Relation between Roots and Coefficient - Advance

In algebra, a quadratic equation is any equation that can be rearranged in standard form as where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no term.

Q1.  If roots of an equation xn-1 = 0 are 1,a1,a2,a3,a3,a4,.........,an-1 , then the value of (1-a1)(1-a2)(1-a3)........(1-an-1)will be

•   n
•  n2
•  n3
•  0

n

Q2.  If the sum of the roots of the equation qx2+2x+3q=0 is equal to their product, then the value of q is equal to

•  t2-2/3
•  t23/2
•  t2-3/2
•  t2None of these

-2/3

Q3.  If A, G, H be respectively, the A.M., G.M. and H.M. of three positive number a, b, c then the equation whose roots are these number is given by

•   x3+3Ax2-3(G3/H)x-G3
•  x3-3Ax2+3(G3/H)x-G3
•  -x3+3Ax2+3(G3/H)x+G3
•   None of these

x3-3Ax2+3(G3/H)x-G3

Q4.  Let A, G and H are the A.M., G.M. and H.M. respectively of two unequal positive integers. Then the equation Ax2-|G|x-H=0 has

•   Both roots as fractions
•   At least one root which is a negative fraction
•  At least one root which is an integer
•  None of these

At least one root which is a negative fraction

Q5.  P, q, r and s are integers. If the A.M. of the roots of x2-px+q2=0 and G.M. of the roots of x2-rx+s2=0 are equal then

•  q is an odd integer
•   r is an even integer
•  p is an even integer
•  s is an odd integer

p is an even integer

Q6.  If the roots of 4x2+5k=(5k+1)x differ by unity then the negative value of k is

•   -3
•   -1/5
•   -3/5
•  None of these

-1/5

Q7.  If the roots of ax2-bx-c=0 change by the same quantity then the expression in a, b, c that does not change is

•   -b2-4ac/a2
•  b2+2ac/a2
•  b2+4ac/a2
•  None of these

b2+4ac/a2

Q8.  The quadratic equation whose roots are the A.M. and H.M. of the roots of the equation x2+7x-1=0 is

•  -14x2-45x+14=0
•   7x2+45x-14=0
•   14x2+45x-14=0
•   None of these

14x2+45x-14=0

Q9.  If x2-bx+c=0 has equal integral roots then

•  b and c are integers
•  b and c are even integers
•   b is an even integer and c is a perfect square of a positive integer
•  None of these

b is an even integer and c is a perfect square of a positive integer

Q10.  If the roots of the equation ax2-4x+a2=0 are imaginary and the sum of the roots is equal to their product then a is

•   -2
•   0
•   2
•  6, –1

2

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