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...