Question
The value of p for which both the roots of the quadratic equation, are less than 2 lies in :
Hint:
Here we have given a quadratic equation, we have to find the value of p, where it lies. Find first D and relate inequality with p. Here roots are less than 2 and also find here inequality of p. Here f(2) > 0 so find p here and compare with those and look where it lies
The correct answer is: ![left parenthesis negative straight infinity comma negative 1 right parenthesis](data:image/png;base64,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)
Here, we have to find the value of p and where it lies.
Firstly, we have quadratic equation,
.
Also, the value of both roots is less than 2.
=0
Now for discriminant,
![D greater or equal than 0 left square bracket D equals b squared – 4 a c right square bracket](data:image/png;base64,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)
![rightwards double arrow left parenthesis negative 20 p right parenthesis squared minus 4 cross times 4 left parenthesis 25 p squared plus 15 p minus 66 right parenthesis greater or equal than 0](data:image/png;base64,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)
Solve this,
![rightwards double arrow 16 left square bracket 25 p squared minus 25 p squared minus 15 p plus 66 right square bracket greater or equal than 0
rightwards double arrow negative 15 p plus 66 greater or equal than 0
p less or equal than 66 over 15
p less or equal than 22 over 5 space space space................ open parentheses 1 close parentheses](data:image/png;base64,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)
Now, we know that the root of the value is always less than 2 so we can write,
![rightwards double arrow fraction numerator negative b over denominator 2 a end fraction space less than space 2 space
fraction numerator 20 p over denominator 8 end fraction space less than space 2
rightwards double arrow p space less than fraction numerator space 16 over denominator 20 end fraction space
rightwards double arrow space p space less than 4 over 5 space minus left parenthesis 2 right parenthesis](data:image/png;base64,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)
So, at x = 2 the quadratic equation is positive, we can write,
f(2)>0
![rightwards double arrow 16 minus 40 p plus 25 p squared plus 15 p minus 66 greater than 0
rightwards double arrow 25 p squared minus 25 p minus 50 greater than 0
p squared minus p minus 2 greater than 0](data:image/png;base64,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)
solving this by factorization, we have
(p−2) (p+1) >0
p > 2 and p < - 1
Hence,
p∈ (−∞, −1) ∪ (2, ∞) ---(3)
From (1), (2) and (3), we get
p belongs to (- −∞, −1)
Therefore, the correct answer is (- −∞, −1)
In this question, we have to find where the p lies. Here, we use discriminant, which is . if D > 0 and D = 0 then real solution but if D < 0 then imaginary solution. Here we also us Factorization of quadratic equations.
Related Questions to study
If
where
determine b –
Just like we can change the base b for the exponential function, we can also change the base b
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is defined so that
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Statement 1:The two Fe atoms in
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Statement 2:
ions decolourise
solution
Statement 1:The two Fe atoms in
have different oxidation numbers
Statement 2:
ions decolourise
solution
, where
and
is
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and
is
In this question, we have to find the . Here Solve first B. In B, rationalize it means multiply it denominator by sign change in both numerator and denominator
Statement 1:The equivalence point refers the condition where equivalents of one species react with same number of equivalent of other species.
Statement 2:The end point of titration is exactly equal to equivalence point
Statement 1:The equivalence point refers the condition where equivalents of one species react with same number of equivalent of other species.
Statement 2:The end point of titration is exactly equal to equivalence point
If and
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Statement 1:Iodimetric titration are redox titrations.
Statement 2:The iodine solution acts as an oxidant to reduce the reductant
Statement 1:Iodimetric titration are redox titrations.
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Statement 1:Diisopropyl ketone on reaction with isopropyl magnesium bromide followed by hydrolysis gives alcohol
Statement 2:Grignard reagent acts as a reducing agent
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Statement 2:Grignard reagent acts as a reducing agent
The first noble gas compound obtained was:
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Given that and
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These four basic properties all follow directly from the fact that logs are exponents.
logb(xy) = logbx + logby.
logb(x/y) = logbx - logby.
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Given that and
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These four basic properties all follow directly from the fact that logs are exponents.
logb(xy) = logbx + logby.
logb(x/y) = logbx - logby.
logb(xn) = n logbx.
logbx = logax / logab.
Statement 1:Change in colour of acidic solution of potassium dichromate by breath is used to test drunk drivers.
Statement 2:Change in colour is due to the complexation of alcohol with potassium dichromate.
Statement 1:Change in colour of acidic solution of potassium dichromate by breath is used to test drunk drivers.
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Total number of solutions of sin{x} = cos{x}, where {.} denotes the fractional part, in [0, 2] is equal to
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Statement 2:The colour change is due to the oxidation of potassium chromate.
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Statement 2:The colour change is due to the oxidation of potassium chromate.
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Statement 2:Nickel is bonded to neutral ligand carbonyl.
Statement 1:Oxidation number of Ni in is zero.
Statement 2:Nickel is bonded to neutral ligand carbonyl.