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Mathematics

Let the solution curve of the differential equation \( xdy = \left( \sqrt{x^2 + y^2} + y \right) dx, \ x > 0, \) intersect the line \( x = 1 \) at \( y = 0 \) and the line \( x = 2 \) at \( y = \alpha \). Then the value of \( \alpha \) is:

62.

Considering only the principal values of the inverse trigonometric functions, the domain of the function \( f(x)=\cos^{-1}\!\left(\dfrac{x^2-4x+2}{x^2+3}\right) \) is:

63.

Let the vectors \( \vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k}, \ \vec{b}=(1-t)\hat{i}+(1+t)\hat{j}+2\hat{k} \) and \( \vec{c}=t\hat{i}-t\hat{j}+\hat{k}, \ t\in \mathbb{R} \) be such that for \( \alpha,\beta,\gamma \in \mathbb{R}, \ \alpha\vec{a}+\beta\vec{b}+\gamma\vec{c}=\vec{0} \Rightarrow \alpha=\beta=\gamma=0 \). Then, the set of all values of \( t \) is:

64.

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation \( \cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x) \) is equal to:

65.

Let the operation \(\ast, \odot \in \{\land, \lor\}\). If \( (p * q)\\odot\ (p \odot \sim q) \) is a tautology, then the ordered pair \( (*,\ \odot) \) is:

66.

Let a vector \( \vec{a} \) has magnitude \( 9 \). Let a vector \( \vec{b} \) be such that for every \( (x,y)\in \mathbb{R}\times\mathbb{R}-\{(0,0)\} \), the vector \( (x\vec{a}+y\vec{b}) \) is perpendicular to the vector \( (6y\vec{a}-18x\vec{b}) \). Then the value of \( \left|\vec{a}\times\vec{b}\right| \) is equal to:

67.

For \( t \in (0, 2\pi) \), if \( ABC \) is an equilateral triangle with vertices \( A(\sin t, -\cos t) \), \( B(\cos t, \sin t) \) and \( C(a, b) \) such that its orthocenter lies on a circle with centre \( \left(1, \dfrac{1}{3}\right) \), then \( (a^2 - b^2) \) is equal to:

68.

For \( \alpha \in \mathbb{N} \), consider a relation \( R \) on \( \mathbb{N} \) given by \( R=\{(x,y)\}: 3x+\alpha y \text{ is a multiple of } 7\} \). The relation \( R \) is an equivalence relation if and only if:

69.

Out of \( 60\% \) female and \( 40\% \) male candidates appearing in an exam, \( 60\% \) candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is:

70.

If \( y = y(x), \ x \in (0,\pi/2) \) be the solution curve of the differential equation \( (\sin^2 2x)\dfrac{dy}{dx} + (8\sin^2 2x + 2\sin 4x)y = 2e^{-4x}(2\sin 2x + \cos 2x), \) with \( y(\pi/4)=e^{-\pi} \), then \( y(\pi/6) \) is equal to:

71.

If the tangents drawn at the points \( P \) and \( Q \) on the parabola \( y^2 = 2x-3 \) intersect at the point \( R(0,1) \), then the orthocenter of the triangle \( PQR \) is:

72.

Let \( C \) be the centre of the circle \( x^2+y^2-x+2y=\dfrac{11}{4} \) and \( P \) be a point on the circle. A line passes through the point \( C \), makes an angle of \( \dfrac{\pi}{4} \) with the line \( CP \) and intersects the circle at the points \( Q \) and \( R \). Then the area of the triangle \( PQR \) (in unit\( ^2 \)) is:

73.

The remainder when \( 7^{2022} + 3^{2022} \) is divided by \( 5 \) is:

74.

Let the matrix \(\displaystyle A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}\) and the matrix \( B_0 = A^{49} + 2A^{98} \). If \( B_n = \operatorname{Adj}(B_{n-1}) \) for all \( n \ge 1 \), then \( \det(B_4) \) is equal to:

75.

Let \( S_1 = \{\, z_1 \in \mathbb{C} : |z_1 - 3| = \dfrac{1}{2} \,\} \) and \( S_2 = \{\, z_2 \in \mathbb{C} : |z_2 - |z_2 + 1|| = |z_2 + |z_2 - 1|| \,\} \). Then, for \( z_1 \in S_1 \) and \( z_2 \in S_2 \), the least value of \( |z_2 - z_1| \) is:

76.

The foot of the perpendicular from a point on the circle \( x^2 + y^2 = 1, \, z = 0 \) to the plane \( 2x + 3y + z = 6 \) lies on which one of the following curves?

77.

If the minimum value of \( f(x)=\dfrac{5x^2}{2}+\dfrac{\alpha}{x^5}, \; x>0 \), is \( 14 \), then the value of \( \alpha \) is equal to:

78.

Let \( \alpha, \beta \) and \( \gamma \) be three positive real numbers. Let \( f(x)=\alpha x^5+\beta x^3+\gamma x, \, x\in \mathbb{R} \) and \( g:\mathbb{R}\to\mathbb{R} \) be such that \( g\big(f(x)\big)=x \) for all \( x\in \mathbb{R} \). If \( a_1,a_2,a_3,\ldots,a_n \) be in arithmetic progression with mean zero, then the value of \( f\!\left(g\!\left(\dfrac{1}{n}\displaystyle\sum_{r=1}^{n} f(a_i)\right)\right) \) is equal to:

79.

Consider the sequence \( a_1, a_2, a_3, \ldots \) such that \( a_1 = 1, a_2 = 2 \) and \( a_{n+2} = \dfrac{2}{a_{n+1}} + a_n \) for \( n = 1, 2, 3, \ldots \). If
\( \left( \dfrac{a_1 + \dfrac{1}{a_2}}{a_3} \right)\left( \dfrac{a_2 + \dfrac{1}{a_3}}{a_4} \right)\left( \dfrac{a_3 + \dfrac{1}{a_4}}{a_5} \right) \ldots \left( \dfrac{a_{30} + \dfrac{1}{a_{31}}}{a_{32}} \right) = 2^{\alpha}\left( {}^{61}C_{31} \right) \) , then \( \alpha \) is equal to:

80.

The minimum value of the twice differentiable function \( f(x) = \displaystyle\int\limits_{0}^{x} e^{\,x-t} f'(t) \, dt - (x^{2} - x + 1) e^{x}, \ x \in \mathbb{R} \) is:

81.

Let \( S \) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \( \{A,B,C,D,E\} \) or a number from \( \{1,2,3,4,5\} \) with the repetition of characters allowed. If the number of passwords in \( S \) whose at least one character is a number from \( \{1,2,3,4,5\} \) is \( \alpha \times 5^{6} \), then \( \alpha \) is equal to ________.

82.

Let \( P(-2,-1,1) \) and \( Q\!\left( \dfrac{56}{17}, \dfrac{43}{17}, \dfrac{111}{17} \right) \) be the vertices of the rhombus \( PRQS \). If the direction ratios of the diagonal \( RS \) are \( \alpha, -1, \beta \), where both \( \alpha \) and \( \beta \) are integers of minimum absolute values, then \( \alpha^{2} + \beta^{2} \) is equal to __________.

83.

Let \( f : [0,1] \to \mathbb{R} \) be a twice differentiable function in \( (0,1) \) such that \( f(0) = 3 \) and \( f(1) = 5 \). If the line \( y = 2x + 3 \) intersects the graph of \( f \) at only two distinct points in \( (0,1) \), then the least number of points \( x \in (0,1) \), at which \( f''(x) = 0 \), is __________.

84.

If \( \displaystyle \int\limits_{0}^{\sqrt{3}} \frac{15x^{3}}{\sqrt{\,1+x^{2}+\sqrt{(1+x^{2})^{3}}\,}}\,dx=\alpha \sqrt{2}+\beta \sqrt{3} \), where \( \alpha ,\beta \) are integers, then \( \alpha +\beta \) is equal to ________.

85.

Let \(\displaystyle A = \begin{bmatrix} 1 & -1 \\ 2 & \alpha \end{bmatrix}\) and \(\displaystyle B = \begin{bmatrix} \beta & 1 \\ 1 & 0 \end{bmatrix}, \alpha, \beta \in R.\) Let \( \alpha_{1} \) be the value of \( \alpha \) which satisfies \(\displaystyle (A + B)^2 = A^2 + \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}\) an \( \alpha_{2} \) be the value of \( \alpha \) which satisfies \( (A+B)^{2}=B^{2} \). Then \( |\alpha_{1}-\alpha_{2}| \) is equal to ________.

86.

For \( p,q\in \mathbb{R} \), consider the real valued function \( f(x)=(x-p)^{2}-q, \, x\in \mathbb{R} \) and \( q>0 \). Let \( a_{1},a_{2},a_{3} \) and \( a_{4} \) be in an arithmetic progression with mean \( p \) and positive common difference. If \( |f(a_{i})|=500 \) for all \( i=1,2,3,4 \), then the absolute difference between the roots of \( f(x)=0 \) is __________.

87.

For the hyperbola \( H:x^{2}-y^{2}=1 \) and the ellipse \( E:\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1, \, a>b>0 \), let the
(\( 1 \)) eccentricity of \( E \) be reciprocal of the eccentricity of \( H \), and
\( 2 \)) the line \( y=\sqrt{\dfrac{5}{2}}\,x+K \) be a common tangent of \( E \) and \( H \).
Then \( 4\!\left(a^{2}+b^{2}\right) \) is equal to __________.

88.

Let \( x_{1},x_{2},x_{3},\ldots ,x_{20} \) be in geometric progression with \( x_{1}=3 \) and the common ratio \( \dfrac{1}{2} \). A new data is constructed replacing each \( x_{i} \) by \( (x_{i}-i)^{2} \). If \( \bar{x} \) is the mean of new data, then the greatest integer less than or equal to \( \bar{x} \) is __________.

89.

\( \displaystyle \lim_{x\to 0}\left(\frac{(x+2\cos x)^{3}+2(x+2\cos x)^{2}+3\sin (x+2\cos x)}{(x+2)^{3}+2(x+2)^{2}+3\sin (x+2)}\right)^{\dfrac{100}{x}} \) is equal to __________.

90.

The sum of all real values of \( x \) for which \( \dfrac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \dfrac{5x^2 - 7x + 19}{3x^2 + 5x + 12} \) is equal to __________.

JEE Main 2022 28th July Morning Shift

Physics Chemistry Mathematics

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Let a vector \( \vec{a} \) has magnitude \( 9 \). Let a vector \( \vec{b} \) be such that for every \( (x,y)\in \mathbb{R}\times\mathbb{R}-\{(0,0)\} \), the vector \( (x\vec{a}+y\vec{b}) \) is perpendicular to the vector \( (6y\vec{a}-18x\vec{b}) \). Then the value of \( \left|\vec{a}\times\vec{b}\right| \) is equal to:

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JEE Mains JEE Advance BITSAT
JEE Mains | PYQ