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प्रश्न
Choose the correct answer for the following question.
The roots of x2 + kx + k = 0 are real and equal, find k.
विकल्प
0
4
0 or 4
2
उत्तर
Given quadratic equation is x2 + kx + k = 0
For real and equal roots, D = 0
\[b^2 - 4ac = 0\]
\[ \Rightarrow k^2 - 4 \times 1 \times k = 0\]
\[ \Rightarrow k^2 - 4k = 0\]
\[ \Rightarrow k\left( k - 4 \right) = 0\]
\[ \Rightarrow k = 0, k = 4\]
Hence, the correct answer is 0 or 4.
संबंधित प्रश्न
If one root of the equation `2x^2+ax+6=0` 2 then a=?
(a)7 (b)-7 (c) 7/2 (d)-7/2
The roots of a quadratic equation are 5 and -2. Then, the equation is
(a)`x^2-3x+10=0` (b)`x^2-3x-10=0` (c)x^2+3x-10=0 (d)`x^2+3x+10=0`
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
For what value of k, the equation `kx^2-6x2=0` has real roots?
(a) `k≤-9/2` (b)`k≥-9/2`
(c)` k≤-2` (d) None of these
If one zero of the polynomial `x^2-4x+1 is (2+sqrt3)` write the other zero.
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3m2 = 2m2 – 9
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Vivek is older than Kishor by 5 years. The sum of the reciprocals of their ages is \[\frac{1}{6}\] Find their present ages.
If 460 is divided by a natural number, quotient is 6 more than five times the divisor and remainder is 1. Find quotient and divisor.
Which is the following equation quadratic?
x2 + 2x + 11 = 0
Solve the following quadratic equation.
m2 + 5m + 5 = 0
Which of the following is a quadratic equation ?
If in an A. P., d = 10, find t6 - t2.
Convert 15 : 20 into percentage.
Find c if the system of equations cx+3y+(3 - c ) = 0, 12x + cy - c = 0 has infinitely many solutions?
Find c if the system of equations cx + 3y + (3 – c) = 0; 12x + cy – c = 0 has infinitely many solutions?
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
The ratio of fruit trees and vegetable trees in an orchard is 3:4. If 6 more trees of each type are planted, the ratio of trees would be 6:7. Find the number of fruit trees and vegetable trees in the orchard.
The ratio of fruit trees and vegetable trees = 3:4
So, let the number of fruit trees= 3x and the number of vegetable trees = `square`
From the given condition,
`(3x + square)/(square + square) = square/square`
`square (3x + square) = square (square + square)`
`square + square = square + square`
`square - square = square - square`
`- square = - square`
`square = square`
x = `square`
∴ Number of fruit trees in the orchard = 3x = 3 × `square` = `square` and number of vegetable trees in the orchard = 4x = 4 × `square` = `square`
Hence, the number of fruit trees and vegetable trees in the orchard are `square` and `square` respectively.
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 = ______
