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JEE Mains | PYQ

Let \( S_n \) be the sum of the first \( n \) terms of an arithmetic progression. If \( S_{3n}=3S_{2n} \) then the value of \( \dfrac{S_{4n}}{S_{2n}} \) is:

Let \( f:[0,\infty)\rightarrow [0,\infty) \) be defined as \( f(x)=\displaystyle\int\limits_{0}^{x}[ y]r \,dy \), where \( [ x] \) is the greatest integer less than or equal to \( x \). Which of the following is true?

Let the foot of perpendicular from a point \( P(1,2,-1) \) to the straight line \( L:\ \dfrac{x}{1}=\dfrac{y}{0}=\dfrac{z}{-1} \) be \( N \). Let a line be drawn from \( P \) parallel to the plane \( x+y+2z=0 \) which meets \( L \) at point \( Q \). If \( \alpha \) is the acute angle between the lines \( PN \) and \( PQ \), then \( \cos\alpha \) is equal to ____________.

Let \( f:\mathbb{R}\to\mathbb{R} \) be defined as

Where \( [x] \) is the greatest integer less than or equal to \( x \). If \( f \) is continuous at \( x=2 \), then \( \lambda+\mu \) is equal to:

The value of the definite integral \( \displaystyle\int\limits_{\pi/24}^{5\pi/24} \frac{dx}{1+\sqrt[3]{\tan 2x}} \) is:

The sum of all values of \( x \) in \( [0,2\pi] \) for which \( \sin x+\sin 2x+\sin 3x+\sin 4x=0 \) is equal to:

The number of real roots of the equation \( e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^{x}+1=0 \) is:

The values of \( a \) and \( b \) for which the system of equations
\( 2x+3y+6z=8 \)
\( x+2y+az=5 \)
\( 3x+5y+9z=b \)
has no solution, are:

Let an ellipse \( E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a^{2}>b^{2} \), passes through \( \left(\dfrac{\sqrt{3}}{2},1\right) \) and has eccentricity \( \dfrac{1}{\sqrt{3}} \). If a circle, centered at focus \( F(\alpha,0),\ \alpha>0 \) of \( E \) and radius \( \dfrac{2}{\sqrt{3}} \) intersects \( E \) at two points \( P \) and \( Q \), then \( PQ^{2} \) is equal to:

If \( b \) is very small as compared to the value of \( a \), so that the cube and other higher powers of \( \dfrac{b}{a} \) can be neglected in the identify
\( \dfrac{1}{a-b} + \dfrac{1}{a-2b} + \dfrac{1}{a-3b} + \ldots + \dfrac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3 \), then the value of \( \gamma \) is:

Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} = 1 + x e^{y-x}, \\ -\sqrt{2} < x < \sqrt{2}, \\ y(0) = 0 \) then, the minimum value of \( y(x), x \in (-\sqrt{2 }, 2 )\) is equal to:

Let \( g : \mathbb{N} \to \mathbb{N} \) be defined as
\( g(3n+1) = 3n + 2, \)
\( g(3n+2) = 3n + 3, \)
\( g(3n+3) = 3n + 1, \\ \text{for all } n \ge 0. \)
Then which of the following statements is true?

Let a parabola \( P \) be such that its vertex and focus lie on the positive \( x \)-axis at a distance \( 2 \) and \( 4 \) units from the origin, respectively. If tangents are drawn from \( O(0,0) \) to the parabola \( P \) which meet \( P \) at \( S \) and \( R \), then the area (in sq. units) of \( \triangle S O R \) is equal to:

The Boolean expression \( (p \Rightarrow q) \land (q \Rightarrow \sim p) \) is equivalent to:

The area (in sq. units) of the region, given by the set \( \{(x,y) \in R \times R \mid x \ge 0, 2x^2 \le y \le 4 - 2x\} \) is:

The locus of the centroid of the triangle formed by any point \( P \) on the hyperbola \( 16x^{2}-9y^{2}+32x+36y-164=0 \) and its foci is:

A spherical gas balloon of radius \( 16 \) meter subtends an angle \( 60^\circ \) at the eye of the observer \( A \) while the angle of elevation of its center from the eye of \( A \) is \( 75^\circ \). Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is:

Let \( 9 \) distinct balls be distributed among \( 4 \) boxes, \( B_1,B_2,B_3 \) and \( B_4 \). If the probability that \( B_3 \) contains exactly \( 3 \) balls is \( k\left(\dfrac{3}{4}\right)^{9} \) then \( k \) lies in the set:

Let \( f(x)=3\sin^{4}x+10\sin^{3}x+6\sin^{2}x-3, \ x \in \left[-\dfrac{\pi}{6},\dfrac{\pi}{2}\right] \). Then, \( f \) is:

The ratio of the coefficient of the middle term in the expansion of \( (1+x)^{20} \) and the sum of the coefficients of two middle terms in expansion of \( (1+x)^{19} \) is:

Let \(\displaystyle M = \left\{ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \{\pm 3, \pm 2, \pm 1, 0\} \right\}\). Define \( f : M \to \mathbb{Z} \), as \( f(A)=\det(A) \), for all \( A \in M \), where \( \mathbb{Z} \) is set of all integers. Then the number of \( A \in M \) such that \( f(A)=15 \) is equal to ____________ .

There are \( 5 \) students in class \( 10 \), \( 6 \) students in class \( 11 \) and \( 8 \) students in class \( 12 \). If the number of ways, in which \( 10 \) students can be selected from them so as to include at least \( 2 \) students from each class and at most \( 5 \) students from the total \( 11 \) students of class \( 10 \) and \( 11 \) is \( 100k \), then \( k \) is equal to ____________ .

Let \( y=y(x) \) be solution of the following differential equation \( e^{y}\dfrac{dy}{dx}-2e^{y}\sin x+\sin x \cos^{2} x=0 , \; y\!\left(\dfrac{\pi}{2}\right)=0 \).
If \( y(0)=\log_{e}\!\left(\alpha+\beta e^{-2}\right) \), then \( 4(\alpha+\beta) \) is equal to____________ .

If the value of \( \left(1+\dfrac{2}{3}+\dfrac{6}{3^{2}}+\dfrac{10}{3^{3}}+\ldots \text{ up to } \infty \right)^{\log_{(0.25)}\!\left(\dfrac{1}{3}+\dfrac{1}{3^{2}}+\dfrac{1}{3^{3}}+\ldots \text{ up to } \infty \right)} \) is \( l \) then \( l^{2} \) is equal to __________ .

Consider the following frequency distribution:

If the sum of all frequencies is \( 584 \) and median is \( 45 \), then \( \left| \alpha - \beta \right| \) is equal to __________ .

Let \(\displaystyle S = \left\{ n \in \mathbb{N} \;\middle|\; \begin{pmatrix} 0 & i \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \forall a, b, c, d \in \mathbb{R} \right\}\), where \( i=\sqrt{-1} \). Then the number of \( 2 \)-digit numbers in the set \( S \) is __________ .

The term independent of ' \( x \) ' in the expansion of \( \left( \dfrac{x+1}{x^{2/3}-x^{1/3}+1}-\dfrac{x-1}{x-x^{1/2}} \right)^{10} \), where \( x \neq 0,1 \) is equal to __________ .

Let \( \vec{p}=2\hat{i}+3\hat{j}+\hat{k} \) and \( \vec{q}=\hat{i}+2\hat{j}+\hat{k} \) be two vectors. If a vector \( \vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}) \) is perpendicular to each of the vectors \( (\vec{p}+\vec{q}) \) and \( (\vec{p}-\vec{q}) \) and \( |\vec{r}|=\sqrt{3} \), then \( |\alpha|+|\beta|+|\gamma| \) is equal to __________ .

If \( \alpha ,\beta \) are roots of the equation \( x^{2}+5(\sqrt{2})\,x+10=0 \), \( \alpha>\beta \) and \( P_{n}=\alpha^{n}-\beta^{n} \) for each positive integer \( n \), then the value of \( \left( \dfrac{P_{17}P_{20}+5\sqrt{2}\,P_{17}P_{19}}{P_{18}P_{19}+5\sqrt{2}\,P_{18}^{2}} \right) \) is equal to __________ .