Question 1 :
The centre of a regular hexagon is at the pointz = i. If one of its vertices is at 2 + i, then the adjacent vertices of 2 + i are at the points
Question 2 :
If α, β be the roots of the quadratic equation x<sup>2</sup> + x + 1 = 0, then the equation whose roots are α<sup>19</sup>, β<sup>7</sup> is
Question 3 :
If x is real, then expression $\frac{x + 2}{2x^{2} + 3x + 6}$ takes all values in the interval
Question 4 :
If α, β are the roots of the equation ax<sup>2</sup> + bx + c = 0, then the value of $\frac{1}{\text{aα} + b} + \frac{1}{\text{aβ} + b}$ is equal to
Question 5 :
The real part of (1−cosθ + 2isin θ)<sup> − 1</sup> is
Question 6 :
If $\left( \frac{1 + i}{1 - i} \right)^{x} = 1$, then
Question 8 :
The value of x in the given equation $4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}\ $is
Question 9 :
If z<sub>r</sub>(r = 0, 1, 2, …, 6) be the roots of the equation (z+1)<sup>7</sup> + z<sup>7</sup> = 0, then $\sum_{r = 0}^{6}{\text{Re\ }\left( z_{r} \right) =}$
Question 10 :
$\sqrt{- 1 - \sqrt{1 - \sqrt{1 - ...\infty}}\ }$ is equal to
Question 11 :
If x<sup>2</sup> + px + 1 is a factor of the expression ax<sup>3</sup> + bx + c, then
Question 12 :
For all $`x^{'},$ x<sup>2</sup> + 2ax + (10 − 3a) > 0, then the interval in which $`a'$ lies, is
Question 13 :
If $|\mathcal{z} - i\text{Re}\ \left( \mathcal{z} \right) = \left| \mathcal{z} - \text{Im}\ \left( \mathcal{z} \right) \right|\ \left( \text{where}\ i = \sqrt{- 1} \right)$, then 𝓏 lies on
Question 14 :
If α<sub>0</sub>, α<sub>1</sub>, α<sub>2</sub>, …, α<sub>n − 1</sub> be the n<sup>th</sup> roots of unity, then the value of $\sum_{i = 0}^{n - 1}\frac{\alpha_{i}}{3 - \alpha_{i}}$ is equal to
Question 15 :
The smallest positive integer n for which $\left( \frac{1 + i}{1 - i} \right)^{n} = 1$, is
Question 16 :
If ω is a complex cube root of unity, then the value of ω<sup>99</sup> + ω<sup>100</sup> + ω<sup>101</sup> is
Question 17 :
If ω is an imaginary cube root of unity, then (1+ω−ω<sup>2</sup>)<sup>7</sup> equals
Question 18 :
All the values of m for which both roots of the equation x<sup>2</sup> − 2mx + m<sup>2</sup> − 1 = 0 are greater than -2 but less than 4 lie in the interval
Question 19 :
$\left| \frac{1}{2}\left( z_{1} + z_{2} \right) + \sqrt{z_{1}z_{2}} \right| + \left| \frac{1}{2}\left( z_{1} + z_{2} \right) - \sqrt{z_{1}z_{2}} \right|$ is equal to
Question 20 :
The value of x<sup>4</sup> + 9x<sup>3</sup> + 35x<sup>2</sup> − x + 4 for $x = - 5 + 2\sqrt{- 4}$ is
Question 21 :
If $\sin^{- 1}{x - \cos^{- 1}{x = \frac{\pi}{6},}}$ then x is
Question 22 : <p>The sum of the infinite series</p> <p>$\sin^{- 1}{\left( \frac{1}{\sqrt{2}} \right) + \sin^{- 1}\left( \frac{\sqrt{2} - 1}{\sqrt{6}} \right)} + \sin^{- 1}{\left( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{12}} \right) + ...}$</p> <p>$+ ... + \sin^{- 1}{\left( \frac{\sqrt{n} - \sqrt{(n - 1)}}{\sqrt{\{ n\left( n + 1 \right)\}}} \right) + ...}$ is</p>
Question 23 :
The value of cos 480<sup>∘</sup>.sin 150<sup>∘</sup> + sin 600<sup>∘</sup>.soc 390<sup>∘</sup> is equal to
Question 24 :
If $A = \left\{ x:\frac{\pi}{6} \leq x \leq \frac{\pi}{3} \right\}$ and f(x) = cos x − x(1 + x), then f(A) is equal to
Question 25 : <p>If $\cos\frac{x}{2}.\cos\frac{x}{2^{2}}\ldots..\cos\frac{x}{2^{n}} = \frac{\sin x}{2^{n}\sin{\ \frac{x}{2^{n}}}}$ , then</p> <p>$\frac{1}{2}\tan{\frac{x}{2} + \frac{1}{2^{2}}\tan{\frac{x}{2^{2}} + ... + \frac{1}{2^{n}}\tan\frac{x}{2^{n}}}}$ is</p>
Question 26 :
If y = (1+tanA)(1 − tan B), where $A - B = \frac{\pi}{4}$, then (y+1)<sup>y + 1</sup> is equal to
Question 27 :
If $\sin^{- 1}{x + \sin^{- 1}{y = \frac{\pi}{2}}}$, then cos<sup> − 1</sup>x + cos<sup> − 1</sup>y is equal to
Question 28 :
If $\left( \tan^{- 1}x \right)^{2} + \left( \cot^{- 1}x \right)^{2} = \frac{5\pi^{2}}{8}$, then x equals
Question 29 :
If twice the square of the diameter of a circle is equal to half the sum of the squares of the sides of incribed triangle ABC, then sin<sup>2</sup>A + sin<sup>2</sup>C is equal to
Question 30 :
If the interior angles of a polygon are in A.P. with common difference 5<sup>∘</sup> and the smallest angle 120<sup>∘</sup>, then the number of sides of the polygon is
Question 31 :
If {tex} a _ { 1 } , a _ { 2 } , a _ { 3 } , \dots , a _ { 2 n + 1 } {/tex} are in A.P., then<br>{tex} \frac { a _ { 2 n + 1 } - a _ { 1 } } { a _ { 2 n + 1 } + a _ { 1 } } + \frac { a _ { 2 n } - a _ { 2 } } { a _ { 2 n } + a _ { 2 } } + \cdots + \frac { a _ { n + 2 } - a _ { n } } { a _ { n + 2 } + a _ { n } } {/tex} is equal to<br>
Question 32 :
If ${p}^{th}$, ${q}^{th}$, ${r}^{th}$ and ${s}^{th}$ terms of an $A.P$ are in $G.P$ then $p-q, q-r, r-s$ are in $G.P$.
Question 33 :
If one geometric mean G and two arithmetic means p and q be inserted between two numbers, then G<sup>2</sup> is equal to
Question 34 :
If $|a|,|b|,|c| < 1$ and $a,b,c\:\epsilon A.P$. then $(1+ a + a^{2} +...\infty),(1+b+b^{2} +...\infty),(1+c+c^{2} +...\infty)$ are in
Question 35 :
If 1, log<sub>9</sub>(3<sup>1-x</sup> + 2) and log<sub>3</sub>(4.3<sup>x</sup> -1) are in A.P. ,then x is equal to -
Question 36 :
Coefficient of {tex} x ^ { 18 } {/tex} in {tex} \left( 1 + x + 2 x ^ { 2 } + 3 x ^ { 3 } + \cdots + 18 x ^ { 18 } \right) ^ { 2 } {/tex} is equal to
Question 37 :
If {tex} a , b , c \in R ^ {+ } {/tex} then the minimum value of {tex} a \left( b ^ { 2 } + c ^ { 2 } \right) + b \left( c ^ { 2 } + a ^ { 2 } \right) {/tex} {tex} + c \left( a ^ { 2 } + b ^ { 2 } \right) {/tex} is equal to
Question 38 :
If $s_{n}=\sum_{n=1}^{n}\frac{1+2+2^{2}+... to\ n \ terms}{2^{n}}$ then $s_{n}$ is equal to
Question 39 :
If the first and the (2n-1)th terms of an A.P., G.P. and H.P. are equal and their nth terms are a, b, c respectively , then
Question 40 :
If {tex} a , b , {/tex} and {tex} c {/tex} are in A.P., {tex} p , q , {/tex} and {tex} r {/tex} are in H.P. and {tex} a p , b q , {/tex} and cr are in G.P., then {tex} \frac { p } { r } + \frac { r } { p } {/tex} is equal to
Question 41 :
Concentric circles of radii {tex} 1,2,3 , \ldots , 100 \mathrm { cm } {/tex} are drawn. The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. Then, the total area of the green regions in sq. cm is equal to
Question 42 :
If the product of n positive numbers is unity , then their sum is
Question 43 :
The sum to 50 terms of the series {tex} \frac { 3 } { 1 ^ { 2 } } + \frac { 5 } { 1 ^ { 2 } + 2 ^ { 2 } } + \frac { 7 } { 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } } {/tex} {tex} + \cdots {/tex} is
Question 45 :
If a, b and c are distinct positive real numbers and a<sup>2</sup> +b<sup>2</sup> +c<sup>2</sup> =1, then ab + bc +ca is
Question 46 :
If {tex} b _ { i } = 1 - a _ { i } ,n a = \sum _ { i = 1 } ^ { n } a _ { i } n b = \sum _ { i = 1 } ^ { n } b _ { i } {/tex}, then {tex} \sum _ { i = 1 } ^ { n } a _ { i } b _ { i } + \sum _ { i = 1 } ^ { n } \left( a _ { i } - a \right) ^ { 2 } = {/tex}
Question 47 :
If {tex} \log _ { 2 } \left( 5 \times 2 ^ { x } + 1 \right) , \log _ { 4 } \left( 2 ^ { 1 - x } + 1 \right) {/tex} and {tex}1 {/tex} are in A.P., then {tex} x {/tex} equals
Question 48 :
{tex} f ( x ) = \frac { ( x - 2 ) ( x - 1 ) } { ( x - 3 ) } , \forall x > 3 . {/tex} The minimum value of {tex} f ( x ) {/tex} is
Question 49 :
The sum of an infinite G.P. is 57 and the sum of their cubes is {tex} 9747 , {/tex} then common ratio of the G.P. is
Question 50 :
If x, y, z are in HP, then the value of expression log<sub>e</sub> (x + z) + log<sub>e</sub> (x - 2y + z) will be -
Question 52 :
Consider the ten numbers {tex} a r , a r ^ { 2 } , a r ^ { 3 } , \ldots , a r ^ { 10 } . {/tex} If their sum is 18 and the sum of their reciprocals is 6 then the product of these ten numbers, is
Question 53 :
If {tex} a , b , {/tex} and {tex} c {/tex} are in G.P. and {tex} x , y , {/tex} respectively, be arithmetic means between {tex} a , b {/tex} and {tex} b , c , {/tex} then the value of {tex} \frac { a } { x } + \frac { c } { y } {/tex} is
Question 54 :
If {tex} S {/tex} denotes the sum to infinity and {tex} S _ { n } {/tex} the sum of {tex} n {/tex} terms of the series {tex} 1 + \frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \cdots , {/tex} such that {tex} S - S _ { n } < \frac { 1 } { 1000 } , {/tex} then the least value of {tex} n {/tex} is
Question 55 : The first three terms of a geometric sequence are x, y, z and these have the sum equal to 42. If the middle term y is multiplied by 5/4, the numbers x, <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e8728a2e6d3604eaa92f812' height='36' width='21' >, z now form an arithmetic sequence. The largest possible value of x, is-
Question 56 :
Let {tex} \alpha , \beta \in R . {/tex} If {tex} \alpha , \beta ^ { 2 } {/tex} be the roots of quadratic equation {tex} x ^ { 3 } - p x {/tex} {tex} + 1 = 0 {/tex} and {tex} \alpha ^ { 2 } , \beta {/tex} be the roots of quadratic equation {tex} x ^ { 2 } - q x + 8 {/tex} {tex} = 0 , {/tex} then the value of {tex} ^ { \prime } r ^ { \prime } {/tex} if {tex} \frac { r } { 8 } {/tex} be arithmetic mean of {tex} p {/tex} and {tex} q {/tex} is
Question 57 :
If the sum of the series $1+\displaystyle \frac{2}{x}$+$\displaystyle \frac{4}{x^{2}}$+$\displaystyle \frac{8}{x^{3}}$+...to $\infty $ is the finite number then
Question 58 :
If ${S}_{n}=\sum _{ r=1 }^{ n }{ \cfrac { 1+2+{ 2 }^{ 2 }+..Sum\quad to\quad r\quad terms }{ { 2 }^{ r } } } $, then ${S}_{n}$ is equal to
Question 59 :
After striking the floor, a certain ball rebounds {tex} ( 4 / 5 ) ^ { \text {th } } {/tex} of height from which it has fallen. Then the total distance that it travels before coming to rest, if it is gently dropped from a height of {tex} 120 \mathrm { m } {/tex} is
Question 60 :
In a G.P. the first, third and fifth terms may be considered as the first, fourth and sixteenth terms of an A.P. Then the fourth term of the A.P., knowing that its first term is 5 is
Question 61 :
{tex} \underset{{ x \rightarrow 0 }}\lim \frac { \sqrt { \frac { 1 } { 2 } ( 1 - \cos 2 x ) } } { x } = {/tex}
Question 62 :
{tex}\underset { x \rightarrow 0 }\lim \frac { \sin x + \log ( 1 - x ) } { x ^ { 2 } } {/tex} is equal to
Question 63 : <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870b8219f8d44d3a17f7a4' height='43' width='147' > , given that f'(2) = 6 and f'(1) = 4
Question 65 :
Let f : R → R be a function defined by f(x) = max {x, x<sup>3</sup>}. The set of all points where f(x) is NOT differentiable is
Question 66 : If f(x) = <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870b4619f8d44d3a17f6e8' height='45' width='121' >, then fof(x) is given by
Question 67 :
The function {tex} y = e ^ { - | x | } {/tex} is
Question 68 : The range of the function <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870d1719f8d44d3a17fcb4' height='32' width='112' > is [2004]
Question 70 : Let f(x) = <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870c9419f8d44d3a17fb0a' height='60' width='100' > be continuous and differentiable every where. The a and b are -
Question 71 :
The function {tex} y = | \sin x | {/tex} is continuous for any {tex} x {/tex} but it is not differentiable at
Question 72 :
{tex}\underset{ x \rightarrow 0 } \lim \frac { 1 + \sin x - \cos x + \log ( 1 - x ) } { x ^ { 3 } } {/tex} equals
Question 73 : y = <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870cff19f8d44d3a17fc67' height='37' width='55' > where t =<img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870d00e6d3604eaa92eeb5' height='36' width='31' >, then the number of points of discontinuities of y = f(x), x ∈ R is-
Question 74 : Let ƒ(x) = x - [x], where [x] denotes the greatest integer ≤ x and g(x) =<img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870a8e75ed294f2c7c3be7' height='28' width='29' ><img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870cdf75ed294f2c7c42b3' height='44' width='72' > , then g(x) is equal to -
Question 75 : If f is a real valued differentiable function satisfying <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e870d42e6d3604eaa92ef8a' height='29' width='402' >, then f(1) equals
Question 76 :
The line y = x - 6 is a normal to the parabola y<sup>2</sup>=8x. Then the foot of the normal is
Question 77 :
The angle between the tangents drawn at the end points of the latus rectum of parabola {tex} y ^ { 2 } = 4 a x {/tex} is
Question 78 :
The equation of the common tangent to the curves y<sup>2</sup> = 8x and xy = -1 is
Question 79 :
The centre of the circle r<sup>2</sup> = 2 - 4r cos θ + 6r sin θ is
Question 80 :
If the line {tex} y = m x + c {/tex} is a tangent to the parabola {tex} y ^ { 2 } = 4 a ( x + a ) {/tex} then {tex} m a + \frac { a } { m } {/tex} is equal to
Question 81 :
Radius of the circle having centre (3, 4) and touching the circle x<sup>2</sup> + y<sup>2</sup> = 4 can be
Question 82 : The locus of the point which moves such that the ratio of its distance from two fixed point in the plane is always a constant <img style='object-fit:contain' src="https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5ea7bc22399925718ac6a20c"> is
Question 83 :
The curve {tex} 16 x ^ { 2 } + 8 x y + y ^ { 2 } - 74 x - 78 y + 212 = 0 {/tex} represents
Question 84 :
The vertex of the parabola {tex} 3 x - 2 y ^ { 2 } - 4 y + 7 = 0 {/tex} is
Question 85 :
If the normal to the parabola y<sup>2</sup> = 4ax at the point (at<sup>2</sup>, 2at) cuts the parabola again at (aT<sup>2</sup>, 2aT), then
Question 86 :
If the minor axis of an ellipse subtends an angle of 60<sup>∘</sup> at each focus of the ellipse, then its eccentricity is
Question 87 :
<font>If hyperbola x</font><sup><font>2</font></sup><font> - y</font><sup><font>2</font></sup><font> = a</font><sup><font>2</font></sup><font> and xy = c</font><sup><font>2</font></sup><font> are of same size, then:</font></p>
Question 88 :
The normal at (ap<sup>2</sup>, 2 ap) on y<sup>2</sup> = 4 ax, meets the curve again at (aq<sup>2</sup>, 2 aq) then
Question 89 : <font>The locus of the center of the circle for which one end of the diameter is (3, 3) while the other end lies on the line x + y = 4 is </font><font color="#000000"><font>-</font></font></p> <p align="justify"> <font color="#000000">
Question 90 :
If a point (x, y) = (tan θ + sin θ, tan θ − sin θ), then locus of (x, y) is
