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Mathematics

Let the system of linear equations
\( 4x + \lambda y + 2z = 0 \)
\( 2x - y + z = 0 \)
\( \mu x + 2y + 3z = 0, \lambda, \mu \in \mathbb{R} \).
Has a non-trivial solution. Then which of the following is true?

62.

Let the centroid of an equilateral triangle \( ABC \) be at the origin. Let one of the sides of the equilateral triangle be along the straight line \( x + y = 3 \). If \( R \) and \( r \) be the radius of circumcircle and incircle respectively of \( \triangle ABC \), then \( (R + r) \) is equal to:

63.

Let in a series of \( 2n \) observations, half of them are equal to \( a \) and remaining half are equal to \( -a \). Also by adding a constant \( b \) in each of these observations, the mean and standard deviation of new set becomes \( 5 \) and \( 20 \), respectively. Then the value of \( a^2 + b^2 \) is equal to:

64.

Let in a Binomial distribution, consisting of \( 5 \) independent trials, probabilities of exactly \( 1 \) and \( 2 \) success be \( 0.4096 \) and \( 0.2048 \) respectively. Then the probability of getting exactly \( 3 \) successes is equal to:

65.

Let \( f:\mathbb{R}\to\mathbb{R} \) be a function defined as .
If \( f \) is continuous at \( x=0 \), then the value of \( a+b \) is equal to:

66.

If \( P \) and \( Q \) are two statements, then which of the following compound statement is a tautology?

67.

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be \( \dfrac{\pi}{3} \). If the radius of the circumcircle of \( \triangle ABC \) is \( 2 \), then the height of the pole is equal to:

68.

Let \( y = y(x) \) be the solution of the differential equation \( \dfrac{dy}{dx} = (y+1)\left((y+1)e^{x^{2}/2} - x\right) \), \( 0 < x < 2.1 \), with \( y(2) = 0 \). Then the value of \( \dfrac{dy}{dx} \) at \( x = 1 \) is equal to:

69.

Let a tangent be drawn to the ellipse \( \dfrac{x^{2}}{27} + y^{2} = 1 \) at \( \left(3\sqrt{3}\cos\theta,\ \sin\theta\right) \) where \( \theta \in \left(0,\ \dfrac{\pi}{2}\right) \). Then the value of \( \theta \) such that the sum of intercepts on axes made by this tangent is minimum is equal to:

70.

Let \( S_{1} \) be the sum of first \( 2n \) terms of an arithmetic progression. Let \( S_{2} \) be the sum of first \( 4n \) terms of the same arithmetic progression. If \( (S_{2} - S_{1} )\) is \( 1000 \), then the sum of the first \( 6n \) terms of the arithmetic progression is equal to:

71.

Let \( g(x) = \displaystyle\int\limits_{0}^{x} f(t)\,dt \), where \( f \) is continuous function in \( [0,3] \) such that \( \dfrac{1}{3} \le f(t) \le 1 \) for all \( t \in [0,1] \) and \( 0 \le f(t) \le \dfrac{1}{2} \) for all \( t \in [1,3] \). The largest possible interval in which \( g(3) \) lies is:

72.

Let a complex number be \( w = 1 - \sqrt{3} i \). Let another complex \( z \) be such that \( |zw| = 1 \) and \( \arg(z) - \arg w = \dfrac{\pi}{2} \). Then the area of the triangle with vertices origin, \( z \) and \( w \) is equal to:

73.

Consider a hyperbola \( H : {x^{2}} - {2y^{2}} = 4 \). Let the tangent at a point \( P(4,\sqrt{6}) \) meet the x-axis at \( Q \) and latus rectum at \( R(x_{1},y_{1}), x_{1} > 0 \). If \( F \) is a focus of \( H \) which is nearer to the point \( P \), then the area of \( \triangle QFR \) is equal to:

74.

In a triangle \( ABC \), if \( | \overrightarrow{BC} | = 8, | \overrightarrow{CA} | = 7, | \overrightarrow{AB} | = 10 \), then the projection of the vector \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to:

75.

Define a relation \( R \) over a class of \( n \times n \) real matrices \( A \) and \( B \) as “\( ARB \) iff there exists a non-singular matrix \( P \) such that \( P A P^{-1} = B \)”. Then which of the following is true?

76.

Let \( \vec{a} \) and \( \vec{b} \) be two non-zero vectors perpendicular to each other and \( |\vec{a}| = |\vec{b}| \). If \( |\vec{a} \times \vec{b}| = |\vec{a}| \), then the angle between the vectors \( (\vec{a} + \vec{b} +( \vec{a} \times \vec{b})) \) and \( \vec{a} \) is equal to:

77.

Let \( S_{1} : x^{2} + y^{2} = 9 \) and \( S_{2} : (x - 2)^{2} + y^{2} = 1 \). Then the locus of center of a variable circle \( S \) which touches \( S_{1} \) internally and \( S_{2} \) externally always passes through the points:

78.

The area bounded by the curve \( 4y^2 = x^2 (4 - x)(x - 2) \) is equal to:

79.

If \( 15\sin^4\alpha + 10\cos^4\alpha = 6 \), for some \( \alpha \in \mathbb{R} \), then the value of \( 27\sec^6\alpha + 8\,\text{cosec}^6\alpha \) is equal to:

80.

Let \( f : \mathbb{R} - \{3\} \to \mathbb{R} - \{1\} \) be defined by \( f(x) = \dfrac{x - 2}{x - 3} \). Let \( g : \mathbb{R} \to \mathbb{R} \) be given as \( g(x) = 2x - 3 \). Then, the sum of all the values of \( x \) for which \( f^{-1}(x) + g^{-1}(x) = \dfrac{13}{2} \) is equal to:

81.

Let the mirror image of the point \( (1, 3, a) \) with respect to the plane \( \vec{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) - b = 0 \) be \( (-3, 5, 2) \). Then, the value of \( |\, a + b \,| \) is equal to:

82.

Let \( I \) be an identity matrix of order \( 2 \times 2 \) and \(\displaystyle P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}\). Then the value of \( n \in \mathbb{N} \) for which \( P^n = 5I - 8P \) is equal to:

83.

Let \( P(x) \) be a real polynomial of degree \( 3 \) which vanishes at \( x = -3 \). Let \( P(x) \) have local minima at \( x = 1 \), local maxima at \( x = -1 \) and \( \displaystyle\int\limits_{-1}^{1} P(x)\,dx = 18 \), then the sum of all the coefficients of the polynomials \( P(x) \) is equal to:

84.

Let \( P \) be a plane containing the line \(\displaystyle \frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2} \) and parallel to the line \(\displaystyle \frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7} \). If the point \(( 1,-1,\alpha) \) lies on the plane \( P \), then the value of \( |5\alpha| \) is equal to:

85.

Let \( {}^{n}C_{r} \) denote the binomial coefficient of \( x^{r} \) in the expansion of \( (1+x)^{n} \). If \( \displaystyle\sum_{k=0}^{10} (2^{2}+3k)\,{}^{n}C_{k}=\alpha.\,3^{10}+\beta.\,2^{10},\ \alpha,\beta\in \mathbb{R} \), then \( \alpha+\beta \) is equal to:

86.

Let \( f:\mathbb{R}\to\mathbb{R} \) satisfy the equation \( f(x+y)=f(x)\cdot f(y) \) for all \( x,y\in\mathbb{R} \) and \( f(x)\neq 0 \) for any \( x\in\mathbb{R} \). If the function \( f \) is differentiable at \( x=0 \) and \( f'(0)=3 \), then \( \displaystyle\lim_{h\to 0}\frac{1}{h}\big(f(h)-1\big) \) is equal to:

87.

If \( f(x) \) and \( g(x) \) are two polynomials such that the polynomial \( P(x)=f(x^{3})+x\,g(x^{3}) \) is divisible by \( x^{2}+x+1 \), then \( P(1) \) is equal to:

88.

If \( \displaystyle \sum_{r=1}^{10} r!\,\big(r^{3}+6r^{2}+2r+5\big)=\alpha\,(11!) \), then the value of \( \alpha \) is equal to:

89.

The term independent of \( x \) in the expansion of \(\displaystyle \left[ \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x - 1}{x - x^{1/2}} \right]^{10}, x \neq 1 \), is equal to:

90.

Let \( y = y(x) \) be the solution of the differential equation \( x\,dy - y\,dx = \sqrt{(x^2 - y^2)}\,dx, \; x \geq 1 \), with \( y(1) = 0 \).
If the area bounded by the line \( x = 1 \), \( x = e^{\pi} \), \( y = 0 \) and \( y = y(x) \) is \( \alpha e^{2\pi} + \beta \), then the value of \( 10(\alpha + \beta) \) is equal to:

JEE Main 2021 18th March Evening Shift

Physics Chemistry Mathematics

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The area bounded by the curve \( 4y^2 = x^2 (4 - x)(x - 2) \) is equal to:

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JEE Mains | PYQ