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JEE MAIN 2017 PYQ

JEE MAIN PYQ 2017
Let $a,b,c\in \mathbb{R}$. If $f(x)=ax^{2}+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy,\ \forall x,y\in \mathbb{R}$, then $\displaystyle \sum_{n=1}^{10} f(n)$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

JEE MAIN PYQ 2017
A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

JEE MAIN PYQ 2017
If for a positive integer $n$, the quadratic equation $x(x+1) + (x+1)(x+2) + \ldots + (x+n-1)(x+n) = 10n$ has two consecutive integral solutions, then $n$ is equal to :





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Solution

JEE MAIN PYQ 2017
If $S$ is the set of distinct values of $b$ for which the following system of linear equations

$x + y + z = 1$

$x + ay + z = 1$

$ax + by + z = 0$
has no solution, then $S$ is :





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Solution

JEE MAIN PYQ 2017
If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$, then $\operatorname{adj}(3A^{2} + 12A)$ is equal to :





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Solution

JEE MAIN PYQ 2017
The function $f:\mathbb{R}\to\left[-\dfrac12,\dfrac12\right]$ defined as $f(x)=\dfrac{x}{1+x^{2}}$, is





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Solution

JEE MAIN PYQ 2017
If $5\left(\tan^{2}x-\cos^{2}x\right)=2\cos2x+9$, then the value of $\cos4x$ is:





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Solution

JEE MAIN PYQ 2017
Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If $\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7} \end{vmatrix} = 3k$, then $k$ is equal to :





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Solution

JEE MAIN PYQ 2017
If two different numbers are taken from the set {0,1,2,3,...,10} then the probability that their sum as well as absolute difference are both multiple of 4, is :





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Solution

JEE MAIN PYQ 2017
For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = $\dfrac{1}{4}$ and P(All the three events occur simultaneously) =$ \dfrac{1}{16}$. Then the probability that at least one of the events occurs, is :





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Solution

JEE MAIN PYQ 2017
Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. Let $\vec{c}$ be a vector such that $|\vec{c}-\vec{a}|=3$, $|(\vec{a}\times\vec{b})\times\vec{c}|=3$ and the angle between $\vec{c}$ and $\vec{a}\times\vec{b}$ is $30^\circ$. Then $\vec{a}\cdot\vec{c}$ is equal to :





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Solution

JEE MAIN PYQ 2017
Let k be an integer such that the triangle with vertices (k,-3k), (5,k) and (-k,2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :





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Solution

JEE MAIN PYQ 2017
If $(2+\sin x)\dfrac{dy}{dx}+(y+1)\cos x=0$ and $y(0)=1$, then $y\left(\dfrac{\pi}{2}\right)$ is equal to :





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Solution

JEE MAIN PYQ 2017
The area (in sq. units) of the region ${(x,y): x\ge 0,\ x+y\le 3,\ x^{2}\le 4y\ \text{and}\ y\le 1+\sqrt{x}}$ is :





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Solution

JEE MAIN PYQ 2017
Let $I_n=\int \tan^{n}x,dx,\ (n>1)$. If $I_4+I_6=a\tan^{5}x+bx^{5}+C$, where $C$ is a constant of integration, then the ordered pair $(a,b)$ is equal to :





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Solution

JEE MAIN PYQ 2017
The integral $\displaystyle \int_{\pi/4}^{3\pi/4}\dfrac{dx}{1+\cos x}$ is equal to :





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Solution

JEE MAIN PYQ 2017
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :





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Solution

JEE MAIN PYQ 2017
If for $x\in\left(0,\dfrac14\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt{x}}{1-9x^{3}}\right)$ is $\sqrt{x}\cdot g(x)$, then $g(x)$ equals :





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Solution

JEE MAIN PYQ 2017
$\lim_{x\to \frac{\pi}{2}} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$ equals :





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Solution

JEE MAIN PYQ 2017
The number of real values of $\lambda$ for which the system of linear equations
$2x+4y-\lambda z=0$
$4x+\lambda y+2z=0$
$\lambda x+2y+2z=0$
has infinitely many solutions, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
If (27)999 is divided by 7, then the remainder is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
Let $A$ be any $3\times 3$ invertible matrix. Then which one of the following is not always true?





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Solution

JEE MAIN PYQ 2017
Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x-1 and it leaves remainder 6 when divided by x+1, then:





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Solution

JEE MAIN PYQ 2017
If the arithmetic mean of two numbers a and b, a>b>0, is five times their geometric mean, then $\dfrac{a+b}{a-b}$ is equal to :





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Solution

JEE MAIN PYQ 2017
If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is :





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Solution

JEE MAIN PYQ 2017
Let f(x) = 210.x + 1 and g(x)=310.x $-$ 1. If (fog) (x) = x, then x is equal to :





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Solution

JEE MAIN PYQ 2017
Let $z\in\mathbb{C}$, the set of complex numbers. Then the equation $2|z+3i|-|z-i|=0$ represents :





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Solution

JEE MAIN PYQ 2017
Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\dfrac{3}{5}$ and the distance between its foci is $6$, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices at the vertices of the ellipse, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The value of $\tan^{-1}\left[\dfrac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right]$, $|x|<\dfrac{1}{2}$, $x\neq0$, is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
If

$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{ 0 & {\cos x} & { - \sin x} \cr {\sin x} & 0 & {\cos x} \cr {\cos x} & {\sin x} & 0 \cr } } \right| = 0} \right\},$

then $\sum\limits_{x \in S} {\tan \left( {{\pi \over 3} + x} \right)} $ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are $\dfrac{3}{4},\ \dfrac{1}{2}$ and $\dfrac{5}{8}$ respectively, then the probability that the target is hit by $P$ or $Q$ but not by $R$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is :





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Solution

JEE MAIN PYQ 2017
The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat{i}-6\hat{j}$ and $3\hat{i}+4\hat{j}-12\hat{k}$, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $\cos^{-1}\left(\dfrac{1}{7}\right)$ and $\sec^{-1}(7)$ at the center respectively, then the distance between these chords, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The integral $\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx$ equals :





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Solution

JEE MAIN PYQ 2017
The area (in sq. units) of the smaller portion enclosed between the curves $x^2 + y^2 = 4$ and $y^2 = 3x$, is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The locus of the point of intersection of the straight lines, $tx-2y-3t=0$, $x-2ty+3=0\ (t\in\mathbb{R}),$ is :





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Solution

JEE MAIN PYQ 2017
The curve satisfying the differential equation $y,dx-(x+3y^{2}),dy=0$ and passing through the point (1,1), also passes through the point :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
If y = ${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$ then (x2 $-$ 1) ${{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$ is equal to :





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Solution

JEE MAIN PYQ 2017
If a point P has co-ordinates (0,-2) and Q is any point on the circle $x^{2}+y^{2}-5x-y+5=0$, then the maximum value of $(PQ)^{2}$ is :





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Solution

JEE MAIN PYQ 2017
The integral

$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $

$\left( {0 < x < {\pi \over 2}} \right)$ is equal to :

(where C is a constant of integration)





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Solution

JEE MAIN PYQ 2017
$ \displaystyle \lim_{x\to 3} \frac{\sqrt{3x}-3}{\sqrt{2x}-\sqrt{6}} $ is equal to:





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Solution

JEE MAIN PYQ 2017
If three positive numbers $a, b$ and $c$ are in A.P. such that $abc = 8$, then the minimum possible value of $b$ is :





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Solution

JEE MAIN PYQ 2017
The number of ways in which $5$ boys and $3$ girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never sit adjacent to each other, is :





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Solution

JEE MAIN PYQ 2017
The coefficient of $x^{-5}$ in the binomial expansion of $\left( \dfrac{x+1}{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1} ;-; \dfrac{x-1}{x - x^{\frac{1}{2}}} \right)^{10}$, where $x \neq 0,1$, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
For two $3 \times 3$ matrices $A$ and $B$, let $A + B = 2B^T$ and $3A + 2B = I_3$, where $B^T$ is the transpose of $B$ and $I_3$ is $3 \times 3$ identity matrix. Then:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The function $f : \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x - 5\left\lfloor \dfrac{\pi x}{5} \right\rfloor$, where $\mathbb{N}$ is the set of natural numbers and $\lfloor x \rfloor$ denotes the greatest integer $\le x$, is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is $60^\circ$. If the area of the quadrilateral is $4\sqrt{3}$, then the perimeter of the quadrilateral is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The equation $\operatorname{Im}\left( \dfrac{iz - 2}{z - i} \right) + 1 = 0,; z \in \mathbb{C},; z \neq i$ represents a part of a circle having radius equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The sum of all real values of $x$ satisfying the equation $2^{(x-1)(x^{2} + 5x - 50)} = 1$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
A value of $x$ satisfying the equation $\sin!\big(\cot^{-1}(1+x)\big) = \cos!\big(\tan^{-1} x\big)$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happen is $\dfrac{1}{12}$ and the probability that neither $E$ nor $F$ happens is $\dfrac{1}{2}$. Then a value of $\dfrac{P(E)}{P(F)}$ is:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
The sum of $100$ observations and the sum of their squares are $400$ and $2475$, respectively. Later on, three observations, $3,4$ and $5$, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is:





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Solution

JEE MAIN PYQ 2017
If the vector $\vec{b} = 3\vec{j} + 4\vec{k}$ is written as the sum of a vector $\vec{b_1}$ parallel to $\vec{a} = \vec{i} + \vec{j}$ and a vector $\vec{b_2}$ perpendicular to $\vec{a}$, then $\vec{b_1} \times \vec{b_2}$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
From a group of $10$ men and $5$ women, four-member committees are to be formed, each of which must contain at least one woman. Then the probability for these committees to have more women than men is:





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Solution

JEE MAIN PYQ 2017
The eccentricity of an ellipse having centre at the origin, axes along the coordinate axes, and passing through the points $(4,-1)$ and $(-2,2)$ is:





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Solution

JEE MAIN PYQ 2017
If $\displaystyle \int_{1}^{2} \frac{dx}{(x^{2} - 2x + 4)^{\tfrac{3}{2}}} = \frac{k}{k+5}$, then $k$ is equal to:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
If $2x = y^{\tfrac{1}{5}} + y^{-\tfrac{1}{5}}$ and $(x^{2} - 1)\dfrac{d^{2}y}{dx^{2}} + \lambda x\dfrac{dy}{dx} + ky = 0$, then $\lambda + k$ is equal to:





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Solution

JEE MAIN PYQ 2017
A line drawn through the point $P(4,7)$ cuts the circle $x^{2} + y^{2} = 9$ at the points $A$ and $B$. Then $PA \cdot PB$ is equal to:





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Solution

JEE MAIN PYQ 2017
Let $f$ be a polynomial function such that $f(3x) = f'(x)\cdot f''(x)$ for all $x \in \mathbb{R}$. Then:





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
A square, of each side $2$, lies above the $x$-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^\circ$ with the positive direction of the $x$-axis, then the sum of the $x$-coordinates of the vertices of the square is:





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Solution

JEE MAIN PYQ 2017
If $\displaystyle f\left(\frac{3x-4}{3x+4}\right) = x + 2,; x \ne -\frac{4}{3}$ and $\displaystyle \int f(x),dx = A\ln|1-x| + Bx + C,$ then the ordered pair $(A,B)$ is equal to (where $C$ is a constant of integration):





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Solution

JEE MAIN PYQ 2017
The value of k for which the function

$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$

is continuous at x = ${\pi \over 2},$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

JEE MAIN PYQ 2017
Let $a,b\in\mathbb{R}$, $(a\neq 0)$. If the function $f$ defined as $f(x)= \begin{cases} \dfrac{2x^{2}}{a}, & 0\le x<1 \\ a, & 1\le x<\sqrt{2} \\ \dfrac{2b^{2}-4b}{x^{3}}, & \sqrt{2}\le x<\infty \end{cases}$ is continuous in the interval $[0,\infty)$, then an ordered pair $(a,b)$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution


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