Find whether the following equations have real roots. If real roots exist, find them.
(i) We Know that b^{2}-4ac > 0 in the case of real roots and b^{2} - 4ac < 0 in the case of imaginary roots.
For the equation
Compare with where
a=8,b=2,c=-3
Hence the equation has real roots
Roots of the equation are
(ii) We Know that b^{2}-4ac > 0 in the case of real roots and b^{2} - 4ac < 0 in the case of imaginary roots.
For the equation
Compare with where
a=-2,b=3,c=2
Hence the equation has real roots
Roots of the equation are
(iii) We Know that b^{2}-4ac > 0 in the case of real roots and b^{2} - 4ac < 0 in the case of imaginary roots.
For the equation
Compare with where
a=5,b=-2,c=-10
Hence the equation has real roots
Roots of the equation are
(iv) We Know that b^{2}-4ac > 0 in the case of real roots and b^{2} - 4ac < 0 in the case of imaginary roots.
For the equation
Compare with where
a=2,b=-16,c=23
Hence the equation has real roots
Roots of the equation are
(v) We Know that b^{2}-4ac > 0 in the case of real roots and b^{2} - 4ac < 0 in the case of imaginary roots.
For the equation
Compare with where
Hence the equation has real roots
Roots of the equation are
View Full Answer(1)Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number
12
Solution
Let the natural number = x
It is given that when the square of x is diminished by 84 then it is equal to thrice of 8 more than x
x = 12, – 9
We can’t take – 9 because it is given that x is a natural number.
Hence x = 12 is the required natural number.
View Full Answer(1)A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.
8
Solution
Let the natural number = x
It is given that x is increased by 12, equal to 160 times its reciprocal
We can’t take x = – 20 because it is given that x is a natural number.
Hence x = 8 is the required natural number.
View Full Answer(1)A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.
45 km per hour.
Solution
Let the original speed of the train = x km per hour.
Total distance = 360 km
If train’s speed increased by 5 then speed = (x + 5) km per hour. Time taken by train to cover 360 km at a speed of (x + 5) km per hour=
Now it is given that the difference between the time taken at a speed of x and at a speed of x + 5 is 48 min.
Or
(divide by 4)
We can’t take x = – 50 because speed is always positive.
Hence original speed of train is 45 km per hour.
View Full Answer(1)If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?
14
Solution
Let zeba’s current age = x
According to question
x = 14
we can’t take 1 because if we take 5 years younger of 1 then it is equal to – 4 and age cannot be negative.
Hence x = 14 i.e., the age of zeba is 14 years
View Full Answer(1)At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.
5, 27
Solution
Let Nisha’s present age = x
At present Asha is 2 more than the square of Nisha’s age.
Hence, Asha present age = x^{2} + 2
According to question
x^{2} + 2 - x = 10x – 1 - ( x^{2} + 2 )
x^{2} + 2 – x = 10x – 1 – x^{2} - 2
x^{2} + 2 – x - 10x + x^{2} + 3=0
2x^{2}-11x+5=0
2x(x-5)-1(x-5)=0
(x-5)(2x-1)=0
?
If Nisha’s age is 5
Asha age = (5)^{2}+2=27
If Nisha’s age is 1/2 then Asha’s age is 2.25 which is not possible.
View Full Answer(1)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 Fig. 4.1]. Find the length and breadth of the pond.
length = 34m, breadth = 24m
Solution
Let the distance between the pond and loan boundary is x.
Hence the length of pond =50-x-x
Then breadth of pond =40-x-x
From the figure
Area of lawn = Area of pond + Area of grass
50 × 40=(50-2x)(40-2x)+1184 (using Area of rectangle = l × B)
2000-1184=2000-100x-80x+4x^{2}
4x^{2}-180x+2000-2000+1184=0
4x^{2}-180x+1184=0
Divide by 4
x^{2}-45x+296=0
x^{2} - 37x - 8x + 296 = 0
x(x-37)-8(x-37)=0
(x-37)(x-8)=0
x = 37, 8
If x = 37
Length of pond =50-2(37)
=50-74
=-24
Which is not possible because length can’t be negative.
If x = 8
Length of pond =50-2(8)
=50-16=34m
Breadth of pond =40-2(8)
=40-16=24m
View Full Answer(1)At t minutes past 2 pm, the time needed by the minutes hand of a clock to show 3 pm was found to be 3 minutes less than minutes. Find t.
14
Solution
At t minutes past 2 pm, the time need by minute hand to show 3 pm = 60 – t
According to question, it is given that (60 – t) is equal to 3 minutes less than
Time can’t equal to negative term.
Hence t = 14 minutes
View Full Answer(1)The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation of two variables and draw the graph of the same by taking the constant mass equal to 6 kg. Read from the graph, the force required when the acceleration produced is 5 m/sec^{2}.
The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation of two variables and draw the graph of the same by taking the constant mass equal to 6 kg. Read from the graph, the force required when the acceleration produced is 6 m/sec^{2}.
It is given that force is directly proportional to acceleration
i.e.,
F = ma {Here m is proportionality constant and also mass of body}
It is also given that m = 6
F | 6 | 12 |
a | 1 | 2 |
The given equation is F = 6a
When acceleration is 5 m/sec^{2}
Put a = 5
kgm/sec^{2}
Hence the required answer is 30 kgm/sec^{2}
It is given that force is directly proportional to acceleration
i.e.,
F = ma {Here m is proportionality constant and also mass of body}
It is also given that m = 6
F | 6 | 12 |
a | 1 | 2 |
The given equation is
F = 6a
When acceleration is 6 m/sec^{2}
Put a = 6
kgm/sec^{2}
Hence the required answer is 36 kgm/sec^{2}
View Full Answer(1)If the temperature of a liquid can be measured in Kelvin units as or in Fahrenheit units as , the relation between the two systems of measurement of temperature is given by the linear equation .Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is .
If the temperature of a liquid can be measured in Kelvin units as or in Fahrenheit units as , the relation between the two systems of measurement of temperature is given by the linear equation .If the temperature is , then find the temperature in Kelvin.
Answer:
Solution:
Given that,
Put the value of x = in equation (1), we get
y = 72 + 32
Answer:
343K
Solution:
Given that,
Put in equation (1) we get
790 – 160 = 9x – 2457
630 + 2457 = 9x
3087 = 9x
343 = x
Hence answer is 343K
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