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Mathematics

Consider the function \( f : R \rightarrow R \) defined by . Then \( f \) is :

Let \( L \) be a tangent to the parabola \( y^{2} = 4x - 20 \) at \( (6, 2) \). If \( L \) is also a tangent to the ellipse \( \dfrac{x^{2}}{2} + \dfrac{y^{2}}{b} = 1 \), then the value of \( b \) is equal to:

63.

If the equation of plane passing through the mirror image of a point \( (2, 3, 1) \) with respect to line \( \dfrac{x + 1}{2} = \dfrac{y - 3}{1} = \dfrac{z - 2}{-1} \) and containing the line \( \dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z + 1}{1} \) is \( \alpha x + \beta y + \gamma z = 24 \), then \( \alpha + \beta + \gamma \) is equal to:

64.

If \( x, y, z \) are in arithmetic progression with common difference \( d \), \( x \neq 3d \), and the determinant of the matrix \(\displaystyle \begin{bmatrix} 3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z \end{bmatrix}\) is zero, then the value of \( k^{2} \) is:

65.

Let \( S_{1}, S_{2} \) and \( S_{3} \) be three sets defined as :
\( S_{1} = \{ z \in \mathbb{C} : |z - 1| \le \sqrt{2} \} \)
\( S_{2} = \{ z \in \mathbb{C} : \mathrm{Re}((1-i)z) \ge 1 \} \)
\( S_{3} = \{ z \in \mathbb{C} : \mathrm{Im}(z) \le 1 \} \)
Then the set \( S_{1} \cap S_{2} \cap S_{3} \) :

66.

The number of solutions of the equation \( x + 2 \tan x = \dfrac{\pi}{2} \) in the internal \( [0, 2\pi] \) is :

67.

Let \( O \) be the origin. Let \( \overrightarrow{OP} = x \hat{i} + y \hat{j} - \hat{k} \) and \( \overrightarrow{OQ} = -\hat{i} + 2 \hat{j} + 3x \hat{k} \), \( x, y \in \mathbb{R}, \, x > 0 \), be such that \( |\overrightarrow {PQ}| = \sqrt{20} \) and the vector \( \overrightarrow{OP} \) is perpendicular to \( \overrightarrow{OQ} \). If \( \overrightarrow{OQ} = 3 \hat{i} + z \hat{j} - 7 \hat{k} \), \( z \in \mathbb{R} \), is coplanar with \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \), then the value of \( x^{2} + y^{2} + z^{2} \) is equal to:

68.

In the integral \( \displaystyle \int_{0}^{10} \frac{[\sin 2\pi x]}{e^{\,x-[ x ]}} \, dx = \alpha e^{-1} + \beta e^{-1/2} + \gamma \), where \( \alpha, \beta, \gamma \) are integer and \( [x] \) denotes the greatest integer less than or equal to \( x \), then the value of \( \alpha + \beta + \gamma \) is equal to:

69.

Let a computer program generate only the digits \( 0 \) and \( 1 \) to form a string of binary numbers with probability of occurrence of \( 0 \) at even places be \( \frac{1}{2} \) and probability of occurrence of \( 0 \) at the odd place be \( \frac{1}{3} \). Then the probability that '\( 10 \)' is followed by '\( 01 \)' is equal to:

70.

Let the tangent to the circle \( x^{2} + y^{2} = 25 \) at the point \( R(3,4) \) meet \( x \)-axis and \( y \)-axis at points \( P \) and \( Q \), respectively. If \( r \) is the radius of the circle passing through the origin \( O \) and having centre at the incentre of the triangle \( OPQ \), then \( r^{2} \) is equal to:

71.

The value of \( \displaystyle \lim_{n \to \infty} \frac{[r] + [2r] + \ldots + [nr]}{n^{2}} \), where \( r \) is a non-zero number and \( [r] \) denotes the greatest integer than or equal to \( r \), is equal to:

72.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = e^{-x} \sin x \). If \( F : [0,1] \to \mathbb{R} \) is a differentiable function such that \( F(x) = \displaystyle\int\limits_{0}^{x} f(t)\, dt \), the value of \( \displaystyle \int\limits_{0}^{1} \left(F^{\prime}(x) + f(x)\right)e^{x} dx \) lies in the interval.

73.

If the series \( AB \), \( BC \) and \( CA \) of a triangle \( ABC \) have \( 3 \), \( 5 \) and \( 6 \) interior points respectively, then the total number of triangles that can be constructed using these as vertices, is equal to:

74.

The value of the limit \( \displaystyle\lim_{\theta \to 0} \dfrac{\tan\!\left(\pi \cos^{2}\theta\right)}{\sin\!\left(2\pi \sin^{2}\theta\right)} \) is equal to:

75.

Two tangents are drawn from a point \( P \) to the circle \( x^{2}+y^{2}-2x-4y+4=0 \), such that the angle between these tangent is \( \tan^{-1}\!\left(\dfrac{12}{5}\right) \), where \( \tan^{-1}\!\left(\dfrac{12}{5}\right)\in(0,\pi) \). If the centre of the circle is denotes by \( C \) and these tangents touch the circle at points \( A \) and \( B \), then the ratio of the areas of \( \triangle PAB \) and \( \triangle CAB \) is:

76.

Let \( y=y(x) \) be the solution of the differentiable equation \( \cos x\, (3\sin x+\cos x+3)\,dy=(1+y\sin x\,(3\sin x+\cos x+3))\,dx, \; 0\le x\le \dfrac{\pi}{2}, \; y(0)=0 \). Then, \( y\!\left(\dfrac{\pi}{3}\right) \) is equal to:

77.

The value of \( \displaystyle\sum_{r=0}^{6} \left({}^{6}C_{r}\cdot {}^{6}C_{6-r}\right) \) is equal to:

78.

If the Boolean expression \( (p \land q){*}\,(p \times q) \) is a tautology, then \( * \) and \( \times \) are respectively given by:

79.

In the curve \( y = y(x) \) is the solution of the differentiable equation
\( 2\left(x^{2}+x^{5/4}\right)dy - y\left(x+x^{1/4}\right)dx = 2x^{9/4}dx, \; x>0 \) which passes through the point \( \left(1,\,1-\dfrac{4}{3}\log_e 2\right) \), then the value of \( y(16) \) is equal to :

80.

The number of solutions of the equation \( \sin^{-1}\left[ x^{2}+\dfrac{1}{3}\right]+\cos^{-1}\left[ x^{2}-\dfrac{2}{3}\right]=x^{2} \), for \( x\in[-1,1] \) and \( [ x ] \) denotes the greatest integer less than or equal to \( x \), is:

81.

Let \( P \) be an arbitrary point having sum of the squares of distances from the planes \( x+y+z=0 \), \( lx-nz=0 \) and \( x-2y+z=0 \), equal to \( 9 \). If the locus of the point \( P \) is \( x^{2}+y^{2}+z^{2}=9 \), then the value of \( l-n \) is equal to ________.

82.

Let \(\displaystyle A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \text{ and } B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) such that \( AB=B \) and \( a+d=2021 \), then the value of \( ad-bc \) is equal to ________.

83.

Consider a set of \( 3n \) numbers having variance \( 4 \). In this set, the mean of first \( 2n \) numbers is \( 6 \) and the mean of the remaining \( n \) numbers is \( 3 \). A new set is constructed by adding \( 1 \) into each of first \( 2n \) number, and subtracting \( 1 \) from each of the remaining \( n \) numbers. If the variance of the new set is \( k \), then \( 9k \) is equal to ________.

84.

If \( 1 \), \( \log_{10}(4^{x}-2) \) and \( \log_{10}\!\left(4^{x}+\dfrac{18}{5}\right) \) are in arithmetic progression for a real number \( x \), then the value of determinant \(\displaystyle \begin{vmatrix} 2\left(x - \dfrac{1}{2}\right) & x - 1 & x^2 \\ 1 & 0 & x \\ x & 1 & 0 \end{vmatrix}\)is equal to:

85.

Let \( f:[-1,1]\rightarrow \mathbb{R} \) be defined as \( f(x)=ax^{2}+bx+c \) for all \( x\in [-1,1] \), where \( a,b,c\in \mathbb{R} \) such that \( f(-1)=2 \), \( f'(-1)=1 \) and for \( x\in [-1,1] \) the maximum value of \( f''(x) \) is \( \frac{1}{2} \). If \( f(x)\le \alpha,\, x\in[-1,1] \), then the least value of \( \alpha \) is equal to ________.

86.

Let \( \vec{x} \) be a vector in the plane containing vectors \( \vec{a}=2\hat{i}-\hat{j}+\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}-\hat{k} \). If the vector \( \vec{x} \) is perpendicular to \(( 3\hat{i}+2\hat{j}-\hat{k} )\) and its projection on \( \vec{a} \) is \( \dfrac{17\sqrt{6}}{2} \), then the value of \( |\vec{x}|^{2} \) is equal to ________.

87.

Let the coefficients of third, fourth and fifth terms in the expansion of \( \left(x+\dfrac{a}{x^{2}}\right)^{n},\ x\neq 0, \) be in the ratio \( 12:8:3 \). Then the term independent of \( x \) in the expansion, is equal to ________.

88.

Let \( I_{n}=\displaystyle\int_{1}^{e}x^{19}(\log |x|)^{n}\,dx \), where \( n\in \mathbb{N} \). If \( (20)\,I_{10}= \alpha I_{9}+\beta I_{8} \) for natural numbers \( \alpha \) and \( \beta \), then \( \alpha-\beta \) equals to ________.

89.

Let \( \tan\alpha, \tan\beta \) and \( \tan\gamma \,;\ \alpha,\beta,\gamma \neq \dfrac{(2n-1)\pi}{2},\ n\in \mathbb{N} \) be the slopes of three line segments \( OA \), \( OB \) and \( OC \), respectively, where \( O \) is origin. If circumcentre of \( \triangle ABC \) coincides with origin and its orthocentre lies on \( y \)-axis, then the value of \( \left(\dfrac{\cos 3\alpha+\cos 3\beta+\cos 3\gamma}{\cos\alpha\cos\beta\cos\gamma}\right)^{2} \) is equal to ________.

90.

Let \( f:[-3,1]\to R \) be given as If the area bounded by \( y=f(x) \) and x-axis \( A \), then the value of \( 6A \) is equal to _______.

JEE Main 2021 17th March Evening Shift

Physics Chemistry Mathematics

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If the Boolean expression \( (p \land q){*}\,(p \times q) \) is a tautology, then \( * \) and \( \times \) are respectively given by:

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