Mathematics - Quadratic Equations
Exam Duration: 45 Mins Total Questions : 30
If 1,a1,a2,a3,....an-1 are the nth roots of unity,then value of (1 - a1) (1 - a2)(1 - a3)...(1 - an-1),is
- (a)
n
- (b)
n2
- (c)
n3
- (d)
0
The equation \(\sqrt { x+1 } \)- \(\sqrt { x-1 } \) = \(\sqrt { 4x-1 } \), has
- (a)
no solution
- (b)
one solution
- (c)
two solutions
- (d)
more than two solutions
If a,b,c are real numbers.such that a2+b2+c2 = 1,then ab +bc + ca lies in the interval
- (a)
\([\frac{1}{2},2]\)
- (b)
[-1,2]
- (c)
\([-\frac{1}{2},1]\)
- (d)
\([-{1},\frac{1}{2}]\)
The roots of the equation x2+2(3a+5)x+2(9a2+25)=0 and real, when a equals
- (a)
\(\frac { 3 }{ 5 } \)
- (b)
\(\frac { 5 }{ 3 } \)
- (c)
\(-\frac { 3 }{ 5 } \)
- (d)
\(-\frac { 5 }{ 3 } \)
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
One of the roots of the equation ax2+bx+c=0 is reciprocal of one of the roots of a' x2+b' x+c'=0, if
- (a)
(aa'-cc')2=(bc'-ab')(b'c-a'b)
- (b)
(ab'-a'b)2=(bc'-b'c)(ca'-c'a)
- (c)
(bc'-b'c)2=(ca'-c'a)(ab'-a'b)
- (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
If one root of the equation x2+lx+12=0 is 4,while the equation x2+lx+m=0 has equal roots,then value of m is
- (a)
\(\frac { 49 }{ 4 } \)
- (b)
\(\frac { 4 }{ 49 } \)
- (c)
4
- (d)
None of these
If one root of the equation 8x2-6x-a-3=0 is the square of the other,then a=
- (a)
4,-24
- (b)
4,24
- (c)
-4,-24
- (d)
-4,24
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 value of 'a' for which one root of the quadratic equation (a2-5a+3)x2+(3a-1)x+2=0 is twice as large as the other, is
- (a)
\(\frac { 2 }{ 3 } \)
- (b)
\(-\frac { 2 }{ 3 } \)
- (c)
\(\frac { 1 }{ 3 } \)
- (d)
\(-\frac { 1 }{ 3 } \)
If a,b,c are the roots of \({ x }^{ 3 }+qx+r=0\)then value of \(\sum { \frac { a }{ b+c } } \)is
- (a)
3
- (b)
q+r
- (c)
q/r
- (d)
-3
Ram and Shyam solve a quadratic equation.In solving the equation Ram commits a mistake in reading the constant term and finds the roots as 8 and 2.Shyam commits a mistake while reading cofficient of x and finds roots as -9 and -1. The correct roots are
- (a)
2,-8
- (b)
9,1
- (c)
9,-1
- (d)
-8,-2
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
The expression \({ x }^{ 2 }+2bx+c\) has the positive value, if
- (a)
\({ b }^{ 2 }-4c>0\)
- (b)
\({ b }^{ 2 }-4c<0\)
- (c)
\({ c }^{ 2 }
- (d)
\({ b }^{ 2 }
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 \(2+i\sqrt { 3 } \) is a root of the equation \({ x }^{ 2 }+px+q=0\), where p and q are real, then (p, q) is equal to
- (a)
(-4, 7)
- (b)
(4, -7)
- (c)
(4, 7)
- (d)
(- 4, - 7)
If \(\alpha \) and \(\beta \) are the roots of the equation ax2 + bx + c = 0, then the quadratic equation whose roots are \(\frac { \alpha }{ 1+\alpha } \) and \(\frac { \beta }{ 1+\beta } \) is
- (a)
\({ ax }^{ 2 }-b(1-x)+c{ (1-x) }^{ 2 }=0\)
- (b)
\({ ax }^{ 2 }-b(x-1)+c{ (x-1) }^{ 2 }=0\)
- (c)
\({ ax }^{ 2 }+b(1-x)+c{ (1-x) }^{ 2 }=0\)
- (d)
\({ ax }^{ 2 }+b(x+1)+c{ (1+x) }^{ 2 }=0\)
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
If P(x) = ax2 + bx + c and Q(x) = - ax2 + dx + c, where \(ac\neq 0\) then P(x).Q(x) = 0 has
- (a)
exactly one real root
- (b)
atleast two real roots
- (c)
exactly three real roots
- (d)
all four real roots
The value of 'x' satisfying the equation \({ x }^{ 4 }-2\left[ xsin\left( \frac { \pi }{ 2 } x \right) \right] ^{ 2 }+1=0\)
- (a)
\(\pm 1\)
- (b)
2
- (c)
0
- (d)
No value of 'x'
Find the value of P such that the difference of the roots of the equation x2-pPx+8=0 is 2
- (a)
土6
- (b)
土3
- (c)
土5
- (d)
土4
solve \(\sqrt{5}\)x2+x+\(\sqrt{5}\)=0
- (a)
\(\pm \frac { \sqrt { 19 } }{ 5 } i\quad \)
- (b)
\(\pm \frac { \sqrt { 19i } }{ 2 } \quad \)
- (c)
\(\frac { -1\pm \sqrt { 19i } }{ 2\sqrt { 5 } } \)
- (d)
\(\frac { -1\pm \sqrt { 19i } }{ \sqrt { 5 } } \quad \)
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
The equations ax2+bx+a=0(a,b∈R) and x3-2x2+2x-1=0 have 2 roots common, Then a+b must be eqaul to
- (a)
1
- (b)
-1
- (c)
0
- (d)
None of these
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
If 2+i is a root of equation x3-5x2+9x-5=0, then the other roots are to
- (a)
1 and 2-i
- (b)
-1 and 3+i
- (c)
0 and 1
- (d)
-1 and i-2
If ∝ and β are the roots of the equation ax2+bx+c=0 then the value of \(\frac { 1 }{ a\alpha +b } +\frac { 1 }{ a\beta +b } \) equals
- (a)
\(\frac{ac}{b}\)
- (b)
1
- (c)
\(\frac{ab}{c}\)
- (d)
\(\frac{b}{ac}\)