Important Quadratic Equations Questions for CBSE Class 11th

If α , γ are roots of the equation A x 2 − 4 x + 1 = 0 , and β , δ are roots of the equation of B x 2 − 6 x + 1 = 0 , then the values of A and B such that α , β , γ and δ are in H.P. are

The greatest value of a non-negative real number λ for which both the equations 2 x 2 + ( λ − 1 ) x + 8 = 0 and x 2 − 8 x + λ + 4 = 0 have real roots is

If the roots of the equation x 2 − 2 c x + a b = 0 are real and unequal, the roots of the equation x 2 − 2 ( a + b ) x + a 2 + b 2 + 2 c 2 = 0 are

If α and β are the roots of c x 2 + b x + a = 0 , then the roots of equation a ( x + 1 ) 2 + b ( x + 1 ) + c = 0 are

tan ⁡ α and tan ⁡ β are roots of the equation x 2 + a x + b = 0 , then the value of sin 2 ⁡ ( α + β ) + a sin ⁡ ( α + β ) ⋅ cos ⁡ ( α + β ) + b cos 2 ⁡ ( α + β ) is equal to

The curve y = ( λ + 1 ) x 2 + 2 intersects the curve y = λx + 3 in exactly one point, it λ equals

If one root of the equation x 2 + (1+3i)x-2(1-i) = 0 is -1 + i, then the other root is

If a , b ∈ R , a ≠ 0 and the quadratic equation ax 2 − bx + 1 = 0 has imaginary roots, then (a + b + 1) is

If α , β are the roots of the equation x 2 − 3 x + 5 = 0 and γ , δ are the roots of the equation x 2 + 5 x − 3 = 0 , then the equation whose roots are αγ + βδ and αδ + βγ is

Number of real roots of the equation x + x − ( 1 − x ) = 1 is

The roots of the equation ( a + b ) x 2 − 15 + ( a − b ) x 2 − 15 = 2 a , where a 2 − b = 1 , are

Let ( sin ⁡ a ) x 2 + ( sin ⁡ a ) x + 1 − cos ⁡ a = 0 The set of values of a for which roots of this equation are real and distinct, is

The roots of the equation ( a + b ) x 2 − 15 + ( a − b ) x 2 − 15 = 2 a , where a 2 − b = 1 , are

If tan 25° and tan 20° are roots of the quadratic equation x 2 + 2 p x + q = 0 , then 2 p − q is equal to

Let a , b , c be non-zero real number such that ∫ 0 1 1 + cos 8 ⁡ x a x 2 + b x + c d x = ∫ 0 2 1 + cos 8 ⁡ x a x 2 + b x + c d x Then the quadratic equation a x 2 + b x + c = 0 has

Number of possible value (s) integer ‘a’ for which the quadratic equation x 2 + a x + 16 = 0 has integral roots, is

The number of integral values of m for which the equation 1 + m 2 x 2 − 2 ( 1 + 3 m ) x + ( 1 + 8 m ) = 0 has no real roots is

If the equations 2 x 2 + k x − 5 = 0 and x 2 − 3 x − 4 = 0 have one root in common, then k =

If the roots of the equation a x 2 − 4 x + a 2 = 0 are imaginary and the sum of the roots is equal to their product, then a =

If α , β are the roots of x 2 − 3 x + a = 0 , a ∈ R and α < 1 < β , then

If the sum of two roots of the equation x 3 − p x 2 + q x − r = 0 is zero, then

If p , q , r are real and p ≠ q then the roots of the equation ( p − q ) x 2 + 5 ( p + q ) x − 2 ( p − q ) = r are

If x = c is a root of order 2 of a polynomial f ( x ) , then x = c is also a root of the polynomial

if α , β are roots of the equation x 2 + x + 1 = 0 , then the equation whose roots are α β and β α , is

If roots of x 2 − ( a − 3 ) x + a = 0 are such that at least one of them is greater than 2 , then

If α and β are the roots of x 2 − p ( x + 1 ) − c = 0 then the values of ( α + 1 ) ( β + 1 ) and α 2 + 2 α + 1 α 2 + 2 α + c + β 2 + 2 β + 1 β 2 + 2 β + c a r e

If the roots of the equation ( a − 1 ) x 2 + x + 1 2 = ( a + 1 ) x 4 + x 2 + 1 are real and distinct then the value of a ∈

If m r , 1 m r ; r = 1 , 2 , 3 , 4 are four pairs of values of x and y that satisfy the equation x 2 + y 2 + 2 g x + 2 f y + c = 0 , then value of m 1 ⋅ m 2 ⋅ m 3 ⋅ m 4

When an unknown polynomial is divided by ( x − 1 ) and ( x − 2 ) we obtain the remainder 2 and 1 , respectively. Then the remainder resulting from the division of this polynomial by ( x − 1 ) ( x − 2 ) is

The value of k for which ( a + 2 b ) where a , b ≠ 0 is a factor of a 4 + 32 b 4 + a 3 b ( k + 3 ) is

Number of real solutions of the equation x − 1 x + 1 − 1 x = x is

Let conditions C 1 and C 2 be defined as follows: C 1 : b 2 − 4 a c ≥ 0 , C 2 : a , − b , c are of same sign. The roots of a x 2 + b x + c = 0 are real and positive , if

If α , β are the roots of a x 2 + c = b x , then the equation ( a + c y ) 2 = b 2 y in y has the roots

If the absolute value of the difference of roots of the equation x 2 + p x + 1 = 0 exceeds 3 p then

If e 1 and e 2 are roots of the equation x 2 − a x + 2 = 0 where e 1 , e 2 are eccentricities of ellipse and hyperbola respectively then range of a is

If m is selected at random from set { 1 , 2 … … .10 } and the probability that the quadratic equation 2 x 2 + 2 m x + m + 1 = 0 has real roots, is

Sum of solutions of the equation | x | 3 − 4 | x | 2 + 3 | x | = 0 is

If α and β are the roots of the equation 2 x ( 2 x + 1 ) = 1 ,then β is equal to

The number of solutions for the equation log 4 ⁡ 2 x 2 + x + 1 − log 2 ⁡ ( 2 x − 1 ) = 1 , is

If one root of the equation x 2 + ( 1 − 3 i ) x − 2 ( 1 + i ) = 0 is -1 + i, then the other root is

There is only one real value of a’ for which the quadratic equation ax 2 + ( a + 3 ) x + a − 3 = 0 has two positive integral solutions. The solutions is

Let a, b and c be real numbers such that a+2b+c = 4. Find the maximum value of (ab +bc + ca).

If a, b, c are in H.P., then the roots of the equation a ( b − c ) x 2 + b ( c − a ) x + c ( a − b ) = 0

If c, d are the roots of the equation (x – a) (x – b) – k = 0, then the roots of the equation (x – c) (x – d) + k = 0 are

For a > 0, the roots of the equation log ax ⁡ a + log x ⁡ a 2 + log a 2 x ⁡ a 3 = 0 are given by:

If x is real, the expression x 2 + 2 x − 11 x − 3 takes all real values except those which lie between a and b, then a and b are

The number of real solutions of x + x + x − 2 = 3 is

The number of rational roots of 81 2 x − 5 3 x + 1 4 − 45 2 x − 5 3 x + 1 2 + 4 = 0 , x ≠ 1 / 3 is

Product of roots of the equation x − 3 x − 6 = 2 is

The number of roots of the equation x x − 3 + x − 3 x = 5 2 , x ≠ 0 , x ≠ 3 is

If α , β are the roots of a x 2 + b x + c = 0 and α + h , β + h are the roots of p x 2 + q x + r = 0 , then h =

If the roots of the quadratic equation x 2 + p x + y = 0 are are tan 30° and tan 15° respectively, then the value of 2 + q − p is,

If α and β are the roots of the equation x 2 + a x + b = 0 , and α 4 and β 4 are the roots of x 2 – p x + q = 0 , the roots x 2 − 4 b x + 2 b 2 − p = 0 are always

if α , β are the roots of a x 2 + b x + c = 0 ; α + h , β + h are the roots of p x 2 + q x + r = 0 ; and D 1 , D 2 the respective discriminants of these equations, then D 1 : D 2 =

Let α and β be the roots of equation x 2 − 6 x − 2 = 0 . if a n = α n − β n for n ≥ 1 , then the value a 10 − 2 a 8 2 a 9 is equal to

If α , β ∈ R are the roots of the a x 2 + b x + c = 0 , k ∈ R lies between α and β , if

The real values of a for which the quadratic equation 2 x 2 − a 3 + 8 a − 1 x + a 2 − 4 a = 0 possesses roots of opposite signs are given by

The quadratic equation those roots are reciprocal of the roots of the equation a x 2 + b x + c = 0 is

If the roofs of the equation 1 x + p + 1 x + q = 1 r are equal in magnitude and opposite’ in sign, then the product of root, is

If a ∈ R and a 1 , a 2 , a 3 … , a n ∈ R then x − a 1 2 + x − a 2 2 + … + x − a n 2 assumes its least value at x =

Let x 1 , x 2 be the roots of the equation x 2 − 3 x + p = 0 and let x 3 , x 4 be the roots of the equation x 2 − 12 x + q = 0 .If the number x 1 , x 2 , x 3 , x 4 (in order) form an increasing G.P., then

If a , b , c are real and x 3 − 3 b 2 x + 2 c 3 is divided by x – a and x – b , then

If the roots of the equation x 2 − p x + q = 0 differ by unity, then

If α , β , γ are the roots of the equation x 3 + x + 1 = 0 then the value of x 3 + β 3 + γ 3 is

If x is real then x 2 − 2 x + 4 x 2 + 2 x + 4 takes values in the interval

If 6 lies between the roots of the equation x 2 + 2 ( a − 3 ) x + 9 = 0 then

If the equations a x 2 + b x + c = 0 and 2 x 2 + 3 x + 4 = 0 a common root, then a : b : c

If a x 2 + c y + a ′ x 2 + c ′ = 0 and x is a rational function of y and a c is negative, then

If the roots of x 3 − 12 x 2 + 39 x − 28 = 0 are in A.P. , then their common difference, is

The set of possible values of a for which x 2 − a 2 − 5 a + 5 x + 2 a 2 − 3 a − 4 = 0 has roots whose sum and product are both less than 1, is

If one root of the equation ( a − b ) x 2 + a x + 1 = 0 be double the other and if a ∈ R , then the greatest value of 8 b , is

If b and c are odd integers, then the equation x 2 + b x + c = 0 has

Given that tan A and tan B are the roots of x 2 − p x + q = 0 , the value of sin 2 ⁡ ( A + B ) , is

The roots of the equation ( a + b ) x 2 − 15 + ( a − b ) x 2 − 15 = 2 a , where a 2 − b = 1 , are

Real roots of the equation x 2 + 5 | x | + 4 = 0 are

If b > a , the the equation ( x − a ) ( x − b ) − 1 = 0 has

If the roots of the equation a x 2 + b x + c = 0 are real and distinct, then

The condition that one root of the equation a x 2 + b x + c = 0 may be double of the other, is

If 0 < a < b < c , and the roots α , β of the equation a x 2 + b x + c = 0 are imaginary, then

For the equation 3 x 2 + p x + 3 = 0 , p > 0 If one of the roots is square of the other, then p is equal to

Let p, q be integers and let α , β be the roots of x 2 − x − 1 = 0 , where α ≠ β . For n = 0 , 1 , 2 , … Let a n = p α n + q β 2 Then,

If the sum of the roots of the equation a x 2 + b x + c = 0 is equal to the sum of the squares of their reciprocals, then

The roots α , β and γ of an equation x 3 − 3 a x 2 + 3 b x − c = 0 are in H.P. Then,

The roots of the equation x 2 − x − 6 = x + 2 are

Both the roots of the equation ( x − a ) ( x − b ) + ( x − b ) ( x − c ) + ( x − c ) ( x − a ) = 0 are always

If a, b, c are positive real numbers such that the equations a x 2 + b x + c = 0 and b x 2 + c x + a = 0 have a common root then

Let α and β be the roots of the equation x 2 − x − 1 = 0 . If P k = α k + β k , k ≥ 1 ,then which one of the following statement is not true?

If a , b , c , d are the roots of the equation x 4 – 2 π x – 2019 = 0 , then the product of a + b + c a b c , b + c + d b c d , c + d + a c d a , d + a + b a b d is equal to

If a < 0 and a – b + c = a – 3 b + 9 c = 0 , then the quadratic expression y = a x 2 + b x + c will attains its maximum value of x = k , the k is:

Number of real solution(s) of the equation | x – 3 | 3 x 2 – 10 x + 3 = 1 is

Number of integral values of x for which 8 + 2 x − x 2 > 6 − 3 x is true is

If the roots of equation x 2 + q x + p = 0 are each 1 less than the roots of the equation x 2 + p x + q = 0 , then ( p + q ) is equal to

Consider the equation x 2 − 2 x + m x 2 − 2 x + n = 0 (where m and n are real numbers). Let the roots α , β , γ , δ ( α < β < γ < δ ) of the equation form an A.P. with first term equal to 1 4 . Then the value of m n is

If the roots α , β of the equation p x 2 + q x + r = 0 are real and of opposite signs (where p , q , r are real coefficients), then the roots of the equation α ( x − β ) 2 + β ( x − α ) 2 = 0 a r e

α , β are the roots of the equation x 2 − 2 x + 3 = 0 . Then the equation whose roots are P = α 3 − 3 α 2 + 5 α − 2 and Q = β 3 − β 2 + β + 5 is

If the system of equations r 2 + s 2 = t and r + s + t = k − 3 2 has exactly one real solution, then the value of k is

Number of real values of p for which the equations x 2 − p x + 8 = 0 and x 2 + p = 0 have two integral solutions is

If a , b , c , d ∈ R then the equation x 2 + a x − 3 b x 2 − c x + b) x 2 − d x + 2 b = 0 has

Consider the equation x 2 + 2 x − n = 0 , where n ∈ N and n ∈ [ 5 , 100 ] . Total number of different values of n so that the given equation has integral roots is

If a x 2 + ( b − c ) x + a − b − c = 0 has unequal real roots for all c ∈ R then

If a and b ( ≠ 0 ) are roots of the equation x 2 + a x + b = 0 , then the least value of x 2 + a x + b ( x ∈ R ) is

If the equation x 2 + 2 | a | x + 4 = 0 has integral roots, then minimum value of a is

If the roots of the quadratic equation x − m m x + 1 = x + n n x + 1 are reciprocal to each other, then

α 1 , β 1 are the roots of a x 2 + b x + c = 0 and α 2 , β 2 are the roots of p x 2 + q x + r = 0 . If α 1 α 2 = β 1 β 2 = 1 , then

If α , β , γ are roots of the cubic x 3 − 2 x + 3 = 0 , then the value of 1 α 3 + β 3 + 6 + 1 β 3 + γ 3 + 6 + 1 γ 3 + α 3 + 6 equals to

The set of all values of ‘a’ for which the roots of the equation ( a + 1 ) x 2 − 3 a x + 4 a = 0 ( a ≠ − 1 ) are real and greater than 1 is

If p ( q − r ) x 2 + q ( r − p ) x + r ( p − q ) = 0 has equal roots then

The solution of the inequality x + 7 x − 5 + 3 x + 1 2 ≥ 0 is

The value of the expression x 4 − 8 x 3 + 18 x 2 − 8 x + 2 , when x = 2 + 3 , is

The sum of the non-real roots of x 2 + x − 2 x 2 + x − 3 = 12 is

The number of irrational roots of the equation 4 x x 2 + x + 3 + 5 x x 2 − 5 x + 3 = − 3 2 is

If x = 1 + 1 3 + 1 2 + 1 3 + 1 2 … ∞ , then value of x is

If a ∈ ( − 1 , 1 ) then roots of the quadratic equation ( a − 1 ) x 2 + ax + 1 − a 2 = 0 are

If ax 2 + c y + a ′ x 2 + c ′ = 0 and x is a rational function of y and ac is negative, then

x 2 − xy + y 2 − 4 x − 4 y + 16 = 0 represents

If the roots of equation 1 ( a − 1 ) x 2 + x + 1 2 = ( a + 1 ) x 4 + x 2 + 1 are real and distinct, then the value of a ∈

Suppose A,B,C are defined as A = a 2 b + ab 2 − a 2 c − ac 2 , B = b 2 c + bc 2 − a 2 b − ab 2 and C = a 2 c + ac 2 − b 2 c − bc 2 , where a > b > c > 0 and the equation Ax 2 + Bx + C = 0 has equal roots, then a, b, c are in

The quadratic x 2 + ax + b + 1 = 0 has roots which are positive integers, then a 2 + b 2 canbe equal to

If α , β arc the roots of ax 2 + c = bx , then the equation ( a + cy ) 2 = b 2 y in y has the roots

The sum of all real values of x satisfring the equation x 2 − 5 x + 5 x 2 + 4 x − 60 = 1 is

If the equation ax 2 + 2 bx + 3 c = 0 and 3 x 2 + 8 x + 15 = 0 have a common root, where a,band c are the lengths of the sides of a △ ABC , then sin 2 ⁡ A + sin 2 ⁡ B + sin 2 ⁡ C is equal to

If the equation x 2 + 2 ( λ + 1 ) x + λ 2 + λ + 7 = 0 has only negative roots, then least value of λ equals

If the roots of the equation 8 x 3 − 14 x 2 + 7 x − 1 = 0 are in GP, then the roots are

If the roots of a 2 + b 2 x 2 − 2 ( b c + a d ) x + c 2 + d 2 = 0 are equal, then

The solution set of the equation pqx 2 − ( p + q ) 2 x + ( p + q ) 2 = 0 is

If sin α and cos α are the roots of the equation ax 2 +bx+c=0, then

If x 2 + x + 1 is a factor of ax 3 + bx 2 + cx + d the real roots of ax 3 + bx 2 + cx + d = 0 is

If x 1 and x 2 are the arithmetic and harmonic means of the roots of the equation ax 2 + bx + c = 0 , the equation whose quadratic roots are x 1 and x 2 , is

The sum of the roots of the equation 2 33 x − 2 + 2 11 x + 2 = 2 22 x + 1 + 1 is

The number of pairs (x, y) which will satisfy the equation x 2 − xy + y 2 = 4 ( x + y − 4 ) is

If x and y are positive integers such that xy+x+y = 71, x 2 y +xy 2 =880, then x 2 +y 2 is equal to

The number of roots of the equation 1 x + 1 ( 1 − x 2 ) = 35 12 is

If α , β are the roots of the equation λ ( x 2 − x ) + x + 5 = 0 . If λ 1 and λ 2 are two values of . λ for which the roots α , β are related by α β + β α = 4 5 find the value of λ 1 λ 2 + λ 2 λ 1

The product of the roots of the equation ( x − 2 ) 2 − 3 | x − 2 | + 2 = 0 is

The number of values of the pair (a, b ) for which the equation α ( x + 1 ) 2 + β ( x 2 − 3 x − 2 ) + x + 1 = 0 , ∀ x ∈ R is identity

If sin q and cos q are the roots of the equation ax 2 +bx + c = 0, then

If a , b , c ∈ R and the equations ax 2 + bx + c = 0 and x 3 + 3 x 2 + 3 x + 2 = 0 have two roots in common, then

Solution of 2 x + 2 | x | ≥ 2 2 is

If the ratio of the roots of the equation x 2 + b x + c = 0 is the same as that of the ratio of the roots of x 2 + q x + r = 0 , then

Solution set of 3 − x = − x 2 − x − 1 , x ∈ R is

Number of solutions of x − 5 x − 2 = 2 − 5 x − 2 is

If the product of the roots of the equation x 2 − 5 k x + 2 e 4 ln ⁡ k − 1 = 0 is 31 then sum of the root is

If a ∈ R and both the roots of x 2 − 6 a x + 9 a 2 + 2 a − 2 = 0 exceed 3, then a lies in the interval

If [ x ] denotes the greatest integer ≤ x , and a , b are two odd integers, then number of solutions of x 2 + a x + b = 0 is

If x 2 − 3 x + 2 is a factor of x 4 − a x 2 + b = 0 then the equation whose roots are a and b is

Suppose a ∈ R . If 3 x 2 + 2 a 2 + 1 x + a 2 − 3 a + 2 ) = 0 possesses roots of opposite signs, then a lies in the interval:

Let α , β , γ be distinct real numbers lying in ( 0 , π / 2 ) , then the equation 1 x − sin ⁡ α + 1 x − sin ⁡ β + 1 x − sin ⁡ γ = 0 , has

Two complex numbers a and β are such that α + β = 2 and α 4 + β 4 = 272 , then the quadratic equation whose roots are α and β can be

if 1 – p is a root of the quadratic equation x 2 + p x + 1 − p = 0 then its roots are

The number of real solutions of x 2 + 5 | x | + 4=0 is

The sum of all the real roots of the equation | x − 2 | 2 + | x − 2 | − 2 = 0 is

The equation x − x 2 − 1 = 2 x − 3 − x 2 has

If α , β are the roots of ( x − a ) ( x − b ) + c = 0 , c ≠ 0 ,then roots of ( α β − c ) x 2 + ( α + β ) x + 1 = 0 are

Value of x = 6 + 6 + 6 + ⋯ up ⁡ t o ∞ is

If x = 7 − 4 3 , then x + 1 x is equal to:

If sin ⁡ α , cos ⁡ α are the roots of the equation a x 2 + b x + c = 0 , ( a ≠ 0 ) ,then

If α , β are the roots of the equation a x 2 + b x + c = 0 , then the value of α 3 + β 3 is

The value of a for which one root of the quadratic equation. a 2 − 5 a + 3 x 2 + ( 3 a − 1 ) x + 2 = 0 (1) is twice the other, is

The integral values of a for which the quadratic equation (x – a) (x – 10) + 1 = 0 has integeral roots are

The equation e sin ⁡ x − e − sin ⁡ x = 4 has:

Range of function f ( x ) = x 2 + x + 2 x 2 + x + 1 x ∈ R is

Suppose a , b , c are three non-zero real numbers. The equation x 2 + ( a + b + c ) x + a 2 + b 2 + c 2 = 0 has

If a and β are the roots of the equation a x 2 + b x + c = 0 , then roots of a x 2 − b x ( x − 1 ) + c ( x − 1 ) 2 = 0 are

The number of real roots of the equation x 2 − 3 | x | + 2 = 0

Two non-integer roots of x 2 − 5 x 2 − 7 x 2 − 5 x + 6 = 0 are

The number of negative roots of 9 x + 2 − 6 3 x + 1 + 1 = 0 is

Let f ( x ) be a quadratic expression which is positive for all x. If g ( x ) = f ( x ) + f ′ ( x ) + f ′′ ( x ) then for all real x,

The number irrational roots of x 2 + 3 x + 2 2 − 8 x 2 + 3 x − 4 = 0 is

The number of irrational roots of the equation ( x − 1 ) ( x − 2 ) ( 3 x − 2 ) ( 3 x + 1 ) = 21 is

Irrational roots of the equation 2 x 4 + 9 x 3 + 8 x 2 + 9 x + 2 = 0 are

Let f ( x ) = x 2 + bx + c where b , c ∈ R . If f(x) is a factor of both x 4 + 6 x 2 + 25 and 3 x 4 + 4 x 2 + 28 x + 5 , then the least value of f(x) is

If e cos ⁡ x − e − cos ⁡ x = 4 then the value of cos x , is

If sin ⁡ θ , sin ⁡ α , cos ⁡ θ are in G.P., then the roots of x 2 + 2 x cot ⁡ α + 1 = 0 are always

If α is a root of the equation 2 x ( 2 x + 1 ) = 1 , then the other root, is

If sin ⁡ α and cos ⁡ α are roots of the equation p x 2 + q x + r = 0 then ,

If the roots of the equation x 3 − p x 2 + q x − r = 0 are in A.P., then

The number of roots of the equation x − 2 x − 1 = 1 − 2 x − 1 , is

If a , b , c are real numbers in G.P. such that a and c are positive, then the roots of the equation a x 2 + b x + c = 0

The value of k for which the equation 3 x 2 + 2 x k 2 + 1 + k 2 − 3 k + 2 = 0 has roots of opposite signs, lies in the interval

Let α , β be the roots of a x 2 + b x + c = 0 ; γ , δ be the roots of p x 2 + q x + r = 0 ; and D 1 , D 2 the respective discriminants of these equations. If α , β , γ and δ are in AP., then D 1 : D 2 =

If a + b = 2 and a 4 + b 4 = 272 then a quadratic equation whose roots are a and b is

If α , β , γ , σ are the roots of the equation x 4 + 4 x 3 − 6 x 2 + 7 x − 9 = 0 then the value of 1 + α 2 1 + β 2 1 + γ 2 1 + σ 2 is

The number of real roots of ( 6 − x ) 4 + ( 8 − x ) 4 = 16 , is

In a triangle PQR, ∠ R = π / 2 . If tan ⁡ ( P / 2 ) and tan ⁡ ( Q / 2 ) are the roots of the equation a x 2 + b x + c = 0 ( a ≠ 0 ) then

The quadratic equation whose roots are A.M and H.M. between the roots of the equation a x 2 + b x + c = 0 , is

The values of ‘ a ‘ for which the roots of the equation x 2 + x + a = 0 are real and exceed ‘ a ’ are

If the equation ( 3 x ) 2 + 27 × 3 1 / p − 15 x + 4 = 0 has equal roots, then p =

The set of values of ‘ a ‘ for which x 2 + a x + sin − 1 ⁡ x 2 − 4 x + 5 + cos − 1 ⁡ x 2 − 4 x + 5 = 0 has at least one real root is given by

If the roots of the equation ( x − b ) ( x − c ) + ( x − c ) ( x − a ) + ( x − a ) ( x − b ) = 0 are equal then

The product of the real roots of the equation | 2 x + 3 | 2 − 3 | 2 x + 3 | + 2 = 0 , is

If x 2 + x + 1 is a factor of a x 3 + b x 2 + c x + d then the real root of a x 3 + b x 2 + c x + d = 0 is,

If α , β are roots of a x 2 + b x + c = 0 then the equation a x 2 − b x ( x − 1 ) + c ( x − 1 ) 2 = 0 has roots

If a , b , c ∈ R and the quadratic equation x 2 + ( a + b ) x + c = 0 has no real roots, then

The sum of the real roots of the equation | x − 2 | 2 + | x − 2 | − 2 = 0 , is

If the sum of two roots of the equation x 3 − p x 2 + q x − r = 0 is zero, then

If roots of a x 2 + b x + c = 0 , a , b , c ∈ R , a ≠ 0 are imaginary, then

If α ≠ β and α 2 = 5 α − 3 , β 2 = 5 β − 3 , then the equation having α / β and β / α as its roots, is

If the equations x 2 − a x + b = 0 and x 2 + b x − a = 0 have a common root, then

If 7 log ⁡ 7 x 2 − 4 x + 5 = x − 1 , x may have value

If a , b , c , d are four consecutive terms of an increasing A.P., then the roots of the equation ( x − a ) ( x − c ) + 2 ( x − b ) ( x − d ) = 0 , are

A value of b for which the equations x 2 + b x – 1 = 0 and x 2 + x + b = 0 have one root in common, is

If a, b, c are in H.P., then the roots of the equation a b − c x 2 + b c − a x + c a − b = 0 are

Minimum value of 4 x 2 − 4 x sin θ − cos 2 θ is

The least positive value of ‘a’ for which the equation 2 x 2 + a − 10 x + 33 2 = 2 a has real roots is

The number of real roots of the equation e 4 x + e 3 x – 4 e 2 x + e x + 1 = 0 is:

Sum of the values of x satisfying the equation 2 x + 2 x + 4 = 4 i s

Given α and β are the roots of the quadratic equation x 2 − 4 x + k = 0 ( k ≠ 0 ) . If α β , α β 2 + α 2 β , α 3 + β 3 are in geometric progression, then the value of k is

The number of solutions of the equation ( 3 | x | − 3 ) 2 = | x | + 7 which belong to the domain of x ( x − 3 ) is

Consider f ( x ) = x 2 − 3 x + a + 1 a , a ∈ R − { 0 } , such that f ( 3 ) > 0 and f ( 2 ) ≤ 0 . If α and β are the roots of equation f ( x ) = 0 then the value of α 2 + β 2 is equal to

Let α , β are roots of x 2 − b x = b ( b > 0 ) and | α | , | β | are roots of x 2 + p x + q = 0 . The minimum value of p 2 − 8 q is equal to

If the roots of the quadratic equation a x 2 + b x − b = 0 ; where a , b ∈ R , such that a ⋅ b > 0 , are α and β then the value of log β – 1 α − 1 is

Let cos 2 ⁡ θ + b and sin 2 ⁡ θ + b are roots of the equation x 2 + 4 x + 61 16 = 0 . The equation whose roots are tan 2 ⁡ θ and cot 2 ⁡ θ is

If α , β are the roots of the equation x 2 + p x − r = 0 and α 3 , 3 β are the roots of the equation x 2 + q x − r = 0 , then r eqaul

Number of ordered pairs ( x , y ) of real numbers satisfying the equation 2 x 4 − 2 x 2 + 3 y 4 − 3 y 2 + 4 = 7 , is

The possible value of p for which graph of the function f ( x ) = 2 p 2 − 3 p tan ⁡ x + tan 2 ⁡ x + 1 does not lie below x -axis for all x ∈ − π 2 , π 2 is

If e λ and e − λ are the roots of 3 x 2 − ( a + b ) x + 2 a = 0 , a , b , λ ∈ R , λ ≠ 0 then least integral value of b is

If the equation f ( x ) = ( a + b − c ) x 2 + ( b + c − a ) x + ( c + a − b ) = 0 has only one root equal to one. Then the value of 3 b + a c is

If the roots of the equation a x 2 − b x + c = 0 are α , β then the roots of the equation b 2 c x 2 − a b 2 x + a 3 = 0 are

If the roots of the equation x 2 − a x + b = 0 are real and differ by a quantity which is less than c ( c > 0 ) , then b lies between

If the sum of two of the roots of x 3 + p x 2 + q x + r = 0 is zero, then p q =

If α and β ( α < β ) are the roots of the equation x 2 + b x + c = 0 , where c < 0 < b , then

The values of p for which one root of the equation x 2 − ( p + 1) x + p 2 + p = 8 exceeds 2 and the other is lesser than 2 are given by

The greatest integral value of c so that both the roots of the equation ( c − 5 ) x 2 − 2 c x + ( c − 4 ) = 0 are positive, one root is less than 2 and other root is lying between 2 and 3 is

If x 2 − 2 x + a = 0 , where [ · ] represents greatest integer function,has no solution then,

Number of real solutions of 2 x − 4 − x + 5 = 1 is

The number of solutions of 3 x 2 + x + 5 = x − 3 is

The number of irrational roots of the equation 4 x x 2 + x + 3 + 5 x x 2 − 5 x + 3 = − 3 2 is

The value of 2 + 1 2 + 1 2 + 1 2 + … … ∞ is

If a , b , c are three distinct positive real numbers then the number of real roots of a x 2 + 2 b | x | − c = 0 is

Number of distinct real solutions of the equation x 2 + x x − 1 2 = 8 is

If x + 1 is a factor of x 4 + ( p − 3 ) x 3 − ( 3 p − 5 ) x 2 + ( 2 p − 9 ) x + 6 , then which of the following is not the other factor?

If roots of an equation x n − 1 = 0 are 1 , a 1 , a 2 , … , a n − 1 , then the value of 1 − a 1 1 − a 2 1 − a 3 …………(1-a n – 1 ) will be

If x = 2 + 2 2 / 3 + 2 1 / 3 then the value of x 3 − 6 x 2 + 6 x is

The least value of the expression x 2 + 4 y 2 + 3 z 2 − 2 x − 12 y − 6 z + 14 is

If f ( x ) = x 2 + 3 x + 2 x 2 − 7 x + a and g ( x ) = x 2 − x − 12 x 2 + 5 x + b , then the values of a and b , if ( x + 1 ) ( x − 4 ) is HCF of f ( x ) and g ( x ) , are

If p , q ∈ { 1 , 2 , 3 , 4 } , the number of equations of the form p x 2 + q x + 1 = 0 having real roots is

If a + b + c = 0 then the roots of the equation 4 a x 2 + 3 b x + 2 c = 0 where a , b , c ∈ R are

For a , b , c non-zero, real distinct, the equation, a 2 + b 2 x 2 − 2 b ( a + c ) x + b 2 + c 2 = 0 has non-zero real roots. One of these roots is also the root of the equation:

The roots of the equation a 4 + b 4 x 2 + 4 a b c d x + c 4 + d 4 = 0

Let a , b and c he real numbers such that 4 a + 2 b + c = 0 and a b > 0 . Then the equation a x 2 + b x + c = 0 has

If the graph of f ( x ) = 2 x 3 + a x 2 + b x ( a , b ∈ N ) cuts the x -axis at three distinct points, then minimum value of a + b is

If a , b , c , d are four consecutive terms of an increasing A.P. then the roots of the equation ( x − a ) ( x − c ) + 2 ( x − b ) ( x − d ) = 0 are

If a + b + c = 0 , then the roots of the equation c 2 − a b x 2 − 2 a 2 − b c x + b 2 − a c = 0 are

The roots of a x 2 + 2 b x + c = 0 and b x 2 − 2 a c x + b = 0 are simultaneously real then

If f ( x ) = a x 2 + b x + c , g ( x ) = − a x 2 + b x + c where a c ≠ 0 then f ( x ) g ( x ) = 0 has

If the expression x 2 − ( 5 m − 2 ) x + 4 m 2 + 10 m + 25 can be expressed as a perfect square, then m =

If a x 2 + c y + a ′ x 2 + c ′ = 0 and x is a rational function of y and a c is negative, then

If x 2 − x y + y 2 − 4 x − 4 y + 16 = 0 an equation in x and y has only one real solution, then values of x and y

If A.M. of the roots of a quadratic equation is 8 / 5 and A.M. of their reciprocals is 8 / 7 , then the equation is

The coefficient of x in the equation x 2 + p x + q = 0 was wrongly written as 17 in place of 13 and the roots thus found was − 2 and − 15 . Then the roots of the correct equation are

Let α , β be the roots of x 2 + b x + 1 = 0 . Then the equation whose roots are − α + 1 β and − β + 1 α is

If b < 0 , then the roots x 1 and x 2 of the equation 2 x 2 + 6 x + b = 0 satisfy the condition x 1 / x 2 + x 2 / x 1 < k where k =

If α and β are the roots of the equation x 2 − a x + b = 0 and A n = α n + β n , then which of the following is true?

If α , β are the non-zero roots of a x 2 + b x + c = 0 and α 2 , β 2 are the roots of a 2 x 2 + b 2 x + c 2 = 0 then a , b , c are in

The equations 2 x 2 + b x + ( c + 1 ) = 0 where b , c ∈ Q and x 2 + 3 x − 5 = 0 have a common root, then the value of b − c is

The number of negative integral values of x satisfying the inequality log 0.6 log 6 x 2 + x x + 4 < 0 is

If a < b < c and x 2 + 3 x + 5 = 0 , a x 2 + b x + c = 0 have a common root, then the value of a – 2b + c is

If the product of roots of equation, x 2 − 5 kx + 2 e 4 ln ⁡ k − 1 = 0 is 31 then the sum of roots is (k is real)

If α , β are the roots of x 2 – x + 1 = 0 , then α 5 + β 5 =

If a , b, c are three distinct positive real numbers, then the number of real roots of ax 2 + 2 b | x | − c = 0 is

If the expression x 2 + 2 ( a + b + c ) x + 3 ( bc + ca + ab ) is a perfect square, then

Let a , b, and c be real numbers such that 4 a + 2 b + c = 0 and a b > 0 . Then the equation a x 2 + bx + c = 0 as

The integral values of m for which the roots of the equation mx 2 + ( 2 m − 1 ) x + ( m − 2 ) = 0 are rational are given by the expression [where n is integer]

If the roots of the equation x 2 + 2 ax + b = 0 are real and distinct and they differ by at most 2m, then b lies in the interval

If x is real, then x x 2 − 5 x + 9 lies between

If x 2 + ax − 3 x − ( a + 2 ) = 0 has real and distinct roots, then the minimum value of a 2 + 1 a 2 + 2 is

If a , b , c , d ∈ R then the equation x 2 + a x − 3 b x 2 − c x + b x 2 − d x + 2 b = 0

If α , β are the roots of x 2 − px + q = 0 and α ′ , β ′ are the roots of x 2 − p ′ x + q ′ = 0 then the value of α − α ′ 2 + β − α ′ 2 + α − β ′ 2 + β − β ′ 2 is

If α and β arc roots of the equation ax 2 + bx + c = 0 , then the roots of the equation a ( 2 x + 1 ) 2 − b ( 2 x + 1 ) ( 3 − x ) + c ( 3 − x ) 2 = 0 are

Suppose a ∈ I and the equation (x-a) (x-5) = 3 has integral roots, then the set of values which ‘a ’ can take is:

If the equation x 2 + 5 + 4 cos ⁡ ( αx + β ) = 2 x has at least one solution where α , β ∈ [ 2 , 5 ] then the value of α + β equal to

Let α and β are roots x 2 − 7 .32 x − 3 = 0 and let A n = α n + β n , then A 37 − 3 A 35 A 36 is equal to

If α , β , γ are the real roots of the equation x 3 − 3 px 2 + 3 qx − 1 = 0 then the centroid of the triangle with vertices α , 1 α , β , 1 β and γ , 1 γ is at the point

The root of the equation 2 ( 1 + i ) x 2 − 4 ( 2 − i ) x − 5 − 3 i = 0 , where i = − 1 , which has greater modulus, is

Two non-integer roots of 3 x − 1 2 x + 3 4 − 5 3 x − 1 2 x + 3 2 + 4 = 0 are

If the roots of the equation x 2 + px + q = 0 are in the same ratio as those of the equation x 2 + lx + m = 0, then p 2 m is equal to

If α and p are the roots of ax 2 + bx 2 c = 0 then the equation ax 2 – bx(x – 1) + c(x -1) 2 = 0 has roots

Let α and β be the roots of the equation x 2 − x − 1 = 0 . If p k = ( α ) k + ( β ) k , k ≥ 1 , then which one of the following statements is not true?

If α and β be two roots of the equation x 2 − 64 x +256=0. Then, value of α 3 β 5 1 / 8 + β 3 α 5 1 / 8 is

Let α = − 1 + i 3 2 . if a = ( 1 + α ) ∑ k = 0 100 α 2 k & b = ∑ k = 0 100 α 3 k then a and b are the roots of the quadratic equation

If α and β are the roots of the quadratic equation, x 2 + xsin ⁡ θ − 2 sin ⁡ θ = 0 , θ ∈ 0 , π 2 , then α 12 + β 12 α − 12 + β − 12 ( α − β ) 24 isequalto

If α and β are the roots of equation 2x 2 – 5x + 7 = 0, then the equation whose roots are 2 α + 3 β and 3 β + 2 β , is

If α and β are the roots of the equation, 7x 2 -3x – 2 = 0,then the value of α 1 − α 2 + β 1 − β 2 is equal to

If m is chosen in the quadratic equation (m 2 + 1)x 2 – 3x + (m 2 + 1) 2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is

If α and β be the roots of x 2 + px + q = 0 then, ωα + ω 2 β ω 2 α + ωβ α 2 β + β 2 α isequal to

If a ≠ b, then the roots of the equation 2 a 2 + b 2 x 2 + 2 ( a + b ) x + 1 = 0 are

In writing an equation of the form ax 2 + bx + c = 0 ;the coefficient of x is written incorrectly and roots are found to be equal. Again, io writing the same equation the constant term is written incorrectly and it is found that one root is equal to those of the previous wrong equation while the other is double of it. If α and β be the roots of correct equation, then ( α − β ) 2 is equal to

All the values of m for which both the roots of the equation x 2 – 2mx + m 2 – 1 – 0 are greater than -2 but less than 4,then m is in the interval

Let α and β be the roots of equation X 2 – 6x – 2 = O. If a n = α n − β n , for n ≥ 1 ,then the value a 10 − 2 a 8 2 a 9 as is equal to

If(a 2 -14a+ 13)x 2 +(a + 2)x – 2=0 does not have two distinct real roots, and maximum value of a 2 – 15a is k, then IKI is equal to

If α and β are the roots of the equation ax 2 + bx + c = 0 and S n = α n + β n , then aS n + 1 + bS n + cS n − 1 is equal to

The roots of the equation | 2 x − 1 | 2 − 3 | 2 x − 1 | + 2 = 0 are

The number of real roots of the equation e sin ⁡ x − e − sin ⁡ x − 4 = 0 are

How many roots of the equation x − 2 x − 1 = 1 − 2 x − 1 have ?

The number of real solutions of the equation x 2 + 4 x + 3 + 2 x + 5 = 0 are

The roots of the equation x 4 − 8 x 2 − 9 = 0 are

The number of real roots of 3 2 x 2 − 7 x + 7 = 9 is

Rational roots of the equation 2 x 4 + x 3 − 11 x 2 + x + 2 = 0 are

The roots of the given equation ( p − q ) x 2 + ( q − r ) x + ( r − p ) = 0 are

If α , β , γ are the roots of x 3 + 2 x 2 − 3 x − 1 = 0 , then α − 2 + β − 2 + γ − 2 is equal to

If α β and γ are the roots of the equation x 3 − 7 x + 7 = 0 , then 1 α 4 + 1 β 4 + 1 γ 4 is

If the roots of the equation 1 x + p + 1 x + q = 1 r are equal in magnitude but opposite in sign, then the product of the roots will be

The harmonic mean of the roots of the equation ( 5 + 2 ) x 2 − ( 4 + 5 ) x + 8 + 2 5 = 0 is

lf a + b + c=0, then the roots of the equation 4 ax 2 + 3 bx + 2 c = 0 are

If x 2 +2x +2xy + my – 3 -0 has two rational factors then the values of m will be

If a < b < c < d, then the roots of the equation (x-a)(x -c) +2(x – b)(x – d) =0 are

If roots of the equation (a – b)x 2 + (c – a)x+(b – c) = 0 are equal, then a, b and c are in

Let α and β be the roots of equation px 2 + qx+ r = 0 p ≠ 0 If p ,q and r, n AP and 1 α + 1 β = 4 then the value of | α − β | is

lf a, b and c are the sides of △ ABC such that a ≠ b ≠ c and x 2 − 2 ( a + b + c ) x + 3 λ ( ab + bc + ca ) = 0 has real roots, then

Let α , β be the roots of x 2 − 2 xcos ⁡ ϕ + 1 = 0 then the equation whose roots are α n and β n is

Let α and α 2 be the roots of x 2 + x + 1 = 0 then the equation whose roots are α 31 and α 62 , is

If α and β are the roots of the equation ax 2 + bx + c = 0 , then the equation whose roots are α + 1 β and β + 1 α , is

The equation whose roots are the square of the roots of the equation 2 x 2 + 3 x + 1 = 0

If the equation 2 x 2 + 3 x + 5 λ = 0 and x 2 + 2 x + 3 λ = 0 have a common root, then λ s equal to

If each pair of the equation x 2 + ax + b = 0 , x 2 + bx + c = 0 and x 2 + cx + a = 0 has a common root, then product of all common roots is

If at least one root of 2 x 2 + 3 x + 5 = 0 and ax 2 + bx + c = 0 , a , b , c ∈ N is common, then the maximum value of a+ b +c is

If the equations 2 ax 2 − 3 bx + 4 c = 0 and 3 x 2 − 4 x + 5 = 0 have a common root, then ( a + b ) / ( b + c ) is equal to ( a , b , c ∈ R )

A value of b for which the equations x 2 + bx − 1 = 0 and x 2 + x + b = 0 have one root in common, is

The values of ‘a’ for which a 2 − 1 x 2 + 2 ( a − 1 ) x + 2 is positive for any x, are

If the roots of the equation x 2 − 2 ax + a 2 + a − 3 = 0 are real and less than 3, then

If α and β ( α < β ) are the roots of the equation x 2 + bx + c = 0 , where c < 0 < b , then

One lies between the roots of the equation − x 2 + ax + a = 0 , a ∈ R if and only if a lies in the interval

Let α , β , γ be the roots of the equation ( x − a ) ( x − b ) ( x − c ) = d , d ≠ 0 the roots of the equation ( x − α ) ( x − β ) ( x − γ ) + d = 0 are

If the roots of the quadratic equation ( 4 p − p 2 − 5 ) x 2 − ( 2 p − 1 ) x + 3 p = 0 lie on either side of unity, the number of integral values of p is

If x 2 + 3 x + 5 = 0 and ax 2 + bx + c = 0 have a common root and a , b , c ∈ N the minimum value of a +b +c is

The sum of all the real roots of the equation | x − 2 | 2 + | x − 2 | − 2 = 0 is

If a∈R and a 1 , a 2 , a 3 ⋯ ⋯ , a n ∈ R then ( x − a 1 ) 2 + ( x − a 2 ) 2 + … . + ( x − a n ) 2 assumes its least value at x =

If a, b, c are in H.P., then the roots of the equation a ( b − c ) x 2 + b ( c − a ) x + c ( a − b ) = 0

If α , β are the roots of ax 2 + bx + c = 0 ; α + h , β + h are the roots of px 2 + qx + r = 0 ; and D 1 , D 2 the respective discriminants of these equation then D 1 : D 2 =

If α , β are roots of the equation 2 x 2 + 6 x + b = 0 ( b < 0 ) , then α β + β α is less than

The coefficient of x 99 in the expansion of (x-1) (x-2)……..(x-100), is

Number of possible value(s) of integer ‘a for which the quadratic equation x 2 + ax +16 =0 has integral roots, is

The roots of the equation ( a + b ) x 2 − 15 + ( a − b ) x 2 − 15 = 2 a , where a 2 − b = 1 , are

If a + b + c = 0 and a, b, c are rational, then the roots of the equation (b + c – a) x 2 + (c + a – b) x + (a + b– c) = 0 are

For all real values of x , | 12 x 4 x 2 + 9 |

If ( 7 − 4 3 ) x 2 − 4 x + 3 + ( 7 + 4 3 ) x 2 − 4 x + 3 = 14 then the value of x is given by

The real values of x for which 3 72 1 3 x 1 3 x > 1 are

If P ( x ) = a x 2 + b x + c and Q ( x ) = − a x 2 + d x + c , where a , b , c ∈ R , then P ( x ) Q ( x ) = 0 has

If 1 3 − 4 i is a root of a x 2 + b x + c = 0 , ( a , b c ∈ R , a ≠ 0 ) , then

Let a , β be the roots of the equation x 2 – p x + r = 0 and α 2 , 2 β be the roots of the equation x 2 − q x + r = 0 . Then the value of r is :

Suppose p ∈ R . If the roots of 3 x 2 + 2 x + p ( 1 − p ) = 0 are of opposite signs, then p must lie in the interval

If α , β are roots x 2 + p x + q = 0 , then value of α 3 + β 3 is

If p , q are roots of x 2 + p x + q = 0 , then

The equation x + 1 − x − 1 = 4 x − 1 , ( x ∈ R )

Suppose α , β are roots of a x 2 + b x + c = 0 then roots of a ( x − 2 ) 2 − b ( x − 2 ) ( x − 3 ) + c ( x − 3 ) 2 = 0 are

If x , y , z ∈ R then the least value of the expression E = 3 x 2 + 5 y 2 + 4 z 2 − 6 x + 20 y − 8 z − 3 is:

If x + 1 is a factor of x 4 + ( p − 3 ) x 3 − ( 3 p − 5 ) x 2 + ( 2 p + 9 ) x + 12 then value of P is.

Sum of the roots of the equation x 2 + | 2 x − 3 | − 4 = 0 is

If the quadratic equation a ( b − c ) x 2 + b ( c − a ) x + c ( a − b ) = 0 , where a , b , c are distinct real numbers and a b c ≠ 0 . has equal roots, then a, b, c are in

If α ∈ ( 0 , π / 2 ) , then the expression x 2 + x + tan 2 ⁡ α x 2 + x is always greater than or equal to

If a , b , and c are not all equal and α and β be the roots of the equation a x 2 + b x + c = 0 , then value of 1 + α + α 2 1 + β + β 2 is

If the roots of the equation 1 x + a + 1 x + b = 1 c are equal in magnitude but opposite in sign, then their product is

If α ≠ β and α 2 = 5 α − 3 , β 2 = 5 β − 3 , then the equation whose roots are α β and β α is

The number of solutions of the equation x 2 – 4 | x | – 2 = 0 is

Suppose a a ∈ I and the equation ( x – a ) ( x – 5 ) = 3 has integral roots, then the set of values which a can take is:

The number of solutions of 4 − x + x + 9 = 5 is

Let f ( x ) = x 2 − 2 x + 3 x 2 − 2 x − 8 , x ∈ R − { − 2 , 4 } The range of f is

If the equations x 2 + m x + 1 = 0 and ( b − c ) x 2 + ( c − a ) x + ( a − b ) = 0 have a common root, then

The number of real solution of 2 sin 2 ⁡ x + 5 2 cos 2 ⁡ x = 7 is

If α , β are the roots of the quadratic equation a x 2 + b x + c = 0 then the quadratic equation whose roots are α 3 , β 3 is

The number of values of k for which the equation x 2 – 2 x + k = 0 has two distinct roots lying in the interval (0, 1) is

If a , b , c are non-zero rational numbers such that a + b + c = 0 , then the roots of the equation ( b + c − a ) x 2 + ( c + a − b ) x + ( a + b − c ) = 0 are

If 4 x − 3 x − 1 / 2 = 3 x + 1 / 2 − 2 2 x − 1 then value of x is equal to

If x 2 + 2 a x + 10 − 3 a > 0 for each x ∈ R then

If roots of the quadratic equation a x 2 + b x + c = 0 are real and are of the form α α − 1 , α + 1 α , then value of ( a + b + c ) 2 is

The number of solutions of x + 1 − x − 1 = 1 ( x ∈ R ) is

Let a > 0 , b > 0 and c > 0 . Then both the roots of the equation 2 a x 2 + 3 b x + 5 c = 0 (1)

If 3 p 2 = 5 p + 2 and 3 q 2 = 5 q + 2 then the equation whose roots 3 p − 2 q and 3 q − 2 p is

The roots of the equation x 2 − x − 6 = x + 2 are

The number of distinct real roots of the equation ( x + 3 ) 4 + ( x + 5 ) 4 = 16 ( 1 ) is

If a , b , c are real, then both the roots of the equation ( x − b ) ( x − c ) + ( x − c ) ( x − a ) + ( x − a ) ( x − b ) = 0 (1) are always

If a , b , c , ∈ R and the equation x 2 + ( a + b ) x + c = 0 has no real roots, then c ( a + b + c ) then more

If roots of the equation x 2 − 2 m x + m 2 − 1 = 0 lie in the interval (– 2, 4), then

The value of 8 + 2 8 + 2 8 + 2 8 + ⋯ is

The number of solutions of | x + 2 | = 2 ( 3 − x ) is

The number of real solutions of the equation 27 1 / x + 12 1 / x = 2 8 1 / x is

The number of real roots of x − 1 x + 1 4 − 13 x − 1 x + 1 2 + 36 = 0 , x ≠ − 1 is

If a x 2 + 2 b x − 3 c = 0 has no real roots and c < 4 3 ( a + b ) then range of c is

The number of irrational roots of the equation 4 x − 1 x 2 + 8 x + 1 x = 29 is

Sum of the roots of the equation 4 x − 1 x 2 − 4 x − 1 x + 1 = 0 is

Product of roots of the equation. 13 − x 2 = x + 5 is

If [y] denote the greatest integer < y , and 2 x 8 2 + 3 x 8 = 20 then x lies in the smallest interval [ a , b ) where b – a is equal to

Let a is a real number satisfying a 3 + 1 a 3 = 18 . Then the value of a 4 + 1 a 4 is .

Let P ( x ) = x 3 − 8 x 2 + cx − d and be a polynomial with real coefficients and with all its roots being distinct positive integers. Then number of possible value of c is .

If the equation 2 x 2 + 4 xy + 7 y 2 − 12 x − 2 y + t = 0 , where t is a parameter has exactly one real solution of the form (x, y), then the sum of x 3 + y 3 is equal to .

If set of values of a for which f ( x ) = ax 2 − ( 3 + 2 a ) x + 6 , a ≠ 0 is positive for exactly three distinct negative integral values of x is (c, d], then the value of (c + d) is equal to .

If exp sin 2 ⁡ x + sin 4 ⁡ x + sin 6 ⁡ x + … upto ∞ ) ln 2 } satisfies the equation x 2 – 17 x + 16 = 0 then value of 2 cos ⁡ x sin ⁡ x + 2 cos ⁡ x ( 0 < x < π / 2 ) is

If a 0 , a 1 , a 2 , a 3 are all positive, then 4 a 0 x 3 + 3 a 1 x 2 + 2 a 2 x + a 3 = 0 has at least one root in (-1, 0) if

If the roots of a x 2 + b x + c = 0 ( a > 0 ) be each greater than unity, then

If a ∈ Z and the equation ( x − a ) ( x − 10 ) + 1 = 0 has integral roots, then the values of a , are

If the difference between the roots of the equation x 2 + a x + 1 = 0 less then 5 , then the set of possible value of a is

If a , b , c ∈ R and a + b + c = 0 , the the quadratic equation 4 a x 2 + 3 b x + 2 c = 0 has

The equation 2 cos 2 ⁡ x 2 sin 2 ⁡ x = x 2 + 1 x 2 , 0 ≤ x ≤ π 2 has

The ratio of the roots of the equation a x 2 + b x + c = 0 is same as the ratio of the roots of the equation p x 2 + q x + r = 0 . If D 1 and D 2 are the discriminants of a x 2 + b x + c = 0 and p x 2 + q x + r = 0 respectively, then D 1 : D 2 =

If a and b are the non-zero distinct roots of x 2 + a x + b = 0 , then the least value of x 2 + a x + b , is

If the roots of the quadratic equation x 2 − 4 x − log 3 ⁡ a = 0 are real, then the least value of a , is

Let α and β the roots of the equation p x 2 + q x + r = 0 , p ≠ 0 If p , q , r are in A .P and 1 α + 1 β = 4 , ,then the value of | α − β | is

If the ratio of the roots of the equation a x 2 + b x + c = 0 is equal to the ratio of the roots of the equation x 2 + x + 1 = 0 , then a , b , c are in

The roots of the quadratic equation ( a + b − 2 c ) x 2 − ( 2 a − b − c ) x + ( a − 2 b + c ) = 0 are

The equation ( a + 2 ) x 2 + ( a − 3 ) x = 2 a − 1 , a ≠ − 2 has rational roots for

If a , b , c arc positive and a = 2 b + 3 c , then roots of the equation a x 2 + b x + c = 0 are real for

If f ( x ) = a x 2 + b x + c , g ( x ) = − a x 2 + b x + c where a c ≠ 0 then f ( x ) g ( x ) = 0 has

If the equation x 3 − 3 x + a = 0 has distinct roots between 0 and 1, then the value of a is

If A , G and Hare respectively the A.M., G.M. and H.M. of three positive numbers a , b and c , then the equation whose roots are a , b , c is

If α , β be the roots of the equation ( x − a ) ( x − b ) + c = 0 ( c ≠ 0 ) , then the roots of the equation ( x − c − α ) ( x − c − β ) = c , are

If α , β are the roots of x 2 + b x + c = 0 and α + h , β + h are the roots of x 2 + q x + r = 0 then, h

If A = { x : f ( x ) = 0 } and B = { x : g ( x ) = 0 } , then A ∩ B will be the set of roots of the equation

The values of ‘a’ for which a 2 − 1 x 2 + 2 ( a − 1 ) x + 2 is positive for any x , are

The number of real solutions of the equation 9 10 x = − 3 + x − x 2 , is

If the absolute value of the difference of the roots of the equation x 2 + a x + 1 = 0 exceeds 3 a , then

The quadratic equations x 2 + a 2 − 2 x − 2 a 2 = 0 and x 2 − 3 x + 2 = 0 have

The equation formed by decreasing each root of a x 2 + b x + c = 0 by 1 is 2 x 2 + 8 x + 2 = 0 , then

The value of a for which the sum of the squares of the roots of the equation x 2 − ( a − 2 ) x − a − 1 = 0 assumes the least value, is

If a , b , c are positive real numbers, then the roots of the equation a x 2 + b x + c = 0

lf the roots of the equation x 3 − p x 2 + q x − r = 0 are in H.P., then

If the roots α , β , γ of the equation x 3 − 3 a x 2 + 3 b x + c = 0 are in H.P., then

For real values of x, the expression ( x − b ) ( x − c ) ( x − a ) will assume all real values provided

The number of real roots of | x | 3 − 3 x 2 + 3 | x | − 2 = 0 is,

The equation x 3 − 6 x 2 + 15 x + 3 = 0 has

If one root of the equation a x 2 + b x + c = 0 is reciprocal of the one root of the equation a 1 x 2 + b 1 x + c 1 = 0 , then

The expression y = a x 2 + b x + c has always the same sign as, c if

The roots of the equation log 2 ⁡ x 2 − 4 x + 5 = ( x − 2 ) are

lf α , β are the roots of x 2 + b x − c − 0 , the the equation whose roots are b and c , is

The number of solutions of the equation 5 x + 5 − x = log 10 ⁡ 25 , x ∈ k is

If 1-2i is a root of the equation z 2 + α z + β = 0 where α , β ∈ R then the value of α − β is

Let α , β be the roots of the equation x 2 − x − 1 = 0 such that P n = α n + β n . If P n + 1 = 29 and P n − 1 = 11 , then P n 2 =

The set of values of a for which each one of the roots of x 2 − 4 a x + 2 a 2 − 3 a + 5 = 0 is greater than 2 , is

If x ∈ R , then the expression 9 x − 3 x + 1 assumes

If α , β are roots of the equation a x 2 + b x + c = 0 then the equation whose roots are 2 α + 3 β and 3 α + 2 β is

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