Mathematics - Theory of Equations

Question - 1

If ∝,β are the roots of equation ax2+bx+c=0 such that β<∝<0 ,then the quadractic equation whose roots are |∝|,|β| is given by

  • A \(|\alpha |{ x }^{ 2 }+|b|x+|c|=0\)
  • B \(a{ x }^{ 2 }-|b|x+|c|=0\)
  • C \(|a|{ x }^{ 2 }-|b|x+|c|=0\)
  • D \(a|{ x| }^{ 2 }-|b|x+|c|=0\)

Question - 2

In a triangle PQR, \(\angle R=\frac { \pi }{ 2 } \).If \(tan\left( \frac { P }{ 2 } \right) \) and \(tan\left( \frac { Q }{ 2 } \right) \)are the roots of the equation \({ ax }^{ 2 }+bx+c=0(a\neq 0)\) then

  • A a+b=c
  • B b+c=a
  • C a+c=b
  • D b=c

Question - 3

Roots of the equations \({ x }^{ 2 }-\sqrt { 5 } x-19=0\) are

  • A real, equal and rational
  • B real, unequal and rational
  • C real, unequal and irrational
  • D complex number

Question - 4

If one root of the equation \({ x }^{ 2 }-\lambda x+12=0\) is even prime while \({ x }^{ 2 }+\lambda x+\mu =0\) has equal roots, then μ is

  • A 8
  • B 16
  • C 24
  • D 32

Question - 5

A car travels 25 km an hour faster than a bus for a journey of 500 km.The bus takes 10h more than the car.If speed of car is p and speed of bus is q,then

  • A p=q2
  • B p=2q
  • C p=3q
  • D p2

Question - 6

Number of real roots of the equation \(\sqrt { x } +\sqrt { x-\sqrt { (1-x) } } =1\) is

  • A 0
  • B 1
  • C 2
  • D 3

Question - 7

If tdenotes nth term of an AP and \({ t }_{ p }=\frac { 1 }{ q } \) and  \({ t }_{ q }=\frac { 1 }{ q } \) then which of the following is necessarily a root of the equation \((p+2q-3r){ x }^{ 2 }+(q+2r-3p)x+(r+2p-3q)=0\) is

  • A tp
  • B tq
  • C tpq
  • D tp+q

Question - 8

The value of a for which the equation x3+ax+1=0 and x4+ax2+1=0 have a common root is

  • A -2
  • B -1
  • C 1
  • D 2

Question - 9

The curve y=(λ+1)x2+2 intersects the curve y=λx+3 in exact one point if λ equals

  • A {-2}
  • B {2}
  • C {-2,2}
  • D {1}

Question - 10

If x2-x-2 is a factor x4-λx2-μ ,then \(\sqrt { (\lambda ^{ 2 }-\mu ^{ 2 }) } \) equals

  • A 1
  • B 3
  • C 5
  • D 7
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