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प्रश्न
A tank fills completely in 2 hours if both the taps are open. If only one of the taps is open at the given time, the smaller tap takes 3 hours more than the larger one to fill the tank. How much time does each tap take to fill the tank completely?
उत्तर
Let the larger tap take x hours.
Smaller tap takes x + 3 hours.
Both fill the tank in 2 hours if they are open together.
Amount of water filled by larger tap in 1 hour = `1/x`
Amount of water filled by smaller tap in 1 hour = `1/(x + 3)`
Amount of water filled in 1 hour = `1/2`
`1/x + 1/(x + 3) = 1/2`
`(x + 3 + x)/(x(x + 3)) = 1/2`
`(2x + 3)/(x^2 + 3x) = 1/2`
2(2x+ 3) = x2 + 3x
4x + 6 = x2 + 3x
0 = x2 + 3x − 4x − 6
x2 − x − 6 = 0
x2 − 3x + 2x − 6 = 0 ...`[(-3 xx 2 = -6),(-3 + 2 = -1)]`
x(x − 3) + 2(x − 3) = 0
(x − 3) (x + 2) = 0
x − 3 = 0 or x + 2 = 0
∴ x = 3 = 0 or x = −2
Thus, the larger tap takes 3 hours while the smaller tap takes 3 + 3 = 6 hours.
संबंधित प्रश्न
(a) `3x-x^2=x^2+5` (b) `(x+2)^2=2(x^2-5)`
(c) `(sqrt2x+3)^2=2x^2+6` (d)` (x-1)^2=3x^2+x-2`
The roots of the equation `ax^2+bx+c=0` will be reciprocal each other if
(a)a=b (b)b=c (c)=a (d)= none of these
If the equation `x^2-5x+1=0` has no real roots then
(a)`k<-2`
(b)`k>2`
(c) `-2<k<2`
(d) None of these
The sum of a number and its reciprocal is `2 1/20` The number is
(a) `5/4 or 4/5` (b)`4/3 or 3/4`
(c) `5/6 or 6/5` (d) `1/6 or 6`
The perimeter of a rectangle is 82m and its area is `400m^2` . The breadth of the rectangle is
(a) 25m (b)20m
(c) 16m (d)9m
The roots of the quadratic equation `2x^2-x-6=0`
(a)`-2, 3/2` (b) `2, -3/2`
(c)` -2, 3/2` (d) `2, 3/2`
Find the roots of the quadratic equation `2x^2-x-6=0`
Write any two quadratic equations.
Decide whether the following equation is quadratic equation or not.
x2 + 5x – 2 = 0
Decide whether the following equation is quadratic equation or not.
(m + 2) (m – 5) = 0
Write the following equation in the form ax2 + bx + c = 0, then write the values of a, b, c for the equation.
(x – 1)2 = 2x + 3
Choose the correct answer for the following question.
Out of the following equations which one is not a quadratic equation?
Choose the correct answer for the following question.
Which of the following quadratic equations has roots 3,5?
If P(y) = y² - 2y + 5, find P(2) .
Two water taps together can fill a tank in `1 7/8` hours. The tap with a longer diameter takes 2 hours less than the tap with a smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.
Complete the following activity to form a quadratic equation.
Activity:
If the value of determinants `|(3, 4),(-2, "x")|` is 23, then find the value of` x.
Form a quadratic equation such that one of its roots is 5. Form a quadratic equation for it and write. (For the formation of word problems you can use quantities like age, rupees, or natural numbers.) (Sample solution for the above example is given below students can take another number to form another example)
Solution:
We need one of the solutions of the quadratic equation as 5.
Then we can take another root as any number like a positive or negative number or zero. Here I am taking another root of the quadratic equation as 2.
Then we can form a word problem as below,
Smita is younger than her sister Mita by 3 years (5 – 2 = 3). If the product of their ages is (5 × 2 = 10). Then find their present ages.
Let the age of Mita be x.
Therefore age of Smita = x – 3
By the given condition,
x(x – 3) = 10
x2 – 3x – 10 = 0
If 460 is divided by a natural number, then quotient is 2 more than nine times the divisor and remainder is 5. Find the quotient and divisor.
In the adjoining fig. `square` ABCD is a trapezium AB || CD and its area is 33 cm2. From the information given in the figure find the lengths of all sides of the `square` ABCD. Fill in the empty boxes to get the solution.
Solution: `square` ABCD is a trapezium.
AB || CD
`"A"(square "ABCD") = 1/2 ("AB" + "CD") xx`______
33 = `1/2 ("x" + 2"x" + 1) xx `______
∴ ______ = (3x + 1) × ______
∴ 3x2 +______ − ______ = 0
∴ 3x(______) + 10(______) = 0
∴ (3x + 10) (______) = 0
∴ (3x + 10) = 0 or ______ = 0
∴ x = `-10/3` or x = ______
But length is never negative.
∴ `"x" ≠ -10/3`
∴ x = ______
AB = ______, CD = ______, AD = BC = ______
