### 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.

### 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...

### 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 $54 \mathrm{~km}=\frac{54}{x} h .$ New speed of the train $=(x+6) \mathrm{km} / \mathrm{h}$ $\therefore$ Time...

### A train covers a distance of at a uniform speed. If the speed had been less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...

### Which of the following is not a quadratic equation?(a) (b) (c) (d)

Answer is (c) $(\sqrt{2} x+3)^{2}=2 x^{2}+6$ $\begin{array}{l} \because(\sqrt{2} x+3)^{2}=2 x^{2}+6 \\ \Rightarrow 2 x^{2}+9+6 \sqrt{2} x=2 x^{2}+6 \end{array}$ $\Rightarrow 6 \sqrt{2} x+3=0$, which...

### 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 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...

### While boarding an aeroplane, a passengers got hurt. The pilot showing promptness and concern, made arrangements to hospitalize the injured and so the plane started late by 30 minutes. To reach the destination, away, in time, the pilot increased the speed by hour. Find the original speed of the plane. Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time?

Let the original speed of the plane be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Actual speed of the plane $=(x+100) \mathrm{km} / \mathrm{h}$ Distance of the journey $=1500 \mathrm{~km}$ Time...

### A person on tour has Rs. 10800 for his expenses. If he extends his tour by 4 days, he has to cut down his daily expenses by Rs. 90 . Find the original duration of the tour.

Let the original duration of the tour be $x$ days. $\therefore \text { Original daily expenses }=\text { γ } \frac{10,800}{x}$ If he extends his tour by 4 days, then his new daily expenses...

### The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by . Find the fraction.

Let the denominator of the required fraction be $x$. Numerator of the required fraction $=x-3$ $\therefore$ Original fraction $=\frac{x-3}{x}$ If 1 is added to the denominator, then the new fraction...

### Divide two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

Let the two natural numbers be $x$ and $y$. According to the question: $\begin{array}{l} x^{2}+y^{2}=25(x+y) \quad \ldots \ldots(i) \\ x^{2}+y^{2}=50(x-y) \end{array}$ From (i) and (ii), we get:...

### If 3 is a root of the quadratic equation ., find the value of so that the roots of the equation are equal.

It is given that 3 is a root of the quadratic equation $x^{2}-x+k=0$. $\begin{array}{l} \therefore(3)^{2}-3+k=0 \\ \Rightarrow k+6=0 \\ \Rightarrow k=-6 \end{array}$ The roots of the equation...

### For what values of are the roots of the equation . real and equal?

The given equation is $4 x^{2}+p x+3=0$. This is of the form $a x^{2}+b x+c=0$, where $a=4, b=p$ and $c=3$. $\therefore D=b^{2}-4 a c=p^{2}-4 \times 4 \times 3=p^{2}-48$ The given equation will have...

### For what value of are the roots of the quadratic equation real and equal.

The given equation is $\begin{array}{l} k x(x-2 \sqrt{5})+10=0 \\ \Rightarrow k x^{2}-2 \sqrt{5} k x+10=0 \end{array}$ This is of the form $a x^{2}+b x+c=0$, where $a=k, b=-2 \sqrt{5} k$ and $c=10$....

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

Given: $\begin{array}{l} 15 x^{2}-28=x \\ \Rightarrow 15 x^{2}-x-28=0 \end{array}$ On comparing it with $a x^{2}+b x+c=0$, we get; $a=25, b=-1$ and $c=-28$ Discriminant $D$ is given by:...

### Find the roots of the given equation:

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

### 10. Solve the following equation:

$10x – 1/x = 3$ Solution:- $10x – 1/x = 3$ $(10x-1)/x=3$ By cross multiplication we get, $10{{x}^{2}}-1=3x$ $10{{x}^{2}}-3x-1=0$ Divided by $10$ for both side of eachterm we get,...

### 15. Solve the following equation:

Solution: - ${{x}^{2}}-(\sqrt{3}+1)x+\sqrt{3}=0$ ${{x}^{2}}-\sqrt{3x}-x+\sqrt{3}=0$ Take out common in each term, $x(x-\sqrt{3})-1(x-\sqrt{3})=0$ $(x-\sqrt{3})(x-1)=0$ Equate both to zero, ...

### 14. Solve the following equation:

Solution: - ${{x}^{2}}-(\sqrt{2}+1)x+\sqrt{2}=0$ ${{x}^{2}}-x-\sqrt{2}x+\sqrt{2}=0$ Take out common in each term, $x(x-1)-\sqrt{2}(x-1)=0$ $(x-1)(x-\sqrt{2})=0$ Equate both to zero,...

### 13. Solve the following equation:

Solution: - ${{a}^{2}}{{x}^{2}}-3abx+2{{b}^{2}}=0$ Divided by${{a}^{2}}$ for both side of each term we get, ${{a}^{2}}{{x}^{2}}/{{a}^{2}}-3abx/{{a}^{2}}+2{{b}^{2}}/{{a}^{2}}=0$...

### 12. Solve the following equation:

Solution: - $\sqrt{2}{{x}^{2}}-3x-2\sqrt{2}$ Divided by $\sqrt{2}$for both side of each term we get, $\sqrt{2}{{x}^{2}}/\sqrt{2}-3x/\sqrt{2}-2\sqrt{2}/\sqrt{2}=0$ ${{x}^{2}}-3x/\sqrt{2}-2=0$...

### 11. Solve the following equation:

Solution: - $2/{{x}^{2}}-5{{x}^{2}}/x+2{{x}^{2}}=0$ Multiply by ${{x}^{2}}$for both side of each term we get, $2{{x}^{2}}/{{x}^{2}}-5{{x}^{2}}/x+2{{x}^{2}}=0$ $2-5x+2{{x}^{2}}=0$ Above equation can...

### 6. Solve the following question:

$\mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}~-\mathbf{11x}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}$ Solution:- $5{{x}^{2}}~-\text{ }11x\text{ }+\text{ }2\text{ }=\text{ }0$...

### 5. Solve the following equation:

$\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~-\text{ }\mathbf{x}\text{ }-\text{ }\mathbf{6}\text{ }=\text{ }\mathbf{0}$ Solution:- $2{{x}^{2}}~-x\text{ }-\text{ }6\text{ }=\text{ }0~~~~~~$ Divided by...

$\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}-\mathbf{3x}-\mathbf{9}\text{ }=\text{ }\mathbf{0}$ Solution:- $2{{x}^{2}}-3x-9\text{ }=\text{ }0$ Divided by $2$for both side of each term we get,...