JEE MAIN Mathematics Previous Year Questions (PYQs) – Page 13 of 275

The values of $\lambda$ and $\mu$ for which the system of linear equations \[ \begin{aligned} x + y + z &= 2,\\ x + 2y + 3z &= 5,\\ x + 3y + \lambda z &= \mu \end{aligned} \] has infinitely many solutions are, respectively:

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity ${1 \over 3}$ and the distance of the nearer focus from this directrix is ${8 \over {\sqrt {53} }}$, then the equation of the other directrix can be :

Let the solution curve $y=f(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{xy}{x^{2}-1}=\dfrac{x^{4}+2x}{\sqrt{1-x^{2}}}$, $x\in(-1,1)$, pass through the origin. Then $\displaystyle \int_{-\sqrt{3}/2}^{\sqrt{3}/2} f(x)\,dx$ is equal to:

The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\frac{x+\pi/4}{\,2-\cos 2x\,}\,dx$ is:

Given that the inverse trigonometric functions assume principal values only. Let $x,y\in[-1,1]$ such that $\cos^{-1}x-\sin^{-1}y=\alpha$, with $-\dfrac{\pi}{2}\le\alpha\le\pi$. Then, the minimum value of $x^{2}+y^{2}+2xy\sin\alpha$ is:

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m$, $n$ are coprime numbers, then $m+n$ is equal to :

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\ \dfrac{x^{2}}{4^{2}}+\dfrac{y^{2}}{3^{2}}=1$ at the points $A$ and $B$. Let the line $y=x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is

Let $a_1,a_2,\dots,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\dots,10$ and $S$ be the set of pairs $(r,k)$, $r,k\in\mathbb{N}$, for which $ \begin{vmatrix} \log_e(a_1^{\,r}a_2^{\,k}) & \log_e(a_2^{\,r}a_3^{\,k}) & \log_e(a_3^{\,r}a_4^{\,k})\\ \log_e(a_4^{\,r}a_5^{\,k}) & \log_e(a_5^{\,r}a_6^{\,k}) & \log_e(a_6^{\,r}a_7^{\,k})\\ \log_e(a_7^{\,r}a_8^{\,k}) & \log_e(a_8^{\,r}a_9^{\,k}) & \log_e(a_9^{\,r}a_{10}^{\,k}) \end{vmatrix} =0. $ Then the number of elements in $S$, is –

Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x $ \in $ R. If f(x) attains maximum value at $\alpha $ and g(x) attains minimum value at $\beta $, then $\mathop {\lim }\limits_{x \to -\alpha \beta } {{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)} \over {{x^2} - 6x + 8}}$ is equal to :

Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equidistant from the two axes then :