JEE Main 2023 Mathematics Question Paper Jan 30 Shift 1- Download Paper with Solution PDF Here

Let
\[ A = \begin{bmatrix} m & n
p & q \end{bmatrix}, \, d = |A| \neq 0, \, and |A - d(Adj A)| = 0. \]
Then:

• (1) \((1 + d)^2 = (m + q)^2\)
• (2) \(1 + d^2 = (m + q)^2\)
• (3) \((1 + d)^2 = m^2 + q^2\)
• (4) \(1 + d^2 = m^2 + q^2\)

The line \( \ell_1 \) passes through the point \( (2, 6, 2) \) and is perpendicular to the plane \( 2x + y - 2z = 10 \). Then the shortest distance between the line \( \ell_1 \) and the line
\[ \frac{x+1}{2} = \frac{y+4}{-3} = \frac{z}{2} \]
is:

• (1) 7
• (2) \( \frac{19}{3} \)
• (3) \( \frac{19}{2} \)
• (4) 9

If an unbiased die, marked with \( -2, -1, 0, 1, 2, 3 \) on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:

• (1) \( \frac{881}{2592} \)
• (2) \( \frac{521}{2592} \)
• (3) \( \frac{440}{2592} \)
• (4) \( \frac{27}{288} \)

Let the system of linear equations

x + y + kz = 2

2x + 3y - z = 1

3x + 4y + 2z = k

have infinitely many solutions. Then the system

(k+1)x + (2k-1)y = 7

(2k+1)x + (k+5)y = 10

has:

• (1) infinitely many solutions
• (2) unique solution satisfying \( x - y = 1 \)
• (3) no solution
• (4) unique solution satisfying \( x + y = 1 \)

If
\[ \tan 15^\circ + \frac{1}{\tan 75^\circ} + \tan 105^\circ + \tan 195^\circ = 2a, \]
then the value of \( a + \frac{1}{a} \) is:

• (1) 4
• (2) \( 4 - 2\sqrt{3} \)
• (3) 2
• (4) \( 5 - 3\sqrt{3} \)

Suppose \( f : \mathbb{R} \to (0, \infty) \) be a differentiable function such that \( 5f(x + y) = f(x) \cdot f(y), \, \forall x, y \in \mathbb{R} \). If \( f(3) = 320 \), then \( \sum_{n=0}^{5} f(n) \) is equal to:

• (1) 6875
• (2) 6575
• (3) 6825
• (4) 6528

If \[ a_n = \frac{-2}{4n^2 - 16n + 15}, \quad then \quad a_1 + a_2 + \dots + a_5 is equal to: \]

• (1) \( \frac{51}{144} \)
• (2) \( \frac{49}{138} \)
• (3) \( \frac{50}{141} \)
• (4) \( \frac{52}{147} \)

If the coefficient of \( x^{15} \) in the expansion of \[ \left(ax^3 + \frac{1}{bx^3}\right)^{15} \]
is equal to the coefficient of \( x^{-15} \) in the expansion of \[ \left(\frac{a}{x^3} - \frac{1}{bx^3}\right)^{15}, \]
where \( a \) and \( b \) are positive real numbers, then for each such ordered pair \( (a, b) \):

• (1) \( a = b \)
• (2) \( ab = 1 \)
• (3) \( a = 3b \)
• (4) \( ab = 3 \)

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are three non-zero vectors and \( \hat{n} \) is a unit vector perpendicular to \( \mathbf{c} \) such that \[ \mathbf{a} = \alpha \mathbf{b} - \hat{n}, \quad (\alpha \neq 0) \]
and \[ \mathbf{\overrightarrow{b}} \cdot \mathbf{\overrightarrow{c}} = 12, \quad then \quad \left| \mathbf{\overrightarrow{c}} \times (\mathbf{\overrightarrow{a} } \times \mathbf{\overrightarrow{b}}) \right| \]
is equal to:

• (1) 15
• (2) 9
• (3) 12
• (4) 6

The number of points on the curve \[ y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x \]
at which the normal lines are parallel to \[ x + 90y + 2 = 0 \]
is:

• (1) 2
• (2) 3
• (3) 4
• (4) 0

Let \[ y = x + 2, \quad 4y = 3x + 6, \quad and \quad 3y = 4x + 1 \]
be three tangent lines to the circle \[ (x - h)^2 + (y - k)^2 = r^2. \]
Then \( h + k \) is equal to:

• (1) 5
• (2) \( 5(1 + \sqrt{2}) \)
• (3) 6
• (4) \( 5\sqrt{2} \)

Let the solution curve \( y = y(x) \) of the differential equation \[ \frac{dy}{dx} - \frac{3x^5 \tan^{-1}(x^3)}{(1+x^6)^{3/2}} y = 2x \] \[exp \frac{x^3-\tan^{-1}x^3}{\sqrt{(1+x)^6}}\]pass through the origin. Then \( y(1) \) is equal to:

• (1) \( \exp\left(\frac{4 - \pi}{4\sqrt{2}}\right) \)
• (2) \( \exp\left(\frac{\pi - 4}{4\sqrt{2}}\right) \)
• (3) \( \exp\left(\frac{1 - \pi}{4\sqrt{2}}\right) \)
• (4) \( \exp\left(\frac{4 + \pi}{4\sqrt{2}}\right) \)

Let a unit vector \( \overrightarrow{OP} \) make angles \( \alpha, \beta, \gamma \) with the positive directions of the coordinate axes \( OX, OY, OZ \) respectively, where \( \beta \in \left( 0, \frac{\pi}{2} \right) \), and \( \overrightarrow{OP} \) is perpendicular to the plane through points \( (1, 2, 3) \), \( (2, 3, 4) \), and \( (1, 5, 7) \). Then which one of the following is true?

• (1) \( \alpha \in \left( \frac{\pi}{2}, \pi \right) \) and \( \gamma \in \left( \frac{\pi}{2}, \pi \right)
• (2) \( \alpha \in \left( 0, \frac{\pi}{2} \right) \) and \( \gamma \in \left( 0, \frac{\pi}{2} \right)
• (3) \( \alpha \in \left( \frac{\pi}{2}, \pi \right) \) and \( \gamma \in \left( 0, \frac{\pi}{2} \right)
• (4) \( \alpha \in \left( 0, \frac{\pi}{2} \right) \) and \( \gamma \in \left( \frac{\pi}{2}, \pi \right) \)

If \([t]\) denotes the greatest integer \(\leq 1\), then the value of \[ \frac{3(e-1)^2}{e} \int_{1}^{2} x^2 e^{[x] + [x^3]} dx \]
is:

• (1) \( e^9 - e \)
• (2) \( e^8 - e \)
• (3) \( e^7 - 1 \)
• (4) \( e^8 - 1 \)

If \( P(h,k) \) be a point on the parabola \( x = 4y^2 \), which is nearest to the point \( Q(0, 33) \), then the distance of \( P \) from the directrix of the parabola \( y^2 = 4(x + y) \) is equal to:

• (1) 2
• (2) 4
• (3) 8
• (4) 6

A straight line cuts off the intercepts \( OA = a \) and \( OB = b \) on the positive directions of the \( x \)-axis and \( y \)-axis, respectively. If the perpendicular from the origin \( O \) to this line makes an angle of \( \frac{\pi}{6} \) with the positive direction of the \( y \)-axis and the area of \( \triangle OAB \) is \( \frac{98}{3} \sqrt{3} \), then \( a^2 - b^2 \) is equal to:

• (1) \( \frac{392}{3} \)
• (2) \( 196 \)
• (3) \( \frac{196}{3} \)
• (4) \( 98 \)

The coefficient of \( x^{301} \) in \[ (1 + x)^{500} + x(1 + x)^{499} + x^2(1 + x)^{498} + \dots + x^{500} \]
is:

• (1) \( ^{501}C_{302} \)
• (2) \( ^{500}C_{301} \)
• (3) \( ^{500}C_{300} \)
• (4) \( ^{501}C_{200} \)

Among the statements:

(S1) \( \left( (p \lor q) \Rightarrow r \right) \Leftrightarrow \left( p \Rightarrow r \right) \)

(S2) \( \left( (p \lor q) \Rightarrow r \right) \Leftrightarrow \left( (p \Rightarrow r) \lor (q \Rightarrow r) \right) \)

Which of the following is true?

• (1) Only (S1) is a tautology
• (2) Neither (S1) nor (S2) is a tautology
• (3) Only (S2) is a tautology
• (4) Both (S1) and (S2) are tautologies

The minimum number of elements that must be added to the relation \[ R = \{(a, b), (b, c)\} \]
on the set \[ \{a, b, c\} \]
so that it becomes symmetric and transitive is:

• (1) 4
• (2) 7
• (3) 5
• (4) 3

If the solution of the equation \[ \log_{\cos x} \cot x + 4 \log_{\sin x} \tan x = 1, \, x \in \left(0, \frac{\pi}{2}\right), \]
is \[ \sin^{-1}\left(\frac{\alpha + \sqrt{\beta}}{2}\right), \]
where \( \alpha, \beta \) are integers, then \( \alpha + \beta \) is equal to:

• (1) \( 3 \)
• (2) \( 5 \)
• (3) \( 6 \)
• (4) \( 4 \)

Let \( S = \{1, 2, 3, 4, 5, 6\} \). Then the number of one-one functions \( f: S \to P(S) \), where \( P(S) \) denotes the power set of \( S \), such that \( f(n) \subset f(m) \) where \( n < m \), is ____.

Let \( \alpha \) be the area of the larger region bounded by the curve \[ y^2 = 8x \]
and the lines \[ y = x \quad and \quad x = 2, \]
which lies in the first quadrant. Then the value of \( 3\alpha \) is equal to:

\(\lambda\)\(_1\) \(<\) \(\lambda\)\(_2\) are two values of \(\lambda\) such that the angle between the planes \[ P_1 : \vec{r} \cdot (3\hat{i} - 5\hat{j} + \hat{k}) = 7 \]
and \[ P_2 : \vec{r} \cdot (\lambda \hat{i} + \hat{j} - 3\hat{k}) = 9 \]
is \(\sin^{-1} \left( \frac{2\sqrt{6}}{5} \right)\), then the square of the length of the perpendicular from the point \((38\lambda, 10\lambda, 2)\) to the plane \(P_1\) is _____.
 

Let \( z = 1 + i \) and \( z_1 = \frac{1 + i\bar{z}}{\bar{z}(1-z) + \frac{1}{z}} \). Then \( \frac{12}{\pi} \, arg(z_1) \) is equal to ____.

\[ \lim_{x \to 0} \frac{48}{x^4} \int_{0}^{x} \frac{t^3}{t^6 + 1} \, dt is equal to \_\_\_\_\_. \]

The mean and variance of 7 observations are 8 and 16, respectively. If one observation 14 is omitted and \( a \) and \( b \) are respectively the mean and variance of the remaining 6 observations, then \( a + 3b - 5 \) is equal to:

If the equation of the plane passing through the point \( (1, 1, 2) \) and perpendicular to the line \[ x - 3y + 2z - 1 = 0, \quad 4x - y + z = 0 \quad is \quad Ax + By + Cz = 1, \]
then \( 140(C - B + A) \) is equal to:

Let \[ \sum_{n=0}^{\infty} \frac{n^3 \big( (2n)! \big) + (2n-1)(n!)}{(n!)(2n)!} = a e + \frac{b}{e} + c, \]
where \(a, b, c \in \mathbb{Z}\) and \(e = \sum_{n=0}^{\infty} \frac{1}{n!}\). Then \(a^2 - b + c\) is equal to ______.
 

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3, and 5 and are divisible by 15 is equal to:

Let \[ f^1(x) = \frac{3x + 2}{2x + 3}, \quad x \in \mathbb{R}, \quad R - \left( -\frac{3}{2} \right). \]
For \( n \geq 2 \), define \( f^n(x) = f^10f^{n-1}(x) \) and if
\[
f^5(x) = \frac{ax + b{bx + a, \quad \gcd(a, b) = 1, \quad \text{then \quad a + b \text{ is equal to: