JEE Main Mathematics - Quadratic Equations
Exam Duration: 60 Mins Total Questions : 30
For the equation 3x2 + px + 3 = 0,p > 0; if one of the roots is square of the other,then p is equal to
- (a)
\(\frac{1}{2}\)
- (b)
1
- (c)
3
- (d)
\(\frac{ 2 }{ 3 }\)
Let \(\alpha,\beta\) be the roots of x2 - x + p = 0 and \(\gamma,\delta\) be roots of x2 - 4x + q.If \(\alpha,\beta,\gamma,\delta\) are in G.P., then the integral values of p and q respectively,are
- (a)
-2,-32
- (b)
-2,3
- (c)
-6,3
- (d)
-6,-32
If \(\alpha\) and \(\beta\) (\(\alpha < \beta\)) are the roots of the equation x2 + bx + c = 0,where c < 0
- (a)
0< \(\alpha\)< \(\beta\)
- (b)
\(\alpha\) <0< \(\beta\) <| \(\alpha\) |
- (c)
\(\alpha\) < \(\beta\) <0
- (d)
\(\alpha\)<0<| \(\alpha\)|< \(\beta\)
The value of \(\lambda \) for which the quadratic equation \(3x^{ 2 }+2(\lambda ^{ 2 }+1)x+({ \lambda }^{ 2 }-3\lambda +2)=0\)has opposite signs lies in the interval
- (a)
\((-\infty ,0)\)
- (b)
\((-\infty ,1)\)
- (c)
\((1,2)\)
- (d)
\(\left( \frac { 3 }{ 2 } ,2 \right) \)
If one of roots of the equation px2+qx+r=0 is unity,then
- (a)
p-q+r=0
- (b)
p-q-r=0
- (c)
p+q+r=0
- (d)
None of these
The difference between the corresponding roots of the equations x2+ax+b=0 and x2+bx+a=0 is same;then
- (a)
a+b-4=0
- (b)
a-b+4=0
- (c)
a+b+4=0
- (d)
None of these
If \(\alpha ,\beta \)are the roots of the equation x2+px+1=0 and \(\gamma ,\delta \) the roots of x2+qx+1=0, then the value of \((\alpha -\gamma )(\beta -\gamma )(\alpha -\delta )(\beta -\delta )\)is
- (a)
(p+q)2
- (b)
(p-q)2
- (c)
p2-q2
- (d)
q2-p2
The value of k for which the number 3 lies between the roots of the equation x2+(1-2k)x+(k2-k-2)=0 is given by
- (a)
k<2
- (b)
2<k<5
- (c)
2<k<3
- (d)
k>5
If \(\sin { \alpha } ,\cos { \alpha } \) are the roots of the equation px2+qx+r=0, then
- (a)
p2-q2+2pr=0
- (b)
(p+r)2 =q2-r2
- (c)
p2+q2-2pr=0
- (d)
(p-r)2=q2+r2
The solution of the inequation 2x2+3x-9\(\underline { < } \)0 is given by
- (a)
-3\(\underline { < } \)x\(\underline { < } \)\(\frac { 3 }{ 2 } \)
- (b)
\(\frac { 3 }{ 2 } \)\(\underline { < } \)x\(\underline { < } \)3
- (c)
-3\(\underline { < } \)x\(\underline { < } \)-\(\frac { 3 }{ 2 } \)
- (d)
None of these
If \(\alpha \) and \(\beta \) are the roots of x2+px+q=0 and \({ \alpha }^{ 4 }\) and \({ \beta }^{ 4 }\)are the roots of x2-rx+s=0, then the equation x2-4qx+2q2-r=0 has always
- (a)
two real roots
- (b)
two passitive roots
- (c)
two negative roots
- (d)
one positive and one negative root
The equation \(\frac { a }{ x-a } +\frac { b }{ x-b } =1\)has root equal in magnitude but opposite in sign, then value of a+b is
- (a)
-1
- (b)
0
- (c)
1
- (d)
None of these
The number of real roots of the equation \(\left| x \right| ^{ 2 }-3\left| x \right| +2=0\)is
- (a)
4
- (b)
1
- (c)
3
- (d)
2
The value of \(\theta \) between 0 and \(2\pi \)which satisfies the equation \(\sin { ^{ 4 }\theta - } 2\sin { ^{ 2 }\theta -1=0 } \)is
- (a)
0
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\pi \)
- (d)
None of these
If a,b,c,d are positive real numbers such that a+b+c+d=2 then M=(a+b)(c+d) satisfies the relation
- (a)
\(0\underline { < } M\underline { < } 1\)
- (b)
\(1\underline { < } M\underline { < } 2\)
- (c)
\(2\underline { < } M\underline { < } 3\)
- (d)
\(3\underline { < } M\underline { < } 4\)
The number of values of k for which the equation x2-3x+k=0, has two distinct real roots lying in the interval (0,1) is
- (a)
0
- (b)
2
- (c)
3
- (d)
Infinitely many
If the roots of the equation ax2+bx+c=0 are in the ratio p:q then
- (a)
pqa2=(p+q)c2
- (b)
pqb2=(p+q)ac
- (c)
pqb2=(p+q)2ac
- (d)
None of these
If \(\alpha \) be a root of the equation, 4x2+2x-1=0, then other root is
- (a)
4\(\alpha \)3+3\(\alpha \)
- (b)
3\(\alpha \)-4\(\alpha \)3
- (c)
\(\alpha \)+4\(\alpha \)3
- (d)
4\(\alpha \)3-3\(\alpha \)
If 2a+3b+6c=0 where a,b,c \(\epsilon \)R, then the equation ax2+bx+c=0 has
- (a)
at least one root in (0,1)
- (b)
at least one root in (1,2)
- (c)
both roots in (0,1)
- (d)
None of these
If the roots of the given equation \(\left( 2k+1 \right) { x }^{ 2 }-\left( 7k+3 \right) x+k+2=0\) are reciprocal to each other, then the value of k will be
- (a)
0
- (b)
1
- (c)
2
- (d)
3
For what value of k will the equation \({ x }^{ 2 }-(3k-1)x+{ 2k }^{ 2 }+2k-11\) have equal roots
- (a)
5
- (b)
9
- (c)
Both (a), (b)
- (d)
0
The number of real solutions of the equation \(\left| { x }^{ 2 }+4x+3 \right| +2x+5=0\) are
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If \(\alpha ,\beta \) are the roots of \({ x }^{ 2 }+px+1=0\) and \(\gamma ,\delta \) are the roots of \({ x }^{ 2 }+qx+1=0\) then \({ q }^{ 2 }-{ p }^{ 2 }\) is equal to
- (a)
\(\left( \alpha -\gamma \right) \left( \beta -\gamma \right) \left( \alpha +\delta \right) \left( \beta +\delta \right) \)
- (b)
\(\left( \alpha +\gamma \right) \left( \beta +\gamma \right) \left( \alpha -\delta \right) \left( \beta -\delta \right) \)
- (c)
\(\left( \alpha +\gamma \right) \left( \beta +\gamma \right) \left( \alpha +\delta \right) \left( \beta +\delta \right) \)
- (d)
None of the above
A real root of the equation log4 \(\left\{ { log }_{ 2 }(\sqrt { x+8 } -\sqrt { x } ) \right\} =0\) is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If \(a,b,c\epsilon R\) and \({ ax }^{ 2 }+bx+c=0\) has no real roots, then
- (a)
c (a + b+ c) > 0
- (b)
c - c (a - b - c) > 0
- (c)
c + c (a - b - c) > 0
- (d)
c (a - b - c) > 0
The number of real values of x for which the equality \(\left| { 3x }^{ 2 }+12x+6 \right| =5x+16\) holds good is
- (a)
4
- (b)
3
- (c)
2
- (d)
1
The real number k for which the equation, \({ 2x }^{ 3 }+3x+k=0\) has two distinct real roots in [0 ,1]
- (a)
lies between 1 and 2
- (b)
lies between 2 and 3
- (c)
lies between -1 and 0
- (d)
does not exist
If 1-i, is a root of the equation x2+ax+b=0, where a,b ∈R, then find the values of a and b.
- (a)
2,2
- (b)
-2,2
- (c)
-2,-2
- (d)
1,2
If ∝ and β are the roots of the equation x2+2x=4=0, then \(\frac { 1 }{ { \alpha }^{ 2 } } +\frac { 1 }{ { \beta }^{ 2 } } \) is equal to
- (a)
-1/2
- (b)
1/2
- (c)
32
- (d)
1/4
If x2+ax+10=0 and x2+bx-10=0 have commen roots, then a2-b2 is equal to
- (a)
10
- (b)
20
- (c)
30
- (d)
40