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If $\sum_{k=1}^{31} \left({}^{31}C_k\right)\left({}^{31}C_{k-1}\right) - \displaystyle \sum_{k=1}^{30} \left({}^{30}C_k\right)\left({}^{30}C_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to:

Let a function \( f : \mathbb{N} \to \mathbb{N} \) be defined by

If the system of linear equations,
\( 2x + 3y - z = -2 \)
\( x + y + z = 4 \)
\( x - y + |\lambda| z = 4\lambda - 4 \)
Where \( \lambda \in \mathbb{R} \), has no solution, then:

Let \( A \) be a matrix of order \( 3 \times 3 \) and \( \det(A) = 2 \). then \( \det\!\big(\det(A)\operatorname{adj}(5\operatorname{adj}(A^{3}))\big) \) is equal to ______.

The total number of \( 5 \)-digit numbers, formed by using the digits \( 1, 2, 3, 5, 6, 7 \) without repetition, which are multiple of \( 6 \), is:

Let \( A_1, A_2, A_3, \ldots \) be an increasing geometric progression of positive real numbers. If \( A_1 A_3 A_5 A_7 = \frac{1}{1296} \) and \( A_2 + A_4 = \frac{7}{36} \), then, the value of \( A_6 + A_8 + A_{10} \) is equal to:

Let \( [t] \) denote the greatest integer less than or equal to \( t \). Then, the value of the integral \( \int_{0}^{1} \left[ -8x^2 + 6x - 1 \right] dx \) is equal to:

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as

Where \( a, b, c \in \mathbb{R} \) and \( [t] \) denotes greatest integer less than or equal to \( t \). Then, which of the following statements is true?

The area of the region \( S = \{(x,y) : y^2 \le 8x,\, y \ge \sqrt{2}x,\, x \ge 1\} \) is:

Let the solution curve \( y = y(x) \) of the differential equation
$\left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] x \frac{dy}{dx} = x + \left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] y$
pass through the points \( (1,0) \) and \( (2\alpha,\alpha) \), \( \alpha > 0 \). Then \( \alpha \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \( x(1-x^{2})\dfrac{dy}{dx} + \left(3x^{2}y - y - 4x^{3}\right) = 0 \), \( x > 1 \), with \( y(2) = -2 \). Then \( y(3) \) is equal to

The number of real solutions of \( x^7 + 5x^3 + 3x + 1 = 0 \) is equal to _______.

Let the eccentricity of the hyperbola \( H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) be \( \sqrt{\frac{5}{2}} \) and length of its latus rectum be \( 6\sqrt{2} \). If \( y = 2x + c \) is a tangent to the hyperbola \( H \), then the value of \( c^2 \) is equal to:

If the tangents drawn at the points \( O(0,0) \) and \( P(1+\sqrt{5}, 2) \) on the circle \( x^2 + y^2 - 2x - 4y = 0 \) intersect at the point \( Q \), then the area of the triangle \( OPQ \) is equal to:

If two distinct points \( Q, R \) lie on the line of intersection of the planes \( -x + 2y - z = 0 \) and \( 3x - 5y + 2z = 0 \) and \( PQ = PR = \sqrt{18} \) where the point \( P \) is \( (1,-2,3) \), then the area of the triangle \( PQR \) is equal to:

The acute angle between the planes \( P_1 \) and \( P_2 \), when \( P_1 \) and \( P_2 \) are the planes passing through the intersection of the planes \( 5x + 8y + 13z - 29 = 0 \) and \( 8x - 7y + z - 20 = 0 \) and the points \( (2,1,3) \) and \( (0,1,2) \), respectively, is:

Let the plane $P : \vec{r} \cdot \vec{a} = d$ contain the line of intersection of two planes $\vec{r} \cdot \left(\hat{i} + 3\hat{j} - \hat{k}\right) = 6$ and $\vec{r} \cdot \left(-6\hat{i} + 5\hat{j} - \hat{k}\right) = 7$. If the plane $P$ passes through the point $\left(2, 3, \frac{1}{2}\right)$, then the value of $\frac{|13\vec{a}|^2}{d^2}$ is equal to:

The probability, that in a randomly selected \( 3 \)-digit number at least \( 2 \) digits are odd, is:

Let \( AB \) and \( PQ \) be two vertical poles, \( 160 \, \text{m} \) apart from each other. Let \( C \) be the middle point of \( B \) and \( Q \), which are feet of these two poles. Let \( \frac{\pi}{8} \) and \( \theta \) be the angles of elevation from \( C \) to \( P \) and \( A \), respectively. If the height of pole \( PQ \) is twice the height of pole \( AB \), then \( \tan^{2}\theta \) is equal to:

Let \( p,q,r \) be three logical statements. Consider the compound statements
$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$ and $S_2 : p \rightarrow (q \vee r)$
Then, which of the following is NOT true?

Let \( R_1 \) and \( R_2 \) be relations on the set \( \{ 1, 2, \ldots, 50 \} \) such that
\( R_1 = \{ (p, p^n) : p \text{ is a prime and } n \ge 0 \text{ is an integer} \} \) and
\( R_2 = \{ (p, p^n) : p \text{ is a prime and } n = 0 \text{ or } 1 \} \).
Then, the number of elements in \( R_1 - R_2 \) is ________.

The number of real solution of the equation \( e^{4x} + 4e^{3x} - 58e^{2x} + 4e^{x} + 1 = 0 \) is ______.

The mean and standard deviation of \( 15 \) observations are found to be \( 8 \) and \( 3 \) respectively. On rechecking it was found that, in the observations, \( 20 \) was misread as \( 5 \). Then, the correct variance is equal to ____________.

If $\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ are coplanar vectors and $\vec{a} \cdot \vec{c} = 5$, $\vec{b} \perp \vec{c}$, then $122(c_1 + c_2 + c_3)$ is equal to ______________.

A ray of light passing through the point \( P(2, 3) \) reflects on the \( x \)-axis at point \( A \) and the reflected ray passes through the point \( Q(5, 4) \). Let \( R \) be the point that divides the line segment \( AQ \) internally into the ratio \( 2\!:\!1 \). Let the co-ordinates of the foot of the perpendicular \( M \) from \( R \) on the bisector of the angle \( PAQ \) be \( (\alpha, \beta) \). Then, the value of \( 7\alpha + 3\beta \) is equal to ________.

Let \( l \) be a line which is normal to the curve \( y = 2x^2 + x + 2 \) at a point \( P \) on the curve. If the point \( Q(6, 4) \) lies on the line \( l \) and \( O \) is origin, then the area of the triangle \( OPQ \) is equal to ________.

Let \( A = \{1,a_1,a_2,\ldots,a_{18},77\} \) be a set of integers with \( 1 < a_1 < a_2 < \ldots < a_{18} < 77 \).
Let the set \( A + A = \{ x + y : x, y \in A \} \) contain exactly \( 39 \) elements. Then, the value of \( a_1 + a_2 + \ldots + a_{18} \) is equal to ________.

The number of positive integers \( k \) such that the constant term in the binomial expansion of \( \left(2x^3 + \frac{3}{x^k}\right)^{12} , \, x \neq 0 \) is \( 2^{8} \).\(\ell\) where \( \ell \) is an odd integer is ________.

The number of elements in the set \( \{ z = a + ib \in \mathbb{C} : a, b \in \mathbb{Z} \text{ and } 1 < | z - 3 + 2i | < 4 \} \) is ________.

Let the lines \( y + 2x = \sqrt{11} + 7\sqrt{7} \) and \( 2y + x = 2\sqrt{11} + 6\sqrt{7} \) be normal to a circle \( C : (x - h)^2 + (y - k)^2 = r^2 \). If the line \( \sqrt{11}y - 3x = \frac{5\sqrt{77}}{3} + 11 \) is tangent to the circle \( C \), then the value of \( (5h - 8k)^2 + 5r^2 \) is equal to ________.

JEE Main 2022 28th June Morning Shift

Physics Chemistry Mathematics

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The number of real solutions of \( x^7 + 5x^3 + 3x + 1 = 0 \) is equal to _______.