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The domain of the function \( f(x)=\sin^{-1}\!\left[\;2x^{2}-3\;\right]+\log_{2}\!\left(\log_{\dfrac{1}{2}}\!\left(x^{2}-5x+5\right)\right) \), where \([t]\) is
the greatest integer functions, is :

62.

Let \( S \) be the set of all \( (\alpha,\beta),\, \pi<\alpha,\beta<2\pi \), for which the complex number \( \dfrac{1-i\sin\alpha}{1+2i\sin\alpha} \) is purely imaginary and \( \dfrac{1+i\cos\beta}{1-2i\cos\beta} \) is purely real. Let \( Z_{\alpha\beta}=\sin 2\alpha + i\cos 2\beta,\, (\alpha,\beta)\in S \). Then \(\displaystyle \sum_{(\alpha,\beta)\in S}\left(iZ_{\alpha\beta}+\frac{1}{i{\bar Z_{\alpha\beta}}}\right) \) is equal to:

63.

The \( \alpha,\beta \) are the roots of the equation
\( x^{2}-\left(5+3^{\sqrt{\log_{3}5}-5\sqrt{\log_{5}3}}\right)+3\left(3^{(\log_{3}5)^{\dfrac{1}{3}}}-5^{(\log_{5}3)^{\dfrac{2}{3}}} -1\right)=0 \), then the equation, whose roots are \( \alpha+\dfrac{1}{\beta} \) and \( \beta+\dfrac{1}{\alpha} \), is:

64.

Let \(\displaystyle A = \begin{pmatrix} 4 & -2 \\ \alpha & \beta \end{pmatrix}\)If \( A^{2}+\gamma A+18I = O \), then \( \det(A) \) is equal to _________.

65.

If for \( p\neq q\neq 0 \), the function \( f(x)=\dfrac{\sqrt[7]{\,p(729+x)\,}-3}{\sqrt[3]{\,729+qx\,}-9} \) is continuous at \( x=0 \), then:

66.

Let \( f(x)=2+|x|-|x-1|+|x+1|,\, x\in \mathbb{R} \).
Consider
\((S1):\;\displaystyle f'\!\left(-\frac{3}{2}\right)+f'\!\left(-\frac{1}{2}\right)+f'\!\left(\frac{1}{2}\right)+f'\!\left(\frac{3}{2}\right)=2\)
\((S2):\; \displaystyle\int\limits_{-2}^{2} f(x)\,dx = 12\)
Then,

67.

Let the sum of an infinite G.P., whose first term is \( a \) and the common ratio is \( r \), be \( 5 \). Let the sum of its first five terms be \( \dfrac{98}{25} \). Then the sum of the first \( 21 \) terms of an A.P, whose first term is \( 10ar \), \( n^{\text{th}} \) term is \( a_{n} \) and the common difference is \( 10ar^{2} \), is equal to:

68.

The area of the region enclosed by \( y \le 4x^{2} \), \( x^{2} \le 9y \) and \( y \le 4 \), is equal to:

69.

The \( \displaystyle\int\limits_{0}^{2}\left(\left|2x^{2}-3x\right|+\left[x-\frac{1}{2}\right]\right)dx \), where \( [\,t\,] \) is the greatest integer function, is equal to:

70.

Consider a curve \( y=y(x) \) in the first quadrant as shown in the figure. Let the area \( A_{1} \) is twice the area \( A_{2} \). Then the normal to the curve perpendicular to the line \( 2x-12y=15 \) does NOT pass through the point.

71.

The equations of the sides \( AB \), \( BC \) and \( CA \) of a triangle \( ABC \) are \( 2x+y=0 \), \( x+py=39 \) and \( x-y=3 \) respectively and \( P(2,3) \) is its circumcentre. Then which of the following is NOT true?

72.

A circle \( C_{1} \) passes through the origin \( O \) and has diameter \( 4 \) on the positive \( x \)-axis. The line \( y=2x \) gives a chord \( OA \) of circle \( C_{1} \). Let \( C_{2} \) be the circle with \( OA \) as a diameter. If the tangent to \( C_{2} \) at the point \( A \) meets the \( x \)-axis at \( P \) and \( y \)-axis at \( Q \), then \( QA:AP \) is equal to:

73.

If the length of the latus rectum of a parabola, whose focus is \( (a,a) \) and the tangent at its vertex is \( x+y=a \), is \( 16 \), then \( |a| \) is equal to:

74.

If the length of the perpendicular drawn from the point \( P(a,4,2), a>0 \) on the line \( \displaystyle\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1} \) is \( 2\sqrt{6} \) units and \( Q(\alpha_1,\alpha_2,\alpha_3) \) is the image of the point \( P \) in this line, then \( a + \displaystyle\sum_{i=1}^{3}\alpha_i \) is equal to:

75.

If the line of intersection of the planes \( ax+by=3 \) and \( ax+by+cz=0,\, a>0 \) makes an angle \( 30^\circ \) with the plane \( y-z+2=0 \), then the direction cosines of the line are:

76.

Let \( X \) have a binomial distribution \( B(n,p) \) such that sum and the product of the mean and variance of \( X \) are \( 24 \) and \( 128 \) respectively. If \( P(X>n-3)=\dfrac{k}{2^n} \), then \( k \) is equal to:

77.

A six faced die is biased such that \( 3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1) \). Let \( X \) be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of \( X \) is:

78.

The angle of elevation of the top \( P \) of a vertical tower \( PQ \) of height \( 10 \) from a point \( A \) on the horizontal ground is \( 45^\circ \). Let \( R \) be a point on \( AQ \) and from a point \( B \), vertically above \( R \), the angle of elevation of \( P \) is \( 60^\circ \). If \( \angle BAQ = 30^\circ, AB = d \) and the area of the trapezium \( PQRB \) is \( \alpha \), then the ordered pair \( (d,\alpha) \) is:

79.

Let \( S = \left\{ \theta \in \left(0, \dfrac{\pi}{2}\right) : \displaystyle\sum_{m=1}^{9} \sec\left(\theta + (m-1)\frac{\pi}{6}\right)\sec\left(\theta + \frac{m\pi}{6}\right) = -\frac{8}{\sqrt{3}} \right\} \). Then

80.

If the truth value pf the statement \( (P \land (\sim R)) \rightarrow ((\sim R) \land Q) \) is \( F \), then the truth value of which of the following is \( F \)?

81.

, where \( \alpha,\beta,\gamma \) are three distinct natural numbers. If \( \dfrac{\det(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32}\times 3^{16} \), then the number of such \( 3 \)- tuples \( (\alpha,\beta,\gamma) \) is ________.

82.

The number of functions \( f \), from the set \( A=\left\{x\in \mathbb{N}:x^{2}-10x+9\le 0\right\} \) to the set \( B=\left\{n^{2}:n\in \mathbb{N}\right\} \) such that \( f(x)\le (x-3)^{2}+1 \), for every \( x\in A \), is ________.

83.

Let for the \( 9^{\text{th}} \) term in the binomial expansion of \( (3+6x)^{n} \), in the increasing powers of \( 6x \), to be the greatest for \( x=\dfrac{3}{2} \), the least value of \( n \) is \( n_{0} \). If \( k \) is the ratio of the coefficient of \( x^{6} \) to the coefficient of \( x^{3} \), then \( k+n_{0} \) is equal to:

84.

\(\displaystyle \frac{2^{3}-1^{3}}{1\times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2\times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3\times 15}+ \ldots + \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots +2^{3}-1^{3}}{15\times 63} \) is equal to ____.

85.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is \( \tan^{-1}\dfrac{3}{4} \). Water is poured in it at a constant rate of \( 6 \) cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is \( 4 \) meters, is _______.

86.

For the curve \( C:\ (x^{2}+y^{2}-3)+(x^{2}-y^{2}-1)^{5}=0 \), the value of \( 3y' - y^{3}y'' \), at the point \( (\alpha,\alpha) \), \( \alpha>0 \), on \( C \), is equal to ________.

87.

Let \( f(x)=\min\{[x-1],[x-2],\ldots,[x-10]\} \) where \( [t] \) denotes the greatest integer \( \le t \). Then \( \displaystyle \int\limits_{0}^{10} f(x)\,dx+\displaystyle \int\limits_{0}^{10} (f(x))^{2}\,dx+\displaystyle \int\limits_{0}^{10} |f(x)|\,dx \) is equal to ________.

88.

Let \( f \) be a differentiable function satisfying \( f(x)=\dfrac{2}{\sqrt{3}}\displaystyle\int\limits_{0}^{\sqrt{3}} f\!\left(\frac{\lambda^{2}x}{3}\right)\,d\lambda,\, x>0 \) and \( f(1)=\sqrt{3} \). If \( y=f(x) \) passes through the point \( (\alpha,6) \), then \( \alpha \) is equal to _________.

89.

A common tangent \( T \) to the curves \( C_{1}:\ \dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1 \) and \( C_{2}:\ \dfrac{x^{2}}{42}-\dfrac{y^{2}}{143}=1 \) does not pass through the fourth quadrant. If \( T \) touches \( C_{1} \) at \( (x_{1},y_{1}) \) and \( C_{2} \) at \( (x_{2},y_{2}) \), then \( |2x_{1}+x_{2}| \) is equal to ______.

90.

Let \( \vec{a},\vec{b},\vec{c} \) be three non-coplanar vectors such that \( \vec{a}\times \vec{b}=4\vec{c},\, \vec{b}\times \vec{c}=9\vec{a} \) and \( \vec{c}\times \vec{a}=\alpha \vec{b} \) and \( \alpha>0 \). If \( |\vec{a}|+|\vec{b}|+|\vec{c}|=\dfrac{1}{36} \), then \( \alpha \) is equal to ________.

JEE Main 2022 27th July Evening Shift

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A six faced die is biased such that \( 3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1) \). Let \( X \) be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of \( X \) is:

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JEE Mains | PYQ