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Mathematics [3 marks sum]

Quadratic Equations · ICSE STD 10 (ICSE - English Medium). 22 questions.

Questions

[3 marks sum]

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22 questions · timed · auto-graded

Question 13 Marks
Solve the following quadratic equation using formula method only
$2 x^2-2 \sqrt{6}+3=0$
Answer
$2 x^2-2 \sqrt{6}+3=0$
$a=2 ; b=-2 \sqrt{6} x ; c=3 $
$D=b^2-4 a c$
$ =(-2 \text { sqrt } 6)^2-4(2)(3)$
$ =12-24 $
$ =-12$
$ x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} $
$ x=\frac{2 \sqrt{6} \pm \sqrt{-12}}{2}$
$ x=\frac{2 \sqrt{6}}{4}$
$ x=\frac{\sqrt{6}}{2}$
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Question 23 Marks
Solve the following quadratic equation using formula method only
$5x^2 - 19x + 17 = 0$
Answer
$5 x^2-19 x+17=0 $
$ a=5 ; b=-19 ; c=17 $
$ D=b^2-4 a c$
$ =(-19)^2-4(5)(17) $
$=361-340 $
$ =21 $
$ x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$
$ x=\frac{19 \pm \sqrt{21}}{10} $
$ x=\frac{19+\sqrt{21}}{10}, x=\frac{19-\sqrt{21}}{10}$
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Question 33 Marks
Solve the following quadratic equation using formula method only
$x^2 - 6x + 4 = 0$
Answer
$x^2-6 x+4=0$
$ a=1 ; b=-6 ; c=4 $
$ D=b^2-4 a c$
$ =(-6)^2-4(1)(4) $
$ =36-16 $
$ =20 $
$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$
$ x=\frac{6 \pm \sqrt{20}}{2} $
$ x=\frac{6+2 \sqrt{5}}{2}, x=\frac{6-2 \sqrt{5}}{2}$
$ x=3+\sqrt{5}, x=3-\sqrt{5}$
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Question 43 Marks
Solve the following quadratic equation using formula method only
$6x^2 + 7x - 10 = 0$
Answer
$6 x^2+7 x-10=0 $
$ a=6 ; b=7 ; c=-10$
$ D=b^2-4 a c $
$ =(7)^2-4(6)(-10) $
$ =49+240 $
$ =289$
$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} $
$x=\frac{-7 \pm \sqrt{289}}{12}$
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Question 53 Marks
Solve the following quadratic equation using formula method only
$x^2 - 4x - 1 = 0$
Answer
$x^2-4 x-1=0 $
$ a=1 ; b=-4 ; c=-1$
$ D=b^2-4 a c $
$ =(-4)^2-4(1)(-1) $
$ =16+4 $
$=20 $
$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} $
$x=\frac{4 \pm \sqrt{20}}{2} $
$ x=\frac{4 \pm 2 \sqrt{5}}{2} $
$x=2+\sqrt{5}, x=2-\sqrt{5}$
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Question 63 Marks
Solve the following quadratic equation using formula method only
$x^2 - 7x - 5 = 0$
Answer
$x^2-7 x-5=0 $
$a=1 ; b=-7 ; c=-5$
$ D=b^2-4 a c $
$=(-7)^2-4(1)(-5) $
$=49+20 $
$=69$
$ x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$
$ x=\frac{7 \pm \sqrt{69}}{2}$
$ x=\frac{7+\sqrt{69}}{2}, x=\frac{7-\sqrt{69}}{2}$
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Question 73 Marks
Solve the following quadratic equation using formula method only
$4x^2 + 12x + 9 = 0$
Answer
$4 x ^2+12 x +9=0 $
$a =4 ; b =12 ; c=9 $
$ D = b ^2-4 ac $
$=(12)^2-4(4)(9)$
$ =144-144 $
$ =0$
$ x =\frac{- b \pm \sqrt{ b ^2-4 ac }}{2 a} $
$ x =\frac{-12 \pm 0}{8} $
$ x =-\frac{3}{2}$
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Question 83 Marks
Solve the following quadratic equation using formula method only
$3 x ^2-5 x +\frac{25}{12}=0$
Answer
$3 x ^2-5 x +\frac{25}{12}=0$
$ a =3 ; b =-5 ; c =\frac{25}{12} $
$ D = b ^2-4 ac$
$ =(-5)^2-4(3)\left(\frac{25}{12}\right) $
$ =25-25 $
$=0 $
$ x =\frac{- b \pm \sqrt{ b ^2-4 ac }}{2 a} $
$x =\frac{-(-5) \pm 0}{6} $
$x =\frac{5}{6}$
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Question 93 Marks
Solve the following quadratic equation using formula method only
$2 x^2-2 \cdot \sqrt{6} x+3=0$
Answer
$2 x^2-2 \cdot \sqrt{6} x+3=0 $
$ a =2 ; b =-2 \sqrt{6} ; c =3 $
$ D = b ^2-4 ac $
$ =(-2 \sqrt{6})^2-4(2)(3) $
$ =24-24 $
$ =0 $
$ x =\frac{- b \pm \sqrt{ b ^2-4 ac }}{2 a} $
$ x =\frac{-(-2 \sqrt{6}) \pm 0}{2 \times 2} $
$x =\frac{\sqrt{6}}{2}$
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Question 103 Marks
Solve the following quadratic equation using formula method only :
$x^2 +10x- 8= 0$
Answer
$x^2+10 x-8=0 $
$ a=1 ; b=10, c=-8 $
$D=b^2-4 a c$
$ =(10)^2-4(1)(-8) $
$ =100+32$
$ =132 $
$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} $
$ x=\frac{-10 \pm \sqrt{132}}{2} $
$ x=-\frac{10}{2} \pm \sqrt{\frac{132}{4}} $
$ x=-5+\sqrt{33}, x=-5-\sqrt{33}$
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Question 113 Marks
Solve the following equation $: 25x (x + 1) = -4$
Answer
$25 x(x+1)=-4 $
$ x^2+x=-\frac{4}{25} $
$x^2+x+\frac{4}{25}=0$
$x^2+\frac{1}{5} x+\frac{4}{5} x+\frac{4}{25}=0 $
$ x\left(x+\frac{1}{5}\right)+\frac{4}{5}\left(x+\frac{4}{25}\right)=0 $
$ \left(x+\frac{1}{5}\right)\left(x+\frac{4}{5}\right)=0 $
$\left(x+\frac{1}{5}\right)=0,\left(x+\frac{4}{5}\right)=0$
$x=-\frac{1}{5}, x=-\frac{4}{5}$
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Question 123 Marks
Solve the following equation: $3x^2 + 25 x + 42 = 0$
Answer
$3 x^2+25 x+42=0$
$ x^2+\frac{25}{3} x+14=0 $
$ x^2+6 x+\frac{7}{3} x+14=0 $
$ x(x+6)+\frac{7}{3}(x+6)=0 $
$ (x+6)\left(x+\frac{7}{3}\right)=0 $
$ (x+6)=0,\left(x+\frac{7}{3}\right)=0$
$ x=-6, x=-\frac{7}{3}$
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Question 133 Marks
Solve the following equation: $4x^2 - 13x - 12 = 0$
Answer
$4 x^2-13 x-12=0$
$x^2-\frac{13}{4} x-3=0$
$x^2-4 x+\frac{3}{4} x-3=0$
$ x(x-4)+\frac{3}{4}(x-4)=0$
$(x-4)\left(x+\frac{3}{4}\right)=0$
$ (x-4)=0,\left(x+\frac{3}{4}\right)=0$
$x=4, x=-\frac{3}{4}$
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Question 143 Marks
Solve the following equation : $5x^2 - 11x + 2 = 0$
Answer
$5 x^2-11 x+2=0$
$5 x^2-10 x-x+2=0 $
$ 5 x(x-2)-1(x-2)=0$
$(x-2)(5 x-1)=0$
$(x-2)=0,(5 x-1)=0$
$x=2, x=\frac{1}{5}$
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Question 153 Marks
Solve the following equation: $2x^2 - x - 6 = 0$
Answer
$2 x^2-x-6=0 $
$x^2-\frac{1}{2} x-3=0$
$x^2-2 x+\frac{3}{2} x-3=0 $
$ x(x-2)+\frac{3}{2}(x-2)=0$
$(x-2)\left(x+\frac{3}{2}\right)=0 $
$ (x-2)=0,\left(x+\frac{3}{2}\right)=0$
$x=2, x=-\frac{3}{2}$
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Question 163 Marks
Solve the following equation: $2x^2 - 3x - 9=0$
Answer
$2 x^2-3 x-9=0 $
$x^2-\frac{3}{2} x-\frac{9}{2}=0 $
$ x^2-3 x+\frac{3}{2} x-\frac{9}{2}=0$
$x(x-3)+\frac{3}{2}(x-3)=0 $
$(x-3)\left(x+\frac{3}{2}\right)=0$
$(x-3)=0,\left(x+\frac{3}{2}\right)=0 $
$x=3, x=-\frac{3}{2}$
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Question 173 Marks
Solve the following equation: $a \left( x ^2+1\right)-x\left( a ^2+1\right)=0$
Answer
$a\left(x^2+1\right)-x\left(a^2+1\right)=0 $
$ a x^2+a-a^2 x-x=0$
$ x^2+1-a x-\frac{1}{a} x=0$
$x^2+1-a x-\frac{1}{a} x+1=0$
$x(x-a)-\frac{1}{a}(x-a)=0 $
$ (x-a)\left(x-\frac{1}{a}\right)=0$
$ x=a ; x=\frac{1}{a}$
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Question 183 Marks
Solve the following equation: $( a + b )^2 x ^2-4 abx -( a - b )^2=0$
Answer
$(a+b)^2 x^2-4 a b x-(a-b)^2=0 $
$ \text { As, }-(a+b)^2+(a-b)^2=-a^2-b^2-2 a b+a^2+b^2-2 a b=-4 a b$
$ (a+b)^2 x^2-\left[(a+b)^2-(a-b)^2\right] x-(a-b)^2=0$
$ (a+b)^2 x^2-(a+b)^2 x+(a-b)^2 x-(a-b)^2=0 $
$ \left\{(a+b)^2 x\right\}(x-1)+\left\{(a-b)^2\right\}(x-1)=0 $
$(x-1)\left[(a+b)^2 x+(a-b)^2\right]=0 $
$ x-1=0 \text { and }(a+b)^2 x+(a-b)^2=0$
$ x=1 \text { and } x=-\frac{(a-b)^2}{(a+b)^2}$
$ x=1 \text { and } x=-\left(\frac{a-b}{a+b}\right)^2$
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Question 193 Marks
Solve the following equation: $7 x+\frac{3}{x}=35 \frac{3}{5}$
Answer
$7 x +\frac{3}{ x }=35 \frac{3}{5} $
$ 7 x ^2+3=\frac{178}{5} x$
$ 7 x ^2-\frac{178}{5} x +3=0 $
$x ^2-\frac{178}{35} x +\frac{3}{7}=0$
$x ^2-5 x -\frac{3}{35} x +\frac{3}{7}=0 $
$x ( x -5)-\frac{3}{35}( x -5)=0$
$( x -5)\left( x -\frac{3}{35}\right)=0$
$ x =5, x =\frac{3}{35}$
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Question 203 Marks
Solve the following equation : $x ^2-4 \sqrt{2} x +6=0$
Answer
$x^2-4 \sqrt{2} x+6=0$
$ x^2-\sqrt{2} x-3 \sqrt{2} x+6=0$
$ x(x-\sqrt{2})-3 \sqrt{2}(x-\sqrt{2})=0$
$(x-\sqrt{2})(x-3 \sqrt{2})=0 $
$ (x-\sqrt{2})=0,(x-3 \sqrt{2})=0 $
$ x=\sqrt{2}, x=3 \sqrt{2}$
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Question 213 Marks
Solve the following equation : $ax ^2+\left(4 a ^2-3 b \right) x -12 ab =0$
Answer
$a x^2+\left(4 a^2-3 b\right) x-12 a b=0 $
$x^2+4 a x-3 \frac{b}{a} x-12 b=0 $
$ x(x+4 a)-3 \frac{b}{a}(x+4 a)=0 $
$ (x+4 a)\left(x-3 \frac{b}{a}\right)=0 $
$ x=-4 a, x=3 \frac{b}{a}$
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Question 223 Marks
Solve the following equation: $x^2-(\sqrt{2}+1) x+\sqrt{2}=0$
Answer
$x^2-(\sqrt{2}+1) x+\sqrt{2}=0 $
$x^2-x-\sqrt{2} x+\sqrt{2}=0$
$ x(x-1)-\sqrt{2}(x-1)=0 $
$ (x-1)(x-\sqrt{2})=0 $
$(x-1)=0,(x-\sqrt{2})=0 $
$ x=1, x=\sqrt{2}$
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[3 marks sum] - Mathematics STD 10 Questions - Vidyadip