EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol. The transformed equation is ( x k) 6( x k) + 6( x k) 5 = Coefficient of x is k 6 = k =. The roots of the equation x 4x + 56x 64 = are in... progression.. Arithmetico-geometric. Harmonic. Arithmetic 4. Geometric Ans : 4 Sol. By verification x = is a factor of given equation. 4 56 64 4 64 (x-) (x -x+) = x -x+= x = 4,8 Roots are,4,8 These are in G.P. 4. If there is a multiple root of order for the equation x x + x =, then the other root is. -.. 4. Ans: Let f(x) = x 4 -x +x- f () = f x = 4x 6x + f = ( ) ( ) f (x) = x -x f () = Roots of given equation are,, Let the other root be α S = + + + α α = - Other root is - 4. The equation whose roots are the negatives of the roots of the equation 7 5 x + x + x x + 7x+ = 7 5 7 5. x + x + x + x 7x+ =. x + x + x + x + 7x =
7 5 7 5. x + x + x x 7x = 4. x + x + x x + 7x = Ans: Sol. f(-x) = (-x) 7 + (-x) 5 +(-x) -(-x) + 7(-x) + = -x 7 -x 5 -x -x -7x+= x 7 +x 5 +x +x +7x-= 5. The biquadratic equation, two of whose roots are + i, is 4 4. x 4x + 5x x =. x 4x 5x + x+ = 4 4. x + 4x 5x + x = 4. x + 4x + 5x x+ = Ans: Sol. The roots of required equation are +i, -i,, + Here S = +i+-i+ + + =4 (sum of the roots) S 4 = (+i) (-i) ( )( + ) (product of the roots) = (-i ) (- ) = - Now verify options. 6. 4 To remove the nd term of the equation x 8x + x x+ = diminished the root of the equation by [ EAMCET- ]... 4. 4 Ans: a ( 8) Sol. h = = = na 4() 5 7. The maximum possible number of real roots of the equation x 6x 4x+ 5= is.. 4. 5 4. Ans: Sol. Let f(x) = x 5 6 x 4x + 5 =, f( x) = x 5 6x + 4x + 5 = Number of positive real roots = Number of changes of signs in f(x) = No. of negative roots = No. of changes of signs in f(-x) = No. of real roots = No. of positive roots + No. of negative roots = = + = 8. If α, β, γ are the roots of the equation x + ax + bx+ c= then α +β +γ = a b c b. c. c. a 4. a Ans: Sol. α + β + γ = + +
= βγ + αγ + αβ = αβγ S b = S c + i 4 9. If is a root of the equation x x + x = then its real roots are.,. -, -., 4., - + i i Sol. If is a roots of the given equation then the other root be roots are Let the remaining roots be α, β Now sum of the roots of given equation = S = + i i + + α + β = + α + β = α + β = By verification roots are,-. If α, β, γ are the roots of x x = then. -.. 4. Ans: Sol. ( Σαβ ) = ( S ) = =. If αβγ,, are the roots of the equation x 4x + + = then ( ) ( ) ( ) α +β + β+γ + γ+α = ) ) ) 4 4) 5 Ans: Sol. α + β + γ = = γ α β = + + α β γ ( α + β ) + ( β + γ ) + ( γ + α ) = ( γ ) + ( α ) + ( β ) EAMCET - αβ + βγ + γα 4 = = = 4 αβγ. Let α and P(x) be a polynomial of degree greater than. If P(x) leaves remainders αand α when divided respectively by x + α and x α then the remainder when P(x) is divided by x α is
) x ) -x ) x 4) x Sol. Let the remainder be R(x), then R(x) = p(x)+q and R(a) = -a Given R(-a) = a pa+q = -a------() - pa + q = a -------() Solving () & (), we get p = -, q = R( x) = x. If the sum of two of the roots of x + px + qx+ r = is zero then pq = ) -r ) r ) r 4) -r Ans: Sol. Let the roots be α, β, γ Given α + β = α + β + γ = p γ = p γ = p is a root of x + px + qx + r = ( p ) + p( p) + q( p) + r = pq = r 4 4. If the roots of the equation 4x x + x + k = are in A.P. Then K = [EAMCET-4] ) - ) ) 4) Ans: Sol. Let the roots be a-d, a, a+d (a-d) + a + (a+d) = 4 a = a = a = is a root of 4x -x +x+k = 4() -() +()+k= +k = k = - 5. αβγ,, are the roots of the equation x x + 7x+ 8= Match the following ) α+β+γ a) ) ) α +β +γ b) 4 4 7 8 + + c) 86 + + 4) βγ γα αβ d) e) ) e, c, a, b ) d, c, a, b ) e, c, b, a 4) e, b, c, a
Sol. Ans: x x + 7x+ 8= Now α + β + γ = ( ) ( ) α + β + γ = α + β + γ αβ + βγ + γα = () (7) = 86 βγ + γα + αβ 7 + + = = αβγ 8 α β α α +β +γ 86 4 + + = = = βγ γα αβ αβγ 8 4 5 6. If f(x) is a polynomial of degree n with rational coefficients and + i, and 5 are three roots of f(x)=, then the least value of n is ) 5 ) 4 ) 4) 6 Ans: Sol. Since +i, and 5 are the some roots of polynomial f(x) of degree n. As we know this conjugate are also the roots of the polynomial is -i, + The least value of n is 5. 7. The roots of the equation x x = are [EAMCET-5] ) -, -, ) -,, - ) -,, - 4) -, -, - Ans Sol. Verify S Here S = By verification the roots are -,-, 8. If αβγ,, are the roots of x + x x = then α +β +γ = ) ) ) 4 4) 5 Ans: α +β +γ = + + α β γ Sol. αβγ = αβ + βγ + γα = αβγ = α β + β γ + γ α = ( αβ + βγ + γα ) = α β + β γ + γ α + αβγ ( α + β + γ ) 9 = α β + β γ + γ α + ()( ) α β + β γ + γ α =
6 α + β + γ = = 9 The difference between two roots of the equation x -x +5x+89= is. Then the roots of the equation are [EAMCET : 6] ) -,5,9 ) -,-7,-9 ),-5,7 4) -,7, 9 Sol. Verify S. If,, x 6x + x + 6 then Σ α β + Σαβ is equal to ) 8 ) 84 ) 9 4) -84 Ans: Sol. Σα β + Σαβ = SS S = (6) () (-6) = 84 4. If,, and 4 are the roots of the equation x + ax + bx + cx+ d =, then a+ b+ c= (E-7) ) -5 ) ) 4) 4 Ans: Sol. (x-)(x-)(x-)(x-4) = x 4 +ax +bx +cx+d x x + 4 x 7x + = x + ax + bx + cx + ( )( ) d 4 4 x x + 5x 5x + 4 = x + ax + bx + cx + d Now a = -, b = 5, c = -5, d = 4 a +b+c=-+(5)-5 =. If α, β, γ are the roots of x x + x 4= then the value of α β + β γ + γ α is ) -7 ) -5 ) - 4) Ans: Sol. α + β + γ =, αβ + βγ + γα =, αβγ = 4 = ( αβ + βγ + γα ) αβγ ( α + β γ ) = ( ) ( 4)( ) = -7 α β + β γ + γ α + EAMCET 8. The cubic equation whose roots are thrice to each of the roots of x -x -4x+= is ) x -6x +6x+7= ) x +6x +6x+7= ) x -6x -6x+7= 4) x +6x -6x+7= x x Sol. x = α f = x x + x 4 + = x + 6x 6x + 7 = 4. The sum of fourth powers of the roots of the equation x + x + = is
) - ) - ) 4) 4 4 4 Sol. Let roots be α, β, γ we have to find α + β + γ Let f(x) = x +x+ f (x) = x + f ( x) ( x + ) Now = f ( x) x + x + 4 4 4 α + β + γ = 5. If α, β, γ are the roots of x +4x+= then the equation whose roots are,, is β + γ γ + α α + β ) x -4x-= ) x -4x+= ) x +4x-= 4) x +4x+= [EAMCET 9] Ans: α α Sol. Let y = = = α = x β + γ α [ α + β + γ = ] Required equation is (-x) +4(-x)+= x + 4x = 6. If f(x)=x 4 -x +ax+b is divisible by x -x+, then (a,b) = ) (-a,-) ) (6,4) ) (9,) 4) (,9) Ans: Sol. x -x+ = (x-)(x-) f()=, f()= -+a+b= -5+a+b= a+b= a+b= Solving () & () we get (a,b) = (9,) 7