$\begin{array}{l} x^{2}-4 a x+4 a^{2}-b^{2}=0 \\ \Rightarrow x^{2}-4 a x+(2 a+b)(2 a-b)=0 \\ \Rightarrow x^{2}-[(2 a+b)+(2 a-b)] x+(2 a+b)(2 a-b)=0 \\ \Rightarrow x^{2}-(2 a+b) x-(2 a-b) x+(2 a+b)(2...

### Solve

$\begin{array}{l} x^{2}+5 x-\left(a^{2}+a-6\right)=0 \\ \Rightarrow x^{2}+5 x-(a+3)(a-2)=0 \\ \Rightarrow x^{2}+[(a+3)-(a-2)] x-(a+3)(a-2)=0 \\ \Rightarrow x^{2}+(a+3) x-(a-2) x-(a+3)(a-2)=0 \\...

### Solve

$\begin{array}{l} 4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0 \\ \Rightarrow 4 \sqrt{3} x^{2}+8 x-3 x-2 \sqrt{3}=0 \\ \Rightarrow 4 x(\sqrt{3} x+2)-\sqrt{3}(\sqrt{3} x+2)=0 \\ \Rightarrow(\sqrt{3} x+2)(4...

### Solve

$\begin{array}{l} \sqrt{3} x^{2}+10 x-8 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x^{2}+12 x-2 x-8 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x(x+4 \sqrt{3})-2(x+4 \sqrt{3})=0 \\ \Rightarrow(x+4 \sqrt{3})(\sqrt{3}...

### Solve

$\begin{array}{l} 2 x^{2}+a x-a^{2}=0 \\ \Rightarrow 2 x^{2}+2 a x-a x-a^{2}=0 \\ \Rightarrow 2 x(x+a)-a(x+a)=0 \\ \Rightarrow(x+a)(2 x-a)=0 \\ \Rightarrow x+a=0 \text { or } 2 x-a=0 \\ \Rightarrow...

### Find the value of so that the quadratic equation has equal roots.

It is given that the quadratic equation $x^{2}-4 k x+k=0$ has equal roots. $\begin{array}{l} \therefore D=0 \\ \Rightarrow(-4 k)^{2}-4 \times 1 \times k=0 \\ \Rightarrow 16 k^{2}-4 k=0 \\...

### If one root of the quadratic equation is reciprocal of the other, find the value of .

Let $\alpha$ and $\beta$ be the roots of the equation $3 x^{2}-10 x+k=0$. $\therefore \alpha=\frac{1}{\beta} \quad$ (Given) $\Rightarrow \alpha \beta=1$ $\Rightarrow \frac{k}{3}=1 \quad \quad$...

### If 1 is a root of the equation . and . then find the value of ab.

It is given that $y=1$ is a root of the equation $a y^{2}+a y+3=0$. $\begin{array}{l} \therefore a \times(1)^{2}+a \times 1+3=0 \\ \Rightarrow a+a+3=0 \\ \Rightarrow 2 a+3=0 \\ \Rightarrow...

### If the roots of the quadratic equation are equal then find the value of .

It is given that the roots of the quadratic equation $2 x^{2}+8 x+k=0$ are equal. $\begin{array}{l} \therefore D=0 \\ \Rightarrow 8^{2}-4 \times 2 \times k=0 \\ \Rightarrow 64-8 k=0 \\ \Rightarrow...

### If is a solution of the quadratic equation . Find the value of .

It is given that $x=\frac{-1}{2}$ is a solution of the quadratic equation $3 x^{2}+2 k x-3=0$ $\begin{array}{l} \therefore 3 \times\left(\frac{-1}{2}\right)^{2}+2 k...

### The sum of two natural numbers is 8 and their product is , Find the numbers.

Let the required natural numbers be $x$ and $(8-x)$. It is given that the product of the two numbers is $15 .$ $\begin{array}{l} \therefore x(8-x)=15 \\ \Rightarrow 8 x-x^{2}=15 \\ \Rightarrow...

### The length of a rectangular field exceeds its breadth by and the area of the field is . The breadth of the field is

(a)

(b)

(c)

(d)

Let the breadth of the rectangular field be $x \mathrm{~m}$. $\therefore$ Length of the rectangular field $=(x+8) m$ Area of the rectangular field $=240 \mathrm{~m}^{2}$ $\therefore(x+8) \times...

### The sum of a number and its reciprocal is . The number is

(a) or

(b) or

(c) or

(d) or 6

Answer is (a) $\frac{5}{4}$ or $\frac{4}{5}$ Let the required number be $x$. According to the question: $\begin{array}{l} x+\frac{1}{x}=\frac{41}{20} \\ \Rightarrow \frac{x^{2}+1}{x}=\frac{41}{20}...

### For what value of , the equation has real roots?

(a)

(b)

(c)

(d)None of these

Answer is (b) $k \geq \frac{-9}{2}$ It is given that the roots of the equation $\left(k x^{2}-6 x-2=0\right)$ are real. $\begin{array}{l} \therefore D \geq 0 \\ \Rightarrow\left(b^{2}-4 a c\right)...

### The roots of the equation are

(a) real, unequal and rational

(b) real, unequal and irrational

(c) real and equal

(d) imaginary

Answer is (b) real, unequal and irrational $\begin{array}{l} \because D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 2 \times 3 \\ =36-24 \\ =12 \end{array}$ 12 is greater than 0 and it is not a...

### The roots of the equation are

(a) real, unequal and rational

(b) real, unequal and irrational

(c) real and equal

(d) imaginary

Answer is (d) imaginary $\begin{array}{l} \because D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 2 \times 7 \\ =36-56 \\ =-20<0 \end{array}$ Thus, the roots of the equation are...

### If the equation has equal roots then value of

(a)

(b)

(c)

(d)

Answer is $(\mathrm{d}) \pm \frac{4}{3}$ It is given that the roots of the equation $\left(4 x^{2}-3 k x+1=0\right)$ are equal. $\begin{array}{l} \therefore\left(b^{2}-4 a c\right)=0 \\...

### If the equation has equal roots then ?

(a) 0 or 0

(b) or 0

(c) 2 or

(d) 0 only

Answer is (c) 2 or $-2$ It is given that the roots of the equation $\left(9 x^{2}+6 k x+4=0\right)$ are equal. $\begin{array}{l} \therefore\left(b^{2}-4 a c\right)=0 \\ \Rightarrow(6 k)^{2}-4 \times...

### The roots of the equation will be reciprocal each other if

(a)

(b)

(c) c=a

(d) none of these

Answer is (c) $\mathrm{c}=\mathrm{a}$ Let the roots of the equation $\left(a x^{2}+b x+c=0\right)$ be $\alpha$ and $\frac{1}{\alpha}$. $\therefore$ Product of the roots $=\alpha \times...

### If the sum of the roots of a quadratic equation is 6 and their product is 6 , the equation is

(a)

(b)

(c)

(d)

Answer is (a) $x^{2}-6 x+6=0$ Given: Sum of roots $=6$ Product of roots $=6$ Thus, the equation is: $x^{2}-6 x+6=0$

### The roots of a quadratic equation are 5 and . Then, the equation is

(a)

(b)

(c)

(d)

Answer is (b) $x^{2}-3 x-10=0$, It is given that the roots of the quadratic equation are 5 and $-2$. Then, the equation is: $\begin{array}{l} x^{2}-(5-2) x+5 \times(-2)=0 \\ \Rightarrow x^{2}-3...

### If one root of the equation is then the other root is

(a)

(b)

(c)

(d) 3

Answer is (d)3. Given: $3 x^{2}-10 x+3=0$ One root of the equation is $\frac{1}{3}$. Let the other root be $\alpha$. Product of the roots $=\frac{c}{a}$ $\begin{array}{l} \Rightarrow \frac{1}{3}...

### If the product of the roots of the equation is then the value of is

(a)

(b)

(c)

(d) 12

Answer is (c) 8 It is given that the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$ The equation can be rewritten as: $x^{2}-3 x+(k-10)=0$ Product of the roots of a quadratic equation...

### If one root of the equation is 2 then ?

(a)

(b)

(c)

(d)

Answer is (b) $-7$ It is given that one root of the equation $2 x^{2}+a x+6=0$ is 2 . $\begin{array}{l} \therefore 2 \times 2^{2}+a \times 2+6=0 \\ \Rightarrow 2 a+14=0 \\ \Rightarrow a=-7...

### Which of the following is a quadratic equation?

(a)

(b)

(c)

(d) None of these

Answer is (b) $x^{3}-x^{2}=(x-1)^{3}$ $\because x^{3}-x^{2}=(x-1)^{3}$ $\Rightarrow x^{3}-x^{2}=x^{3}-3 x^{2}+3 x-1$ $\Rightarrow 2 x^{2}-3 x+1=0$, which is a quadratic equation

### Which of the following is a quadratic equation?

(a)

(b)

(c)

(d)

Answer is (d) $2 x^{2}-5 x=(x-1)^{2}$ A quadratic equation is the equation with degree 2 . $\because 2 x^{2}-5 x=(x-1)^{2}$ $\Rightarrow 2 x^{2}-5 x=x^{2}-2 x+1$ $\Rightarrow 2 x^{2}-5 x-x^{2}+2...

### The length of the hypotenuse of a right-angled triangle exceeds the length of the base by and exceeds twice the length of the altitude by . Find the length of each side of the triangle.

Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...

### The area of right -angled triangle is 165 sq meters. Determine its base and altitude if the latter exceeds the former by 7 meters.

Let the base be $x \mathrm{~m}$. Therefore, the altitude will be $(x+7) m$ $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text { Altitude } \\ \therefore...

### The area of a right triangle is . If the base of the triangle exceeds the altitude by , find the dimensions of the triangle.

Let the altitude of the triangle be $x \mathrm{~cm}$ Therefore, the base of the triangle will be $(x+10) \mathrm{cm}$ $\begin{array}{l} \text { Area of triangle }=\frac{1}{2} x(x+10)=600 \\...

### The length of a rectangle is thrice as long as the side of a square. The side of the square is , more than the width of the rectangle. Their areas being equal, find the dimensions.

Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...

### A rectangular filed in long and wide. There is a path of uniform width all around it, having an area of . Find the width of the path

Let the width of the path be $x \mathrm{~m}$. $\therefore$ Length of the field including the path $=16+x+x=16+2 x$ Breadth of the field including the path $=10+x+x=10+2 x$ Now, (Area of the field...

### The length of rectangle is twice its breadth and its areas is . Find the dimension of the rectangle.

Let the length and breadth of the rectangle be $2 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 2 x \times x=288 \\ \Rightarrow 2 x^{2}=288 \\...

### Two pipes running together can fill a tank in minutes. If on pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

Let the time taken by one pipe to fill the tank be $x$ minutes. $\therefore$ Time taken by the other pipe to fill the tank $=(x+5) \min$ Suppose the volume of the tank be $V$. Volume of the tank...

### A takes 10 days less than the time taken by to finish a piece of work. If both and together can finish the work in 12 days, find the time taken by B to finish the work.

Let B takes $x$ days to complete the work. Therefore, A will take $(x-10)$ days. $\begin{array}{l} \therefore \frac{1}{x}+\frac{1}{(x-10)}=\frac{1}{12} \\ \Rightarrow...

### The speed of a boat in still water is . It can go upstream and downstream is 5 hours. Fid the speed of the stream.

Speed of the boat in still water $=8 \mathrm{~km} / \mathrm{hr}$. Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Speed upstream $=(8-x) \mathrm{km} / \mathrm{hr}$ Speed...

### The distance between Mumbai and Pune is . Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two train differ by

Let the speed of the Deccan Queen be $x \mathrm{~km} / \mathrm{hr}$. According to the question: Speed of another train $=(x-20) \mathrm{km} / \mathrm{hr}$ $\begin{array}{l} \therefore...

### A passenger train takes 2 hours less for a journey of if its speed is increased by from its usual speed. Find its usual speed.

Let the usual speed $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\begin{array}{l} \frac{300}{x}-\frac{300}{(x+5)}=2 \\ \Rightarrow \frac{300(x+5)-300 x}{x(x+5)}=2 \\ \Rightarrow...

### A train travels at a uniform speed. If the speed had been more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...

### The ratio of the sum and product of the roots of the equation is

(a)

(b)

(c)

(d)

Answer is (d) $2: 3$ Given: $7 x^{2}-12 x+18=0$ $\therefore \alpha+\beta=\frac{12}{7}$ and $\beta=\frac{18}{7}$, where $\alpha$ and $\beta$ are the roots of the equation $\therefore$ Ratio of the...

### The sum of the roots of the equation is

(a) 2

(b)

(c) 6

(d)

Answer is (b) -2 Sum of the roots of the equation $x^{2}-6 x+2=0$ is $\alpha+\beta=\frac{-b}{a}=\frac{-(-6)}{1}=6$, where $\alpha$ and $\beta$ are the roots of the equation.

### If is a solution of the equation , then

(a) 11

(b)

(c) 13

(d)

Answer is (b) $-11$ It is given that $x=3$ is a solution of $3 x^{2}+(k-1) x+9=0$; therefore, we have: $\begin{array}{l} 3(3)^{2}+(k-1) \times 3+9=0 \\ \Rightarrow 27+3(k-1)+9=0 \\ \Rightarrow...

### Which of the following is a quadratic equation?

(a)

(b)

(c)

(d) None of these

Answer is (b) $x^{3}-x^{2}=(x-1)^{3}$ $\begin{array}{l} \because x^{3}-x^{2}=(x-1)^{3} \\ \Rightarrow x^{3}-x^{2}=x^{3}-3 x^{2}+3 x-1 \end{array}$ $\Rightarrow 2 x^{2}-3 x+1=0$, which is a quadratic...

### Which of the following is a quadratic equation?

(a)

(b)

(c)

(d)

(d) $2 x^{2}-5 x=(x-1)^{2}$ A quadratic equation is the equation with degree 2 . $\begin{array}{l} \because 2 x^{2}-5 x=(x-1)^{2} \\ \Rightarrow 2 x^{2}-5 x=x^{2}-2 x+1 \\ \Rightarrow 2 x^{2}-5...

### The hypotenuse of a right-angled triangle is 1 meter less than twice the shortest side. If the third side 1 meter more than the shortest side, find the side, find the sides of the triangle.

Let the shortest side be $x \mathrm{~m}$. Therefore, according to the question: Hypotenuse $=(2 x-1) m$ Third side $=(x+1) m$ On applying Pythagoras theorem, we get: $\begin{array}{l} (2...

### The length of the hypotenuse of a right-angled triangle exceeds the length of the base by and exceeds twice the length of the altitude by . Find the length of each side of the triangle.

Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...

### The hypotenuse of a right-angled triangle is 20 meters. If the difference between the lengths of the ther sides be 4 meters, find the other sides.

Let one side of the right-angled triangle be $x \mathrm{~m}$ and the other side be $(x+4) m$. On applying Pythagoras theorem, we have: $\begin{array}{l} 20^{2}=(x+4)^{2}+x^{2} \\ \Rightarrow...

### The area of right-angled triangle is 96 sq meters. If the base is three time the altitude, find the base.

Let the altitude of the triangle be $x \mathrm{~m}$. Therefore, the base will be $3 x \mathrm{~m}$. $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text {...

### A farmer prepares rectangular vegetable garden of area 180 sq meters. With 39 meters of barbed wire, he can fence the three sides of the garden, leaving one of the longer sides unfenced. Find the dimensions of the garden.

Let the length and breadth of the rectangular garden be $x$ and $y$ meter, respectively. Given: $x y=180 s q m.....(i)and$ $\begin{array}{l} 2 y+x=39 \\ \Rightarrow x=39-2 y \end{array}$ Putting the...

### The sum of the areas of two squares is . If the difference in their perimeter be , find the sides of the two square.

Let the length of the side of the first and the second square be $x$ and $y \cdot$ respectively. According to the question: $x^{2}+y^{2}=640$ Also, $\begin{array}{l} 4 x-4 y=64 \\ \Rightarrow x-y=16...

### The perimeter of a rectangular plot is and its area is 288 sq meters. Find the dimension of the plot

Let the length and breadth of the rectangular plot be $x$ and $y$ meter, respectively. Therefore, we have: $\begin{array}{l} \text { Perimeter }=2(x+y)=62 \quad \ldots . .(i) \text { and } \\ \text...

### The length of a rectangular field is three times its breadth. If the area of the field be 147 sq meters, find the length of the field.

Let the length and breadth of the rectangle be $3 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 3 x \times x=147 \\ \Rightarrow 3 x^{2}=147 \\...

### Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time which each tap can separately fill the tank.

Let the tap of smaller diameter fill the tank in $x$ hours. $\therefore$ Time taken by the tap of larger diameter to fill the tank $=(x-9) h$ Suppose the volume of the tank be $V$. Volume of the...

### Two pipes running together can fill a cistern in minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.

Let one pipe fills the cistern in $x$ mins. Therefore, the other pipe will fill the cistern in $(x+3)$ mins. Time taken by both, running together, to fill the cistern $=3 \frac{1}{13} \min...

### A motorboat whose speed is in still water, goes downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Downstream speed $=(9+x) \mathrm{km} / \mathrm{hr}$ Upstream speed $=(9-x) \mathrm{km} / \mathrm{hr}$ Distance covered...

### A motor boat whose speed in still water is , takes 1 hour more to go upstream than to return to the same spot. Find the speed of the stream

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. Given: Speed of the boat $=18 \mathrm{~km} / \mathrm{hr}$ $\therefore$ Speed downstream $=(18+x) \mathrm{km} / \mathrm{hr}$ Speed...

### A passenger train takes 2 hours less for a journey of if its speed is increased by from its usual speed. Find its usual speed.

Let the usual speed $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\begin{array}{l} \frac{300}{x}-\frac{300}{(x+5)}=2 \\ \Rightarrow \frac{300(x+5)-300 x}{x(x+5)}=2 \\ \Rightarrow...

### A train travels at a uniform speed. If the speed had been more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...

### A train travels at a certain average speed for a distanced of and then travels a distance of 63 at an average speed of more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

Let the first speed of the train be $x \mathrm{~km} / \mathrm{h}$. Time taken to cover $54 \mathrm{~km}=\frac{54}{x} h \quad\left(\right.$ Time $\left.=\frac{\text { Distance }}{\text { Speed...

### A truck covers a distance of at a certain average speed and then covers another 200 at an average speed which is per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck.

Let the first speed of the truck be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Time taken to cover $150 \mathrm{~km}=\frac{150}{x} h \quad\left(\right.$ Time $\left.=\frac{\text { Dis tan ce...

### Two years ago, man’s age was three times the square of his son’s age. In three years’ time, his age will be four time his son’s age. Find their present ages.

Let son's age 2 years ago be $x$ years. Then, Man's age 2 years ago $=3 x^{2}$ years $\therefore$ Sons present age $=(x+2)$ years Man's present age $=\left(3 x^{2}+2\right)$ years In three years...

### The product of Tanvy’s age (in years) 5 years ago and her age is 8 years later is 30 . Find her present age.

Let the present age of Meena be $x$ years. According to the question: $\begin{array}{l} (x-5)(x+8)=30 \\ \Rightarrow x^{2}+3 x-40=30 \\ \Rightarrow x^{2}+3 x-70=0 \\ \Rightarrow x^{2}+(10-7) x-70=0...

### The sum of reciprocals of Meena’s ages (in years ) 3 years ago and 5 years hence . Find her present ages.

Let the present age of Meena be $x$ years Meena's age 3 years ago $=(x-3)$ years Meena's age 5 years hence $=(x+5)$ years According to the given condition, $\frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$...

### A dealer sells an article for 75 and gains as much per cent as the cost priced of the article. Find the cost price of the article.

Let the cost price of the article be $x$ $\therefore$ Gain percent $=x \%$ According to the given condition, $\Rightarrow \frac{100 x+x^{2}}{100}=75$ $\Rightarrow x^{2}+100 x=7500$ $\Rightarrow...

### In a class test, the sum of the marks obtained by in mathematics and science is 28 . Had he got 3 more marks in mathematics and 4 marks less in science, the product of marks obtained in the two subjects would have been Find the marks obtained by him in the two subjects separately.

Let the marks obtained by $P$ in mathematics and science be $x$ and $(28-x)$, respectively. According to the given condition, $\begin{array}{l} (x+3)(28-x-4)=180 \\ \Rightarrow(x+3)(24-x)=180 \\...

### If the price of a book is reduced by Rs. 5, a person can buy 4 more books for 300 . Find the original price of the book.

Let the original price of the book be $Rs x$. $\therefore$ Number of books bought at original price for $Rs. 600=\frac{600}{x}$ $\therefore$ Number of books bought at reduced price for...

### Some students planned a picnic. The total budget for food was Rs. But, 5 students failed to attend the picnic and thus the cost for food for each member increased by Rs. How many students attended the picnic and how much did each student pay for the food?

Let $x$ be the number of students who planned a picnic. $\therefore$ Original cost of food for each member $=$ γ $\frac{2000}{x}$ Five students failed to attend the picnic. So, $(x-5)$ students...

### In a class test, the sum of Kamal’s marks in mathematics and English is 40 . Had he got 3 marks more in mathematics and 4 marks less in English, the product of the marks would have been 360 . Find his marks in two subjects separately.

Let the marks of Kamal in mathematics and English be $x$ and $y$, respectively. According to the question: $x+y=40$ Also, $\begin{array}{l} (x+3)(y-4)=360 \\ \Rightarrow(x+3)(40-x-4)=360 \quad[\text...

### The sum of a number and its reciprocal is . Find the number.

Let the required number be $x$. According to the given condition, $\begin{array}{l} x+\frac{1}{x}=2 \frac{1}{30} \\ \Rightarrow \frac{x^{2}+1}{x}=\frac{61}{30} \\ \Rightarrow 30 x^{2}+30=61 x \\...

### The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is . Find the fraction.

Let the numerator be $x$. $\therefore$ Denominator $=x+3$ $\therefore$ Original number $=\frac{x}{x+3}$ According to the question: $\frac{x}{x+3}+\frac{1}{\left(\frac{x}{x+3}\right)}=2 \frac{9}{10}$...

### A two-digit number is 4 times the sum of its digits and twice the product of digits. Find the number.

Let the digits at units and tens places be $x$ and $y$, respectively. Original number $=10 y+x$ According to the question: $\begin{array}{l} 10 y+x=4(x+y) \\ \Rightarrow 10 y+x=4 x+4 y \\...

### The difference of the squares of two natural numbers is 45 . The square of the smaller number is four times the larger number. Find the numbers.

Let the greater number be $x$ and the smaller number be $y$. According to the question: $\begin{array}{l} x^{2}-y^{2}=45 \\ y^{2}=4 x \end{array}$ From (i) and (ii), we get: $x^{2}-4 x=45$...

### Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164 .

Let the larger and smaller parts be $x$ and $y$, respectively. According to the question: $\begin{array}{l} x+y=16 .....(i) \\ 2 x^{2}=y^{2}+164.....(ii) \end{array} \quad$ From (i), we get:...

### Divide 57 into two parts whose product is 680 .

Let the two parts be $x$ and $(57-x)$. According to the given condition, $\begin{array}{l} x(57-x)=680 \\ \Rightarrow 57 x-x^{2}=680 \\ \Rightarrow x^{2}-57 x+680=0 \\ \Rightarrow x^{2}-40 x-17...

### The sum of the squares two consecutive multiples of 7 is 1225 . Find the multiples.

Let the required consecutive multiplies of 7 be $7 x$ and $7(x+1)$. According to the given condition, $(7 x)^{2}+[7(x+1)]^{2}=1225$ $\Rightarrow 49 x^{2}+49\left(x^{2}+2 x+1\right)=1225$...

### The difference of two natural number is 3 and the difference of their reciprocals is . Find the numbers.

Let the required natural numbers be $x$ and $(x+3)$. Now, $x<x+3$ $\therefore \frac{1}{x}>\frac{1}{x+3}$ According to the given condition, $\begin{array}{l}...

(i)

(ii)

Solution: (i) We can write the given question as, $\sin ^{-1} \frac{1}{2}-2 \sin ^{-1} \frac{1}{\sqrt{2}}=\sin ^{-1} \frac{1}{2}-\sin ^{-1}\left(2 \times \frac{1}{\sqrt{2}}...

### The sum of two natural numbers is 9 and the sum of their reciprocals is . Find the numbers.

Let the required natural numbers be $x$ and $(9,-x)$. According to the given condition, $\begin{array}{l} \frac{1}{x}+\frac{1}{9-x}=\frac{1}{2} \\ \Rightarrow \frac{9-x+x}{x(9-x)}=\frac{1}{2} \\...

### Find the tow consecutive positive odd integer whose product s 483 .

Let the two consecutive positive odd integers be $x$ and $(x+2)$. According to the given condition, $\begin{array}{l} x(x+2)=483 \\ \Rightarrow x^{2}+2 x-483=0 \\ \Rightarrow x^{2}+23 x-21 x-483=0...

### Find the principal value of the following:

(i)

(ii)

Solution: (i) It is given that functions can be written as $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\frac{\sqrt{3}}{2 \sqrt{2}}-\frac{1}{2 \sqrt{2}}\right) $Taking $1 /...

### Two natural number differ by 3 and their product is 504 . Find the numbers.

Let the required numbers be $x$ and $(x+3)$. According to the question: $\begin{array}{l} x(x+3)=504 \\ \Rightarrow x^{2}+3 x=504 \\ \Rightarrow x^{2}+3 x-504=0 \\ \Rightarrow x^{2}+(24-21) x-504=0...

### The sum of the squares of two consecutive positive even numbers is 452 . Find the numbers.

Let the two consecutive positive even numbers be $x$ and $(x+2)$. According to the given condition, $\begin{array}{l} x^{2}+(x+2)^{2}=452 \\ \Rightarrow x^{2}+x^{2}+4 x+4=452 \\ \Rightarrow 2...

### The sum of the squares of two consecutive positive integers is 365 . Find the integers.

Let the required two consecutive positive integers be $x$ and $(x+1)$. According to the given condition, $\begin{array}{l} x^{2}+(x+1)^{2}=365 \\ \Rightarrow x^{2}+x^{2}+2 x+1=365 \\ \Rightarrow 2...

### If a and are real and then show that the roots of the equation are equal and unequal.

The given equation is $(a-b) x^{2}+5(a+b) x-2(a-b)=0$. $\begin{array}{l} \therefore D=[5(a+b)]^{2}-4 \times(a-b) \times[-2(a-b)] \\ =25(a+b)^{2}+8(a-b)^{2} \end{array}$ Since a and $\mathrm{b}$ are...

### Find the values of for which the given quadratic equation has real and distinct roots:

(i) .

(ii) .

(i) The given equation is $9 x^{2}+3 k x+4=0$. $\therefore D=(3 k)^{2}-4 \times 9 \times 4=9 k^{2}-144$ The given equation has real and distinct roots if $D>0 .$ $\begin{array}{l} \therefore 9...

### Find the values of for which the given quadratic equation has real and distinct roots:

(i) .

(ii) .

(i) The given equation is $k x^{2}+6 x+1=0$. $\therefore D=6^{2}-4 \times k \times 1=36-4 k$ The given equation has real and distinct roots if $D>0$. $\begin{array}{l} \therefore 36-4 k>0 \\...

### Find the value of a for which the equation has equal roots.

$(\alpha-12) x^{2}+2(\alpha-12) x+2=0$ Here, $a=(\alpha=12), b=2(\alpha-12) \text { and } c=2$ It is given that the roots of the equation are equal; therefore, we have $\begin{array}{l} D=0 \\...

### If the roots of the quadratic equation are real and equal, show that either or

$\left(c^{2}-a b\right) x^{2}-2\left(a^{2}-b c\right) x+\left(b^{2}-a c\right)=0$ Here, $a=\left(c^{2}-a b\right), b=-2\left(a^{2}-b c\right), c=\left(b^{2}-a c\right)$ It is given that the roots of...

### If the quadratic equation has equal roots, prove that

$\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ Here, $a=\left(1+m^{2}\right), b=2 m c \text { and } c=\left(c^{2}-a^{2}\right)$ It is given that the roots of the equation are equal;...

### If is a root of the equation . find the value of for the which the quadratic equation has equal roots.

It is given that $-4$ is a root of the quadratic equation $x^{2}+2 x+4 p=0$ $\begin{array}{l} \therefore(-4)^{2}+2 \times(-4)+4 p=0 \\ \Rightarrow 16-8+4 p=0 \\ \Rightarrow 4 p+8=0 \\ \Rightarrow...

### If is a root of the quadratic equation . and the quadratic equation has equal roots, find the value of .

It is given that $-5$ is a root of the quadratic equation $2 x^{2}+p x- 15=0$ $\therefore 2(-5)^{2}+p \times(-5)-15=0$ $\begin{array}{l} \Rightarrow-5 p+35=0 \\ \Rightarrow p=7 \end{array}$ The...

### Find the value of for which the quadratic equation . has real and equal roots.

The given equation is $(2 p+1) x^{2}-(7 p+2) x+(7 p-3)=0$. This is of the form $a x^{2}+b x+c=0$, where $a=2 p+1, b=-(7 p+2)$ and $c=7 p-3$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =-[-(7...

### Find the nonzero value of for which the roots of the quadratic equation . are real and equal.

The given equation is $9 x^{2}-3 k x+k=0$. This is of the form $a x^{2}+b x+c=0$, where $a=9, b=-3 k$ and $c=k$. $\therefore D=b^{2}-4 a c=(-3 k)^{2}-4 \times 9 \times k=9 k^{2}-36 k$ The given...

### Show that the roots of the equation are real for all real values of and .

$x^{2}+p x-q^{2}=0$ Here, $a=1, b=p$ and $c=-q^{2}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =p^{2}-4 \times 1 \times\left(-q^{2}\right) \\ =\left(p^{2}+4...

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

(i)

(ii) .

(i) The given equation is $12 x^{2}-4 \sqrt{15} x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=12, b=-4 \sqrt{15}$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-4 \sqrt{15})^{2}-4...

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

(i)

(ii)

(i) The given equation is $2 x^{2}-8 x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=2, b=-8$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-8)^{2}-4 \times 2 \times 5=64-40=24>0$...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: and

The given equation is $a^{2} b^{2} x^{2}-\left(4 b^{4}-3 a^{4}\right) x-12 a^{2} b^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=a^{2} b^{2}, B=-\left(4 b^{4}-3 a^{4}\right)$ and $c=-12...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .

The given equation is $4 x^{2}-4 a^{2} x+\left(a^{4}-b^{4}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=-4 a^{2}$ and $C=a^{4}-b^{4}$ $\therefore$ Discriminant, $B^{2}-4 A...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .

The given equation is $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=6$ and $C=-\left(a^{2}+2 a-8\right)$ $\therefore$ Discriminant, $D=$...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:

Given: $x^{2}-2 a x+\left(a^{2}-b^{2}\right)=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=1, B=-2 a$ and $C=\left(a^{2}-b^{2}\right)$ Discriminant $D$ is given by: $\begin{array}{l}...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:

The given equation is $36 x^{2}-12 a x+\left(a^{2}-b^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=36, B=-12 a$ and $C=a^{2}-b^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-12...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:

The given equation is $\begin{array}{l} \frac{m}{n} x^{2} \frac{n}{m}=1-2 x \\ \Rightarrow \frac{m^{2} x^{2}+n^{2}}{m n}=1-2 x \\ \Rightarrow m^{2} x^{2}+n^{2}=m n-2 m n x \\ \Rightarrow m^{2}...

The given equation is $\begin{array}{l} \frac{1}{x}-\frac{1}{x-2}=3, x \neq 0,2 \\ \Rightarrow \frac{x-2-x}{x(x-2)}=3 \\ \Rightarrow \frac{-2}{x^{2}-2 x}=3 \\ \Rightarrow-2=3 x^{2}-6 x \\...

The given equation is $\begin{array}{l} x+\frac{1}{x}=3, x \neq 0 \\ \Rightarrow \frac{x^{2}+1}{x}=3 \\ \Rightarrow x^{2}+1=3 x \\ \Rightarrow x^{2}-3 x+1=0 \end{array}$ This equation is of the form...

The given equation is $3 x^{2}-2 x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=(-2)^{2}-4 \times 3 \times 2=4-24=-20<0$ Hence,...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .

The given equation is $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=-(\sqrt{3}+1)$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a...

The given equation is $x^{2}+x+2=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=1$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=1^{2}-4 \times 1 \times 2=1-8=-7<0$ Hence, the given...

The given equation is $3 x^{2}-2 \sqrt{6} x+2=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2 \sqrt{6}$ and $c=2$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{6})^{2}-4 \times 3...

The given equation is $2 x^{2}+6 \sqrt{3} x-60=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=6 \sqrt{3}$ and $c=-60$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(6 \sqrt{3})^{2}-4 \times 2...

Given: $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=\sqrt{3}, b=10$ and $c=-8 \sqrt{3}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a...

The given equation is $2 x^{2}-2 \sqrt{2} x+1=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=-2 \sqrt{2}$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{2})^{2}-4 \times 2...

Given: $\begin{array}{l} 16 x^{2}+24 x+1 \\ \Rightarrow 16 x^{2}-24 x-1=0 \end{array}$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=16, b=-24$ and $c=-1$ Discriminant $D$ is given by:...

The given equation is $2 x^{2}+x-4=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=1$ and $c=-4$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(1)^{2}-4 \times 2 \times(-4)=1+32=33>0$ So, the...

Given: $x^{2}-4 x-1=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-4$ and $c=-1$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-4)^{2}-4 \times 1...

### Find the discriminant of the given equation:

$\begin{array}{l} \Rightarrow 2 x^{2}-3 x+1=0 \end{array}$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=-3$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-3)^{2}-4 \times 2 \times...

### Find the discriminant of the given equation:

Here, $\begin{array}{l} a=2 \\ b=-5 \sqrt{2} \\ c=4 \end{array}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-5 \sqrt{2})^{2}-4 \times 2 \times 4 \\ =(25 \times 2)-32 \\...

### Find the discriminant of the given equation:

Here, $\begin{array}{l} a=2 \\ b=-7 \\ c=6 \end{array}$ Discriminant $D$ is diven by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-7)^{2}-4 \times 2 \times 6 \\ =49-48 \\ =1 \end{array}$

### Find the roots of the given equation:

$\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x+21=0 \quad$ (Multiplying both sides by $\left.\sqrt{3}\right)$ $\Rightarrow 3 x^{2}+10 \sqrt{3} x=-21$ $\Rightarrow(\sqrt{3}...

### Find the roots of the given equation:

$\begin{array}{l} 5 x^{2}-6 x-2=0 \\ \Rightarrow 25 x^{2}-30 x-10=0 \\ \Rightarrow 25 x^{2}-30 x=10 \end{array}$ $\begin{array}{l} \Rightarrow(5 x)^{2}-2 \times 5 x \times 3+3^{2}=10+3^{2} \\...

### Find the roots of the given equation:

$\begin{array}{l} 7 x^{2}+3 x-4=0 \\ \Rightarrow 49 x^{2}+21 x-28=0 \end{array}$ $\Rightarrow 49 x^{2}+21 x=28$ $\Rightarrow(7 x)^{2}+2 \times 7 x \times...

### Find the roots of the given equation:

$\begin{array}{l} 3 x^{2}-x-2=0 \\ \Rightarrow 9 x^{2}-3 x-6=0 \quad \text { (Multiplying both sides by 3) } \\ \Rightarrow 9 x^{2}-3 x=6 \\ \Rightarrow(3 x)^{2}-2 \times 3 x \times...

### Find the roots of the given equation:

$2 x^{2}+5 x-3=0$ $\Rightarrow 4 x^{2}+10 x-6=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}+10 x=6$ $\Rightarrow(2 x)^{2}+2 \times 2 x \times...

### Find the roots of the given equation:

$\begin{array}{l} 4 x^{2}+4 \sqrt{3} x+3=0 \\ \Rightarrow 4 x^{2}+4 \sqrt{3} x=-3 \\ \Rightarrow(2 x)^{2}+2 \times 2 x \times \sqrt{3}+(\sqrt{3})^{2}=-3+(\sqrt{3})^{2} \end{array}$ $\begin{array}{l}...

### Find the roots of the given equation:

$x^{2}-4 x+1=0$ $\Rightarrow x^{2}-4 x=-1$ $\Rightarrow x^{2}-2 \times x \times 2+2^{2}=-1+2^{2} \quad$ (Adding $2^{2}$ on both sides) $\Rightarrow(x-2)^{2}=-1+4=3$ $\Rightarrow x-2=\pm \sqrt{3}...

### Find the roots of the given equation:

$\begin{array}{l} 2^{2 x}-3.2^{(x+2)}+32=0 \\ \Rightarrow\left(2^{x}\right)^{2}-3.2^{x} \cdot 2^{2}+32=0 \end{array}$ Let $2^{x}$ be $y$. $\begin{array}{l} \therefore y^{2}-12 y+32=0 \\ \Rightarrow...

### Find the roots of the given equation:

$\begin{array}{l} 3^{(x+2)}+3^{-x}=10 \\ 3^{x} .9+\frac{1}{3^{x}}=10 \end{array}$ Let $3^{x}$ be equal to $y$. $\begin{array}{l} \therefore 9 y+\frac{1}{y}=10 \\ \Rightarrow 9 y^{2}+1=10 y \\...

### Find the roots of the given equation:

$\begin{array}{l} \frac{a}{(x-b)}+\frac{b}{(x-a)}=2 \\ \Rightarrow\left[\frac{a}{(x-b)}-1\right]+\left[\frac{b}{(x-b)}-1\right]=0 \\ \Rightarrow \frac{a-(x-b)}{x-b}+\frac{b-(x-b)}{x-b}=0...

### Find the roots of the given equation:

Given: $\left(\frac{4 x-3}{2 x+1}\right)-10\left(\frac{2 x+1}{4 x-3}\right)=3$ Putting $\frac{4 x-3}{2 x+1}=y$, we get: $\begin{array}{l} y-\frac{10}{y}=3 \\ \Rightarrow \frac{y^{2}-10}{y}=3 \\...

### Find the roots of the given equation:

$\begin{array}{l} 3\left(\frac{3 x-1}{2 x+3}\right)-2\left(\frac{2 x+3}{3 x-1}\right)=5, x \neq \frac{1}{3},-\frac{3}{2} \\ \Rightarrow \frac{3(3 x-1)^{2}-2(2 x+3)^{2}}{(2 x+3)(3 x-1)}=5 \\...

### Find the roots of the given equation:

$\begin{array}{l} \frac{1}{(x-2)}+\frac{2}{(x-1)}=\frac{6}{x} \\ \Rightarrow \frac{(x-1)+2(x-2)}{(x-1)(x-2)}=\frac{6}{x} \\ \Rightarrow \frac{3 x-5}{x^{2}-3 x+2}=\frac{6}{x} \\ \Rightarrow 3 x^{2}-5...

### Find the roots of the given equation:

$\begin{array}{l} \frac{x-4}{x-5}+\frac{x-6}{x-7}=3 \frac{1}{3}, x \neq 5,7 \\ \Rightarrow \frac{(x-4)(x-7)+(x-5)(x-6)}{(x-5)(x-7)}=\frac{10}{3} \\ \Rightarrow \frac{x^{2}-11 x+28+x^{2}-11...

### Find the roots of the given equation:

$\begin{array}{l} \frac{x}{x-1}+\frac{x-1}{x}=4 \frac{1}{4}, x \neq 0,1 \\ \Rightarrow \frac{x^{2}+(x-1)^{2}}{x(x-1)}=\frac{17}{4} \\ \Rightarrow \frac{x^{2}+x^{2}-2 x+1}{x^{3}-x}=\frac{17}{4} \\...

### Find the roots of the given equation:

$\begin{array}{l} \frac{1}{2 a+b+2 x}=\frac{1}{2 a}+\frac{1}{b}+\frac{1}{2 x} \\ \Rightarrow \frac{1}{2 a+b+2 x}-\frac{1}{2 x}=\frac{1}{2 a}+\frac{1}{b} \\ \Rightarrow \frac{2 x-2 a-b-2 x}{2 x(2...

### Find the roots of the given equation:

$\begin{array}{l} \frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, x \neq-1, \frac{1}{3} \\ \Rightarrow \frac{3}{x+1}-\frac{2}{3 x-1}=\frac{1}{2} \\ \Rightarrow \frac{9 x-3-2 x-2}{(x+1)(3...

### Find the roots of the given equation:

$\begin{array}{l} \frac{16}{x}-1=\frac{15}{x+1}, x \neq 0,-1 \\ \Rightarrow \frac{16}{x}-\frac{15}{x+1}=1 \\ \Rightarrow \frac{16 x+16-15 x}{x(x+1)}=1 \\ \Rightarrow \frac{x+16}{x^{2}+x}=1 \\...

### Find the roots of the given equation:

$\begin{array}{l} a^{2} b^{2} x^{2}+b^{2} x-a^{2} x-1=0 \\ \Rightarrow b^{2} x\left(a^{2} x+1\right)-1\left(a^{2} x+1\right)=0 \\ \Rightarrow\left(b^{2} x-1\right)\left(a^{2} x+1\right)=0 \\...

### Find the roots of the given equation:

$\begin{array}{l} 4 x^{2}-2\left(a^{2}+b^{2}\right) x+a^{2} b^{2}=0 \\ \Rightarrow 4 x^{2}-2 a^{2} x-2 b^{2} x+a^{2} b^{2}=0 \\ \Rightarrow 2 x\left(2 x-a^{2}\right)-b^{2}\left(2 x-a^{2}\right)=0 \\...

### Find the roots of the given equation:

$\begin{array}{l} a b x^{2}+\left(b^{2}-a c\right) x-b c=0 \\ \Rightarrow a b x^{2}+b^{2} x-a c x-b c=0 \\ \Rightarrow b x(a x+b)-c(a x+b)=0 \\ \Rightarrow(b x-c)(a x+b)=0 \\ \Rightarrow b x-c=0...

### Find the roots of the given equation:

We write, $-(2 b-1) x=-(b-5) x-(b+4) x$ as $\begin{array}{l} x^{2} \times\left(b^{2}-b-20\right)=\left(b^{2}-b-20\right) x^{2}=[-(b-5) x] \times[-(b+4) x] \\ \therefore x^{2}-(2 b-1)...

### Find the roots of the given equation:

We write, $5 x=(a+3) x-(a-2) x$ as $\begin{array}{l} x^{2} \times\left[-\left(a^{2}+a-6\right)\right]=-\left(a^{2}+a-6\right) x^{2}=(a+3) x \times[-(a-2) x] \\ \therefore x^{2}+5...

### Find the roots of the given equation:

We write, $4 b x=2(a+b) x-2(a-b) x$ as $\begin{array}{l} 4 x^{2} \times\left[-\left(a^{2}-b^{2}\right)\right]=-4\left(a^{2}-b^{2}\right) x^{2}=2(a+b) x \times[-2(a-b) x] \\ \therefore 4 x^{2}+4 b...

### Find the roots of the given equation:

$\begin{array}{l} \frac{2}{x^{2}}-\frac{5}{x}+2=0 \\ \Rightarrow 2-5 x+2 x^{2}=0 \\ \Rightarrow 2 x^{2}-5 x+2=0 \\ \Rightarrow 2 x^{2}-(4 x+x)+2=0 \\ \Rightarrow 2 x^{2}-4 x-x+2=0 \\ \Rightarrow 2...

### Find the roots of the given equation:

We write, $-x=-\frac{x}{2}-\frac{x}{2}$ as $2 x^{2} \times \frac{1}{8}=\frac{x^{2}}{4}=\left(-\frac{x}{2}\right) \times\left(-\frac{x}{2}\right)$ $\begin{array}{l} \therefore 2 x^{2}-x+\frac{1}{8}=0...

### Find the roots of the given equation:

We write, $13 x=5 x+8 x$ as $5 x^{2} \times 8=40 x^{2}=5 x \times 8 x$ $\begin{array}{l} \therefore 5 x^{2}+13 x+8=0 \\ \Rightarrow 5 x^{2}+5 x+8 x+8=0 \\ \Rightarrow 5 x(x+1)+8(x+1)=0 \\...

### Find the roots of the given equation:

We write, $3 \sqrt{3} x=5 \sqrt{3} x-2 \sqrt{3} x$ as $x^{2} \times(-30)=-30 x^{2}=5 \sqrt{3} x \times(-2 \sqrt{3} x)$ $\begin{array}{l} \therefore x^{2}+3 \sqrt{3} x-30=0 \\ \Rightarrow x^{2}+5...

### Find the roots of the given equation:

We write, $-3 \sqrt{5} x=-2 \sqrt{5} x-\sqrt{5} x$ as $x^{2} \times 10=10 x^{2}=(-2 \sqrt{5} x) \times(-\sqrt{5} x)$ $\begin{array}{l} \therefore x^{2}-3 \sqrt{5} x+10=0 \\ \Rightarrow x^{2}-2...

### Find the roots of the given equation:

We write, $-2 \sqrt{6} x=-\sqrt{6} x$ ad $3 x^{2} \times 2=6 x^{2}=(-\sqrt{6} x) \times(-\sqrt{6} x)$ $\begin{array}{l} \therefore 3 x^{2}-2 \sqrt{6} x+2=0 \\ \Rightarrow 3 x^{2}-\sqrt{6} x-\sqrt{6}...

### Find the roots of the given equation:

We write, $-6 x=7 x-13 x$ as $\sqrt{7} x^{2} \times(-13 \sqrt{7})=-91 x^{2}=7 x \times(-13 x)$ $\begin{array}{l} \therefore \sqrt{7} x^{2}-6 x-13 \sqrt{7}=0 \\ \Rightarrow \sqrt{7} x^{2}+7 x-13 x-13...

### Find the roots of the given equation:

$\begin{array}{l} \sqrt{3} x^{2}+11 x+6 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x^{2}+9 x+2 x+6 \sqrt{3}=0 \\ \Rightarrow \sqrt{3} x(x+3 \sqrt{3})+2(x+3 \sqrt{3})=0 \end{array}$ $\begin{array}{l}...

### Find the roots of the given equation:

We write: $2 \sqrt{2} x=3 \sqrt{2} x-\sqrt{2} x$ as $x^{2} \times(-6)=-6 x^{2}=3 \sqrt{2} x \times(-\sqrt{2} x)$ $\therefore x^{2}+2 \sqrt{2} x-6=0$ $\Rightarrow x^{2}+2 \sqrt{2} x-\sqrt{2} x-6=0$...

### Find the roots of the given equation:

$\begin{array}{l} 4-11 x=3 x^{2} \\ \Rightarrow 3 x^{2}+11 x-4=0 \\ \Rightarrow 3 x^{2}+12 x-x-4=0 \\ \Rightarrow 3 x(x+4)-1(x+4)=0 \\ \Rightarrow(x+4)(3 x-1)=0 \\ \Rightarrow x+4=0 \text { or } 3...

### Find the roots of the given equation:

$\begin{array}{l} 15 x^{2}-28=x \\ \Rightarrow 15 x^{2}-x-28=0 \\ \Rightarrow 15 x^{2}-(21 x-20 x)-28=0 \\ \Rightarrow 15 x^{2}-21 x+20 x-28=0 \end{array}$ $\begin{array}{l} \Rightarrow 3 x(5...

### Find the roots of the given equation: .

$\begin{array}{l} 6 x^{2}+11 x+3=0 \\ \Rightarrow 6 x^{2}+9 x+2 x+3=0 \\ \Rightarrow 3 x(2 x+3)+1(2 x+3)=0 \\ \Rightarrow(3 x+1)(2 x+3)=0 \\ \Rightarrow 3 x+1=0 \text { or } 2 x+3=0 \\ \Rightarrow...

### Find the roots of the given equation:

$\begin{array}{l} x^{2}+12 x+35=0 \\ \Rightarrow x^{2}+7 x+5 x+35=0 \\ \Rightarrow x(x+7)+5(x+7)=0 \\ \Rightarrow(x+5)(x+7)=0 \\ \Rightarrow x+5=0 \text { or } x+7=0 \\ \Rightarrow x=-5 \text { or }...

### Find the roots of the given equation:

We write, $x=4 x-3 x$ as $2 x^{2} \times(-6)=-12 x^{2}=4 x \times(-3 x)$ $\begin{array}{l} \therefore 2 x^{2}+x-6=0 \\ \Rightarrow 2 x^{2}+4 x-3 x-6=0 \\ \Rightarrow 2 x(x+2)-3(x+2)=0 \\...

### Find the roots of the given equation:

$\begin{array}{l} 4 x^{2}+5 x=0 \\ \Rightarrow x(4 x+5)=0 \\ \Rightarrow x=0 \text { or } 4 x+5=0 \end{array}$ $\Rightarrow x=0$ or $x=-\frac{5}{4}$ So, the roots of the given equation are 0 and...

### Find the roots of the given equation:

$\begin{array}{l} (2 x-3)(3 x+1)=0 \\ \Rightarrow 2 x-3=0 \text { or } 3 x+1=0 \\ \Rightarrow 2 x=3 \text { or } 3 x=-1 \\ \Rightarrow x=\frac{3}{2} \text { or } x=-\frac{1}{3} \end{array}$ As a...

### Find the value of and for which and are the roots of the equation

Given $\frac{3}{4}$ is a root of $a x^{2}+b x-6=0$; therefore, we have: $a \times\left(\frac{3}{4}\right)^{2}+b \times \frac{3}{4}-6=0$ $\begin{array}{l} \Rightarrow \frac{9 a}{16}+\frac{3 b}{4}=6...

### Which of the following are the roots of

(i)

(ii)

(i) $x=(-1)$ $\begin{array}{l} \text { L.H.S. }=x^{2}+2 x-1 \\ =3 \times(-1)^{2}+2 \times(-1)-1 \\ =3-2-1 \\ =0 \\ =\text { R.H.S. } \end{array}$ Thus, $(-1)$ is a root of $\left(3 x^{2}+2...

### Which of the following are quadratic equation in ?

(i)

(ii)

(i) $\begin{array}{l} x^{2}-\frac{2}{x}=x^{2} \\ \Rightarrow x^{2}+2=x^{3} \\ \Rightarrow x^{3}-x^{2}-2=0 \end{array}$ $\left(x^{3}-x^{2}-2\right)$ is not a quadratic polynomial. $\therefore...

### Is it possible to design a rectangular park of perimeter 80 and area 400 m2? If so find its length and breadth.

(i) $2{{x}^{2}}~+~kx~+\text{ }3\text{ }=\text{ }0$ Comparing the given equation with ax2 + bx + c = 0, we get, a = 2, b = k and c = 3 As we know, Discriminant = b2 – 4ac $=\text{ }{{\left( k...

### 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 years was 48.

Solution: Let’s say, the age of one friend be x years. Then, the age of the other friend will be (20 – x) years. Four years ago, Age of First friend = (x – 4) years Age of Second friend = (20 – x –...

### Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.

Solution: Let the breadth of mango grove be l. Length of mango grove will be 2l. Area of mango grove $=\text{ }\left( 2l \right)\text{ }\left( l \right)=\text{ }2{{l}^{2}}$ $2{{l}^{2~}}=\text{...

### Find the values of k for each of the following quadratic equations, so that they have two equal roots. (i) 2×2 + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0

Solutions: $\left( i \right)\text{ }2{{x}^{2}}~+~kx~+\text{ }3\text{ }=\text{ }0$ Comparing the given equation with $a{{x}^{2}}~+~bx~+~c~=\text{ }0$ , we get, $a~=\text{ }2,~b~=\text{ }k\text{...

### Find the nature of the roots of the following quadratic equations. If the real roots exist, find them;(i) 2×2 – 6x + 3 = 0

$\left( i \right)\text{ }2{{x}^{2}}~~6x~+\text{ }3\text{ }=\text{ }0$ Comparing the equation with $a{{x}^{2}}~+~bx~+~c~=\text{ }0$ , we get a = 2, b = -6, c = 3 a = 2, b = -6, c = 3 As we know,...

### Find the nature of the roots of the following quadratic equations. If the real roots exist, find them; (i) 2×2 – 3x + 5 = 0 (ii) 3×2 – 4√3x + 4 = 0

(i) Given, $2{{x}^{2}}~\text{ }3x~+\text{ }5\text{ }=\text{ }0$ Comparing the equation with $a{{x}^{2}}~+~bx~+~c~=\text{ }0$ , we get a = 2, b = -3 and c = 5 We know, Discriminant...

### Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.

Solution: Let the sides of the two squares be x m and y m. Therefore, their perimeter will be 4x and 4y respectively And area of the squares will be ${{x}^{2}}~and~{{y}^{2}}~$ respectively. Given,...

### An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speeds of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

Solution: Let us say, the average speed of passenger train = x km/h. Average speed of express train = (x + 11) km/h Given, time taken by the express train to cover 132 km is 1 hour less than the...

### Two water taps together can fill a tank in 75/8. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Solution: Let the time taken by the smaller pipe to fill the tank = x hr. Time taken by the larger pipe = (x – 10) hr Part of tank filled by smaller pipe in 1 hour = 1/x Part of tank filled by...

### A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train

Solution: Let us say, the speed of the train be x km/hr. Time taken to cover 360 km = 360/x hr. As per the question given, $\Rightarrow \left( x~+\text{ }5 \right)\left( 360-1/x \right)\text{...

### 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.

Solution: Let us say, the larger and smaller number be x and y, respectively. As per the question given, ${{x}^{2~}}~{{y}^{2}}~=\text{ }180\text{ }and~{{y}^{2}}~=\text{ }8x$ $\Rightarrow...

### 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.

Solution: Let us say, the shorter side of the rectangle be x m. Then, larger side of the rectangle = (x + 30) m As given, the length of the diagonal is = x + 30 m Therefore, $\Rightarrow...

### 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

Solution: Let us say, the marks of Shefali in Maths be x. Then, the marks in English will be 30 – x. As per the given question, $\left( x~+\text{ }2 \right)\left( 30\text{ }~x~\text{ }3...

### 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

Solution: Let us say, present age of Rehman is x years. Three years ago, Rehman’s age was $\left( x-3 \right)$ years. Five years after, his age will be $\left( x~+\text{ }5 \right)$ years. Given,...

### Find the roots of the following equations: (i) x-1/x = 3, x ≠ 0 (ii) 1/x+4 – 1/x-7 = 11/30, x = -4, 7

Solution: $\left( i \right)~x-1/x~=\text{ }3$ $\Rightarrow ~{{x}^{2}}~\text{ }3x~-1\text{ }=\text{ }0$ On comparing the given equation with $a{{x}^{2}}~+~bx~+~c~=\text{ }0$ , we get $a~=\text{...

### Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

$\left( \mathbf{iii} \right)\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{4}\surd \mathbf{3x}~+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$ On comparing the given equation...

### Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

$\left( \mathbf{i} \right)\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~~\mathbf{7x}~+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$ On comparing the given equation with $a{{x}^{2}}~+~bx~+~c~=\text{...

### Find the roots of the following quadratic equations, if they exist, by the method of completing the square:(i) 4×2 + 4√3x + 3 = 0 (ii) 2×2 + x + 4 = 0

$\left( \mathbf{iii} \right)\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{4}\surd \mathbf{3x}~+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$ Converting the equation into...

### Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (i) 2×2 – 7x +3 = 0 (ii) 2×2 + x – 4 = 0

Solutions: $\left( \mathbf{i} \right)\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~-\mathbf{7x}~+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$ $\Rightarrow 2{{x}^{2}}~-7x~=\text{ }3$ Dividing by 2...

### A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs.90, find the number of articles produced and the cost of each article.

Solution: Let us say, the number of articles produced be x. Therefore, cost of production of each article = Rs (2x + 3) Given, total cost of production is Rs.90 $\therefore ~x\left( 2x~+\text{ }3...

### The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Solution: Let us say, the base of the right triangle be x cm. Given, the altitude of right triangle = (x – 7) cm From Pythagoras theorem, we know, Base2 + Altitude2 = Hypotenuse2 ∴ x2 + (x – 7)2 =...

### Find two consecutive positive integers, sum of whose squares is 365.

Solution: Let us say, the two consecutive positive integers be x and x + 1. Therefore, as per the given questions, ${{x}^{2}}~+\text{ }{{\left( x~+\text{ }1 \right)}^{2}}~=\text{ }365$ $\Rightarrow...

### Find two numbers whose sum is 27 and product is 182.

Solution: Let us say, first number be x and the second number is 27 – x. Therefore, the product of two numbers x(27 – x) = 182 ⇒ x2 – 27x – 182 = 0 ⇒ x2 – 13x – 14x + 182 = 0 ⇒ x(x – 13) -14(x – 13)...

### 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 production was ` 750. We would like to find out the number of toys produced on that day.

(i) Let us say, the number of marbles John have = x. Therefore, number of marble Jivanti have = 45 – x After losing 5 marbles each, Number of marbles John have = x – 5 Number of marble Jivanti have...

### Find the roots of the following quadratic equations by factorisation: (i) 100×2 – 20x + 1 = 0

Given, 100x2 – 20x + 1=0 Taking LHS, = 100x2 – 10x – 10x + 1 = 10x(10x – 1) -1(10x – 1) = (10x – 1)2 The roots of this equation, 100x2 – 20x + 1=0, are the values of x for which (10x – 1)2= 0 ∴...

### Find the roots of the following quadratic equations by factorisation: (i) √2 x2 + 7x + 5√2 = 0 (ii) 2×2 – x +1/8 = 0

√2 x2 + 7x + 5√2=0 Taking LHS, => √2 x2 + 5x + 2x + 5√2 => x (√2x + 5) + √2(√2x + 5)= (√2x + 5)(x + √2) The roots of this equation, √2 x2 + 7x + 5√2=0 are the values of x for which (x –...

### Find the roots of the following quadratic equations by factorisation: (i) x2 – 3x – 10 = 0 (ii) 2×2 + x – 6 = 0

Solutions: $\left( i \right)\text{ }Given,~{{x}^{2}}~\text{ }3x~\text{ }10\text{ }=0$ Taking LHS, $=>{{x}^{2}}~\text{ }5x~+\text{ }2x~\text{ }10$ $=>x\left( x~\text{ }5 \right)~+\text{...

### Represent the following situations in the form of quadratic equations: (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 find Rohan’s present age. (ii) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken

(iii) Let us consider, Age of Rohan’s = x years Therefore, as per the given question, Rohan’s mother’s age = x + 26 After 3 years, Age of Rohan’s = x + 3 Age of Rohan’s mother will be = x + 26 + 3...

### Represent the following situations in the form of quadratic equations: (i) The area of a rectangular plot is 528 m2. 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.

Solutions: (i) Let us consider, Breadth of the rectangular plot = x m Thus, the length of the plot = (2x + 1) m. As we know, $Area\text{ }of\text{ }rectangle\text{ }=\text{ }length~\times...

### Check whether the following are quadratic equations: (i) (x + 2)3 = 2x (x2 – 1) (ii) x3 – 4×2 – x + 1 = (x – 2)3

$\left( vii \right)\text{ }Given,\text{ }{{\left( x~+\text{ }2 \right)}^{3}}~=\text{ }2x({{x}^{2}}~\text{ }1)$ By using the formula for ${{\left( a+b \right)}^{2~}}=\text{ }{{a}^{2}}+2ab+{{b}^{2}}$...

### Check whether the following are quadratic equations: (i) (2x – 1)(x – 3) = (x + 5)(x – 1) (ii) x2 + 3x + 1 = (x – 2)2

(i) Given, (2x – 1)(x – 3) = (x + 5)(x – 1) By using the formula for (a+b)2=a2+2ab+b2 ⇒ 2x2 – 7x + 3 = x2 + 4x – 5 ⇒ x2 – 11x + 8 = 0 Since the above equation is in the form of ax2 + bx + c = 0....

### Check whether the following are quadratic equations: (i) (x – 2)(x + 1) = (x – 1)(x + 3) (ii) (x – 3)(2x +1) = x(x + 5)

$\left( iii \right)\text{ }Given,\text{ }\left( x\text{ }\text{ }2 \right)\left( x\text{ }+\text{ }1 \right)\text{ }=\text{ }\left( x\text{ }\text{ }1 \right)\left( x\text{ }+\text{ }3 \right)$ By...

### Check whether the following are quadratic equations: (i) (x + 1)2 = 2(x – 3) (ii) x2 – 2x = (–2) (3 – x)

(i) Given, (x + 1)2 = 2(x – 3) By using the formula for (a+b)2 = a2+2ab+b2 ⇒ x2 + 2x + 1 = 2x – 6 ⇒ x2 + 7 = 0 Since the above equation is in the form of ax2 + bx + c = 0. Therefore, the given...

### 1. A takes days less than the time taken by to finish a piece of work. If both and together can finish the work in days, find the time taken by to finish the work.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let’s consider that $B$ takes $x$ days to complete the piece of work....

### 1. The hypotenuse of a right triangle is cm. The difference between the lengths of the other two sides of the triangle is cm. Find the lengths of these sides.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let the length of one side of the right triangle be $x$ cm So, the...

### 3. A fast train takes one hour less than a slow train for a journey of km. If the speed of the slow train is km/hr less than that of the fast train, find the speed of the two trains.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let’s consider the speed of the fast train as x km/hr Then, the speed...

### 2. A train, traveling at a uniform speed for km, would have taken minutes less to travel the same distance if its speed were km/hr more. Find the original speed of the train.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let the original speed of train be $x$ km/hr When increased by $5$,...

### 1. The speed of a boat in still water is km/hr. It can go km upstream and km downstream in hours. Find the speed of the stream.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let the speed of stream be $x$ km/hr Given, speed of boat in still...

### Find the roots of the following quadratic equations (if they exist) by the method of completing the square. 4.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Given equation, $2{{x}^{2}}+x-4=0$ $2\left(...

### Find the roots of the following quadratic equations (if they exist) by the method of completing the square. 3.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Given equation, ${{x}^{2}}+\frac{11}{3}+\frac{10}{3}=0$...

### Find the roots of the following quadratic equations (if they exist) by the method of completing the square. 1.

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Given equation, ${{x}^{2}}-4\sqrt{2x}+6=0$ ${{x}^{2}}-2\times x\times...