Advertisements
Advertisements
рдкреНрд░рд╢реНрди
The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.
Advertisements
рдЙрддреНрддрд░
Let the numbers be x and 8 − x
Given that the sum of these numbers is 8
And 15 times the sum of their reciprocals as 8
`rArr15(1/x+1/(8-x))=8`
`rArr15(((8-x)+x)/(x(8-x)))=8`
⇒ 15 ((8 − ЁЭСе) + ЁЭСе) = 8(ЁЭСе(8 − ЁЭСе))
⇒ 15 [8 − ЁЭСе + ЁЭСе] = 8ЁЭСе(8 − ЁЭСе)
⇒ 120 = 64x − 8x2
⇒ 8ЁЭСе2 − 64ЁЭСе + 120 = 0
⇒ 8[ЁЭСе2 − 8ЁЭСе + 15] = 0
⇒ ЁЭСе2 − 5ЁЭСе − 3ЁЭСе + 15 = 0
⇒ (ЁЭСе − 5) (ЁЭСе − 3) = 0
⇒ x = 5 or x = 3
∴ The two numbers are 5 and 3.
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНрди
Solve for x :
`1/(2x - 3) + 1/(x - 5) = 1 1/9 , X != 3/2, 5`
Solve the following quadratic equations by factorization:
48x2 − 13x − 1 = 0
Solve the following quadratic equations by factorization:
a2x2 - 3abx + 2b2 = 0
Solve the following quadratic equations by factorization:
`(x+3)/(x+2)=(3x-7)/(2x-3)`
Find the consecutive numbers whose squares have the sum 85.
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.
A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for a journey. Find the original speed of the train.
Solve the following quadratic equations by factorization:
`100/x-100/(x+5)=1`
A teacher on attempting to arrange the students for mass drill in the form of solid square found that 24 students were left. When he increased the size of the square by one student, he found that he was short of 25 students. Find the number of students.
The sum of two natural numbers is 20 while their difference is 4. Find the numbers.
Solve the following quadratic equations by factorization: \[\frac{3}{x + 1} - \frac{1}{2} = \frac{2}{3x - 1}, x \neq - 1, \frac{1}{3}\]
Find the values of k for which the roots are real and equal in each of the following equation:\[px(x - 3) + 9 = 0\]
If −5 is a root of the quadratic equation\[2 x^2 + px - 15 = 0\] and the quadratic equation \[p( x^2 + x) + k = 0\] has equal roots, find the value of k.
If 1 is a root of the quadratic equation \[3 x^2 + ax - 2 = 0\] and the quadratic equation \[a( x^2 + 6x) - b = 0\] has equal roots, find the value of b.
Write the number of real roots of the equation x2 + 3 |x| + 2 = 0.
If the roots of the equations \[\left( a^2 + b^2 \right) x^2 - 2b\left( a + c \right)x + \left( b^2 + c^2 \right) = 0\] are equal, then
If y = 1 is a common root of the equations \[a y^2 + ay + 3 = 0 \text { and } y^2 + y + b = 0\], then ab equals
A farmer wishes to grow a 100m2 rectangular vegetable garden. Since he was with him only 30m barbed wire, he fences 3 sides of the rectangular garden letting the compound of his house to act as the 4th side. Find the dimensions of his garden .
Solve the following by reducing them to quadratic form:
`sqrt(x^2 - 16) - (x - 4) = sqrt(x^2 - 5x + 4)`.
In each of the following determine whether the given values are solutions of the equation or not.
x2 + 6x + 5 = 0; x = -1, x = -5
In each of the following determine whether the given values are solutions of the equation or not
2x2 - 6x + 3 = 0; x = `(1)/(2)`
Solve the following equation by factorization
21x2 – 8x – 4 = 0
Solve the following equation by factorization
3(x – 2)2 = 147
Solve the following equation by factorization
`x + (1)/x = 2(1)/(20)`
Solve the following equation by factorization
`x^2/(15) - x/(3) - 10` = 0
Solve the following equation by factorization
`(1)/(2a + b + 2x) = (1)/(2a) + (1)/b + (1)/(2x)`
Solve the following equation by factorization
`sqrt(3x + 4) = x`
A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 square metres, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x.
A man spent Rs. 2800 on buying a number of plants priced at Rs x each. Because of the number involved, the supplier reduced the price of each plant by Rupee 1.The man finally paid Rs. 2730 and received 10 more plants. Find x.
In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m2 [see figure]. Find the length and breadth of the pond.
