JEE Mathematics - Quadratic Equations Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
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 b > a,then the equation (x-a)(x-b)-1 = 0,has
- (a)
both roots in [a,b]
- (b)
both roots in (-\(\alpha\),a)
- (c)
both roots in (b,\(\infty\))
- (d)
one root in (- \(\infty\),a) and other in (b, \(\infty\))
If one root of the quadratic equation ax2+bx+c=0 is equal to the nth power of the other then \(({ ac }^{ n })^{ \frac { 1 }{ n+1 } }+({ a }^{ n }c)^{ \frac { 1 }{ n+1 } }+b\), equals
- (a)
0
- (b)
1
- (c)
2
- (d)
4
If one root of the equation x2+ax+1=0, lies within the unit circle,then the other root
- (a)
lies within the circle
- (b)
lies on the circle
- (c)
lies outside the circle
- (d)
lies within or on the circle
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)
\(\left( \frac { 3 }{ 2 } ,2 \right) \)
Both the roots of the quadratic equation (x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0 are
- (a)
Positive
- (b)
Negative
- (c)
Real
- (d)
None of these
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 quadratic equation whose toots are reciprocal of the roots of the equation x3-3x+2=0 is
- (a)
3x2-2x+1=0
- (b)
2x2-3x+1=0
- (c)
x2-3x+2=0
- (d)
None of these
The equation whose roots are the squares of the roots of the equation x2-x+1=0, is
- (a)
x2-x-1=0
- (b)
x2+x+1=0
- (c)
x2+x-1=0
- (d)
x2-x+1=0
A quadratic equation with rational coefficient can have
- (a)
both roots equal and irrational
- (b)
one root rational and other irrational
- (c)
one root real and other imaginary
- (d)
None of these
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
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
The greatest and the least values of the expression \(\frac { { x }^{ 2 }+2x+1 }{ { x }^{ 2 }+2x+7 } \) for all real values of x,are respectively
- (a)
2,0
- (b)
1,2
- (c)
2,3
- (d)
1,0
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
If the sum of the roots of the equation (a+1)x2+(2a+3)x+(3a+4)=0 is -1 then the product of the roots is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
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
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)
\(-\frac { 1 }{ 3 } \)
The real number x, when added to its inverse gives the minimum value of the sum at x equal to
- (a)
2
- (b)
1
- (c)
-1
- (d)
-2
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 one root of \({ 5x }^{ 2 }+13x+k=0\) is reciprocal of the other, then k is equal to
- (a)
0
- (b)
5
- (c)
\(\frac { 1 }{ 6 } \)
- (d)
6
The equation of the smallest degree with real coefficients having 1+i as one of the root is.
- (a)
\({ x }^{ 2 }+x+1=0\)
- (b)
\({ x }^{ 2 }+2x+2=0\)
- (c)
\({ x }^{ 2 }+2x+2=0\)
- (d)
\({ x }^{ 2 }+2x-2=0\)
If \(\alpha \) and \(\beta \) are the roots of the equation \({ 2x }^{ 2 }-(p+1)x+(p-1)=0\) and \(\alpha -\beta =\alpha \beta \), then what is the value of p
- (a)
1
- (b)
2
- (c)
3
- (d)
- 2
The number of roots of the equation \({ \left| x \right| }^{ 2 }-7\left| x \right| +12=0\) is
- (a)
1
- (b)
2
- (c)
3
- (d)
None of these
If \({ \alpha }\)and \(\beta \) are the roots of the equation \({ x }^{ 2 }-4x+1=0\), then the value of \({ \alpha }^{ 3 }+{ \beta }^{ 3 }\) is
- (a)
76
- (b)
52
- (c)
- 52
- (d)
- 76
The value of p such that the difference of the roots of the equation x2 - px + 8 = 0 is 2, is
- (a)
\(\pm 3\)
- (b)
\(\pm 6\)
- (c)
\(\pm 2\)
- (d)
\(\pm 1\)
If x2 + 5 = 2x - 4 cos (a + b), where \(a,b\epsilon (0,5),\) is satisfied for atleast one real x, then the maximum value of a + b in \([0,2\pi ]\)
- (a)
\(3\pi \)
- (b)
\(2\pi \)
- (c)
\(\pi \)
- (d)
None of the above
Observe the following columns.
Column I |
ColumnII | ||
A. |
If a, b, c, d are four non zero real numbers such that (d + a - b)2 + (d + b - c)2 = 0 and roots of the equation a(b - c)x2 + b (c - a)x + c(a - b) = 0 are real and equal, then |
p. |
\(a+b+c\neq 0\) |
B. |
If a, b, c are three non - zero real numbers such that the roots of the equation (b - c) x2 + (c - a) x + (a - b) = 0 are real and equal, then |
q. |
a, b, c are in AP |
C. |
If the three equations x2 + px + 12 = 0, x2 + qx + 15 = 0 and x2 + (p + q) x + 36 = 0 have a common positive root and a, b, c be their other roots, then |
r. |
a, b, c are in GP |
|
s. |
a, b, c are in HP | |
t. | a = b = c |
- (a)
A B C (p,q,r,s,t) (p,q) (p) - (b)
A B C (p,q,r) (p,q,r,s,t) (p,r) - (c)
A B C (p,q,s) (p,q,r) (q,t) - (d)
None of the above
The product of all the values of x satisfying the equation
\((5+2\sqrt { 6 } )^{ { x }^{ 2 }-3 }+(5-2\sqrt { 6 } )^{ { x }^{ 2 }-3 }=10\) is
- (a)
4
- (b)
6
- (c)
8
- (d)
19
If \(a+b+c>\frac { 9c }{ 4 } \) and equation \({ ax }^{ 2 }+2bx-5c=0\) has non - real complex roots, then
- (a)
a > 0, c > 0
- (b)
a > 0, c < 0
- (c)
a < 0, c < 0
- (d)
a < 0, c > 0
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 roots of the quadratic equation (a + b - 2c) x2 - (2a - b - c)x + (a - 2b + c) = 0 are
- (a)
a + b + c and a - b + c
- (b)
\(\frac { 1 }{ 2 } and\quad a-2b+c\)
- (c)
\(a-2b+c\quad and\quad \frac { 1 }{ (a+b-2c) } \)
- (d)
None of the above
Let \(\alpha \) and \(\beta \) be the roots of equation \({ px }^{ 2 }+qx+r=0,p\neq 0\). If p, q, r are in AP and \(\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } =4\), then the value of \(\left| \alpha -\beta \right| \) is
- (a)
\(\frac { \sqrt { 61 } }{ 9 } \)
- (b)
\(\frac { 2\sqrt { 17 } }{ 9 } \)
- (c)
\(\frac { \sqrt { 34 } }{ 9 } \)
- (d)
\(\frac { 2\sqrt { 13 } }{ 9 } \)
Let \(\alpha ,\beta \) be real and z be a complex number. If \({ z }^{ 2 }+\alpha z+\beta =0\) has two distinct roots on the Re z = 1, then it is necessary that
- (a)
\(\beta \epsilon (-1,0)\)
- (b)
\(\left| \beta \right| =1\)
- (c)
\(\beta \epsilon (1,\infty )\)
- (d)
\(\beta \epsilon (0,1)\)
If the difference between the roots of the equation x2 + ax + 1 = 0 is less than \(\sqrt { 5 } \), then the set of possible values of a is
- (a)
\((-3,3)\)
- (b)
\((-3,\infty )\)
- (c)
\((3,\infty )\)
- (d)
\((-\infty ,-3)\)
If the roots of the quadratic equation x2 + px + q = 0 are tan 30o and tan 15o respectively, then the value of 2 + q - p is
- (a)
3
- (b)
0
- (c)
1
- (d)
2
If the roots of the equation x2 - bx + c = 0 are two consecutive integers, then b2 - 4c equals
- (a)
1
- (b)
2
- (c)
3
- (d)
-2
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 ∝ is one root of the equation 4x2+2x-1=0, then the other root is
- (a)
4∝3+3∝
- (b)
4∝3-3∝
- (c)
∝3+3∝
- (d)
∝3-3∝
The number of real roots of \({ \left( x+\frac { 1 }{ x } \right) }^{ 3 }+{ \left( x+\frac { 1 }{ x } \right) }\)=0 is
- (a)
0
- (b)
2
- (c)
4
- (d)
6
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 the roots of the equation \(\frac{a}{x-a}+\frac{b}{x-b}=1\) are equal in magnitude and opposite in sign, then
- (a)
a=b
- (b)
a+b=1
- (c)
a-b=1
- (d)
a+b=0
The equation of smallest degree with real coefficients having 2+3i as one of the roots is
- (a)
x2+4x+13
- (b)
x2+4x-13=0
- (c)
x2-4x+13=0
- (d)
x2-4x-13=0
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 ∝ 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)
\(\frac{b}{ac}\)
If 3P2=5P+2 and 3q2=5q+2 where P≠q, then the equation whose roots are 3p-2q and 3q-2p is
- (a)
3x2-5x-100=0
- (b)
5x2+3x+100=0
- (c)
3x2-5x+100=0
- (d)
3x2+5x=100=0
The value of x in the given equation \({ 4 }^{ x }-3^{ x-\frac { 1 }{ 2 } }={ 3 }^{ x+\frac { 1 }{ 2 } }-2^{ 2x-1 }\) is a/b where a and b don’t have any common factor. Find ab2.
- (a)
15
- (b)
12
- (c)
13
- (d)
5
If P(x)=ax2+bx+c and Q(x)=-ax2+dx+c where, \(\\ ac\neq 0\) ,then P(x).Q(x)=0 has at least m real roots find m.
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If the roots of the equation x2+x+1=0 are α,β and the roots of the equation x2+px+q=0 are \(\cfrac { \alpha }{ \beta } ,\cfrac { \beta }{ \alpha } \) then p is equal to
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If the roots of ax2+bx+c=0 are α,β and the roots of Ax2+Bx+C=0 are \(\alpha -k,\beta -k\) ,then \( \cfrac { { B }^{ 2 }-4AC }{ { b }^{ 2 }-4ac } \) is equal to Aman.Find value of m – n.
- (a)
5
- (b)
4
- (c)
3
- (d)
2