### 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