## Is it possible to design a rectangular park of perimeter $80 \mathrm{~m}$ and area $400 \mathrm{~m}^{2}$ ?

[question] Question. Is it possible to design a rectangular park of perimeter $80 \mathrm{~m}$ and area $400 \mathrm{~m}^{2} ?$ If so, find its length and breadth. [/question] [solution] Solution: Perimeter of the rectangular park = 80 m $\Rightarrow$ Length $+$ Breath of the park $=\frac{\mathbf{8 0}}{\mathbf{2}} \mathrm{m}=40 \mathrm{~m}$. Let the breadth be x metres, then length = (40 – x) m Here, $x<40$ $x \times(40-x)=400[$ Each $=$ area of the park $]$ i.e., $-x^{2}+40 x-400=0$ i.e., $x^{2...

## Is the following situation possible?

[question] Question. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in year was 48. [/question] [solution] Solution: Let the age of one friend be x years. Age of the other friend will be (20 – x) years. 4 years ago, age of 1st friend = (x – 4) years And, age of 2nd friend = (20 – x – 4) $=(16-x)$ years Given that, $(x-4)(16-x)=48$ $16 x-64-x^{2}+4 x=48$ $-x^{2}+20 x-112=0$ $x^{2}-...

## Is it possible to design a rectangular mango grove whose length is twice its breadth,

[question] Question. Is it possible to design a rectangular mango grove whose length is twice its breadth, and area is $800 \mathrm{~m}^{2}$ ? If so, find its length and breadth. [/question] [solution] Solution: Let x be the breadth and 2x be the length of the rectangle. $x \times 2 x=800$ $\Rightarrow 2 x^{2}=800$ $\Rightarrow x^{2}=400=(20)^{2}$ $\Rightarrow x=20$ Hence, the rectangle is possible and it has breadth $=20 \mathrm{~m}$ and length $=40 \mathrm{~m}$. [/solution]...

## Find the values of k for each of the following quadratic equations,

[question] Question. Find the values of k for each of the following quadratic equations, so that they have two real equal roots. (i) $2 x^{2}+k x+3=0$ (ii) $k x(x-2)+6=0$ [/question] [solution] Solution: (i) $2 x^{2}+k x+3=0$ $\mathrm{a}=2, \mathrm{~b}=\mathrm{k}, \mathrm{c}=3$ $\mathrm{D}=\mathrm{b}^{2}-4 \mathrm{ac}=\mathrm{k}^{2}-4 \times 2 \times 3=\mathrm{k}^{2}-24$ Two roots will be equal if $D=0$, i.e., if $k^{2}-24=0$ i.e., if $k^{2}-24$, i.e., if $k=\pm \sqrt{\mathbf{2 4}}$ i.e., if $\m...

## Find the nature of the roots of the following quadratic equations.

[question] Question. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : (i) $2 x^{2}-3 x+5=0$ (ii) $3 x^{2}-4 \sqrt{3} x+4=0$ (iii) $2 x^{2}-6 x+3=0$ [/question] [solution] Solution: (i) $2 x^{2}-3 x+5=0$ $\mathrm{a}=2, \mathrm{~b}=-3, \mathrm{c}=5$ Discriminant $\mathrm{D}=\mathrm{b}^{2}-4 \mathrm{ac}=9-4 \times 2 \times 5$ $=9-40=-31$ $\Rightarrow \mathrm{D}<0$ Hence, no real root. (ii) $3 x^{2}-4 \sqrt{3} x+4=0$ $a=3, b=-4 \sqrt{3}, c=4$ Di...

## Sum of the area of two squares is $468 \mathrm{~m}^{2}$.

[question] Question. Sum of the area of two squares is $468 \mathrm{~m}^{2}$. If the difference of their perimeters is $24 \mathrm{~m}$, find the sides of the two squares. [/question] [solution] Solution: Let the sides of the two squares be $x \mathrm{~m}$ and $y \mathrm{~m}$. Therefore, their perimeter will be $4 x$ and $4 y$ respectively and their areas will be $x^{2}$ and $y^{2}$ respectively. It is given that $4 x-4 y=24$ $x-y=6$ $x=y+6$ Also, $x^{2}+y^{2}=468$ $\Rightarrow(6+y)^{2}+y^{2}=46...

## A train travels 360 km at a uniform speed.

[question] Question. A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken I hour less for the same journey. Find the speed of the train. [/question] [solution] Solution: Let the speed of the train be x km/hr. We are given that 360 km distance is to be travelled at uniform speed of x km/hr. Time taken to cover the distance of $360 \mathrm{~km}=\frac{\mathbf{3 6 0}}{\mathbf{x}}$ hours. In case, the speed is increased by $5 \mathrm{~km} / \mathrm{hr}$, ...

## The difference of squares of two numbers is 180.

[question] Question. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers [/question] [solution] Solution: Let the larger and smaller number be x and y respectively. According to the given question, $x^{2}-y^{2}=180$ and $y^{2}=8 x$ $\Rightarrow x^{2}-8 x=180$ $\Rightarrow x^{2}-8 x-180=0$ $\Rightarrow x^{2}-18 x+10 x-180=0$ $\Rightarrow x(x-18)+10(x-18)=0$ $\Rightarrow(x-18)(x+10)=0$ $\Rightarrow x=18,-10$ However, ...

## The diagonal of a rectangular field is 60 metres more than the shorter side.

[question] Question. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field. [/question] [solution] Solution: In rectangle ABCD, let the shorter side BC = x metres. Then AB = (x + 30) metres and diagonal AC = (x + 60) metres. By Pythagoras Theorem we have $\mathrm{AB}^{2}+\mathrm{BC}^{2}=\mathrm{AC}^{2}$ $\Rightarrow(x+30)^{2}+x^{2}$ $=(x+60)^{2}$ $\Rightarrow x^{2}+60 x+900+x^{2}$ $=x^{...

## In a class test, the sum of Shefali's marks in Mathematics and English is 30.

[question] Question. In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects. [/question] [solution] Solution: Let the marks in Maths be x. Then, the marks in English will be 30 – x. According to the given question, (x+2)(30-x-3)=210 $(x+2)(27-x)=210$ $\Rightarrow-x^{2}+25 x+54=210$ $\Rightarrow x^{2}-25 x+156=0$ $\Rightarro...

## The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3.

[question] Question. The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age. [/question] [solution] Solution: Let the present age of Rehman be x years. We are given that $\frac{\mathbf{1}}{\mathbf{x}-\mathbf{3}}+\frac{\mathbf{1}}{\mathbf{x}+\mathbf{5}}=\frac{\mathbf{1}}{\mathbf{3}}$ $\Rightarrow \frac{x+5+x-3}{(x-3)(x+5)}=\frac{1}{3}$ $\Rightarrow 3(2 x+2)=(x-3)(x+5)$ $\Rightarrow 6 x+6=x^{2}+2 x-15$ $\Rightarrow x^{2}-4 x-21=0$ $a=1,...

## Find the roots of the following equations :

[question] Question. Find the roots of the following equations : (i) $x-\frac{\mathbf{1}}{\mathbf{x}}=3, x \neq 0$ (ii) $\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}, x \neq-4,7$ [/question] [solution] Solution: (i) $x-\frac{1}{x}=3 \quad \Rightarrow x^{2}-3 x-1=0$ On comparing this equation with $a x^{2}+b x+c=0$ we obtain $\mathrm{a}=1, \mathrm{~b}=-3, \mathrm{c}=-1$ By using quadratic formula, we obtain $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \Rightarrow x=\frac{3 \pm \sqrt{9+4}}{2}$ $\Rightarr...

## Find the roots of the quadratic equations given in

[question] Question. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula. [/question] [solution] Solution: (i) $2 x^{2}-7 x+3=0$ $a=2, b=-7, c=3$ Discriminant $D=(-7)^{2}-4(2)(3)=25$ $\Rightarrow \sqrt{10}=\sqrt{25}=5$ Roots of the given quadratic equation are $\frac{-\mathbf{b} \pm \sqrt{\mathbf{D}}}{\mathbf{2 a}}$ i.e., $\frac{\mathbf{7} \pm \mathbf{5}}{\mathbf{2} \times \mathbf{2}}$, i.e., the roots are 3 and $\frac{\mathbf{1}}{\mathbf{2}}$. (ii) $2 ...

## Find the roots of the following quadratic equations,

[question] Question. Find the roots of the following quadratic equations, if they exist, by the method of completing the square (1) $2 x^{2}-7 x+3=0$ (ii) $2 x^{2}+x-4=0$ (iii) $4 x^{2}+4 \sqrt{3} x+3=0$ (iv) $2 x^{2}+x+4=0$ [/question] [solution] Solution: (i) $2 x^{2}-7 x+3=0$ $\Rightarrow 4 x^{2}-14 x+6=0$ $\Rightarrow(2 x)^{2}-7(2 x)+6=0$ $\Rightarrow y^{2}-7 y+6=0$ where $y=2 x$ $\Rightarrow\left\{\mathbf{y}^{2}-\frac{7}{2} \mathbf{y}-\frac{7}{2} \mathbf{y}+\left(-\frac{7}{2}\right)^{2}\rig...

## The altitude of a right triangle is 7 cm less than its base.

[question] Question. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides. [/question] [solution] Solution: In $\triangle \mathrm{ABC}$, base $\mathrm{BC}=\mathrm{x} \mathrm{cm}$ and altitude $A C=(x-7) \mathrm{cm}$ $\angle \mathrm{ACB}=90^{\circ}$ $\mathrm{AB}=13 \mathrm{~cm}$ By Pythagoras theorem, we have $\mathrm{BC}^{2}+\mathrm{AC}^{2}=\mathrm{AB}^{2}$ $\Rightarrow x^{2}+(x-7)^{2}=13^{2}$ $\Rightarrow x^{2}+x^{2}-14 x+49=169$ $\Ri...

## Find two consecutive positive integers,

[question] Question. Find two consecutive positive integers, sum of whose squares is 365. [/question] [solution] Solution: Let the consecutive positive integers be x and x + 1. Given that $x^{2}+(x+1)^{2}=365$ $\Rightarrow x^{2}+x^{2}+1+2 x=365$ $\Rightarrow 2 x^{2}+2 x-364=0$ $\Rightarrow x^{2}+x-182=0$ $\Rightarrow x^{2}+14 x-13 x-182=0$ $\Rightarrow x(x+14)-13(x+14)=0$ $\Rightarrow(x+14)(x-13)=0$ Either $x+14=0$ or $x-13=0$ i.e., $x=-14$ or $x=13$ Since the integers are positive, $x$ can only...

## Find two numbers whose sum is 27

[question] Question. Find two numbers whose sum is 27 and product is 182. [/question] [solution] Solution: Let one number be x, then second number = 27 – x $x \times(27-x)=182$ $\Rightarrow 27 x-x^{2}=182$ $\Rightarrow x^{2}-27 x+182=0$ $\Rightarrow x^{2}-14 x-13 x+182=0$ $\Rightarrow x(x-14)-13(x-14)=0$ $\Rightarrow(x-13)(x-14)=0$ $\Rightarrow x=13$ or 14 $\Rightarrow 27 x=14$ or 13 Hence, the two marbles are 13 and 14 . [/solution]...

## Represent the following situations mathematically.

[question] Question. Represent the following situations mathematically. (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of ...

## Find the roots of the following quadratic equations by factorisation :

[question] Question. Find the roots of the following quadratic equations by factorisation : (i) $x^{2}-3 x-10=0$ (ii) $2 x^{2}+x-6=0$ (iii) $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$ (iv) $2 x^{2}-x+\frac{1}{8}=0$ (v) $100 x^{2}-20 x+1=0$ [/question] [solution] Solution: (i) $x^{2}-3 x-10=0$ $\Rightarrow x^{2}-5 x+2 x-10=0$ $\Rightarrow x(x-5)+2(x-5)=0$ $\Rightarrow(x+2)(x-5)=0$ $\Rightarrow x+2=0$ or $x-5=0$ $\Rightarrow x=-2$ or $x=5$ Hence, the two roots are $-2$ and 5 . (ii) $2 x^{2}+x-6=0$ $\Rightar...

## Represent the following situations in the form of quadratic equations :

[question] Question. Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528 \mathrm{~m}^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. (ii) The product of two consecutive positive integers is 306. We need to find the integers. (iii)Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to...