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Maths 3 Mark Question

Polynomials · Maharashtra Board STD 9 (MSBSHSE - Semi English Medium). 13 questions.

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3 Mark Question

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13 questions · self-marked practice — reveal the answer and mark yourself.

Question 13 Marks
We have seen the example of expenditure and income (in terms of polynomials) of Govind who is a dry land farmer. He has borrowed rupees one lakh twenty-five thousand from the bank as an agriculture loan and repaid the said loan at 10 p.c.p.a.
He had spent ₹ 10,000 on seeds. The expenses on soyabean crop was ₹ 2000x for fertilizers and pesticides and ₹ $4000 x^2$ was spent on wages and cultivation. He spent $₹ 8000 y$ on fertilizers and pesticides and $₹ 9000 y^2$ on cultivation and wages for cotton and tur crop.
His total income was
$\text { ₹ }\left(14000 x^2+\frac{25000}{3} y^2+16000 y\right)$
By taking $x=2, y=3$ write the income expenditure account of Govind's farming.
Answer
– Credit (Income)
$₹ 1,25,000$ Bank loan
$₹ 56000$ Income from soyabean
$₹ 75000$ Income from cotton
$₹ 48000$ Income from tur
$₹ 304000$ Total income– Debit (Expenses)
$₹ 1,37,000$ loan paid with interest for seeds
$₹ 10000$ For seeds
$₹ 4000$ Fertilizers and pesticides for soyabean
$₹ 16000$ Wages and cultivation charges for soyabean
$₹ 24000$ Fertilizers and pesticides for cotton & tur
$₹ 81000$ Wages and cultivation charges for cotton & tur
$₹ 272000$ Total expenditure
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Question 23 Marks
Read the following passage, write the appropriate amount in the boxes and discuss.
Govind, who is a dry land farmer from Shiralas has a $5$ acre field. His family includes his wife, two children and his old mother. He borrowed one lakh twenty five thousand rupees from the bank for one year as agricultural loan at 10 p.c.p.a. He cultivated soyabean in x acres and cotton and tur in y acres. The expenditure he incurred was as follows :
He spent $₹10,000$ on seeds. The expenses for fertilizers and pesticides for the soyabean crop was$ ₹ 2000x$ and $₹ 4000x^2$ were spent on wages and cultivation of land. He spent ₹ 8000y on fertilizers and pesticides and ₹ 9000y2 for wages and cultivation of land for the cotton and tur crops.Let us write the total expenditure on all the crops by using variables x and y.
$₹ 10000 + 2000x + 4000\times 2 + 8000y + 9000y^2$
He harvested $5 x^2$ quintals soyabean and sold it at $₹ 2800$ per quintal. The cotton crop yield was $\frac{5}{3} y ^2$ quintals which fetched $₹ 5000$ per quintal.
The tur crop yield was $4y$ quintals and was sold at $₹ 4000$ per quintal. Write the total income in rupees that was obtained by selling the entire farm produce, with the help of an expression using variables x and y.
Answer
Total income $=$ income on soyabean crop $+$ income on cotton crop $+$ income on tur crop
$=₹\left(5 x^2 \times 2800\right)+₹\left(\frac{5}{3} y^2 \times 5000\right)+₹(4 y \times 4000)$
$=₹\left(14000 x^2+\frac{25000}{3} y^2+16000 y\right)$
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Question 33 Marks
Factorise: $(x+2)(x-3)(x-7)(x-2)+64$
Answer
$(x+2)(x-3)(x-7)(x-2)+64$
$=(x+2)(x-7)(x-3)(x-2)+64$
$=\left(x^2-5 x-14\right)\left(x^2-5 x+6\right)+64$
$=(m-14)(m+6)+64 \ldots \ldots \ldots\left(\text { putting } x^2-5 x=m\right)$
$=m^2-14 m+6 m-84+64$
$=m^2-8 m-20$
$=(m-10)(m+2)$
$=\left(x^2-5 x-10\right)\left(x^2-5 x+2\right) \ldots .\left(\text { replace } m \text { with } x^2-5 x\right)$
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Question 43 Marks
Factorise : $\left(y^2-3 y\right)^2-5\left(y^2-3 y\right)-50$.
Answer
Let $\left(y^2-3 y\right)=x$
$\therefore\left(y^2-3 y\right) 2-5\left(y^2-3 y\right)-50=x^2-5 x-50$
$=x^2-10 x+5 x-50$
$=x(x-10)+5(x-10)$
$=(x-10)(x+5)$
$=\left(y^2-3 y-10\right)\left(y^2-3 y+5\right)$
$=\left[y^2-5 y+2 y-10\right]\left(y^2-3 y+5\right)$
$=[y(y-5)+2(y-5)]\left(y^2-3 y+5\right)$
$=(y-5)(y+2)\left(y^2-3 y+5\right)$
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Question 53 Marks
If the polynomial $t^3-3 t^2+k t+50$ is divided by $(t-3)$, the remainder is 62 . Find the value of $k$.
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Question 83 Marks
Show that $m -1$ is a factor of $m ^{21}-1$ and $m ^{22}-1$.
Answer
i. $p(m)=m^{21}-1$
Divisor $= m -1$
$\therefore$ take $m =1$
Remainder $=p(1)$
$p(m)=m^{21}-1$
$\therefore P(1)=1^{21}-1=1-1=0$
$\therefore$ By factor theorem, $m -1$ is a factor of $m ^{21}-1$.
ii. $p(m)=m^{22}-1$
Divisor $=m-1$
$\therefore$ take $m =1$
Remainder $= p (1)$
$p(m)=m^{22}-1$
$\therefore P(1)=1^{22}-1=1-1=0$
$\therefore$ By factor theorem, $m -1$ is a factor of $m ^{22}-1$.
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Question 93 Marks
If $(x-2)$ is a factor of $x^3-m x^2+10 x-20$, then find the value of $m$.
Answer
$p(x)=x^3-m x^2+10 x-20 x-2$ is a factor of $x 3-m x^2+10 x-20$.
$\therefore$ By factor theorem,
Remainder $=p(2)=0$
$p ( x )= x ^3- m x ^2+10 x -20$
$\therefore p (2)=(2)^3- m (2)^2+10(2)-20$
$\therefore 0=8-4 m+20-20$
$\therefore 0=8-4 m$
$\therefore 4 m=8$
$\therefore m =2$
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Question 103 Marks
Use factor theorem to determine whether $x+3$ is a factor of $x^2+2 x-3$ or not.
Answer
$p(x)=x^2+2 x-3$
$\text { Divisor }=x+3$
$\therefore \text { take } x=-3$
$\therefore \text { Remainder }=p(-3)$
$p(x)=x^2+2 x-3$
$\therefore p(-3)=(-3)^2+2(-3)-3$
$=9-6-3$
$\therefore p(-3)=0$
$\therefore$ By factor theorem, $x +3$ is a factor of $x ^2+2 x -3$.
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Question 113 Marks
If the polynomial $y^3-5 y^2+7 y+m$ is divided by $y+2$ and the remainder is 50 , then find the value of $m$.
Answer
$p(y)=y^3-5 y^2+7 y+m$
$\text { Divisor }=y+2$
$\therefore \text { take } y=-2$
$\therefore \text { By remainder theorem, }$
$\text { Remainder }=p(-2)=50$
$P(y)=y^3-5 y^2+7 y+m$
$\therefore P(-2)=(-2)^3-5(-2)^2+7(-2)+m$
$\therefore 50=-8-5(4)-14+m$
$\therefore 50=-8-20-14+m$
$\therefore 50=-42+m$
$\therefore m=50+42$
$\therefore m=92$
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Question 123 Marks
If $p(m)=m^3+2 m^2-m+10$, then $P(a)+p(-a)=$ ?
Answer
$p(m)=m^3+2 m^2-m+10$
Put $m = a$ in the given polynomial.
$\therefore p(a)=a^3+2 a^2-a+10$
Put $m=-a$ in the given polynomial.
$p(-a)=(-a)^3+2(-a)^2-(-a)+10$
$\therefore p(-a)=-a^3+2 a^2+a+10 \ldots$
Adding (i) and (ii),
$p(a)+p(-a)=\left(a^3+2 a^2-a+10\right)+\left(-a^3+2 a^2+a+10\right)$
$=a^3-a^3+2 a^2+2 a^2-a+a+10+10$
$\therefore p(a)+p(-a)=4 a^2+20$
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Question 133 Marks
Write the polynomials in index form.
i.$(1, 2, 3)$
ii. $(5, 0, 0, 0 ,-1)$
iii. $(-2, 2, -2, 2)$
Answer
i. Number of coefficients $=3$
$\therefore$ Degree $=3-1=2$
$\therefore$ Taking x as variable, the index form is $x ^2+2 x +3 ii$. Number of coefficients $=5$
$\therefore$ Degree $=5-1=4$
$\therefore$ Taking x as variable, the index form is $5 x ^4+0 x ^3+0 x ^2+0 x -1$
iii. Number of coefficients $=4$
$\therefore$ Degree $=4-1=3$
$\therefore$ Taking x as variable, the index form is $-2 x ^3+2 x ^2-2 x +2$
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