KC Sinha Mathematics Solution Class 11 Chapter 11 द्विघात समीकरण Quadratic Equation) Exercise 11.1

Exercise 11.1

Solve the following equations by factorization method

Question 1

$2 x^{2}+3=0$
Sol :
$2 x^{2}=-3$

$x^{2}=-\frac{3}{2}$

$x=\pm \sqrt{\frac{-3}{2}}=\pm \sqrt{\frac{3}{2}} i$


ALTERNATE METHOD

$2 x^{2}+3=0$

$(\sqrt{2} x)^{2}-(\sqrt{3} i)^{2}=0$

$\left(\sqrt{2} x+\sqrt 3 i\right)(\sqrt{2} x-\sqrt{3} i)=0$

$\sqrt{2} x+\sqrt3 i=0$

$\sqrt 2 x=-\sqrt 3 i$

$x=\frac{-\sqrt{3}}{2}i$

and $\sqrt 2 x-\sqrt 3i=0$

$\sqrt2x=\sqrt3i$

$x=\frac{\sqrt{3}}{2}i$

Question 2

$x^{2}+x+1=0$
Sol :
$x^{2}+2 \cdot x \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}+1-\left(\frac{1}{2}\right)^{2}=0$

$\left(x+\frac{1}{2}\right)^{2}+1-\frac{1}{4}=0$

$\left(x+\frac{1}{2}\right)^{2}+\frac{4-1}{4}=0$

$\left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}=0$

$\left(x+\frac{1}{2}\right)^{2}-\left(\frac{\sqrt{3}}{2}i\right)^{2}=0$

$\left(x+\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)\left(x+\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)=0$

$x+\frac{1}{2}-\frac{\sqrt3}{2} i=0$

$x=-\frac{1}{2}+\frac{\sqrt 3}{2}i$

and $x+\frac{1}{2}+\frac{\sqrt{3}}{2} i=0$

$x=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$

∴ $x=-\frac{1}{2} \pm \frac{\sqrt 3}{2} i$

Question 3

$x^{2}+2 x+5=0$
Sol :
$x^{2}+2 \cdot x \cdot 1+1^{2}+5-1^{2}=0$

$(x+1)^{2}+5-1=0$

$(x+1)^{2}+4=0$

$(x+1)^{2}-(2 i)^{2}=0$

(x+1-2i)(x+1+2i)=0

x+1-2i=0

x=-1+2i

and x+1+2i=0

x=-1-2i

∴ x=-1±2i

ALTERNATE METHOD

$x^{2}+2 \cdot x \cdot 1+1^{2}+5-1^{2}=0$

$(x+1)^{2}+5-1=0$

$(x+1)^{2}+4=0$

$(x+1)^{2}=-4$

$x+1=\pm \sqrt{-4}$

x+1=±2i

x=-1±2i

Question 4

$x^{2}-4 x+7=0$
Sol :
$x^{2}-2 \cdot x \cdot 2+2^{2}+7-2^{2}=0$

$(x-2)^{2}+3=0$

$(x-2)^{2}-(\sqrt{3} i)^{2}=0$

Question 5

$x^{2}-4 x+13=0$

Question 6

$9 x^{2}-12 x+20=0$

Sol :
$(3x)^2-2 \cdot 3 x \cdot 2+2^{2}+20-2^{2}=0$

$(3 x-2)^{2}+11=0$

Question 7

Solve the following equations by using the general expression for roots of a quadratic equation:
$2 x^{2}-\sqrt{3} x+1=0$
Sol :
$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2a}i$

a=2 ,$b=-\sqrt{3}$ , c=1

$=\frac{-(-\sqrt{3}) \pm \sqrt{4 \times 2 \times 1-(-\sqrt{3})^{2}}}{2 \times 2}i$

$x=\frac{\sqrt{3} \pm \sqrt{8-3}}{4}i$

$x=\frac{\sqrt{3} \pm \sqrt{5} i}{4}$

$=\frac{\sqrt{3}}{4} \pm \frac{\sqrt{5}}{4}i$

Question 8

$17 x^{2}-8 x+1=0$
Sol :
a=17 , b=-8 , c=1

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2a}i$

$=\frac{-(-8) \pm \sqrt{4 \times 17 \times 1-(-8)^{2}}}{2 \times 17}i$

$x=\frac{8 \pm \sqrt{68-64}}{34}i$

$2=\frac{8 \pm \sqrt{4}}{34}i$

$x=\frac{8 \pm 2 i}{34}$

$x=\frac{2(4 \pm i)}{34}$

$=\frac{4}{17} \pm \frac{1}{17}$

Question 12

$9 x^{2}+4=0$
Sol :
a=9 , b=0  , c=4

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2 a}i$

$=\frac{-0 \pm \sqrt{4 \times 9 \times 4-0^{2}}}{2 \times 9}i$

$x=\frac{\pm \sqrt{144}}{18}i$

$x=\pm \frac{12 i}{18}$

$=\pm \frac{2}{3} i$

Question 17

$\sqrt{3} x^{2}-\sqrt{2} x+3 \sqrt{3}=0$
Sol :
$a=\sqrt{3}$, $b=-\sqrt{2}$ ,$c=3 \sqrt{3}$

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2 a}i$

$x=\frac{-(-\sqrt{2}) \pm \sqrt{4 \times \sqrt{3} \times 3 \sqrt{3}-(-\sqrt{2})^{2}}}{2 \sqrt{3}}i$

$x=\frac{\sqrt{2} \pm \sqrt{36-2}}{2 \sqrt{3}}i$

$x=\frac{\sqrt{2} \pm \sqrt{34}}{2 \sqrt{3}}i$

Question 27

$x^{2}-2 x+\frac{3}{2}=0$
Sol :
$\frac{2 x^{2}-4 x+3}{2}=0$

$2 x^{2}-4 x+3=0$

a=2 , b=-4 , c=3

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2 a}i$

$x=\frac{-(-4) \pm \sqrt{4 \times 2 \times 3-(-4)^{2}}}{2 \times 2}i$

$x=\frac{4 \pm \sqrt{24-16}}{4}i$

$x=\frac{4 \pm \sqrt{8} i}{4}$

$=\frac{4 \pm 2 \sqrt{2} i}{4}$

$x=\frac{2(2 \pm \sqrt{2} i)}{4}$

$x=\frac{2 \pm \sqrt{2} i}{2}$

$x=1 \pm \frac{\sqrt{2}}{2} i$

Question 28

$x^{2}+\frac{x}{\sqrt{2}}+1=0$
Sol :
$\frac{\sqrt{2} x^{2}+x+\sqrt{2}}{\sqrt{2}}=0$

$\sqrt 2 x^{2}+x+\sqrt2=0$

$a=\sqrt{2}$ , b=1 , $c=\sqrt{2}$

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2 a}i$

$x=\frac{-1 \pm \sqrt{4 \times \sqrt{2} \times \sqrt{2}-1^{2}}}{2 \sqrt{2}}$

$x=\frac{-1 \pm \sqrt{8-1}}{2 \sqrt{2}}i$

$x=\frac{-1 \pm \sqrt{7} i}{2 \sqrt{2}}$

Question 30

$x^{2}+x+\frac{1}{\sqrt{2}}=0$
Sol :
$\frac{\sqrt{2} x^{2}+\sqrt{2} x+1}{\sqrt{2}}=0$

$\sqrt{2} x^{2}+\sqrt{2} x+1=0$

$a=\sqrt{2}$, $b=\sqrt{2}$, c=1

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2 a}i$

$x=\frac{-\sqrt{2} \pm \sqrt{4 \sqrt{2} \times 1-(\sqrt{2})^{2}}i}{2 \sqrt{2}}$

$x=\frac{-\sqrt{2} \pm \sqrt{4 \sqrt{2}-2}}{2 \sqrt{2}}i$

$x=\frac{\sqrt{2} \pm \sqrt{2(2 \sqrt{2}-1)}}{2 \sqrt{2}}i$

$x=\sqrt2\left(\frac{-1 \pm \sqrt{2 \sqrt{2}-1}i}{2 \sqrt{2}}\right)$

$x=-\frac{1 \pm \sqrt{2 \sqrt{2}-1}}{2}$

Question 31

$\sqrt{5} x^{2}+x+\sqrt{5}=0$
Sol :
$a=\sqrt{5}$ ,b=1, $c=\sqrt{5}$

$x=\frac{-b \pm \sqrt{4 a c-b^{2}}}{2 a}i$

$x=\frac{-1 \pm \sqrt{4 \times \sqrt{5} \times \sqrt{5}-1^{2}}i}{2 \times \sqrt{5}}$

$x=\frac{-1 \pm \sqrt{20-1}}{2 \sqrt{5}}i$

$x=\frac{-1 \pm \sqrt{19}}{2 \sqrt{5}}i$