JEE Main 2023 Mathematics Question Paper Jan 30 Shift 2- Download Paper with Solution PDF Here

Let \( x = \left( 8\sqrt{3}+13 \right)^{13} \) and \( y = \left( 7\sqrt{2}+9 \right)^9 \). If \( [t] \) denotes the greatest integer \( \leq t \), then

• (1) \( [x] + [y] \) is even
• (2) \( [x] \) is odd but \( [y] \) is even
• (3) \( [x] \) is even but \( [y] \) is odd
• (4) Both \( [x] \) and \( [y] \) are both odd

A vector \( \mathbf{v} \) in the first octant is inclined to the x-axis at 60°, to the y-axis at 45° and to the z-axis at an acute angle. If a plane passing through the points \( \left( \sqrt{2}, -1, 1 \right) \) and \( (a, b, c) \), is normal to \( \mathbf{v} \), then
 

• (1) \( \sqrt{2}a + b + c = 1 \)
• (2) \( a + b + \sqrt{2}c = 1 \)
• (3) \( a + \sqrt{2}b + c = 1 \)
• (4) \( \sqrt{2}a - b + c = 1 \)

Let f, g and h be the real valued functions defined on \( \mathbb{R} \) as \( f(x) = \left\lbrace \begin{array}{ll} \frac{x}{|x|}, & x \neq 0
\end{array} \right. \)

• (1) \( f \) is continuous at \( x = 0 \)
• (2) \( g \) is continuous at \( x = 0 \)
• (3) \( h \) is continuous at \( x = 0 \)
• (4) \( f, g, h \) are continuous at \( x = 0 \)

The number of ways of selecting two numbers a and b, \( a \in \{2, 4, 6, \dots, 100\}\) and \( b \in \{1, 3, 5, 7, \dots, 99\} \) such that 2 is the remainder when \( a + b \) is divided by 23 is:

• (1) 186
• (2) 54
• (3) 108
• (4) 268

If \( P \) is a 3x3 real matrix such that \( P^T = aP + (a-1)I \), where \( a > 1 \), then:

• (1) P is a singular matrix
• (2) \( |Adj P| > 1 \)
• (3) \( |Adj P| = \frac{1}{2} \)
• (4) \( |Adj P| = 1 \)

Let \( \lambda \in \mathbb{R}, \, \mathbf{a} = \lambda i + 2j - 3k, \, \mathbf{b} = i - \lambda j + 2k \). If \( \left( (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} \times \mathbf{b}) \right) \times (\mathbf{a} - \mathbf{b}) = 8i - 40j - 24k \), then \( |\lambda (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})|^2 \) is equal to ......

• (1) 140
• (2) 132
• (3) 144
• (4) 136

Let \( \mathbf{a} \) and \( \mathbf{b} \) be two vectors. Let \( |\mathbf{a}| = 1, |\mathbf{b}| = 4 \) and \( \mathbf{a} \cdot \mathbf{b} = 2 \). If \( \mathbf{c} = (2\mathbf{a} \times \mathbf{b}) - 3\mathbf{b} \), then the value of \( \mathbf{b} \cdot \mathbf{c} \) is

• (1) -24
• (2) -48
• (3) -84
• (4) -60

Let \( a_1 = 1, a_2, a_3, a_4, \dots \) be consecutive natural numbers. Then
\[ \tan^{-1} \left( \frac{1}{1 + a_1 a_2} \right) + \tan^{-1} \left( \frac{1}{1 + a_2 a_3} \right) + \dots + \tan^{-1} \left( \frac{1}{1 + a_{2021}a_{2022}} \right) \]

is equal to:

• (1) \( \frac{\pi}{4} - \cot^{-1}(2022) \)
• (2) \( \cot^{-1}(2022) - \frac{\pi}{4} \)
• (3) \( \tan^{-1}(2022) - \frac{\pi}{4} \)
• (4) \( \frac{\pi}{4} - \tan^{-1}(2022) \)

The parabolas: \( ax^2 + 2bx + cy = 0 \) and \( dx^2 + 2ex + fy = 0 \) intersect on the line \( y = 1 \). If \( a, b, c, d, e, f \) are positive real numbers and \( a, b, c, d, e, f \) are in G.P., then

• (1) \( d, e, f \) are in A.P.
• (2) \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are in G.P.
• (3) \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are in A.P.
• (4) \( d, e, f \) are in G.P.

If a plane passes through the points \( (-1, k, 0), (2, k, -1), (1, 1, 2) \) and is parallel to the line \( \frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1} \), then the value of \( \frac{k^2+1}{(k-1)(k-2)} \) is:

• (1) \( \frac{17}{5} \)
• (2) \( \frac{5}{17} \)
• (3) \( \frac{6}{13} \)
• (4) \( \frac{13}{6} \)

Let a, b, c > 1, \( a^3, b^3 \) and \( c^3 \) be in A.P., and \( \log_b a, \log_a c \) and \( \log_c b \) be in G.P. If the sum of the first 20 terms of an A.P., whose first term is \( \frac{a + 4b + c}{3} \) and the common difference is \( \frac{a - 8b + c}{10} = -444 \), then \( abc \) is equal to:

• (1) 343
• (2) 216
• (3) \( 343 \)
• (4) \( \frac{125}{8} \)

Let \( S \) be the set of all values of \( a_1 \) for which the mean deviation about the mean of 100 consecutive positive integers \( a_1, a_2, a_3, \dots, a_{100} \) is 25. Then \( S \) is:

• (1) \( \emptyset \)
• (2) \( \{99\} \)
• (3) \( \mathbb{N} \)
• (4) \( \{\} \)

\(\lim_{n \to \infty} \left( 3n \left[ 4 + \left( 2 + \frac{1}{n} \right)^2 + \left( 2 + \frac{2}{n} \right)^2 + \dots + \left( 3 - \frac{1}{n} \right)^2 \right] \right)\)

• (1) 12
• (2) \( \frac{19}{3} \)
• (3) 0
• (4) 19

For \( \alpha, \beta \in \mathbb{R} \), suppose the system of linear equations \( x - y + z = 5 \) \( 2x + 2y + \alpha z = 8 \) \( 3x - y + 4z = \beta \)

• (1) \( x^2 - 10x + 16 = 0 \)
• (2) \( x^2 + 18x + 56 = 0 \)
• (3) \( x^2 - 18x + 56 = 0 \)
• (4) \( x^2 + 14x + 24 = 0 \)

\( 50th root of a number x is 12 and 50th root of another number y is 18 \). Then the remainder obtained on dividing \( (x + y) \) by 25 is .......

Let \( A = \{ 1, 2, 3, 5, 8, 9 \} \). Then the number of possible functions \( f : A \to A \) such that \( f(m \cdot n) = f(m) \cdot f(n) \) for every \( m, n \in A \) with \( m \cdot n \in A \) is equal to ........

Let \( P(a_1, b_1) \) and \( Q(a_2, b_2) \) be two distinct points on a circle with center \( C(\sqrt{2}, \sqrt{3}) \). Let \( O \) be the origin and \( OC \) be perpendicular to both \( CP \) and \( CQ \). If the area of the triangle \( OPC \) is \( \frac{\sqrt{35}}{2} \), then \( a_1^2 + a_2^2 + b_1^2 + b_2^2 \) is equal to .......

The 8th common term of the series \( S_1 = 3 + 7 + 11 + 15 + 19 + \dots \), \( S_2 = 1 + 6 + 11 + 16 + 21 + \dots \),
is .........

Let a line L pass through the point \( P(2, 3, 1) \) and be parallel to the line \( x + 3y - 2z - 2 = 0 \) i.e. \( x - y + 2z = 0 \). If the distance of L from the point \( (5, 3, 8) \) is \( \alpha \), then \( 3\alpha^2 \) is equal to ........

If \( \int \sqrt{\sec 2x - 1} \, dx = \alpha \log_e \left( \cos 2x + \beta \right) + \sqrt{\cos 2x \left( 1 + \cos 2x \right) \left( \frac{1}{\beta} \right)} \) + constant, then \( \beta - \alpha \) is equal to ........

If the value of real number \( a > 0 \) for which \( x^2 - 5ax + 1 = 0 \) and \( x^2 - ax - 5 = 0 \) have a common real root is \( \frac{3}{\sqrt{2}} \), then \( \beta \) is equal to .......

The number of seven-digit odd numbers that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is ........

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is \( p \). Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is \( q \). If \( p : q = m : n \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to ..........

Let A be the area of the region \( \{ (x, y) : y \geq x^2, y \geq (1 - x)^2, y \leq 2x(1 - x) \} \).Then 540A is equal to .......