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Phrases Quadratic Equations PYQ



If the equation |x26x+8|=a has four real solution then find the value of a?





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Between any two real roots of the equation exsinx=1, the equation excosx=1 has





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Number of Roots

Given:

exsinx=1 has two real roots → say x1 and x2

Apply Rolle’s Theorem:

Since f(x)=exsinx is continuous and differentiable, and f(x1)=f(x2), ⇒ There exists c(x1,x2) such that f(c)=0

Compute:

f(x)=ex(sinx+cosx)=0tanx=1

At this point, excosx=1

At least one root



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Let C denote the set of all tuples (x,y) which satisfy x22y=0 where x and y are natural numbers. What is the cardinality of C?





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If α,β are the roots of x2x1=0 and An=αn+βn, the Arithmetic mean of An1 and An is





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If the roots of the quadratic equation x2+px+q=0 are tan 30° and tan 15° respectively, then the value of 2 + p - q is





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The quadratic equation whose roots are  is





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For what value of p, the polynomial  x43x3+2px26 is exactly divisible by (x1)





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α, β are the roots of the an equation x22xcosθ+1=0, then the equation having roots αn and βn is





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The equation (x-a)3+(x-b)3+(x-c)3 = 0 has





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Let f(x) = (x – a)3 + (x – b)3 + (x – c)3.
Then f'(x) = 3{(x – a)2 + (x – b)2 + (x –c)2}
clearly , f'(x) > 0 for all x.
so, f'(x) = 0 has no real roots.
Hence, f(x) = 0 has two imaginary and one real root


Let  and  be the roots f the equation  and  are the roots of the equation , then the value of r,





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If α and β are the two roots of the quadratic equation x2+ax+b=0,(ab0) then the quadratic roots whose roots 1α3+α and 1β3+β is





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If α≠β and α2=5α3,β2=5β3, then the equation whose roots are αβ and βα is 





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Given the equation x+y=1, x2+y2=2, x5+y5=A. Let N be the number of solution pairs (x,y) to this system of equations. Then AN is equal to





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If α and β are the roots of the equation 2x2+2px+p2=0, where p is a non-zero real number, and α4 and β4 are the roots of x2rx+s=0, then the roots of 2x24p2x+4p42r=0 are:





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If x and y are positive real numbers satisfying the system of equations x2+yxy=336 and y2+xxy=112, then x + y is:





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The value of k for which the equation (k2)x2+8x+k+4=0 has both real, distinct and negative roots is





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Roots of equation are ax22bx+c=0 are n and m , then the value of ban2+c+bam2+c is





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If a + b + c = 0, then the value of 





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a,b,c are positive integers such that a2+2b22bc=100 and 2abc2=100. Then the value of a+bc is





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If x2+2ax+103a>0 for all x ∈ R, then





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