Question 1 : If a ∈ (-1, 1), then roots of the quadratic equation (a - 1) x<sup>2</sup> + ax + <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e74642d4b0dee6204206930' height='25' width='45' > = 0 are -<br>
Question 2 :
The equation 3x<sup>2</sup> + 4ax + b = 0 has at least one root in (0,1) if <br>
Question 3 :
If {tex} { 'p' } {/tex} and {tex} ' q ' {/tex} are distinct prime numbers, then the number of distinct imaginary numbers which are {tex} p ^ { \text {th } } {/tex} as well as {tex} q ^ { \text {th } } {/tex} roots of unity are
Question 4 :
If {tex} x ^ { 2 } + 2 a x + b \geq c , \forall x \in R , {/tex} then
Question 5 :
If x<sup>2</sup> - 3x + 2 is a factor of x<sup>4</sup> - px<sup>2</sup> + q, then p,q are -<br>
Question 6 :
If for complex numbers {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } , \arg \left( z _ { 1 } \right) - \arg \left( z _ { 2 } \right) = 0 , {/tex} then {tex} | z _ { 1 } - {/tex} {tex} z _ { 2 } | {/tex} is equal to
Question 7 :
The locus of point {tex} z {/tex} satisfying {tex} \operatorname { Re } \left( \frac { 1 } { z } \right) = k , {/tex} where {tex} k {/tex} is a non- zero real number, is
Question 8 :
If sin θ and cos θ are the roots of the equation <br>lx<sup>2</sup> + mx + n = 0, then -<br>
Question 9 : For real values of x, the expression <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e7464384b0dee620420694e' height='37' width='77' > will assume all real values provided -<br>
Question 10 : The inequality <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e7463d34b0dee6204206872' height='21' width='41' >≥x is satisfied if<br>
Question 11 : The largest negative integer which satisfies <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e7462a24b0dee6204206681' height='39' width='96' > is<br>
Question 12 :
If {tex} | z | = 1 {/tex} then the point representing the complex number {tex} - 1 + {/tex} {tex} 3 z {/tex} will lie on
Question 13 :
If the difference between the roots of the equation {tex} x ^ { 2 } + a x + 1 = 0 {/tex} is less than {tex} \sqrt { 5 } , {/tex} then the set of possible values of {tex} a {/tex} is
Question 14 :
If the equation {tex} a x ^ { 2 } + b x + 6 = 0 {/tex} does not have two distinct real roots, then the least value of {tex} 3 a + b {/tex} is
Question 15 :
If {tex} z = r e ^ { i \theta } , {/tex} then {tex} \left| e ^ { i z } \right| {/tex} is equal to
Question 16 :
For all x ∈ R, if mx<sup>2</sup> - 9mx + 5m + 1 > 0, then m lies in the interval -<br>
Question 17 :
The value of 'a' for which one root of quadratic equation (a<sup>2</sup> - 5a + 3) x<sup>2</sup> + (3a - 1)x + 2 = 0 is twice as large as other is :<br>
Question 18 :
Total number of integral values of a so that <br>x<sup>2</sup> -(a+1) x+a-1 = 0 has integral roots is equal to -<br>
Question 19 :
Let {tex} C _ { 1 } {/tex} and {tex} C _ { 2 } {/tex} are concentric circles of radius {tex}1{/tex} and {tex} 8 / 3 , {/tex} respectively, having centre at {tex} ( 3,0 ) {/tex} on the Argand plane. If the complex number {tex} z {/tex} satisfies the inequality<br>{tex} \log _ { 1 / 3 } \left( \frac { | z - 3 | ^ { 2 } + 2 } { 11 | z - 3 | - 2 } \right) > 1 {/tex} then<br>
Question 20 :
If a, b, c ∈ R and 1 is a root of equation ax<sup>2</sup> + bx + c = 0, then equation 4ax<sup>2</sup> + 3bx + 2c = 0, c ≠ 0 has<br>
Question 21 :
Let {tex} \left| z _ { r } - r \right| \leq r , \forall r = 1,2,3 , \ldots , n . {/tex} Then {tex} \left| \sum _ { r = 1 } ^ { n } z _ { r } \right| {/tex} is less than
Question 22 :
If {tex} z ^ { 2 } + z + 1 = 0 , {/tex} where {tex} z {/tex} is a complex number, then the value of {tex} \left( z + \frac { 1 } { z } \right) ^ { 2 } + \left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 2 } {/tex} {tex} + \left( z ^ { 3 } + \frac { 1 } { z ^ { 3 } } \right) ^ { 2 } + \ldots + \left( z ^ { 6 } + \frac { 1 } { z ^ { 6 } } \right) ^ { 2 } {/tex} is
Question 23 : Solution set of <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e746439cae19162cd5e260a' height='43' width='119' >≥ 0 is <br>
Question 24 :
For the equation 3x<sup>2</sup> + Px + 3 = 0, P > 0 if one root of the equation is square of the other then P equals<br>
Question 25 :
The coefficient of x in the equation x<sup>2</sup>+ px+q =0 was taken as 17 in place of 13, its roots were found to be -2 and -15, The roots of the original equation are -<br>
Question 26 :
The number of complex numbers {tex} z {/tex} satisfying {tex} | z - 3 - i | = | z - 9 - i | {/tex} and {tex} | z - 3 + 3 i | = 3 {/tex} are
Question 27 :
If the roots of the equation ax<sup>2</sup> + bx + c = 0 are in the ratio m : n, then<br>
Question 28 :
The value of a for which the equation {tex} \left( a ^ { 2 } + 4 a + 3 \right) x ^ { 2 } {/tex} {tex} + \left( a ^ { 2 } - a - 2 \right) x + ( a + 1 ) a = 0 {/tex} has more than two roots is
Question 29 :
If {tex} \omega {/tex} is an imaginary cube root of unity then {tex} \left( 1 + \omega - \omega ^ { 2 } \right) ^ { 7 } {/tex} equals
Question 30 :
If 1/i is a root of the equation ax<sup>2</sup> + bx + c = 0, then<br>
Question 31 :
The polynomial (ax<sup>2</sup> + bx + c) (ax<sup>2</sup> -dx -c); ac ≠ 0 has-<br>
Question 32 :
If {tex} k > 0 , | z | = | w | = k {/tex} and {tex} \alpha = \frac { z - \bar { w } } { k ^ { 2 } + z \bar { w } } , {/tex} then {tex} \operatorname { Re } ( \alpha ) {/tex} equals
Question 33 :
The value of 'a' for which the equations x<sup>2</sup> - 3x + a = 0 and x<sup>2</sup> + ax - 3 = 0 have a common root is -<br>
Question 34 :
If {tex} t {/tex} and {tex} c {/tex} are two complex numbers such that {tex} | t | \neq | c | , | t | = 1 {/tex} and {tex} z {/tex} {tex} = ( a t + b ) / ( t - c ) , z = x + i y . {/tex} Locus of {tex} z {/tex} is (where {tex} a , b {/tex} are complex numbers)
Question 35 :
The values of {tex} a {/tex} for which the equation {tex} 2 x ^ { 2 } - 2 ( 2 a + 1 ) x + a ( a - 1 ) = 0 {/tex} has roots {tex} \alpha {/tex} and {tex} \beta {/tex} where {tex} \alpha < a < {/tex} {tex} \beta {/tex} are such that
Question 36 :
If p and q are the roots of the equation x<sup>2</sup> + px + q = 0, then<br>
Question 37 :
If x<sup>2</sup> + 2ax + b ≥ c, ∀ x ∈ R, then
Question 38 :
If roots of the equation 3x<sup>2</sup> + 2(a<sup>2</sup> + 1) x + (a<sup>2</sup> - 3a + 2) = 0 are of opposite signs, then a lies in the interval -<br>
Question 39 :
If {tex} | z - 4 | < | z - 2 | , {/tex} its solution is given by
Question 40 :
The expression y = ax<sup>2</sup> + bx + c has always the same sign as c if -<br>
Question 41 :
If 2x<sup>1/3</sup> + 2x<sup>-1/3</sup> = 5, then x is equal to -<br>
Question 42 :
The minimum value of {tex} a {/tex} for which {tex} a ^ { 2 } x ^ { 2 } + ( 2 a - 1 ) x + 1 {/tex} is non-negative for any real {tex} x {/tex} is
Question 43 :
The locus of the centre of a circle which touches: the circle {tex} \left| z - z _ { 1 } \right| = a {/tex} and {tex} \left| z - z _ { 2 } \right| = b {/tex} externally {tex} ( z , z _ { 1 } \& z _ { 2 } {/tex} are complex numbers) will be
Question 44 :
The equation sin x = x<sup>2</sup> + x + 1 has-<br>
Question 45 :
Product of real roots of the equation<br> {tex}\mathrm{x ^ { 2 } + | x | + 9 = 0}{/tex}
Question 46 :
Let {tex} z , \omega {/tex} be complex numbers such that {tex} z + i \bar { \omega } = 0 {/tex} and {tex} z \omega = \pi {/tex}. Then arg {tex} z {/tex} equals
Question 47 :
If {tex} w = \alpha + i \beta , {/tex} where {tex} \beta \neq 0 {/tex} and {tex} z \neq 1 , {/tex} satisfies the condition that {tex} \left( \frac { w - \overline { W } z } { 1 - z } \right) {/tex} is purely real, then the set of values of {tex} z {/tex} is
Question 50 :
If {tex} z {/tex} is a complex number such that {tex} - \pi / 2 \leq \arg z \leq \pi / 2 , {/tex} then which of the following inequality is true?
