Q1. Decide which are quadratic: (i) \(x^2+5x-2=0\) (ii) \(y^2=5y-10\) (iii) \(y^2+\dfrac1y=2\) (iv) \(x+\dfrac1x=-2\) (v) \((m+2)(m-5)=0\) (vi) \(m^3+3m^2-2=3m^3\).

Ans. Quadratic: (i),(ii) [\(\Rightarrow y^2-5y+10=0\)], (v) [\(\Rightarrow m^2-3m-10=0\)]. Not quadratic as written: (iii),(iv). (vi) reduces to \(-2m^3+3m^2-2=0\) (cubic).

Q2. Convert to \(ax^2+bx+c=0\) and give \(a,b,c\): (i) \((x-1)^2=2x+3\) (ii) \(x^2+5x=-(3-x)\) (iii) \(3m^2=2m^2-9\) (iv) \(P(3+6p)=-5\) (v) \(x^2-9=13\).

Ans. (i) \(x^2-4x-2=0\) → \(1,-4,-2\). (ii) \(x^2+4x+3=0\) → \(1,4,3\). (iii) \(m^2+9=0\) → \(1,0,9\). (iv) \(6p^2+3p+5=0\) → \(6,3,5\). (v) \(x^2-22=0\) → \(1,0,-22\).

Q3. Verify roots: (i) \(x^2+4x-5=0\) at \(x=1,-1\) (ii) \(2m^2-5m=0\) at \(m=2,\tfrac52\).

Ans. (i) \(1+4-5=0\) ✓; \(1-4-5=-8\) ✗. (ii) \(8-10\neq0\); \(2\cdot(25/4)-5(5/2)=0\) ✓.

Q4. If \(x=3\) is a root of \(kx^2-10x+3=0\), find \(k\).

Ans. \(9k-30+3=0\Rightarrow k=3\).

Q5. One root is \(-\dfrac{7}{5}\) of \(5m^2+2m+k=0\). Find \(k\).

Ans. \(5\cdot\dfrac{49}{25}+2\cdot\dfrac{-7}{5}+k=0\Rightarrow \dfrac{49-14}{5}+k=0\Rightarrow 7+k=0\Rightarrow k=-7\).

Q6. Factorise and solve: \(m^2-14m+13=0\).

Ans. \((m-13)(m-1)=0\Rightarrow m=13,1\).

Q7. Factorise and solve: \(3x^2-x-10=0\).

Ans. \((3x+5)(x-2)=0\Rightarrow x=-\dfrac{5}{3},\,2\).

Q8. Solve by completing square: \(x^2+8x-48=0\).

Ans. \((x+4)^2=64\Rightarrow x=-12,\,4\).

Q9. Use formula to solve: \(x^2-2x-3=0\).

Ans. \(\Delta=16\Rightarrow x=\dfrac{2\pm4}{2}=3,-1\).

Q10. Use formula to solve: \(25x^2+30x+9=0\).

Ans. \(\Delta=0\Rightarrow x=\dfrac{-30}{50}=-\dfrac{3}{5}\) (double root).

Q11. Use formula: \(x^2+10x+2=0\).

Ans. \(x=\dfrac{-10\pm\sqrt{92}}{2}=-5\pm\sqrt{23}\).

Q12. Compute discriminant and nature: \(x^2+2x-9=0\).

Ans. \(\Delta=4+36=40>0\Rightarrow\) real, unequal.

Q13. If roots are \(-3,-7\), write the equation.

Ans. \(x^2-(-10)x+21=0\Rightarrow x^2+10x+21=0\).

Q14. Find quadratic if sum of roots \(=10\), product \(=9\).

Ans. \(x^2-10x+9=0\).

Q15. For \(2x^2+6x-5=0\), find \(\alpha+\beta,\alpha\beta\).

Ans. \(-\dfrac{6}{2}=-3,\ \dfrac{-5}{2}=-\dfrac{5}{2}\).

Q16. Show that \(x^2+x-20=0\) has integral roots.

Ans. \(\Delta=1+80=81=9^2\Rightarrow x=4,-5\).

Q17. Determine \(\Delta\) for \(x^2+x+5=0\) and comment.

Ans. \(\Delta=1-20=-19<0\Rightarrow\) no real roots.

Q18. If \(a,b\) are roots of \(x^2+5x-1=0\), find \(a^2+b^2\).

Ans. \((a+b)^2-2ab=25+2=27\).

Q19. If \(a,b\) as above, find \(a^3+b^3\).

Ans. \((a+b)^3-3ab(a+b)=-125-15=-140\).

Q20. Write equation with roots \(0\) and \(7\).

Ans. \(x(x-7)=0\Rightarrow x^2-7x=0\).

Q1. Solve by factorisation: \(x^2-15x+54=0\).

Ans. \((x-6)(x-9)=0\Rightarrow x=6,9\).

Q2. Solve by factorisation: \(x^2+x-20=0\).

Ans. \((x+5)(x-4)=0\Rightarrow x=-5,4\).

Q3. Solve: \(2y^2+27y+13=0\).

Ans. \(D=27^2-104=629\Rightarrow y=\dfrac{-27\pm\sqrt{629}}{4}\).

Q4. Solve: \(5m^2=22m+15\).

Ans. \(5m^2-22m-15=0\Rightarrow m=\dfrac{22\pm\sqrt{484+300}}{10}=\dfrac{22\pm\sqrt{784}}{10}=\dfrac{22\pm28}{10}\Rightarrow m=5,-\dfrac{3}{5}\).

Q5. Solve: \(2x^2-2x+\dfrac12=0\).

Ans. Multiply 2: \(4x^2-4x+1=0=(2x-1)^2\Rightarrow x=\dfrac12\).

Q6. Solve: \(6x-\dfrac{2}{x}=1\).

Ans. Multiply by \(x\): \(6x^2-1= x\Rightarrow 6x^2-x-1=0\Rightarrow x=\dfrac{1\pm\sqrt{1+24}}{12}=\dfrac{1\pm5}{12}\Rightarrow x=\dfrac12,-\dfrac13.\)

Q7. Solve by completing square: \(m^2-5m+3=0\).

Ans. \(\left(m-\tfrac52\right)^2=\tfrac{25}{4}-3=\tfrac{13}{4}\Rightarrow m=\dfrac{5\pm\sqrt{13}}{2}\).

Q8. Solve by formula: \(9y^2-12y+2=0\).

Ans. \(y=\dfrac{12\pm\sqrt{144-72}}{18}=\dfrac{12\pm6\sqrt2}{18}=\dfrac{2\pm\sqrt2}{3}\).

Q9. Solve by formula: \(2y^2+9y+10=0\).

Ans. \(\Delta=81-80=1\Rightarrow y=\dfrac{-9\pm1}{4}=-2,-\dfrac{5}{2}.\)

Q10. Solve: \(5x^2-4x-7=0\).

Ans. \(x=\dfrac{4\pm\sqrt{16+140}}{10}=\dfrac{4\pm2\sqrt{39}}{10}=\dfrac{2\pm\sqrt{39}}{5}\).

Q11. Find nature of roots without solving: \(x^2-4x+4=0\).

Ans. \(\Delta=16-16=0\Rightarrow\) equal real roots \(x=2\).

Q12. If \(\alpha,\beta\) are roots of \(x^2-13x+k=0\) and \(|\alpha-\beta|=7\), find \(k\).

Ans. As in 1-Mark Q20, \(k=30\).

Q13. Construct equation with roots \(\dfrac12,-\dfrac12\).

Ans. Sum \(=0\), product \(=-\dfrac14\Rightarrow x^2-\dfrac14=0\Rightarrow 4x^2-1=0\).

Q14. Show that \(x^2+6x+5=0\) has roots \(-1,-5\).

Ans. \(\Delta=36-20=16\Rightarrow x=\dfrac{-6\pm4}{2}=-1,-5\).

Q15. If \(a,b\) roots of \(2x^2+6x-5=0\), find \(a^2+b^2\).

Ans. \((a+b)^2-2ab=(-3)^2-2(-\tfrac{5}{2})=9+5=14.\)

Q16. Word problem: Rectangle perimeter \(=40\) cm, length \(=2\)×breadth\(+2\).

Ans. \(x+y=20,\ x-2y=2\Rightarrow (x,y)=(14,6)\).

Q17. Word problem: Train speed increased by \(5\) km/h saves \(48\) min over \(360\) km. Find original speed.

Ans. \(\dfrac{360}{x}-\dfrac{360}{x+5}=\dfrac{4}{5}\Rightarrow x^2+5x-2250=0\Rightarrow x=45\) km/h.

Q18. Word problem: Onion storehouse: breadth \(x\), length \(x+7\), diagonal \(x+8\). Find dimensions.

Ans. \(x^2+(x+7)^2=(x+8)^2\Rightarrow x=5\Rightarrow (b,l)=(5,12)\) m.

Q19. If sum of roots \(=-7\) and product \(=5\), write equation and nature of roots.

Ans. \(x^2+7x+5=0\); \(\Delta=49-20=29>0\Rightarrow\) real, unequal.

Q20. If one root is \(2\) of \(x^2+mx-5=0\), find \(m\).

Ans. \(4+2m-5=0\Rightarrow m=\dfrac{1}{2}\).

Q1. Write any two quadratic equations.

Ans. \(x^2-4x-5=0,\quad 3y^2+2y-7=0\).

Q2. Decide which are quadratic: (1) \(x^2+5x-2=0\) (2) \(y^2=5y-10\) (3) \(y^2+\dfrac1y=2\) (4) \(x+\dfrac1x=-2\) (5) \((m+2)(m-5)=0\) (6) \(m^3+3m^2-2=3m^3\).

Ans. Quadratic: (1), (2) ⇒ \(y^2-5y+10=0\); (5) ⇒ \(m^2-3m-10=0\). Not quadratic as written: (3), (4). (6) reduces to \(-2m^3+3m^2-2=0\) (cubic).

Q3. Write in \(ax^2+bx+c=0\) and list \(a,b,c\): (1) \(2y=10-y^2\) (2) \((x-1)^2=2x+3\) (3) \(x^2+5x=-(3-x)\) (4) \(3m^2=2m^2-9\) (5) \(P(3+6p)=-5\) (6) \(x^2-9=13\).

• (1) \(y^2+2y-10=0\Rightarrow (1,2,-10)\)
• (2) \(x^2-4x-2=0\Rightarrow (1,-4,-2)\)
• (3) \(x^2+4x+3=0\Rightarrow (1,4,3)\)
• (4) \(m^2+9=0\Rightarrow (1,0,9)\)
• (5) \(6p^2+3p+5=0\Rightarrow (6,3,5)\)
• (6) \(x^2-22=0\Rightarrow (1,0,-22)\)

Q4. Check given values as roots: (1) \(x^2+4x-5=0\) for \(x=1,-1\) (2) \(2m^2-5m=0\) for \(m=2,\dfrac{5}{2}\).

Ans. (1) \(1\) is a root; \(-1\) is not. (2) \(m=\dfrac{5}{2}\) is a root; \(m=2\) is not.

Q5. Find \(k\) if \(x=3\) is a root of \(kx^2-10x+3=0\).

Ans. \(k=3\).

Q6. One root of \(5m^2+2m+k=0\) is \(-\dfrac{7}{5}\). Find \(k\).

Ans. \(k=-7\).

Q1–Q12. Solve:

• (1) \(x^2-15x+54=0\Rightarrow x=6,9\).
• (2) \(x^2+x-20=0\Rightarrow x=-5,4\).
• (3) \(2y^2+27y+13=0\Rightarrow y=\dfrac{-27\pm\sqrt{629}}{4}\).
• (4) \(5m^2=22m+15\Rightarrow m=5,-\dfrac{3}{5}\).
• (5) \(2x^2-2x+\dfrac{1}{2}=0\Rightarrow x=\dfrac12\) (double).
• (6) \(6x-\dfrac{2}{x}=1\Rightarrow x=\dfrac12,-\dfrac13\).
• (7) \(2x^2+7x+\dfrac{5}{2}=0\Rightarrow x=\dfrac{-7\pm\sqrt{49-20}}{4}=\dfrac{-7\pm\sqrt{29}}{4}\).
• (8) \(3x^2-2\sqrt6\,x+2=0\Rightarrow x=\dfrac{2\sqrt6\pm\sqrt{24-24}}{6}=\dfrac{\sqrt6}{3}\) (double).
• (9) \(2m(m-24)=50\Rightarrow 2m^2-48m-50=0\Rightarrow m=25,-1\).
• (10) \(25m^2=9\Rightarrow m=\pm\dfrac{3}{5}\).
• (11) \(7m^2=21m\Rightarrow m(7m-21)=0\Rightarrow m=0,3\).
• (12) \(m^2-11=0\Rightarrow m=\pm\sqrt{11}\).

Q1–Q6. Solve:

• (1) \(x^2+x-20=0\Rightarrow x=4,-5\).
• (2) \(x^2+2x-5=0\Rightarrow x=-1\pm\sqrt6\).
• (3) \(m^2-5m+3=0\Rightarrow m=\dfrac{5\pm\sqrt{13}}{2}\).
• (4) \(9y^2-12y+2=0\Rightarrow y=\dfrac{2\pm\sqrt2}{3}\).
• (5) \(2y^2+9y+10=0\Rightarrow y=-2,-\dfrac{5}{2}\).
• (6) \(5x^2=4x+7\Rightarrow x=\dfrac{2\pm\sqrt{39}}{5}\).

Q1. Write \(a,b,c\): (1) \(x^2-7x+5=0\) (2) \(2m^2=5m-5\) (3) \(y^2=7y\).

• (1) \(a=1,b=-7,c=5\)
• (2) \(2m^2-5m+5=0\Rightarrow a=2,b=-5,c=5\)
• (3) \(y^2-7y=0\Rightarrow a=1,b=-7,c=0\)

Q2. Solve using formula:

• (1) \(x^2+6x+5=0\Rightarrow x=-1,-5\).
• (2) \(x^2-3x-2=0\Rightarrow x=\dfrac{3\pm\sqrt{17}}{2}\).
• (3) \(3m^2+2m-7=0\Rightarrow m=\dfrac{-1\pm\sqrt{22}}{3}\).
• (4) \(5m^2-4m-2=0\Rightarrow m=\dfrac{2\pm\sqrt{14}}{5}\).
• (5) \(y^2+\dfrac13y-2=0\Rightarrow y=\dfrac{-1\pm\sqrt{73}}{6}\).
• (6) \(5x^2+13x+8=0\Rightarrow x=-1,-\dfrac{8}{5}\).

Q1. Compute \(\Delta\) for \(x^2+7x-1=0\), \(2y^2-5y+10=0\), \(2x^2+\dfrac{4}{3}x+2=0\).

• \(x^2+7x-1=0\Rightarrow \Delta=49+4=53>0\) (real, unequal).
• \(2y^2-5y+10=0\Rightarrow \Delta=25-80=-55<0\) (not real).
• \(2x^2+\dfrac{4}{3}x+2=0\Rightarrow \Delta=\dfrac{16}{9}-16=-\dfrac{128}{9}<0\) (not real).

Q1. If roots sum is \(-7\) and product \(5\), write equation and nature.

Ans. \(x^2+7x+5=0\); \(\Delta=29>0\Rightarrow\) real, unequal.

Q2. Form equations with given roots: (i) \(0,4\) (ii) \(3,-10\) (iii) \(\dfrac12,-\dfrac12\).

• (i) \(x^2-4x=0\)
• (ii) \(x^2+7x-30=0\)
• (iii) \(4x^2-1=0\)

Q3. If \(a,b\) are roots of \(y^2-2y-7=0\), find \(a^2+b^2\) and \(a^3+b^3\).

• \(a+b=2,\ ab=-7\Rightarrow a^2+b^2=(a+b)^2-2ab=4+14=18\).
• \(a^3+b^3=(a+b)^3-3ab(a+b)=8-3(-7)(2)=8+42=50\).

Q1. Pragati’s age: \((x-2)(x+3)=84\). Find present age.

Ans. \(x^2+x-90=0\Rightarrow x=9\) (valid). Present age \(=9\) years.

Q2. Sum of squares of two consecutive even naturals is \(244\).

Ans. Let \(2n,2n+2\). \(8n^2+8n+4=244\Rightarrow n^2+n-30=0\Rightarrow n=5\Rightarrow 10,12\).

Q3. Orange garden: \(rc=150\), rows \(=c+5\).

Ans. \(c(c+5)=150\Rightarrow c=10,\ r=15\).

Q4. Ages reciprocals: \( \dfrac{1}{k}+\dfrac{1}{k+5}=\dfrac{1}{6}\).

Ans. \(k^2+5k=12k+30\Rightarrow k^2-7k-30=0\Rightarrow k=10\). Ages \(=10,15\) yrs.

Q5. Tests: second \(=x+10\), \(5(x+10)=x^2\).

Ans. \(x^2-5x-50=0\Rightarrow x=10\). Scores: \(10,20\).

Q6. Pots: number \(n\), cost/ pot \(c=10n+40\), total \(=600\).

Ans. \(n(10n+40)=600\Rightarrow n^2+4n-60=0\Rightarrow n=6,\ c=100\) (₹).

Q7. Boat: 36 km downstream & 36 km upstream in 8 h, boat \(12\) km/h.

Ans. \( \dfrac{36}{12+v}+\dfrac{36}{12-v}=8\Rightarrow v=6\) km/h.

Q8. Work: Pintu takes 6 days more than Nishu; together 4 days.

Ans. \( \dfrac{1}{x}+\dfrac{1}{x+6}=\dfrac{1}{4}\Rightarrow x=6\). Nishu \(6\) d, Pintu \(12\) d.

Q9. Division: \(460=nq+1,\ q=5n+6\).

Ans. \(5n^2+6n-459=0\Rightarrow n=9\Rightarrow q=51\).

Q1. MCQs: (1)–(7).

• (1) Quadratic: \( \dfrac{1}{2x}(x+2)=x \) simplifies to linear; correct quadratic is (B) \(x(x+5)=2\Rightarrow x^2+5x-2=0\).
• (2) Not quadratic: (A) \(x^2+4x=11+x^2\Rightarrow 4x=11\) (linear).
• (3) \(x^2+kx+k=0\) real & equal ⇒ \(\Delta=0\Rightarrow k^2-4k=0\Rightarrow k=0,4\) ⇒ (C).
• (4) For \(2x^2-5x+2=0\): \(\Delta=25-16=9\) ⇒ (B) \(17\) is wrong; correct value is \(9\). If options are as given, mark the one matching \(\Delta=9\).
• (5) Roots \(3,5\Rightarrow x^2-8x+15=0\) ⇒ (B).
• (6) Sum \(-5\Rightarrow x^2+5x+\_ =0\) ⇒ options: (C) \(x^2+3x-5=0\) has sum \(-3\); (B) \(x^2-5x+3=0\) has sum \(5\); (D) \(3x^2+15x+3=0\) has sum \(-5\) ⇒ (D).
• (7) \(5m^2-5m+5=0\Rightarrow \Delta=25-100=-75<0\) ⇒ (C) not real.
• (8) Root \(2\) in \(x^2+mx-5=0\Rightarrow 4+2m-5=0\Rightarrow m=\tfrac12\) ⇒ (C).

Q2. Which are quadratic? (1) \(x^2+2x+11=0\) (2) \(x^2-2x+5=x^2\) (3) \((x+2)^2=2x^2\).

Ans. (1) Yes. (2) Reduces to \(-2x+5=0\) (linear). (3) Expands to \(x^2+4x+4-2x^2=0\Rightarrow -x^2+4x+4=0\) (quadratic).

Q3. Compute \(\Delta\): (1) \(2y^2-y+2=0\) (2) \(5m^2-m=0\) (3) \(5x^2-x-5=0\).

• (1) \((-1)^2-16=-15\) (not real).
• (2) \((-1)^2-4\cdot5\cdot0=1>0\) (real, unequal; one is \(0\)).
• (3) \(1+100=101>0\) (real, unequal).

Q4. One root of \(2x^2+kx-2=0\) is \(-2\). Find \(k\).

Ans. \(8-2k-2=0\Rightarrow k=3\).

Q5. Frame equations for roots: (1) \(10,-10\) (2) \(1\pm\sqrt{5}\) (3) \(0,7\).

• (1) \(x^2-0x-100=0\Rightarrow x^2-100=0\).
• (2) Sum \(=2\), product \(=-4\Rightarrow x^2-2x-4=0\).
• (3) \(x^2-7x=0\).

Q6. Nature of roots: (1) \(3x^2-5x+7=0\) (2) \(3x^2+2x-2\sqrt{3}=0\) (3) \(m^2-2m+1=0\).

• (1) \(\Delta=25-84=-59<0\) not real.
• (2) \(\Delta=4+24=28>0\) real unequal.
• (3) \(\Delta=4-4=0\) real equal (\(m=1\)).

Q7. Solve: (1) \(\dfrac{1}{5}x=\dfrac{1}{2x}\) (2) \(x^2-\dfrac{3}{10}x-\dfrac{1}{10}=0\) (3) \((2x+3)^2=25\) (4) \(m^2+5m+5=0\) (5) \(5m^2+2m+1=0\) (6) \(x^2-4x-3=0\).

• (1) \(x^2=\dfrac{5}{2}\Rightarrow x=\pm\dfrac{\sqrt{10}}{2}\).
• (2) \(x=\dfrac{3\pm\sqrt{9+40}}{20}=\dfrac{3\pm7}{20}\Rightarrow x=\dfrac12,-\dfrac{1}{5}\).
• (3) \(2x+3=\pm5\Rightarrow x=1,-4\).
• (4) \(m=\dfrac{-5\pm\sqrt{25-20}}{2}=\dfrac{-5\pm\sqrt5}{2}\).
• (5) \( \Delta=4-20=-16<0\Rightarrow\) not real.
• (6) \(x=2\pm\sqrt{7}\).

Q8. Find \(m\) so that \((m-12)x^2+2(m-12)x+2=0\) has real and equal roots.

Ans. For \(a\ne0\), \(\Delta=0\): \(4(m-12)^2-8(m-12)=0\Rightarrow 4(m-12)[(m-12)-2]=0\Rightarrow m=12\) (degenerate) or \(m=14\). Real & equal (non-degenerate) ⇒ \(m=14\).

Q9. Sum of two roots is \(5\) and sum of their cubes is \(35\). Find the equation.

Ans. Let roots \(\alpha,\beta\). \(\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=35\Rightarrow 125-15\alpha\beta=35\Rightarrow \alpha\beta=6\). Equation: \(x^2-5x+6=0\).

Q10. Find quadratic whose roots are \((\alpha+\beta)^2\) and \((\alpha-\beta)^2\) for \(2x^2+2(p+q)x+p^2+q^2=0\).

Ans. Here \(\alpha+\beta=-\dfrac{2(p+q)}{2}=-(p+q)\), \(\alpha\beta=\dfrac{p^2+q^2}{2}\). Then required roots: \((p+q)^2\) and \((\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=(p+q)^2-2(p^2+q^2)\). Sum \(S=2(p+q)^2-2(p^2+q^2)=4pq\). Product \(P=(p+q)^2[(p+q)^2-2(p^2+q^2)]=(p^2+2pq+q^2)(-p^2+2pq-q^2)=-(p^2-q^2)^2+4pq(p^2+2pq+q^2)=4p^2q^2\). Equation: \(x^2-4pq\,x+4p^2q^2=0\Rightarrow (x-2pq)^2=0\).

Q11. Mukund & Sagar have amounts differing by ₹50; product ₹15000. Find amounts.

Ans. Let Sagar ₹\(x\), Mukund ₹\(x+50\). \(x(x+50)=15000\Rightarrow x^2+50x-15000=0\Rightarrow x=100\). Amounts: ₹100 and ₹150.

Q12. Squares of two numbers: difference \(120\); square of smaller \(=2\times\) greater.

Ans. Let greater \(=g\), smaller \(=s\). \(s^2=2g,\ g^2-s^2=120\Rightarrow g^2-2g-120=0\Rightarrow g=12\) (positive), \(s=\sqrt{24}=2\sqrt6\).

Q13. \(540\) oranges to be distributed. If \(30\) students more, each gets \(3\) less. Find students.

Ans. Let \(n\) students. \(\dfrac{540}{n}-\dfrac{540}{n+30}=3\Rightarrow 540\cdot\dfrac{30}{n(n+30)}=3\Rightarrow n(n+30)=5400\Rightarrow n^2+30n-5400=0\Rightarrow n=60\).

Q14. Farm: length \(=2b+10\), pond side \(=b/3\), area farm \(=20\times\) area pond. Find \(b\).

Ans. \((2b+10)b=20\cdot(b/3)^2\Rightarrow 2b^2+10b=\dfrac{20b^2}{9}\Rightarrow 18b^2+90b=20b^2\Rightarrow 2b^2-90b=0\Rightarrow b=45\). Length \(=100\). Pond side \(=15\).

Q15. Two taps fill tank together in 2 h; small tap takes 3 h more than large tap alone. Find times.

Ans. Let large \(=x\) h, small \(=x+3\) h. \( \dfrac{1}{x}+\dfrac{1}{x+3}=\dfrac{1}{2}\Rightarrow 2x+3=x(x+3)/2\Rightarrow x^2+x-6=0\Rightarrow x=2\). Large \(2\) h, small \(5\) h.