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JEE Mains | PYQ

Let \( z_1, z_2 \in \mathbb{C} \) be the distinct solutions of the equation \( z^2 + 4z - (1 + 12i) = 0 \). Then \( |z_1|^2 + |z_2|^2 \) is equal to:

If \(f : N \to Z\) is defined by  and \(\displaystyle \sum_{n=1}^{k} f(n) = 98\), then \(k\) is equal to:

Let \( M \) be a \( 3\times 3 \) matrix such that then \( x+y+z \) equals:

If the sum of the first \( 10 \) terms of the series \( \dfrac{1}{1+1^{4}\times 4}+\dfrac{2}{1+2^{4}\times 4}+\dfrac{3}{1+3^{4}\times 4}+\dfrac{4}{1+4^{4}\times 4}+\cdots \) is \( \dfrac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to:

Let \( A_1, A_2, A_3, \ldots, A_{39} \) be \( 39 \) arithmetic means between the numbers \( 59 \) and \( 159 \). Then the mean of \( A_{25}, A_{28}, A_{31} \) and \( A_{36} \) is equal to:

The coefficient of \( x^2 \) in the expansion of \( \left(2x^2 + \frac{1}{x}\right)^{10} \), \( x \neq 0 \), is:

The probabilities that players \( A \) and \( B \) of a team are selected for the captaincy for a tournament are \( 0.6 \) and \( 0.4 \), respectively. If \( A \) is selected the captain, the probability that the team wins the tournament is \( 0.8 \) and if \( B \) is selected the captain, the probability that the team wins the tournament is \( 0.7 \). Then the probability that the team wins the tournament is:

A box contains \( 5 \) blue, \( 6 \) yellow and \( 4 \) red balls. The number of ways, of drawing \( 8 \) balls containing at least two balls of each colour, is:

A variable \( X \) takes values \( 0, 0, 2, 6, 12, 20, \ldots , n(n-1) \) with frequencies \( {}^{n}C_{0}, {}^{n}C_{1}, {}^{n}C_{2}, {}^{n}C_{3}, {}^{n}C_{4}, {}^{n}C_{5}, \ldots , {}^{n}C_{n} \), respectively. If the mean of this data is \( 60 \), then its median is:

Let the point \( P \) be the vertex of the parabola \( y = x^{2} - 6x + 12 \). If a line passing through the point \( P \) intersects the circle \( x^{2} + y^{2} - 2x - 4y + 3 = 0 \) at the point \( R \) and \( S \), then the maximum value of \( (PR + PS)^{2} \) is:

Let the directrix of the parabola \( P: y^{2} = 8x \), cut x-axis at the point \( A \). Let \( B(\alpha, \beta),\ \alpha > 1 \), be a point on \( P \) such that the slope of \( AB \) is \( \frac{3}{5} \). If \( BC \) is a focal chord of \( P \), then six times the area of \( \triangle ABC \) is:

Let the eccentricity of a hyperbola satisfy the equation \( 6e^{2} - 11e + 3 = 0 \). If the foci of the hyperbola are \( (3, 5) \) and \( (3, -4) \), then the length of its latus rectum is:

Let a triangle \( \triangle PQR \) be such that \( P \) and \( Q \) lie on the line \( \frac{x+3}{8} = \frac{y-4}{2} = \frac{x+1}{2} \) and are at a distance of \( 6 \) units from \( R(1, 2, 3) \). If \( (\alpha, \beta, \gamma) \) is the centroid of \( \triangle PQR \), then \( \alpha + \beta + \gamma \) is equal to:

If the distance of the point \( (a, 2, 5) \) from the image of the point \( (1, 2, 7) \) in the line \( \frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2} \) is \( 4 \), then the sum of all possible value of \( a \) is equal to:

Let \( O \) be the origin, \( \overrightarrow{OP} = \vec{a} \) and \( \overrightarrow{OQ} = \vec{b} \). If \( R \) is the point on \( \overrightarrow{OP} \) such that \( \overrightarrow{OP} = 5\,\overrightarrow{OR} \), and \( M \) is the point such that \( \overrightarrow{OQ} = 5\,\overrightarrow{RM} \), then \( \overrightarrow{PM} \) is equal to:

Let \( f(x) =\displaystyle \lim_{y \to 0} \dfrac{(1 - \cos(xy)) \tan(xy)}{y^{3}} \). Then the number of solutions of the equation \( f(x) = \sin x, \, x \in \mathbb{R} \) is:

Let \( \left(2^{1-a}+2^{1+a}\right), f(a), \left(3^a+3^{-a}\right) \) be in A.P. and \( \alpha \) be the minimum value of \( f(a) \). Then the value of the integral \( \displaystyle\int\limits_{\log_{e}(\alpha-1)}^{\log_{e}(\alpha)} \dfrac{dx}{\left(e^{2x}-e^{-2x}\right)} \) is:

Let \( f:[1,\infty)\to \mathbb{R} \) be a differentiable function defined as \( f(x)=\int_{1}^{x} f(t)\,dt+(1-x)(\log_{e}x-1)+e \). Then the value of \( f(f(1)) \) is:

Let \( f(x) \) and \( g(x) \) be twice differentiable functions satisfying \( f''(x)=g''(x) \) for all \( x\in \mathbb{R} \), \( f'(1)=2g'(1)=4 \) and \( g(2)=3f(2)=9 \). Then \( f(25)-g(25) \) is equal to:

Let \( A=\{1,4,7\} \) and \( B=\{2,3,8\} \). Then the number of elements, in the relation \( R=\{((a_{1},b_{1}),(a_{2},b_{2}))\in(A\times B)\times(A\times B):a_{1}+b_{2}\text{ divides }a_{2}+b_{1}\} \) is ______________ .

From the point \( (-1,-1) \), two rays are sent making along of \( 45^\circ \) with line \( x+y=0 \). These rays get reflected from the mirror \( x+2y=1 \). If the equations of the reflected rays are \( ax+by=9 \) and \( cx+dy=7 \), \( a,b,c,d\in \mathbb{Z} \), then the value of \( ad+bc \) is ______________ .

If \( S=\left\{\theta\in[-\pi,\pi]:\cos\theta\cos\dfrac{50}{2}=\cos70\cos\dfrac{70}{2}\right\} \), then \( n(S) \) is equal to ______________ .

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f(x) + 3 f\!\left(\frac{\pi}{2} - x\right) = \sin x, \; x \in \mathbb{R} \). Let the maximum value of \( f \) on \( \mathbb{R} \) be \( \alpha \). If the area of the region bounded by the curves \( g(x) = x^2 \) and \( h(x) = \beta x^3, \; \beta > 0 \), is \( \alpha^2 \), then \( 30\beta^3 \) is equal to ________.

Let \( y=y(x) \) be the solutions of the differential equation \( (\tan x)^{\tfrac{1}{2}}\,dy=\left(\sec^3 x-(\tan x)^{\tfrac{3}{2}}\,y\right)dx \),\(0 < x < \frac{\pi}{2},\)\(y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}.\) If \(y\left(\frac{\pi}{3}\right) = \frac{4}{5}\alpha,\)   \(\text{then } \alpha^4\) equals_______________