = \( \Large \frac{1}{ \beta } \)

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

Comments

Similar Questions

1). Both the roots of the given equation. \( \Large \left(x-a\right) \left(x-b\right)+ \left(x-b\right) \left(x-c\right)+ \left(x-c\right) \left(x-a\right)=0 \) are always

A). positive

B). negative

C). real

D). imaginary

2). If the difference of roots of the equation \( \Large x^{2}-bx+c=0 \) be 1, then;

A). \( \Large b^{2}-4c-1=0 \)

B). \( \Large b^{2}-4c=0 \)

C). \( \Large b^{2}-4c+1=0 \)

D). \( \Large b^{2}+4c-1=0 \)

3). If \( \Large 2+i\sqrt{3} \) is a root of the equation \( \Large x^{2}+px+q=0 \) where p and q are real, then \( \Large p, q \) is equal to:

A). (-4, 7)

B). (4,-7)

C). (4, 7)

D). (-4, -7)

4). If \( \Large x=\sqrt{1+\sqrt{1+\sqrt{1}+.....}} \), the x is equal to

A). \( \Large \frac{1+\sqrt{5}}{2} \)

B). \( \Large \frac{1-\sqrt{5}}{2} \)

C). \( \Large \frac{1\pm \sqrt{5}}{2} \)

D). none of these .

5). If \( \Large \alpha , \beta \) are the roots of equation \( \Large ax^{2}+bx+c=0 \), then \( \Large \frac{ \alpha }{ \alpha \beta +b}+\frac{ \beta }{ \alpha x+b} \) is equal to

A). 2/a

B). 2/b

C). 2/c

D). -2/a

6). If the roots of the equation \( \Large \frac{ \alpha }{x- \alpha }+\frac{ \beta }{x- \beta }=1 \) be equal in magnitude but opposite in sign, then \( \Large \alpha + \beta \) is equal to:

A). 0

B). 1

C). 2

D). none of these

7). If the roots of the equation \( \Large px^{2}+2qx+r=0 \) and \( \Large qx^{2}-2\sqrt{pr}x+q=0 \) be real, then:

A). \( \Large p=q \)

B). \( \Large q^{2}=pr \)

C). \( \Large p^{2}=qr \)

D). \( \Large r^{2}=pq \)

8). If one root of the quadratic equation \( \Large ax^{2}+bx+c=0 \) is equal to nth power of the other root, then the value of \( \Large \left(ac^{n}\right)^{\frac{1}{n+1}} + \left(a^{n}c\right)^{\frac{1}{n+1}} \) is equal to

A). b

B). -b

C). \( \Large b^{1/n+1} \)

D). \( \Large -b^{1/n+1} \)

9). The quadratic in t, such that AM of its roots is A and GM is G is:

A). \( \Large t^{2}-2At+G^{2}=0 \)

B). \( \Large t^{2}-2At-G^{2}=0 \)

C). \( \Large t^{2}+2At+G^{2}=0 \)

D). none of these.

10). If \( \Large n^{2}px+1 \) is a factor of expression \( \Large ax^{3}+bx+c \) then:

A). \( \Large a^{2}+c^{2}=-ab \)

B). \( \Large a^{2}-c^{2}=-ab \)

C). \( \Large a^{2}-c^{2}=ab \)

D). none of these