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JEE Mains | PYQ

Let \( |\vec{a}|=2 \), \( |\vec{b}|=3 \) and the angle between the vectors \( \vec{a} \) and \( \vec{b} \) be \( \dfrac{\pi}{4} \). Then \( \left|(\vec{a}+2\vec{b})\times(2\vec{a}-3\vec{b})\right|^{2} \) is equal to:

Let \( N \) be the foot of perpendicular from the point \( P(1,-2,3) \) on the line passing through the points \( (4,5,8) \) and \( (1,-7,5) \). Then the distance of \( N \) from the plane \( 2x-2y+z+5=0 \) is:

If \(\displaystyle \lim_{x \to 0} \dfrac{e^{ax} - \cos(bx) - \dfrac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17\), then \(5a^2+b^2\) is equal to:

The area of the region \( \{(x,y): x^{2}\le y\le |x^{2}-4|,\, y\ge 1\} \) is:

If the system of equations
\( 2x+y-z=5 \)
\( 2x-5y+\lambda z=\mu \)
\( x+2y-5z=7 \)
has infinitely many solutions, then \( (\lambda+\mu)^{2}+(\lambda-\mu)^{2} \) is equal to:

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

The coefficient of \(x^5\) in the expansion of \(\left(2x^3-\dfrac{1}{3x^2}\right)^5\) is :

The value of \(\displaystyle \dfrac{e^{-\dfrac{\pi}{4}} + \displaystyle\int\limits_{0}^{\dfrac{\pi}{4}} e^{-x} \tan^{50}x \, dx}{\displaystyle\int\limits_{0}^{\pi/4} e^{-x} \left(\tan^{49}x + \tan^{51}x\right) dx}\) is :

Let \(S=\left\{z\in\mathbb{C}:\ \overline z=i\big(z^2+\operatorname{Re}(\overline z)\big)\right\}\). Then \(\displaystyle\sum_{z\in S}\lvert z\rvert^2\) is equal to :

Let for a triangle \(ABC\),
\(\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}\)
\(\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}\)
\(\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta\hat{k}\).
If \(\delta>0\) and the area of the triangle \(ABC\) is \(5\sqrt{6}\), then \(\overrightarrow{CB}\cdot\overrightarrow{CA}\) is equal to :

The range of \(f(x)\)  \(=4\sin^{-1}\left(\dfrac{x^2}{x^2+1}\right)\) is  :

Let \( \alpha , \beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x+2=0 \). Then \( \alpha^{14}+\beta^{14} \) is equal to:

, \( |A|=2 \). . If \( \left|\,2\,\operatorname{adj}\!\left(2\,\operatorname{adj}(2A)\right)\!\right|=32^{\,n} \), then \( 3n+\alpha \) is equal to:

Let \( a_{1},a_{2},a_{3},\ldots \) be a G.P. of increasing positive numbers. Let the sum of its \( 6^{\text{th}} \) and \( 8^{\text{th}} \) terms be \( 2 \) and the product of its \( 3^{\text{rd}} \) and \( 5^{\text{th}} \) terms be \( \dfrac{1}{9} \). Then \( 6(a_{2}+a_{4})(a_{4}+a_{6}) \) is equal to:

Let the centre of a circle \( C \) be \( (\alpha,\beta) \) and its radius \( r<8 \). Let \( 3x+4y=24 \) and \( 3x-4y=32 \) be two tangents and \( 4x+3y=1 \) be a normal to \( C \). Then \( (\alpha-\beta+r) \) is equal to:

Let \( (\alpha,\beta) \) be the centroid of the triangle formed by the lines \( 15x-y=82 \), \( 6x-5y=-4 \) and \( 9x+4y=17 \). Then \( \alpha+2\beta \) and \( 2\alpha-\beta \) are the roots of the equation:

The statement \( \big(p \land (\sim q)\big) \lor \big((\sim p)\land q\big) \lor \big((\sim p)\land (\sim q)\big) \) is equivalent to ________.

The line, that is coplanar to the line \( \dfrac{x+3}{-3}=\dfrac{y-1}{1}=\dfrac{z-5}{5} \), is:

The plane, passing through the points \( (0,-1,2) \) and \( (-1,2,1) \) and parallel to the line passing through \( (5,1,-7) \) and \( (1,-1,-1) \), also passes through the point:

For \( x\in(-1,1] \), the number of solutions of the equation \( \sin^{-1}x=2\tan^{-1}x \) is equal to __________.

The mean and standard deviation of the marks of \( 10 \) students were found to be \( 50 \) and \( 12 \) respectively. Later, it was observed that two marks \( 20 \) and \( 25 \) were wrongly read as \( 45 \) and \( 50 \) respectively. Then the correct variance is __________.

Let \( A = \{-4, -3, -2, 0, 1, 3, 4\} \) and \( R = \{(a, b) \in A \times A : b = |a| \text{ or } b^2 = a + 1\} \) be a relation on \( A \). Then the minimum number of elements, that must be added to the relation \( R \) so that it becomes reflexive and symmetric, is _________.

Let \( f_n=\displaystyle\int\limits_{0}^{\pi/2}\left(\sum_{k=1}^{n}\sin^{k-1}x\right)\left(\sum_{k=1}^{n}(2k-1)\sin^{k-1}x\right)\cos x \, dx,\; n\in \mathbb{N} \). Then \( f_{21}-f_{20} \) is equal to ______.

Let \( [\alpha] \) denote the greatest integer \( \le \alpha \). Then \( [\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \ldots + [\sqrt{120}] \) is equal to _____.

If \( y=y(x) \) is the solution of the differential equation \( \dfrac{dy}{dx}+\dfrac{4x}{(x^{2}-1)}y=\dfrac{x+2}{(x^{2}-1)^{5/2}},\, x>1 \) such that \( y(2)=\dfrac{2}{9}\log_{e}(2+\sqrt{3}) \) and \( y(\sqrt{2})=\alpha \log_{e}(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma},\; \alpha,\beta,\gamma \in \mathbb{N} \), then \( \alpha\beta\gamma \) is equal to __________.

The remainder, when \( 7^{103} \) is divided by \( 17 \), is __________.

Let \( f(x)=\displaystyle\sum_{k=1}^{10}k\,x^{k},\; x\in \mathbb{R} \). If \( 2f(2)+f'(2)=119(2)^{n}+1 \) then \( n \) is equal to __________.

Total number of \( 3 \)-digit numbers that are divisible by \( 6 \) and can be formed by using the digits \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 5 \) with repetition, is _________.

The foci of a hyperbola are \( (\pm 2, 0) \) and its eccentricity is \( \dfrac{3}{2} \). A tangent, perpendicular to the line \( 2x + 3y = 6 \), is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x and y-axes are \( a \) and \( b \) respectively, then \( |6a| + |5b| \) is equal to _________.