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Standard X

Mathematics

Discriminant

State the fol...

Question

State the following statement is True or False
If $k^{2} x^{2} - 2 (k - 1) x + 4 = 0$ has equal roots, then $k = - 1$ or $k = \frac{1}{3}$

True

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False

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Solution

The correct option is A True
Given equation is,
$k^{2} x^{2} - 2 (k - 1) x + 4 = 0$
Since, the roots of the equation are equal, hence discriminant, $D = 0$
$D = b^{2} - 4 a c = 0$
$a = k^{2}, b = - 2 (k - 1), c = 4$
$∴ (2 (k - 1))^{2} - 4 \times 4 \times k^{2} = 0$
$4 k^{2} - 8 k + 4 - 16 k^{2} = 0$
$- 12 k^{2} - 8 k + 4 = 0$
$3 k^{2} + 2 k - 1 = 0$
$3 k^{2} + 3 k - k - 1 = 0$
$(3 k - 1) (k + 1) = 0$
$∴ k = - 1, \frac{1}{3}$

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(ii)

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(iii)

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(v)

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(xiii)

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State the following statement is True or FalseIf k2x2−2(k−1)x+4=0 has equal roots, then k=−1 or k=13

The correct option is A TrueGiven equation is,k2x2−2(k−1)x+4=0Since, the roots of the equation are equal, hence discriminant, D=0D=b2−4ac=0a=k2,b=−2(k−1),c=4∴(2(k−1))2−4×4×k2=04k2−8k+4−16k2=0−12k2−8k+4=03k2+2k−1=03k2+3k−k−1=0(3k−1)(k+1)=0∴k=−1,13

State the following statement is True or False
If $k^{2} x^{2} - 2 (k - 1) x + 4 = 0$ has equal roots, then $k = - 1$ or $k = \frac{1}{3}$