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Maths 1 Marks Question

P-1 Quadratic Equations · Maharashtra Board STD 10 (MSBSHSE - English Medium). 50 questions.

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Question 11 Mark
$(x + 2)^2 = 2x^2$
Answer
$
(x+2)^2=2 x^2 \Rightarrow x^2+4 x+4=2 x^2
$
$x^2-4 x-4=0$ is a quadractic equation because it is the form of $a x^2+b c+c=0$ and it has degree 2
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Question 21 Mark
$x^2-2 x+5=x^2$
Answer
$x^2-2 x+5=x^2$
$-2 x +5=0 \therefore$ it is not a quadratic equation because it is not in the form of $ax ^2+ bc + c =0$ and it doesn't have degree 2 .
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Question 31 Mark
$x^2+2 x+11=0$
Answer
$x ^2+2 x -11=0$ is a quadractic equation because it is the form of $a x ^2+ bc + c =0$ and it has degree 2.
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Question 71 Mark
Two roots of quadratic equations are given ; frame the equation.
0 and 7
Answer
$\therefore \text { Let } \alpha=0 \text { and } \beta=7 $
$ \therefore \alpha+\beta=0+7=7 \text { and } \alpha \beta=0 \times 7=0$
$ \therefore \text { and quadratic equation is, } x ^2-(\alpha+\beta) x +\alpha \beta=0$
$ \therefore x ^2-(27) x +(0)=0$
$ \therefore x ^2-7 x =0$
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Question 81 Mark
Two roots of quadratic equations are given ; frame the equation.
1–3√5 and 1 + 3√5
Answer
$:  { Let } \alpha=1-3 \sqrt{5} \text { and } \beta=1+3 \sqrt{5} $
$ \therefore \alpha+\beta=1-3 \sqrt{5}+1+3 \sqrt{5}=2 \text { and } \alpha \beta=(1-3 \sqrt{5}) \times(1+3 \sqrt{5}) $
$ =1-45=-44 $
$ \therefore \text { and quadratic equation is, } x^2-(\alpha+\beta) x+\alpha \beta=0 $
$ \therefore x^2-(2) x+(-44)=0 $
$ \therefore x^2-2 x-44=0$
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Question 91 Mark
$2 m^2-5 m=0, m=2, \frac{5}{2}$
Answer
$2 m^2-5 m=0 $
$\text { Put } m =2 \Rightarrow 2(2)^2-5 \times 2 \Rightarrow 2 \times 4-10 \Rightarrow 8-10 \Rightarrow-2 $
$\text { Put } m =\frac{5}{2} \Rightarrow 2\left(\frac{5}{2}\right)^2-5 \times \frac{5}{2} \Rightarrow 2 \times \frac{25}{4}-\frac{25}{2} \Rightarrow \frac{25}{2}-\frac{25}{2}=0$
$\therefore m =2$ is not root of the equation and $m =\frac{5}{2}$ is a root of the equation.
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Question 101 Mark
$x^2 + 4x – 5 = 0, x = 1, -1$
Answer
$x^2+4 x-5=0$
Put $x =1$
$\Rightarrow 1^2+4 \times 1-5 $
$\Rightarrow 1+4-5=0$
Put $x =-1$
$\Rightarrow(-1)^2+4(-1)-5 $
$\Rightarrow 1-4-5=-8$
$\therefore x =1$ is a root of the equation and $x =-1$ is not a root of the equation.
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Question 111 Mark
Two roots of quadratic equations are given ; frame the equation.
10 and -10
Answer
Let $\alpha=10$ and $\beta=-10$
$\therefore \alpha+\beta=10-10$
$=0$
$\alpha \beta=10(-10)$
$=-100$
$\therefore$ and quadratic equation is, $x^2-(\alpha+\beta) x+\alpha \beta=0$
$\Rightarrow x^2-0(x)-100=0$
$\Rightarrow x^2-100=0$
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Question 121 Mark
Write the following equation in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for the equation.
$x^2-9=13$
Answer
$ \Rightarrow x^2-9-13=0$
$\Rightarrow x^2-22=0$
$\Rightarrow x^2+0 x-22=0$
So, it is of the form $a x^2+b x+c=0$ where $a=1, b=0$ and $c=-22$.
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Question 151 Mark
$x^2+5 x=-(3-x)$
Answer
$x^2+5 x=-(3-x) \Rightarrow x^2+5 x=-3+x $
$ \Rightarrow x^2+5 x-x+3=0 $
$ \Rightarrow x^2+4 x+3=0$
$a=1, b=4, c=3$
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Question 161 Mark
$(x-1)^2=2 x+3$
Answer
$(x-1)^2=2 x+3 $
$\Rightarrow x^2-2 x+1=2 x+3 $
$ \Rightarrow x^2-2 x-2 x+1-3=0 $
$ \Rightarrow x^2-4 x-2=0 $
$ a=1, b=-4, c=-2$
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Question 181 Mark
$m^3+3 m^2-2=3 m^3$
Answer
$m^3+3 m^2-2=3 m^3$
$\Rightarrow m^3+3 m^2-2-3 m^3=0 \Rightarrow-2 m^3+3 m^2-2=0$
$\therefore$ it is not a quadratic equation because it is not in the form of $ax ^2+ bc + c =0$
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Question 191 Mark
(m + 2)(m–5) = 0
Answer
$( m +2)( m -5)=0$
$\Rightarrow m ( m -5)+2(m-5) \Rightarrow m ^2-5 m+2 m-10 \Rightarrow m ^2-3 m-10=0$
$\therefore$ it is a quadratic equation because it is the form of $ax ^2+ bc + c =0$
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Question 201 Mark
$x+\frac{1}{x}=-2 $
Answer
$x+\frac{1}{x}=-2 $
$x^2+1=-2 x \Rightarrow x^2+2 x+1=0$
$\therefore$ it is a quadratic equation because it is the form of $ax ^2+ bc + c =0$
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Question 211 Mark
$y ^2+\frac{1}{ y }=2 $
Answer
$y ^2+\frac{1}{ y }=2 $
$\Rightarrow y ^3+1=2 y \Rightarrow y ^3-2 y +1$
$\therefore$ it is not a quadratic equation because it is not in the form of $ax ^2+ bc + c =0$
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Question 221 Mark
$y^2=5 y-10$
Answer
$y^2=5 y-10$
$y ^2-5 y +10=0 \therefore$ it is a quadratic equation because it is the form of $ax ^2+ bc + c =0$
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Question 251 Mark
Which of the following are quadratic equations?
(i) $x-\frac{5}{x}=3 x-9$
(ii) $(x+3)(x-4)=0$
(iii) $\frac{5}{x}-3=x^2$
(iv) $n^3-n+4=n^3$
(v) $x-3=4 x^2$
Answer
(i), (ii) and (v) are quadratic equations
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Question 261 Mark
Two roots of quadratic equations are given ; frame the equation.
0 and 7
Answer
Let $\alpha=0$ and $\beta=7$
$\therefore \alpha+\beta=0+7=7$ and $\alpha \beta=0 \times 7=0$
$\therefore$ and quadratic equation is, $x ^2-(\alpha+\beta) x +\alpha \beta=0$
$\therefore x ^2-(27) x +(0)=0$
$\therefore x ^2-7 x =0$
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Question 271 Mark
Two roots of quadratic equations are given ; frame the equation.
1–3√5 and 1 + 3√5
Answer
Let $\alpha=1-3 \sqrt{5}$ and $\beta=1+3 \sqrt{5}$
$\begin{array}{l}
\therefore \alpha+\beta=1-3 \sqrt{5}+1+3 \sqrt{5}=2 \text { and } \alpha \beta=(1-3 \sqrt{5}) \times(1+3 \sqrt{5}) \\
=1-45=-44
\end{array}$
$\therefore$ and quadratic equation is, $x^2-(\alpha+\beta) x+\alpha \beta=0$
$\begin{array}{l}
\therefore x ^2-(2) x +(-44)=0 \\
\therefore x ^2-2 x -44=0
\end{array}$
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Question 281 Mark
Two roots of quadratic equations are given ; frame the equation.
10 and -10
Answer
Let $\alpha=10$ and $\beta=-10$
$\begin{array}{l}
\therefore \alpha+\beta=10-10 \\
=0 \\
\alpha \beta=10(-10) \\
=-100
\end{array}$
$\therefore$ and quadratic equation is, $x ^2-(\alpha+\beta) x +\alpha \beta=0$
$\begin{array}{l}
\Rightarrow x^2-0(x)-100=0 \\
\Rightarrow x^2-100=0
\end{array}$
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Question 291 Mark
Compare the given quadratic equation to the general form and write values of a,b, c : y2 = 7y
Answer

$\begin{array}{l}y^2-7 y+0=0 \text { and } a x^2+b x+c=0 \\ a=1, b=-7, c=0\end{array}$
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Question 301 Mark
Compare the given quadratic equation to the general form and write values of a,b, c : 2m2 = 5m – 5
Answer

$\begin{array}{l}2 m^2-5 m+5=0 \text { and } a x^2+b x+c \\ a=2, b=-5, c=5\end{array}$
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Question 311 Mark
Compare the given quadratic equation to the general form and write values of a,b, c : x2 – 7x + 5 = 0
Answer

$\begin{array}{l}x^2-7 x+5=0 \text { and } a x^2+b x+c=0 \\ a=1, b=-7, c=5\end{array}$
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Question 321 Mark
is the following equation quadratic : (x + 2)2 = 2x2
Answer

$(x+2)^2=2 x^2 \Rightarrow x^2+4 x+4=2 x^2
$
$x^2-4 x-4=0$ is a quadractic equation because it is the form of $a x^2+b c+c=0$ and it has degree 2
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Question 331 Mark
is the following equation quadratic : x2 – 2x + 5 = x2
Answer

$x^2-2 x+5=x^2$
$-2 x +5=0 \therefore$ it is not a quadratic equation because it is not in the form of $ax ^2+ bc + c =0$ and it doesn't have degree 2.
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Question 341 Mark
is the following equation quadratic : x2 + 2x + 11 = 0
Answer

$x^2+2 x-11=0$ is a quadractic equation because it is the form of $a x^2+b c+c=0$ and it has degree 2.
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Question 351 Mark
Determine whether the values given against each of the quadratic equation are theroots of the equation : $2 m^2-5 m=0, m=2, \frac{5}{2}$
Answer

$\begin{array}{l}2 m ^2-5 m =0 \\ \text { Put } m =2, \Rightarrow 2(2)^2-5 \times 2 \Rightarrow 2 \times 4-10 \Rightarrow 8-10 \Rightarrow-2 \\ \text { Put } m =\frac{5}{2}, \Rightarrow 2\left(\frac{5}{2}\right)^2-5 \times \frac{5}{2} \Rightarrow 2 \times \frac{25}{4}-\frac{25}{2} \Rightarrow \frac{25}{2}-\frac{25}{2}=0 \\ \therefore m =2 \text { is not root of the equation and } m =\frac{5}{2} \text { is a root of the equation.| }\end{array}$
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Question 361 Mark
Determine whether the values given against each of the quadratic equation are theroots of the equation : x2 + 4x – 5 = 0, x = 1, -1
Answer

$x^2+4 x-5=0$
Put $x=1$
$\begin{array}{l}
\Rightarrow 1^2+4 \times 1-5 \\
\Rightarrow 1+4-5=0
\end{array}$
Put $x=-1$
$\begin{array}{l}
\Rightarrow(-1)^2+4(-1)-5 \\
\Rightarrow 1-4-5=-8
\end{array}$
$\therefore X =1$ is a root of the equation and $x =-1$ is not a root of the equation.
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Question 371 Mark
Write the following equation in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for the equation.
$x^2-9=13$
Answer

$\begin{array}{l}
\Rightarrow x^2-9-13=0 \\
\Rightarrow x^2-22=0 \\
\Rightarrow x^2+0 x-22=0
\end{array}
$
So, it is of the form $a x^2+b x+c=0$ where $a=1, b=0$ and $c=-22$.
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Question 381 Mark
Write the following equation in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for each equation : P(3 + 6p) = - 5
Answer

$\begin{array}{l}p(3+6 p)=-5 \Rightarrow 3 p+6 p^2+5=0 \\ 6 p^2+3 p+5=0 \\ a=6, b=3, c=5\end{array}$
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Question 391 Mark
Write the following equation in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for each equation : 3m2 = 2m2 - 9
Answer

$\begin{array}{l}3 m^2=2 m^2-9 \Rightarrow 3 m^2-2 m^2+9=0 \\ m^2+0 m+9=0 \\ a=1, b=0, c=9\end{array}$
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Question 401 Mark
Write the following equation in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for each equation : x2 + 5x = - (3-x)
Answer

$\begin{array}{l}x^2+5 x=-(3-x) \Rightarrow x^2+5 x=-3+x \\ \Rightarrow x^2+5 x-x+3=0 \\ \Rightarrow x^2+4 x+3=0 ; \\ a=1, b=4, c=3\end{array}$
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Question 411 Mark
Write the following equations in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for each equation : (x-1)2 = 2x + 3
Answer

$\begin{array}{l}(x-1)^2=2 x+3 \\ \Rightarrow x^2-2 x+1=2 x+3 \\ \Rightarrow x^2-2 x-2 x+1-3=0 \\ \Rightarrow x^2-4 x-2=0 ; \\ a=1, b=-4, c=-2\end{array}$
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Question 421 Mark
Write the following equation in the form $a x^2+b x+c=0$, then write the values of $a, b, c$ for each equation : 2y = 10 – y2
Answer

$\begin{array}{l}2 y=10-y^2 \\ \Rightarrow 2 y+y^2-10=0 \\ y^2+2 y-10=0 ; \\ a=1, b=2, c=-10\end{array}$
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Question 431 Mark
Decide whether the following equation is quadratic equation or not. : m3 + 3m2 – 2 = 3 m3
Answer

$\begin{array}{l}
m^3+3 m^2-2=3 m^3 \\
\Rightarrow m^3+3 m^2-2-3 m^3=0 \\
\Rightarrow-2 m^3+3 m^2-2=0
\end{array}$
$\therefore$ it is not a quadratic equation because it is not in the form of $ax ^2+ bc + c =0$
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Question 441 Mark
Decide whether the following equation is quadratic equation or not : (m + 2)(m–5) = 0
Answer

$\begin{array}{l}
(m+2)(m-5)=0 \\
\Rightarrow m(m-5)+2(m-5) \Rightarrow m^2-5 m+2 m-10 \Rightarrow m^2-3 m-10=0
\end{array}$
$\therefore$ it is a quadratic equation because it is the form of $ax ^2+ bc + c =0$
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Question 451 Mark
Decide whether the following equation is quadratic equation or not : $x+\frac{1}{x}=-2$
Answer

$\begin{array}{l}
x+\frac{1}{x}=-2 \\
x^2+1=-2 x \Rightarrow x^2+2 x+1=0
\end{array}$
$\therefore$ it is a quadratic equation because it is the form of $ax ^2+ bc + c =0$
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Question 461 Mark
Decide whether the following equation is quadratic equation or not. : $y^2+\frac{1}{y}=2$
Answer

$\begin{array}{l}
y^2+\frac{1}{y}=2 \\
\Rightarrow y^3+1=2 y \Rightarrow y^3-2 y+1
\end{array}$
$\therefore$ it is not a quadratic equation because it is not in the form of $ax ^2+ bc + c =0$
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Question 471 Mark
Decide whether the following equation is quadratic equation or not. : y2 = 5y - 10
Answer

$\begin{array}{l}
y^2=5 y-10 \\
y^2-5 y+10=0
\end{array}$
$\therefore$ it is a quadratic equation because it is the form of
$a x^2+b c+c=0$
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Question 481 Mark
Decide whether the following equation is quadratic equation or not : x2 + 5x – 2 = 0
Answer

$x^2+5 x-2=0$ is a quadractic equation because it is the form of $a x^2+b c+c=0$
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Question 501 Mark
Which of the following are quadratic equations?
(i) $x-\frac{5}{x}=3 x-9$
(ii) $(x+3)(x-4)=0$
(iii) $\frac{5}{x}-3=x^2$
(iv) $n^3-n+4=n^3$
(v) $x-3=4 x^2$
Answer
(i), (ii) and (v) are quadratic equations
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