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Exam Countdown: JEE Main 2026
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Q.) If ( b1.b2 = 2(c1 + c2) ), then at least one of the equations ( x^2 + b1x + c1 = 0 ) and ( x^2 + b2x + c2 = 0 ) has:
[A]
imaginary roots
[B]
real roots
[C]
purely imaginary roots
[D]
none of these
[A] imaginary roots
[B] real roots
[C] purely imaginary roots
[D] none of these
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Number System Inequalities and Theory of Equations 📚
1
If x = 2 + 2 ^ (1/3) + 2 ^ (2/3), then the value of (x^3 - 6x^2 + 6x) is ...
2
The least value of the expression (x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14) is: ...
3
Number of positive integers (n) for which (n^2 + 96) is a perfect square is: ...
4
If x, y ∈ R satisfy the equation x² + y²-4x-2y+5=0, then the value of the expression [(√x− y^.5)²+4√ ...
5
The number of real roots of the equation (x^2 - 3|x| + 2 = 0) is: ...
6
If (x = 1 + i) is a root of the equation (x^3 -ix+1-1= 0), then the other real root is: ...
7
The number of roots of the equation ( x - 2)^.5(x^2 - 4x + 3) = 0 ) is: ...
8
If the expression (x^2 + 2(a + b + c)x + 3(bc + ca + ab)) is a perfect square, then: ...
9
Let ( p ) and ( q ) be roots of the equation ( x^2 - 2x + A = 0 ) and let ( r ) and ( s ) be the roo ...
10
The number of irrational roots of the equation {4x}/{x^2+x+3} + {5x}/{x^2-5x+3} = -{3}/{2} ) is: ...
11
Let a, b, and c be real numbers such that 4a + 2b + c = 0 and ab > 0. Then the equation ax² + bx + c ...
12
If ( a, b ) are the roots of the equation ( x^2 - 2x + 3 = 0 ), then the equation whose roots are ( ...
13
If ( alpha, beta ) are the roots of the equation ( 2x^2 - 35x + 2 = 0 ), then the value of ( (2alpha ...
14
If a, b, c are three distinct positive real numbers, then the number of real roots of ( ax^2 + 2b|x| ...
15
If ( p(q-r)x^2 + q(r-p)x + r(p-q) = 0 ) has equal roots, then ( {2}/{q} ) equals: ...
16
Let ( a notequal 0 ) and ( p(x) ) be a polynomial of degree greater than 2. If ( p(x) ) leaves remai ...
17
The quadratic ( x^2 + ax + b + 1 = 0 ) has roots which are positive integers. Then ( a^2 + b^2 ) can ...
18
The sum of values of ( x ) satisfying the equation ( 31+8(15)^.5) ^(x^2-3) = (32+8(15)^.5 )^x^2-3is: ...
19
If ( alpha, beta ) be the roots of the equation ( ax^2 + bx + c = 0 ), then the value of ( a alpha^2 ...
20
If ( a, b, c, d in ε{R} ), then the equation ( (x^2 + ax -3b)(x^2 - cx + d)(x^2 - cx + b)(x^2 -dx +2 ...
21
If ( alpha ) and( beta ) are the roots of the equation ( x^2 + px + q = 0 ), and ( alpha )^4 and (be ...
22
If ( alpha ) and ( beta ) are the roots of the equation ( x^2 + px + q = 0 ), and ( alpha )^4 and ( ...
23
If the roots of the equation ( (a-1)(x^2 + x + 1)^2 = (a + 1)(x^4 + x^2 + 1) ) are real and distinct ...
24
If ( b1.b2 = 2(c1 + c2) ), then at least one of the equations ( x^2 + b1x + c1 = 0 ) and ( x^2 + b2x ...
25
The integral values of ( m ) for which the roots of the equation ( mx^2 + (2m-1)x + (m-2) = 0 ) are ...
26
Suppose ( A, B, C ) are defined as ( A = a^2b + ab^2 - a^2c - ac^2 ), ( B = b^2c + bc^2 - a^2b-ab^2 ...
27
The coefficient of ( x ) in the equation ( x^2 + px + q = 0 ) was wrongly written as 17 in place of ...
28
If ( a(p+q)^2 + 2bpq + c = 0 ) and ( a(p+r)^2 + 2bpr + c = 0 ) (( a notequal 0 )), then: ...
29
If the roots of the equation ( ax^2 -bx + c = 0 ) are ( α, β ), then the roots of the equation ( b ...
30
If ( α) and ( β ), ( α ) and (gamma ), ( α ) and ( delta ) are the roots of the equations( ax^2 + ...
31
The expression ( x^2 - xy + y^2 - 4x - 4y + 16 =0 ) represents: ...
32
If (α ) and ( β) are the non-zero roots of ( ax^2 + bx + c = 0 ) and ( α^2), ( β^2 ) are the roots o ...
33
If the roots of the equation ( ax^2 + bx + c = 0 ) are of the form ( (k+1)/k ) and (k+2}/k+1), then ...
34
If ( α, β ) are the roots of ( ax^2 + bx + c = 0 ) and ( α + h, β + h ) are the roots of ( px^2 + qx ...
35
If ( α, β) are the roots of the equation ( (x-a)(x-b) + c = 0 ) (( c notequal to 0, then the roots ...
36
If ( α, β ) are the roots of ( ax^2 + c = bx ), then the equation ( (a + cy)^2 = b^2 y ) in ( y ) ha ...
37
If the roots of the equation ( x^2 + 2ax+b = 0 ) are real and distinct and they differ by at most ( ...
38
If the ratio of the roots of ( ax^2 + 2bx+c=0 ) is the same as the ratio of the roots of ( px^2+2qx+ ...
39
If one root of ( x^2 - x - k = 0 ) is the square of the other, then ( k ) equals: ...
40
If (α) and (β ) are the roots of the equation ( x^2 - px - 1/{2p^2} = 0 ) where ( p in belongs{R} ), ...
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