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Mathematics

If \( y = y(x) \) is the solution of the differential equation \( \dfrac{dy}{dx} + (\tan x) \, y = \sin x, \, 0 \le x \le \dfrac{\pi}{3} \) with \( y(0)=0 \), then \( y\!\left(\dfrac{\pi}{4}\right) \) equal to:

62.

Let \( p(x)=x^{2}+bx+c \) be a quadratic polynomial with real coefficients such that \( \int\limits_{0}^{1} p(x)\,dx = 1 \) and \( p(x) \) leaves remainder \( 5 \) when it is divided by \( (x-2) \). Then the value of \( 9(b+c) \) is equal to:

63.

The maximum value of

64.

If \( (x,y,z) \) be an arbitrary point lying on a plane \( P \) which passes through the points \( (42,0,0) \), \( (0,42,0) \) and \( (0,0,42) \), then the value of the expression \(\displaystyle 3 + \frac{x-11}{(y-19)^{2}(z-12)^{2}} + \frac{y-19}{(x-11)^{2}(z-12)^{2}} + \frac{z-12}{(x-11)^{2}(y-19)^{2}} + \frac{x+y+z}{14(x-11)(y-19)(z-12)} \) is equal to:

65.

Let \( A=\{2,3,4,5,\ldots,30\} \) and '\( \simeq \)' be an equivalence relation on \( A\times A \), defined by \( (a,b)\simeq (c,d) \), if and only if \( ad=bc \). Then the number of ordered pairs which satisfy this equivalence relation with ordered pair \( (4,3) \) is equal to:

66.

Let \( A(-1,1),\, B(3,4) \) and \( C(2,0) \) be given three points. A line \( y=mx,\, m>0 \), intersects lines \( AC \) and \( BC \) at point \( P \) and \( Q \) respectively. Let \( A_{1} \) and \( A_{2} \) be the area of \( \triangle ABC \) and \( \triangle PQC \) respectively, such that \( A_{1}=3A_{2} \), then the value of \( m \) is equal to:

67.

Let the lengths of intercepts on x-axis and y-axis made by the circle \( x^2 + y^2 + ax + 2ay + c = 0, \ (a < 0) \) be \( 2\sqrt{2} \) and \( 2\sqrt{5} \), respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line \( x + 2y = 0 \), is equal to:

68.

If the foot of the perpendicular from point \( (4, 3, 8) \) on the line \( L_1 : \dfrac{x-a}{1}=\dfrac{y-2}{3}=\dfrac{z-b}{4} \),\(\displaystyle 1 \neq 0\) is \( (3,5,7) \), then the shortest distance between the line \( L_1 \) and line \( L_2 : \dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} \) is equal to:

69.

Given that the inverse trigonometric functions take principal values only. Then the number of real values of \( x \) which satisfy \( \sin^{-1}\!\left(\dfrac{3x}{5}\right)+\sin^{-1}\!\left(\dfrac{4x}{5}\right)=\sin^{-1}x \) is equal to:

70.

Let \( C \) be the locus of the mirror image of a point on the parabola \( y^{2}=4x \) with respect to the line \( y=x \). Then the equation of tangent to \( C \) at \( P(2,1) \) is:

71.

Let \( \vec{a}=\hat{i}+2\hat{j}-3\hat{k} \) and \( \vec{b}=2\hat{i}-3\hat{j}+5\hat{k} \). If \( \vec{r}\times \vec{a}=\vec{b}\times \vec{r},\ \vec{r}\cdot(\alpha \hat{i}+2\hat{j}+\hat{k})=3 \) and \( \vec{r}\cdot(2\hat{i}+5\hat{j}-\alpha \hat{k})=-1, \ \alpha \in \mathbb{R} \), then the value of \( \alpha+|\vec{r}|^{2} \) is equal to:

72.

Consider the integral \( I=\displaystyle\int\limits_{0}^{10}\dfrac{[x] \, e^{[x]}}{e^{x-1}}\,dx \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then the value of \( I \) is equal to:

73.

If the points of intersection of the ellipse \( \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 \) and the circle \( x^2 + y^2 = 4b,\, b>4 \) lie on the curve \( y^2 = 3x^2 \), then \( b \) is equal to:

74.

Let \( C_1 \) be the curve obtained by the solution of differential equation \( 2xy \, \dfrac{dy}{dx} = y^2 - x^2, \, x>0 \). Let the curve \( C_2 \) be the solution of \( \dfrac{2xy}{x^2-y^2} = \dfrac{dy}{dx} \). If both the curves pass through \( (1,1) \), then the area enclosed by the curves \( C_1 \) and \( C_2 \) is equal to:

75.

Let \( a \in \mathbb{R} \) be such that the function is continuous at \( x=0 \), where \( \{x\} = x - [ x ], \, [ x ] \) is the greatest integer less than or equal to \( x \). Then:

76.

The least value of \( |z| \) where \( z \) is complex number which satisfies the inequality \( \displaystyle \exp\left(\frac{(|z| + 3)(|z| - 1)}{||z| + 1|}\log_e 2\right) \ge \log_{\sqrt{2}}|5\sqrt{7} + 9i|, \, i = \sqrt{-1} \), is equal to:

77.

Let \( A \) denote the event that a \( 6 \)-digit integer formed by \( 0,1,2,3,4,5,6 \) without repetitions, be divisible by \( 3 \). Then probability of event \( A \) is equal to:

78.

Consider a rectangle \( ABCD \) having \( 5,\,7,\,6,\,9 \) points in the interior of the line segments \( AB,\, CD,\, BC,\, DA \) respectively. Let \( \alpha \) be the number of triangles having these points from different sides as vertices and \( \beta \) be the number of quadrilaterals having these points from different sides as vertices. Then \(( \beta - \alpha )\) is equal to:

79.

Let \( f : S \to S \) where \( S = (0,\infty) \) be a twice differentiable function such that \( f(x+1)=x f(x) \). If \( g:S \to \mathbb{R} \) be defined as \( g(x)=\log_e f(x) \), then the value of \( \left|\, g''(5)-g''(1)\,\right| \) is equal to:

80.

Let \( f \) be a real valued function, defined on \( \mathbb{R}-\{-1,1\} \) and given by \( f(x)=3\log_e \left|\dfrac{x-1}{x+1}\right|-\dfrac{2}{x-1} \). Then in which of the following intervals, function \( f(x) \) is increasing?

81.

Let \( n \) be a positive integer. Let \( A=\displaystyle\sum_{k=0}^{n}(-1)^k \,{{}^n C_ k}\left[\left(\dfrac{1}{2}\right)^k+\left(\dfrac{3}{4}\right)^k+\left(\dfrac{7}{8}\right)^k+\left(\dfrac{15}{16}\right)^k+\left(\dfrac{31}{32}\right)^k\right] \). If \( 63A=1-\dfrac{1}{2^{30}} \), then \( n \) is equal to ____________.

82.

If the distance of the point \( Q(1,-2,3) \) from the plane \( x+2y-3z+10=0 \) measured parallel to the line \( \dfrac{x-1}{3}=\dfrac{2-y}{m}=\dfrac{z+3}{1} \) is \( \sqrt{\dfrac{{7}}{{2}}} \), then the value of \( |m| \) is equal to ____________.

83.

Let \( \vec{c} \) be a vector perpendicular to the vectors \( \vec{a}=\hat{i}+\hat{j}-\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}+\hat{k} \). If \( \vec{c}\cdot(\hat{i}+\hat{j}+3\hat{k})=8 \) then the value of \( \vec{c}\cdot(\vec{a}\times\vec{b}) \) is equal to ________ .

84.

Let \(\displaystyle A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} \text{ and } B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}\)be two \( 2\times 1 \) matrices with real entries such that \( A=XB \), where \(\displaystyle X = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}\)and \( k\in\mathbb{R} \). If \( a_{1}^{2}+a_{2}^{2}=\dfrac{2}{3}\left(b_{1}^{2}+b_{2}^{2}\right) \) and \( (k^{2}+1)b_{2}^{2}\ne-2b_{1}b_{2} \), then the value of \( k \) is ______ .

85.

Consider the statistics of two sets of observations as follows:

If the variance of the combined set of these two observations is \( \dfrac{17}{9} \), then the value of \( n \) is equal to ____.

86.

Let \( \dfrac{1}{16}, \, a \) and \( b \) be in G.P. and \( \dfrac{1}{a}, \, \dfrac{1}{b}, \, 6 \) be in A.P., where \( a, b > 0 \). Then \( 72(a+b) \) is equal to ________.

87.

Let \( f:\mathbb{R}\to\mathbb{R} \) and \( g:\mathbb{R}\to\mathbb{R} \) be defined as where \( a, b \) are non-negative real numbers. If \( (g\circ f)(x) \) is continuous for all \( x\in\mathbb{R} \), then \( a+b \) is equal to ____________.

88.

Let \(\displaystyle S_n(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \dots\dots \) upto \( n \)-terms, where \( a>1 \). If \( S_{24}(x)=1093 \) and \( S_{12}(2x)=265 \), then value of \( a \) is equal to __________.

89.

In \( \triangle ABC \), the lengths of sides \( AC \) and \( AB \) are \( 12 \, \text{cm} \) and \( 5 \, \text{cm} \), respectively. If the area of \( \triangle ABC \) is \( 30 \, \text{cm}^{2} \) and \( R \) and \( r \) are respectively the radii of circumcircle and incircle of \( \triangle ABC \), then the value of \( 2R+r \) (in \( \text{cm} \)) is equal to ____________.

90.

For real numbers \( \alpha, \beta, \gamma \) and \( \delta \), if
\( \int \dfrac{(x^{2}-1) + \tan^{-1}\!\left(\dfrac{x^{2}+1}{x}\right)}{(x^{4}+3x^{2}+1)\,\tan^{-1}\!\left(\dfrac{x^{2}+1}{x}\right)} \, dx = \alpha \log_{e}\!\left( \tan^{-1}\!\left(\dfrac{x^{2}+1}{x}\right) \right) + \beta \tan^{-1}\!\left( \dfrac{\gamma (x^{2}-1)}{x} \right) + \delta \tan^{-1}\!\left( \dfrac{x^{2}+1}{x} \right) + C \) where \( C \) is an arbitrary constant, then the value of \( 10(\alpha+\beta\gamma+\delta) \) is equal to ________.

JEE Main 2021 16th March Evening Shift

Physics Chemistry Mathematics

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If the foot of the perpendicular from point \( (4, 3, 8) \) on the line \( L_1 : \dfrac{x-a}{1}=\dfrac{y-2}{3}=\dfrac{z-b}{4} \),\(\displaystyle 1 \neq 0\) is \( (3,5,7) \), then the shortest distance between the line \( L_1 \) and line \( L_2 : \dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} \) is equal to:

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JEE Mains JEE Advance BITSAT
JEE Mains | PYQ