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Let \( (\alpha, \beta, \gamma) \) be the image of the point \( P (2, 3, 5) \) in the plane \( 2x + y - 3z = 6 \). Then \( \alpha + \beta + \gamma \) is equal to:

62.

For any vector \( \vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \), with \( 10|a_i|<1, \ i=1,2,3 \), consider the following statements:
(A) : \( \max\{|a_1|,|a_2|,|a_3|\}\le |\vec{a}| \)
(B) : \( |\vec{a}|\le 3\max\{|a_1|,|a_2|,|a_3|\} \)

63.

Let sets \( A \) and \( B \) have \( 5 \) elements each. Let the mean of the elements in sets \( A \) and \( B \) be \( 5 \) and \( 8 \) respectively and the variance of the elements in sets \( A \) and \( B \) be \( 12 \) and \( 20 \) respectively. A new set \( C \) of \( 10 \) elements is formed by subtracting \( 3 \) from each element of \( A \) and adding \( 2 \) to each element of \( B \). Then the sum of the mean and variance of the elements of \( C \) is ________ .

64.

Let \( f(x)=[x^2-x]+|-x+[x]| \), where \( x\in \mathbb{R} \) and \( [t] \) denotes the greatest integer less than or equal to \( t \). Then, \( f \) is:

65.

An organization awarded \( 48 \) medals in event ‘\( A \)’, \( 25 \) in event ‘\( B \)’ and \( 18 \) in event ‘\( C \)’. If these medals went to total \( 60 \) men and only \( 5 \) men got medals in all the three events, then, how many received medals in exactly two of three events?

66.

The number of integral solutions \( x \) of \( \log_{\left(x+\dfrac{7}{2}\right)}\left(\dfrac{x-7}{2x-3}\right)^2 \ge 0 \) is:

67.

Consider ellipses \( E_k : kx^2 + k^2y^2 = 1, \, k = 1, 2, \ldots, 20 \). Let \( C_k \) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \( E_k \). If \( r_k \) is the radius of the circle \( C_k \), then the value of \( \displaystyle \sum_{k=1}^{20} \frac{1}{r_k^2} \) is:

68.

The value of integral \( \displaystyle \int_{-\log_e 2}^{\log_e 2} e^x \Big( \log_e( e^x + \sqrt{1 + e^{2x}} ) \Big) \, dx \) is equal to:

69.

The number of triplets \( (x, y, z) \), where \( x, y, z \) are distinct non negative integers satisfying \( x + y + z = 15 \), is:

70.

The number of elements in the set \( S = \left\{ \theta \in [0, 2\pi] : 3\cos^4 \theta - 5\cos^2 \theta - 2\sin^6 \theta + 2 = 0 \right\} \) is:

71.

Let \( y = y(x) \) be a solution curve of the differential equation, \( (1 - x^2y^2)\,dx = y\,dx + x\,dy \). If the line \( x = 1 \) intersects the curve \( y = y(x) \) at \( y = 2 \) and the line \( x = 2 \) intersects the curve \( y = y(x) \) at \( y = \alpha \), then a value of \( \alpha \) is:

72.

Let \( S = \{ M = [a_{ij}],\ a_{ij} \in \{0, 1, 2\},\ 1 \le i, j \le 2 \} \) be a sample space and \( A = \{ M \in S : M \text{ is invertible} \} \) be an event. Then \( P(A) \) is equal to:

73.

Let \( R \) be a rectangle given by the lines \( x=0 \), \( x=2 \), \( y=0 \) and \( y=5 \). Let \( A(\alpha, 0) \) and \( B(0, \beta) \), \( \alpha \in [0, 2] \) and \( \beta \in [0, 5] \), be such that the line segment \( AB \) divides the area of the rectangle \( R \) in the ratio \( 4:1 \). Then, the mid-point of \( AB \) lies on a :

74.

Let \( A \) be a \( 2 \times 2 \) matrix with real entries such that \( A' = \alpha A + I \), where \( \alpha \in \mathbb{R} - \{-1, 1\} \). If \( \det \left(A^{2} - A\right) = 4 \), then the sum of all possible values of \( \alpha \) is equal to:

75.

Let \( w_{1} \) be the point obtained by the rotation of \( z_{1} = 5 + 4i \) about the origin through a right angle in the anticlockwise direction, and \( w_{2} \) be the point obtained by the rotation of \( z_{2} = 3 + 5i \) about the origin through a right angle in the clockwise direction. Then the principal argument of \( w_{1} - w_{2} \) is equal to:

76.

Let \( f : [2, 4] \to \mathbb{R} \) be a differentiable function such that \( (x \log_{e} x) f^{\prime}(x) + (\log_{e} x) f(x) + f(x) \ge 1, \, x \in [2,4] \) with \( f(2) = \dfrac{1}{2} \) and \( f(4) = \dfrac{1}{4} \). Consider the following two statements:
\( (A): \quad f(x) \le 1, \text{ for all } x \in [2, 4] \)
\( (B): \quad f(x) \ge \dfrac{1}{8}, \text{ for all } x \in [2, 4] \)
Then,

77.

Let \( x_{1}, x_{2}, \ldots, x_{100} \) be in an arithmetic progression, with \( x_{1} = 2 \) and their mean equal to \( 200 \). If \( y_{i} = i(x_{i}-i), 1 \le i \le 100 \), then the mean of \( y_{1}, y_{2}, \ldots, y_{100} \) is:

78.

If equation of the plane that contains the point \( (-2, 3, 5) \) and is perpendicular to each of the planes \( 2x + 4y + 5z = 8 \) and \( 3x - 2y + 3z = 5 \) is \( \alpha x + \beta y + \gamma z + 97 = 0 \) then \( \alpha + \beta + \gamma = \)

79.

Let \( \vec{a} \) be a non-zero vector parallel to the line of intersection of the two planes described by \( \hat{i}+\hat{j}, \hat{i}+\hat{k} \) and \( \hat{i}-\hat{j}, \hat{j}-\hat{k} \). If \( \theta \) is the angle between the vector \( \vec{a} \) and the vector \( \vec{b}=2\hat{i}-2\hat{j}+\hat{k} \) and \( \vec{a}\cdot\vec{b}=6 \), then the ordered pair \( \left(\theta, \left|\vec{a}\times\vec{b}\right|\right) \) is equal to:

80.

Area of the region \( \{(x,y): x^{2}+(y-2)^{2}\le 4,\; x^{2}\ge 2y\} \) is:

81.

, where \( a,c\in\mathbb{R} \). If \( A^{3}=A \) and the positive value of \( a \) belongs to the interval \( (n-1,n] \), where \( n\in\mathbb{N} \), then \( n \) is equal to ________ .

82.

Let a line \( l \) pass through the origin and be perpendicular to the lines \( l_{1}:\ \vec{r}=(\hat{i}-11\hat{j}-7\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k}),\ \lambda\in\mathbb{R} \) and \( l_{2}:\ \vec{r}=(-\hat{i}+\hat{k})+\mu(2\hat{i}+2\hat{j}+\hat{k}),\ \mu\in\mathbb{R} \). If \( P \) is the point of intersection of \( l \) and \( l_{1} \), and \( Q(\alpha,\beta,\gamma) \) is the foot of perpendicular from \( P \) on \( l_{2} \), then \( 9(\alpha+\beta+\gamma) \) is equal to __________ .

83.

Let \( H_{n}:\ \dfrac{x^{2}}{1+n}-\dfrac{y^{2}}{3+n}=1,\ n\in\mathbb{N} \). Let \( k \) be the smallest even value of \( n \) such that the eccentricity of \( H_{k} \) is a rational number. If \( l \) is the length of the latus rectum of \( H_{k} \), then \( 21l \) is equal to __________ .

84.

The mean of the coefficients of \( x, x^{2}, \ldots, x^{7} \) in the binomial expansion of \( (2+x)^{9} \) is __________ .

85.

For \( m,n>0 \), let \( \alpha(m,n) = \displaystyle\int_{0}^{2} t^{m}(1+3t)^{n} dt \) .If \( 11\alpha(10,6)+18\alpha(11,5)=p(14)^{6} \), then \( p \) is equal to ______.

86.

If \( a \) and \( b \) are the roots of the equation \( x^{2}-7x-1=0 \), then the value of \( \dfrac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}} \) is equal to ______.

87.

In an examination, \( 5 \) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is ______.

88.

The number of ordered triplets of the truth values of \( p \), \( q \) and \( r \) such that the truth value of the statement \( (p\vee q)\wedge(p\vee r)\Rightarrow(q\vee r) \) is true, is equal to ______.

89.

Let \( S=109+\dfrac{108}{5}+\dfrac{107}{5^{2}}+\ldots+\dfrac{2}{5^{107}}+\dfrac{1}{5^{108}} \). Then the value of \( \left(16S-(25)^{-54}\right) \) is equal to ______.

90.

The number of integral terms in the expansion of \( \left(3^\dfrac{1}{{2}}+5^\dfrac{1}{{4}}\right)^{680} \) is equal to:

JEE Main 2023 11th April Morning Shift

Physics Chemistry Mathematics

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In an examination, \( 5 \) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is ______.

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JEE Mains | PYQ