Chapter 2 · Polynomials

NCERT Exemplar Class 10 Maths Exercise 2.3

All 10 questions of NCERT Exemplar Class 10 Maths Exercise 2.3 (Polynomials), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Find the zeroes of ​the ​following polynomials by factorisation method and verify the relations ​between the ​zeroes ​and the ​coefficients of the polynomials: 1. $4x^2-3x-1$ 2. $3x^2+4x-4$ 3. $5t^2+12t+7$ 4. $t^3-2t^2-15t$ 5. $2x^2+\frac{7}{2}x+\frac{3}{4}$ 6. $4x^2+5\sqrt{2}x-3$ ​7. $2s^2-(1+2\sqrt{2})s+\sqrt{2}$ 8. $v^2+4\sqrt{3}v-15$ 9. $y^2+\frac{3}{2}\sqrt{5}y-5$ ​10. $7y^2-\frac{11}{3}y-\frac{2}{3}$
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   Zeroes: (1) -1/4, 1; (2) 2/3, -2; (3) -7/5, -1; (4) 0, 5, -3; (5) -3/2, -1/4; (6) $\sqrt{2}/4$, $-3\sqrt{2}/2$; (7) 1/2, $\sqrt{2}$; (8) $-5\sqrt{3}$, $\sqrt{3}$; (9) $-2\sqrt{5}$, $\sqrt{5}/2$; (10) -1/7, 2/3. All verified against the coefficient relations.

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2. Q2
   Find ​the zeroes ​of the following polynomials ​by factorisation method ​and verify the ​relations between the zeroes ​and the coefficients of the ​polynomials: $3x^2+4x-4$
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   The zeroes of $3x^2+4x-4$ are $\alpha =-2$ and $\beta =\frac{2}{3}$. Verified: sum of zeroes $=-2+\frac{2}{3}=-\frac{4}{3}=-\frac{b}{a}$, and product of zeroes $=(-2)\times \frac{2}{3}=-\frac{4}{3}=\frac{c}{a}$.

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3. Q3
   Find ​the zeroes of the ​following ​polynomials by ​factorisation method and verify ​the relations between the zeroes and the ​coefficients of the polynomials: $5t^2+12t+7$
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   The zeroes of $5t^2+12t+7$ are $\alpha =-1$ and $\beta =-\frac{7}{5}$. Verified: sum of zeroes $=-1-\frac{7}{5}=-\frac{12}{5}=-\frac{b}{a}$, and product of zeroes $=(-1)\times \left(-\frac{7}{5}\right)=\frac{7}{5}=\frac{c}{a}$.

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4. Q4
   Find ​the zeroes of ​the ​following polynomials by factorisation method ​and verify the relations ​between ​the zeroes and ​the coefficients of the ​polynomials: $t^3-2t^2-15t$
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   The zeroes of $t^3-2t^2-15t$ are $t=0$, $t=5$ and $t=-3$. Verified: $\alpha +\beta +\gamma =2=-\frac{b}{a}$, $\alpha \beta +\beta \gamma +\gamma \alpha =-15=\frac{c}{a}$, and $\alpha \beta \gamma =0=-\frac{d}{a}$.

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5. Q5
   Find the zeroes of the ​following polynomials ​by factorisation ​method and ​verify the relations ​between the zeroes and the coefficients ​of the ​polynomials: $2x^2+\frac{7}{2}x+\frac{3}{4}$
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   The zeroes of $2x^2+\frac{7}{2}x+\frac{3}{4}$ are $x=-\frac{3}{2}$ and $x=-\frac{1}{4}$. Verified: sum of zeroes $=-\frac{3}{2}-\frac{1}{4}=-\frac{7}{4}=-\frac{b}{a}$, and product of zeroes $=\left(-\frac{3}{2}\right)\times \left(-\frac{1}{4}\right)=\frac{3}{8}=\frac{c}{a}$.

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6. Q6
   Find the ​zeroes of the following polynomials by factorisation method and ​verify the ​relations between the zeroes ​and the coefficients ​of ​the ​polynomials: $4x^2+5\sqrt{2}x-3$
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   The zeroes of $4x^2+5\sqrt{2}x-3$ are $x=-\frac{3\sqrt{2}}{2}$ and $x=\frac{\sqrt{2}}{4}$. Verified: sum of zeroes $=-\frac{3\sqrt{2}}{2}+\frac{\sqrt{2}}{4}=-\frac{5\sqrt{2}}{4}=-\frac{b}{a}$, and product of zeroes $=\left(-\frac{3\sqrt{2}}{2}\right)\times \frac{\sqrt{2}}{4}=-\frac{3}{4}=\frac{c}{a}$.

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7. Q7
   Find ​the ​zeroes of the following polynomials by ​factorisation ​method and verify ​the ​relations between the zeroes ​and the ​coefficients of the polynomials: $2s^2-(1+2\sqrt{2})s+\sqrt{2}$
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   The zeroes are $s=\frac{1}{2}$ and $s=\sqrt{2}$. Verification: sum of zeroes $=\frac{1}{2}+\sqrt{2}=\frac{1+2\sqrt{2}}{2}=-\frac{b}{a}$ and product of zeroes $=\frac{1}{2}\times \sqrt{2}=\frac{\sqrt{2}}{2}=\frac{c}{a}$.

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8. Q8
   Find the ​zeroes of the following polynomials ​by factorisation method and ​verify the ​relations between the ​zeroes and the ​coefficients of the polynomials: $v^2+4\sqrt{3}v-15$
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   The zeroes are $v=\sqrt{3}$ and $v=-5\sqrt{3}$. Verification: sum of zeroes $=\sqrt{3}-5\sqrt{3}=-4\sqrt{3}=-\frac{b}{a}$ and product of zeroes $=\sqrt{3}\times (-5\sqrt{3})=-15=\frac{c}{a}$.

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9. Q9
   Find the ​zeroes of the following ​polynomials ​by factorisation method ​and verify the ​relations between the zeroes and the coefficients of the polynomials: $y^2+\frac{3}{2}\sqrt{5}y-5$
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   The zeroes are $y=-2\sqrt{5}$ and $y=\frac{\sqrt{5}}{2}$. Verification: sum of zeroes $=-2\sqrt{5}+\frac{\sqrt{5}}{2}=-\frac{3\sqrt{5}}{2}=-\frac{b}{a}$ and product of zeroes $=(-2\sqrt{5})\times \frac{\sqrt{5}}{2}=-5=\frac{c}{a}$.

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10. Q10
    Find the zeroes ​of the following polynomials by factorisation method and ​verify the relations ​between ​the zeroes and ​the coefficients of the ​polynomials: $7y^2-\frac{11}{3}y-\frac{2}{3}$
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    The zeroes are $y=\frac{2}{3}$ and $y=-\frac{1}{7}$. Verification: sum of zeroes $=\frac{2}{3}-\frac{1}{7}=\frac{11}{21}=-\frac{b}{a}$ and product of zeroes $=\frac{2}{3}\times \left(-\frac{1}{7}\right)=-\frac{2}{21}=\frac{c}{a}$.

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