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Mathematics

Consider a triangle \( \Delta \) whose two sides lie on the \( x \)-axis and the line \( x + y + 1 = 0 \). If the orthocenter of \( \Delta \) is \( (1, 1) \), then the equation of the circle passing through the vertices of the triangle \( \Delta \) is:

40.

The area of the region \( \{(x,y): 0 \le x \le \frac{9}{4},\, 0 \le y \le 1,\, x \ge 3y,\, x + y \ge 2\} \) is:

41.

Consider three sets \( E_1=\{1,2,3\} \), \( F_1=\{1,3,4\} \) and \( G_1=\{2,3,4,5\} \). Two elements are chosen at random, without replacement, from the set \( E_1 \) and let \( S_1 \) denote the set of these chosen elements. Let \( E_2=E_1-S_1 \) and \( F_2=F_1 \cup S_1 \). Now two elements are chosen at random, without replacement, from the set \( F_2 \) and let \( S_2 \) denote the set of these chosen elements.
Let \( G_2=G_1 \cup S_2 \). Finally, two elements are chosen at random, without replacement, from the set \( G_2 \) and let \( S_3 \) denote the set of these chosen elements.
Let \( E_3=E_2 \cup S_3 \). Given that \( E_1=E_3 \), let \( p \) be the conditional probability of the event \( S_1=\{1,2\} \). Then the value of \( p \) is:

42.

Let \( \theta_1,\theta_2,\ldots,\theta_{10} \) be positive valued angles (in radian) such that \( \theta_1+\theta_2+\ldots+\theta_{10}=2\pi \). Define the complex numbers \( z_1=e^{i\theta_1},\, z_k=z_{k-1}e^{i\theta_k} \) for \( k=2,3,\ldots,10 \), where \( i=\sqrt{-1} \). Consider the statements P and Q given below:
P: \( |z_2-z_1|+|z_3-z_2|+\ldots+|z_{10}-z_9|+|z_1-z_{10}| \le 2\pi \)
Q: \( |z_2^2-z_1^2|+|z_3^2-z_2^2|+\ldots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2| \le 4\pi \)
Then,

43.

Three numbers are chosen at random, one after another with replacement, from the set \( S=\{1,2,3,\ldots,100\} \). Let \( p_1 \) be the probability that the maximum of chosen numbers is at least \( 81 \) and \( p_2 \) be the probability that the minimum of chosen numbers is at most \( 40 \).

The value of \( \frac{625}{4}\, p_1 \) is ______.

44.

Three numbers are chosen at random, one after another with replacement, from the set \( S = \{ 1, 2, 3, \ldots, 100 \} \). Let \( p_1 \) be the probability that the maximum of chosen numbers is at least \( 81 \) and \( p_2 \) be the probability that the minimum of chosen numbers is at most \( 40 \).

The value of \( \frac{125}{4} \, p_2 \) is _______.

45.

Let \( \alpha, \beta \) and \( \gamma \) be real numbers such that the system of linear equations
\( x + 2y + 3z = \alpha \)
\( 4x + 5y + 6z = \beta \)
\( 7x + 8y + 9z = \gamma - 1 \)
is consistent. Let \( |M| \) represent the determinant of the matrix

Let \( P \) be the plane containing all those \( (\alpha, \beta, \gamma) \) for which the above system of linear equations is consistent, and \( D \) be the square of the distance of the point \( (0, 1, 0) \) from the plane \( P \).
The value of \( |M| \) is _______.

46.

Let \( \alpha, \beta \) and \( \gamma \) be real numbers such that the system of linear equations
\( x + 2y + 3z = \alpha \)
\( 4x + 5y + 6z = \beta \)
\( 7x + 8y + 9z = \gamma - 1 \)
is consistent. Let \( |M| \) represent the determinant of the matrix

Let \( P \) be the plane containing all those \( (\alpha, \beta, \gamma) \) for which the above system of linear equations is consistent, and \( D \) be the square of the distance of the point \( (0, 1, 0) \) from the plane \( P \).
The value of \( D \) is _______.

47.

Consider the lines \( L_1 \) and \( L_2 \) defined by
\( L_1 : x\sqrt{2} + y - 1 = 0 \) and \( L_2 : x\sqrt{2} - y + 1 = 0 \)
For a fixed constant \( \lambda \) , let \( C \) be the locus of a point \( P \) such that the product of the distance of \( P \) from \( L_1 \) and the distance of \( P \) from \( L_2 \) is \( \lambda^{2} \). The line \( y = 2x + 1 \) meets \( C \) at two points \( R \) and \( S \), where the distance between \( R \) and \( S \) is \( \sqrt{270} \) .
Let the perpendicular bisector of \( RS \) meet \( C \) at two distinct points \( R' \) and \( S' \). Let \( D \) be the square of the distance of the distance between \( R' \) and \( S' \)
The value of \( \lambda^{2} \) is _____

48.

Consider the lines \( L_1 \) and \( L_2 \) defined by
\( L_1 : x\sqrt{2} + y - 1 = 0 \) and \( L_2 : x\sqrt{2} - y + 1 = 0 \)
For a fixed constant \( \lambda \) , let \( C \) be the locus of a point \( P \) such that the product of the distance of \( P \) from \( L_1 \) and the distance of \( P \) from \( L_2 \) is \( \lambda^{2} \). The line \( y = 2x + 1 \) meets \( C \) at two points \( R \) and \( S \), where the distance between \( R \) and \( S \) is \( \sqrt{270} \) .
Let the perpendicular bisector of \( RS \) meet \( C \) at two distinct points \( R' \) and \( S' \). Let \( D \) be the square of the distance of the distance between \( R' \) and \( S' \)
The value of \( D \) is _____

49.

For any \( 3 \times 3 \) matrix \( M \), let \( |M| \) denote the determinant of \( M \). Let

If \( Q \) is a nonsingular matrix of order \( 3 \times 3 \), then which of the following statements is (are) TRUE?

50.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = \dfrac{x^{2} - 3x - 6}{x^{2} + 2x + 4} \). Then which of the following statements is (are) TRUE?

51.

Let \( E, F \) and \( G \) be three events having probabilities
\( P(E)=\dfrac{1}{8} \), \( P(F)=\dfrac{1}{6} \) and \( P(G)=\dfrac{1}{4} \) and let \( P(E \cap F \cap G) = \dfrac{1}{10} \).
For any event \( H \), \( H^{c} \) denotes its complement, then which of the following statements is(are) TRUE?

52.

For \( 3 \times 3 \) matrix \( M \), let \( |M| \) denote the determinant of \( M \). Let \( I \) be the \( 3 \times 3 \) identity matrix. Let \( E \) and \( F \) be two \( 3 \times 3 \) matrices such that \( (I - EF) \) is invertible. If \( G = (I - EF)^{-1} \), then which of the following statements is(are) TRUE?

53.

For any positive integer \( n \), let \( S_{n} : (0,\infty) \to \mathbb{R} \) be defined by
\( S_{n}(x) = \displaystyle \sum_{k=1}^{n} \cot^{-1}\!\left(\dfrac{1 + k(k+1)x^{2}}{x}\right) \)
Where for any \( x \in \mathbb{R} \), \( \cot^{-1}(x) \in (0,\pi) \) and \( \tan^{-1}(x) \in \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) \). Then which of the following statements is (are) TRUE?

54.

For any complex number \( w = c + i d \), let \( \arg(w) \in (-\pi, \pi) \), where \( i = \sqrt{-1} \). Let \( \alpha \) and \( \beta \) be real numbers such that for all complex numbers \( z = x + i y \) satisfying \( \arg\!\left(\frac{z + \alpha}{z + \beta}\right) = \frac{\pi}{4} \), the ordered pair \( (x,y) \) lies on the circle \( x^2 + y^2 + 5x - 3y + 4 = 0 \). Then which of the following statements is (are) TRUE?

55.

For \( x \in \mathbb{R} \), the number of real roots of the equation \( 3x^2 - 4\lvert x^2 - 1 \rvert + x - 1 = 0 \) is ______.

56.

In a triangle \( ABC \), let \( AB = \sqrt{23} \), \( BC = 3 \) and \( CA = 4 \). Then the value of \( \dfrac{\cot A + \cot C}{\cot B} \) is _____.

57.

Let \( \vec{u}, \vec{v} \) and \( \vec{\omega} \) be vectors in three-dimensional space, where \( \vec{u} \) and \( \vec{v} \) are unit vectors which are not perpendicular to each other and \( \vec{u}\!\cdot\!\vec{w} = 1 \), \( \vec{v}\!\cdot\!\vec{w} = 1 \), \( \vec{w}\!\cdot\!\vec{w} = 4 \). If the volume of the parallelepiped, whose adjacent sides are represented by the vectors \( \vec{u}, \vec{v} \) and \( \vec{w} \) is \( \sqrt{2} \), then the value of \( \lvert 3\vec{u} + 5\vec{v} \rvert \) is ________.

JEE Advanced 2021 Paper - 1

Physics Chemistry Mathematics

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For any complex number \( w = c + i d \), let \( \arg(w) \in (-\pi, \pi) \), where \( i = \sqrt{-1} \). Let \( \alpha \) and \( \beta \) be real numbers such that for all complex numbers \( z = x + i y \) satisfying \( \arg\!\left(\frac{z + \alpha}{z + \beta}\right) = \frac{\pi}{4} \), the ordered pair \( (x,y) \) lies on the circle \( x^2 + y^2 + 5x - 3y + 4 = 0 \). Then which of the following statements is (are) TRUE?

JEE MainsJEE AdvanceBITSATJEE Advance | PYQ

JEE Mains JEE Advance BITSAT
JEE Advance | PYQ