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JEE Mains | PYQ

If \( S \) and \( S' \) are the foci of the ellipse \( \frac{x^{2}}{18} + \frac{y^{2}}{9} = 1 \) and \( P \) be a point on the ellipse, then \( \min(SP \cdot S'P) + \max(SP \cdot S'P) \) is equal to:

Let \( a_{1},a_{2},a_{3},\ldots \) be in an A.P. such that \(  \displaystyle \sum_{k=1}^{12} 2a_{2k-1} = -\frac{72}{5} a_{1},\ a_{1} \ne 0 \). If \(  \displaystyle \sum_{k=1}^{n} a_{k} = 0 \), then \( n \) is:

For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \( \lim_{x \to 0} \dfrac{x^{2}\sin(\alpha x) + (\gamma-1)e^{\ x^{2}}}{\sin 2x - \beta x} = 3 \), then \( \beta + \gamma - \alpha \) is equal to:

Let the focal chord \( PQ \) of the parabola \( y^{2} = 4x \) make an angle of \( 60^\circ \) with the positive \( x \)-axis, where \( P \) lies in the first quadrant. If the circle, whose one diameter is \( PS \), \( S \) being the focus of the parabola, touches the \( y \)-axis at the point \( (0, \alpha) \), then \( 5\alpha^{2} \) is equal to:

The number of sequences of ten terms, whose terms are either \( 0 \) or \( 1 \) or \( 2 \), that contain exactly five \( 1 \)s and exactly three \( 2 \)s, is equal to:

Let \( z \) be a complex number such that \( |z| = 1 \). If \( \dfrac{2 + k^2 {z}}{k + \bar{z}} = kz, \; k \in \mathbb{R} \), then the maximum distance of \( k + i k^2 \) from the circle \( |z - (1 + 2i)| = 1 \) is:

If \( \theta \in [-2\pi, 2\pi] \), then the number of solutions of \( 2\sqrt{2}\cos^{2}\theta + (2-\sqrt{6})\cos\theta - \sqrt{3} = 0 \), is equal to:

The term independent of \( x \) in the expansion of \( \left( \dfrac{x+1}{(x^{2/3}+1-x^{1/3})} - \dfrac{x-1}{(x-x^{1/2})} \right)^{10} , \; x>1, \) is:

Let the vertices \( Q \) and \( R \) of the triangle \( PQR \) lie on the line \( \dfrac{x+3}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} \), \( QR = 5 \) and the coordinates of the point \( P \) be \( (0, 2, 3) \). If the area of the triangle \( PQR \) is \( \dfrac{m}{n} \) then:

If \( \vec{a} \) is a nonzero vector such that its projections on the vectors $2\hat{i} - \hat{j} + 2\hat{k}, \hat{i} + 2\hat{j} - 2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along \( \vec{a} \) is:

The largest \( n \in \mathbb{N} \) such that \( 3^{n} \) divides \( 50! \) is:

If the system of linear equations
\( 3x + y + \beta z = 3 \)
\( 2x + \alpha y - z = -3 \)
\( x + 2y + z = 4 \)
has infinitely many solutions, then the value of \( 22\beta - 9\alpha \) is:

Let one focus of the hyperbola \( H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be at \( (\sqrt{10}, 0) \) and the corresponding directrix be \( x = \frac{9}{\sqrt{10}} \). If \( e \) and \( l \) respectively are the eccentricity and the length of the latus rectum of \( H \), then \( 9(e^2 + l) \) is equal to:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( (\sin x \cos y)(f(2x+2y)-f(2x-2y))=(\cos x \sin y)(f(2x+2y)+f(2x-2y)) \), for all \( x, y \in \mathbb{R} \). If \( f'(0)=\frac{1}{2} \), then the value of \( 24\, f'\!\left(\frac{5\pi}{3}\right) \) is:

Let \( P_n=\alpha^n+\beta^n,\, n\in \mathbb{N} \). If \( P_{10}=123,\, P_9=76,\, P_8=47 \) and \( P_1=1 \), then the quadratic equation having roots \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:

Let \( ABCD \) be a tetrahedron such that the edges \( AB \), \( AC \) and \( AD \) are mutually perpendicular. Let the areas of the triangles \( ABC \), \( ACD \) and \( ADB \) be \( 5 \), \( 6 \) and \( 7 \) square units respectively. Then the area (in square units) of the \( \triangle BCD \) is equal to:

Let \( A \) be the set of all functions \( f : Z \to Z \) and \( R \) be a relation on \( A \) such that \( R=\{(f,g): f(0)=g(1) \text{ and } f(1)=g(0)\} \). Then \( R \) is:

Let \( a \in \mathbb{R} \) and \( A \) be a matrix of order \( 3 \times 3 \) such that \( \det(A) = -4 \) and  , where \( I \) is the identity matrix of order \( 3 \times 3 \). If \( \det\!\big((a+1)\operatorname{adj}((a-1)A)\big) \) is \( 2^{m}\,3^{n} \), \( m, n \in \{  0 \), \( 1 \), \( 2 \), \( \ldots \), \( 20 )\} \), then \( m+n \) is equal to:

If the function \( f(x)=2x^3-9ax^2+12a^2x+1 \), where \( a>0 \), attains its local maximum and local minimum values at \( p \) and \( q \), respectively, such that \( p^2=q \), then \( f(3) \) is equal to:

Let $[\cdot]$ denote the greatest integer function. If $\int_{0}^{e^3} \left[\frac{1}{e^{x-1}}\right] dx = \alpha - \log_e 2$, then $\alpha^3$ is equal to _______.

Let \( f:\mathbb{R}\rightarrow \mathbb{R} \) be a thrice differentiable odd function satisfying \( f'(x)\ge 0 \), \( f''(x)=f(x) \), \( f(0)=0 \), \( f'(0)=3 \). Then \( 9\,f(\log_{e}3) \) is equal to __________.

If the area of the region \( \{(x,y): |4-x^{2}|\le y \le x^{2},\, y\le 4,\, x\ge 0\} \) is \( \left(\frac{80\sqrt{2}}{\alpha}-\beta\right) \), \( \alpha, \beta \in \mathbb{N} \), then \( \alpha+\beta \) is equal to __________.

The absolute difference between the squares of the radii of the two circles passing through the point \( (-9,\,4) \) and touching the lines \( x+y=3 \) and \( x-y=3 \), is equal to __________.

Three distinct numbers are selected randomly from the set \( (1, 2, 3, \ldots, 40) \). If the probability, that the selected numbers are in an increasing G.P., \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to __________.