Recent questions in JEEMAIN PAST PAPERS

If the circles $x^2 + y^2 + 2ax + cy + a = 0 $ and $x^2 + y^2-3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $Q$, then the line $5x+by-a = 0$ passes through $P$ and $Q$ for :

asked Nov 5, 2018 by pady_1

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If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of a triangle is :

asked Nov 5, 2018 by pady_1

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If non-zero numbers $a, b, c$ are in HP, then the straight line $\frac{x}{a} +\frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point. That point is :

asked Nov 5, 2018 by pady_1

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For any vector $\overrightarrow{a}$, the value of $(\overrightarrow{a} \times \hat{i})^2+ (\overrightarrow{a} \times \hat{j})^2 + ( \overrightarrow{a} \times \hat{k})^2$ is equal to :

asked Nov 5, 2018 by pady_1

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The distance between the line $\overrightarrow{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda (\hat{i} - \hat{j} + 4 \hat{k}) $ and the plane $\overrightarrow{r} . (\hat{i} + 5 \hat{j} + \hat{k}) = 5$ is :

asked Nov 5, 2018 by pady_1

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If the plane $2ax - 3ay + 4az + 6 = 0$ passes through the mid point of the line joining the centres of the spheres $x^2 + y^2 +z^2 + 6x - 8y - 2z = 13$ and $ x^2 + y^2 + z^2 - 10x +4y - 2z = 8 $, than $a$ equals :

asked Nov 5, 2018 by pady_1

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The parabolas $y^2 = 4x$ and $x^2 = 4y$ divide the square region bounded by the lines $x = 4, y = 4$ and the co-ordinate axes. If $S_1, S_2, S_3$ are respectively the areas of these parts numbered from top to bottom, then $S_1 : S_2 : S_3$ is :

asked Nov 5, 2018 by pady_1

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The area enclosed between the curve $y= \log_e (x+e)$ and the co-ordinate axes is :

asked Nov 5, 2018 by pady_1

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If $\begin {align}I_1 = \int_0^1 2^{x^2} \; \; dx , I_2 = \int_0^1 2 ^{x^3} \;\; dx, I_3 = \int_1^2 2^{x^2} \; \; dx \; and \; I_4 = \int_1^2 2^{x^3} \;\; dx, \end{align}$ then :

asked Nov 5, 2018 by pady_1

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Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x-axis$ and the ordinates $x=\frac{\pi}{4}$ and $x=\beta > \pi/4$ is $( \beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta)$. Then $f(\frac{\pi}{2})$ is :

asked Nov 5, 2018 by pady_1

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Let $f:R \to R$ be a differentiable function having $f(2) = 6, f'(2) =(\frac{1}{48}) $. Then $\displaystyle\lim_{x \to 2} \int_6^{f(x)} \frac{4t^3}{x-2} dt$ equals :

asked Nov 5, 2018 by pady_1

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$\begin{align}\int \left\{\frac{(\log x -1)}{1 + (\log x)^2}\right\}^2 dx \end{align}$ is equal to :

asked Nov 5, 2018 by pady_1

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A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\; cm^2/min$. When the thickness of ice is 15 cm, then the rate at which the thickness of ice decreases, is :

asked Nov 5, 2018 by pady_1

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The line parallel to the $x-axis$ and passing through the intersection of the lines $ax+2by+3b =0$ and $bx-2ay - 3a = 0$, where $(a,b) \neq (0,0)$ is :

asked Nov 5, 2018 by pady_1

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If $x \frac{dy}{dx} = y (\log y - \log x + 1)$, then the solution of the equation is :

asked Nov 5, 2018 by pady_1

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Let $\alpha$ and $\beta$ be the distinct roots of $ax^2 + bx + c = 0 $, then $\displaystyle{\lim_{x \to \alpha} \frac{1- \cos (ax^2 + bx + c)}{(x-\alpha)^2}}$ is equal to :

asked Nov 5, 2018 by pady_1

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A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched ?

asked Nov 5, 2018 by pady_1

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The normal to the curve $x = a (\cos \theta + \theta \sin \theta), y = a (\sin \theta - \theta \cos \theta)$ at any point $'\theta'$ is such that :

asked Nov 5, 2018 by pady_1

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If in a $\Delta ABC$, the altitudes from the vertices $A, B, C$ on opposite sides are in HP, then $\sin A, \sin B, \sin C$ are in :

asked Nov 5, 2018 by pady_1

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If $\cos^{-1} x- \cos^{-1} \frac{y}{2} = \alpha$, then $4x^2 - 4xy \cos \alpha + y^2$ is equal to :

asked Nov 5, 2018 by pady_1

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In a triangle ABC, let $\angle{C} = \pi / 2$, if $r$ is the inradius and $R$ is the circumradius of the triangle ABC, then $2(r+R)$ equals :

asked Nov 5, 2018 by pady_1

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If $x = \displaystyle\sum_{n=0}^{\alpha} a^n,y=\displaystyle\sum_{n=0}^{\alpha} b^n,z= \displaystyle\sum_{n=0}^{\alpha} c^n$ where $a, b, c$ are in AP and $|a| < 1, |b| < 1, |c|<1$, then $x, y, z$ are in :

asked Nov 5, 2018 by pady_1

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If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3/2} - (1+\frac{1}{2}x)^3}{(1-x)^{1/2}}$ may be approximated as :

asked Nov 5, 2018 by pady_1

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If $f$ is a real-valued differentiable function satisfying $|f(x) - f(y)| \leq (x-y)^2, x, y \in R$ and $f(0) = 0$, then $f(1)$ equals :

asked Nov 5, 2018 by pady_1

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Let $f$ be differentiable for all $x$. If $f(1) = -2$ and $f'(x) \geq 2$ for $x \in [1,6]$, then :

asked Nov 5, 2018 by pady_1

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Suppose $f(x)$ is differentiable at $x=1$ and $ \displaystyle \lim_{h \to 0} \frac{1}{h} f(1+h) =5 $, then $f'(1)$ equals :

asked Nov 5, 2018 by pady_1

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If the roots of the equation $x^2 - bx + c = 0$ be two consecutive integers, then $b^2 - 4c$ equals :

asked Nov 5, 2018 by pady_1

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The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a-2)x -a - 1 = 0$ assume the least value is :

asked Nov 5, 2018 by pady_1

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The system of equations $\alpha x + y + z = \alpha -1$ $x + \alpha y + z = \alpha -1 $ $x + y + \alpha z = \alpha - 1$ has no solution, if $\alpha$ is :

asked Nov 5, 2018 by pady_1

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If $a^2+b^2+c^2 = -2$ and $f(x) = \begin{vmatrix} 1+a^2x & (1+b^2)x & (1+c^2) x \\ (1+a^2)x & 1+b^2x & (1+c^2)x \\ (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}$, then $f(x)$ is a polynomial of degree :

asked Nov 5, 2018 by pady_1

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If $w = \frac{z}{z - \frac{1}{3} i } $ and $|w| = 1$, then $z$ lies on :

asked Nov 5, 2018 by pady_1

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If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2| = |z_1| + |z_2|$, then arg $z_1$ - arg $z_2$ is equal to :

asked Nov 5, 2018 by pady_1

1 answer

Let $ f : (-1, 1) \to B$ be a function defined by$f(x) = \tan^{-1} \frac{2x}{1-x^2}$, then $f$ is both one-one and onto when $B$ is the interval :

asked Nov 5, 2018 by pady_1

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If the coefficient of $x^7$ in $\begin{bmatrix} ax^2 + \begin{pmatrix} \frac{1}{bx} \end{pmatrix} \end{bmatrix}^{11} $ equals the coefficient of $x^{-7}$ in $\begin{bmatrix} ax^2 - \begin{pmatrix} \frac{1}{bx} \end{pmatrix} \end{bmatrix}^{11} $, then $a$ and $b$ satisfy the relation :

asked Nov 5, 2018 by pady_1

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If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then which one of the following holds for all $n \geq 1$, by the principle of mathematical induction ?

asked Nov 5, 2018 by pady_1

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The value of $^{50}C_4 $ $\displaystyle\sum_{r = 1}^{6}$ $^{56-r}C_3$ is :

asked Nov 5, 2018 by pady_1

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If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number :

asked Nov 5, 2018 by pady_1

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In a triangle PQR, $\angle{R} = \frac{\pi}{2}$. If $\tan (\frac{P}{2})$ and $\tan (\frac{Q}{2})$ are the roots of $ax^2 + bx + c = 0, a \neq 0$, then :

asked Nov 5, 2018 by pady_1

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If the coefficients of $r^{th}$, $(r+1)^{th}$ and $(r+2)^{th}$ terms in the binomial expansion of $(1 + y)^m$ are in AP, then $m$ and $r$ satisfy the equation :

asked Nov 5, 2018 by pady_1

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ABC is a triangle. Forces $\overrightarrow{P}, \overrightarrow{Q}, \overrightarrow{R}$ acting along $IA, IB$ and $IC$ respectively are in equilibrium, where $I$ is the incentre of $\Delta ABC$. Then $\overrightarrow{P} : \overrightarrow{Q} : \overrightarrow{R}$ is :

asked Nov 5, 2018 by pady_1

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The differential equation representing the family of curves $y^2 = 2c (x + \sqrt{c})$, where $c > 0$, is a parameter, is of order and degree as follows:

asked Nov 5, 2018 by pady_1

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Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ is :

asked Nov 5, 2018 by pady_1

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$ \displaystyle{\lim_{x \to \infty} [\frac{1}{n^2} \sec^2 \frac{1}{n^2} + \frac{2}{n^2} \sec^2 \frac{4}{n^2} + . . . . . . + \frac{n}{n^2} \sec^2 1]}$

asked Nov 5, 2018 by pady_1

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If the cube roots of unity are 1, $\omega, \omega^2$, then the roots of the equation $(x-1)^3 + 8 = 0$, are :

asked Nov 5, 2018 by pady_1

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If $A^2 - A + I = 0$, then the inverse of A is :

asked Nov 5, 2018 by pady_1

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Let $R = \{ (3,3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6) \}$ be a relation on the set $A = \{ 3, 6, 9, 12\}$. The relation is :

asked Nov 5, 2018 by pady_1

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If in a frequency distribution, the Mean and Medium are 21 and 22 respectively, then its Mode is approximately :

asked Nov 5, 2018 by pady_1

1 answer

Let P be the point (1, 0) and Q a point on the locus $y^2=8x$. The locus of mid point of PQ is :

asked Nov 5, 2018 by pady_1

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If $C$ is the mid point of AB and P is any point outside AB, then :

asked Nov 5, 2018 by pady_1

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$t_{1/4}$ can be taken as the time taken for the concentration of a reactant to drop to $\frac{3}{4}$ of its initial value. If the rate constant for a first order reaction is $k$, the $t_{1/4}$ can be written as :

asked Nov 5, 2018 by pady_1

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