JEE Main 2023 Mathematics Question Paper Jan 31 Shift 1- Download Paper with Solution PDF Here

JEE Main 2023 Mathematics Questions with Solutions

Mathematics
Section – A

Question 1:

If the maximum distance of normal to the ellipse \[ \frac{x^2}{4} + \frac{y^2}{b^2} = 1, \, b < 2, from the origin is 1, then the eccentricity of the ellipse is: \]

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

For all \( z \in \mathbb{C} \) on the curve \( C \) such that \( | z | = 1 \), let the locus of the point \( z + \frac{1}{z} \) be the curve \( C_1 \). Then:

• (1) the curves \( C_1 \) and \( C_2 \) intersect at 4 points
• (2) the curves \( C_1 \) lies inside \( C_2 \)
• (3) the curves \( C_1 \) and \( C_2 \) intersect at 2 points
• (4) the curves \( C_2 \) lies inside \( C_1 \)

A wire of length 20 m is to be cut into two pieces. A piece of length \( \ell_1 \) is bent to make a square of area \( A_1 \), and the other piece of length \( \ell_2 \) is made into a circle of area \( A_2 \). If \( 2A_1 + 3A_2 \) is minimum, then \( \frac{\ell_1}{\ell_2} \) is equal to:

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

For the system of linear equations:

\( x + y + z = 6 \)

\( \alpha x + \beta y + 7z = 3 \)

\( x + 2y + 3z = 14 \)

Which of the following is NOT true?

• (1) If \( \alpha = \beta \), then the system has no solution
• (2) If \( \alpha = \beta \) and \( \alpha \neq 7 \), then the system has a unique solution.
• (3) There is a unique point \( (\alpha, \beta) \) on the line \( x + 2y + 18 = 0 \) for which the system has infinitely many solutions.
• (4) For every point \( (\alpha, \beta) \neq (7, 7) \) on the line \( x = 2y + 7 \), the system has infinitely many solutions.

Let the shortest distance between the lines \[ L: \frac{x - 5}{2} = \frac{y - \lambda}{0} = \frac{z + 1}{1}, \quad \lambda \geq 0 \quad and \quad L_1: x + 1 = y - 1 = 4 - z = 2\sqrt{6} \]
If \( (\alpha, \beta, \gamma) \) lies on \( L \), then which of the following is NOT possible?

• (1) \( \alpha + 2\gamma = 24 \)
• (2) \( 2\alpha + \gamma = 7 \)
• (3) \( 2\alpha - \gamma = 9 \)
• (4) \( \alpha - 2\gamma = 19 \)

Let \( y = f(x) \) represent a parabola with focus \( \left( -\frac{1}{2}, 0 \right) \) and directrix \( y = -\frac{1}{2} \).

Then \[ S = \left\{ x \in \mathbb{R} : \tan^{-1} \left( \sqrt{f(x)} + \sin \left( \sqrt{f(x) + 1} \right) \right) = \frac{\pi}{2} \right\} \]
contains:

• (1) Exactly two elements
• (2) Exactly one element
• (3) An infinite set
• (4) An empty set

Let \[ A = \begin{pmatrix} 1 & 0 & 0
0 & 4 & -1
0 & 12 & -3 \end{pmatrix} \]
Then the sum of the diagonal elements of the matrix \( (A + I)^{11} \) is equal to:

• (1) 6144
• (2) 4094
• (3) 4097
• (4) 2050

Let \( R \) be a relation on \( \mathbb{N} \times \mathbb{N} \) defined by \[ (a, b) \, R \, (c, d) \quad if and only if \quad ad(b - c) = bc(a - d). \]
Then \( R \) is:

• (1) Symmetric but neither reflexive nor transitive
• (2) Transitive but neither reflexive nor symmetric
• (3) Reflexive and symmetric but not transitive
• (4) Symmetric and transitive but not reflexive

Let \[ y = f(x) = \sin^3 \left( \frac{\pi}{3} \cos \left( \frac{\pi}{3\sqrt{2}} \left( -4x^3 + 5x^2 + 1 \right)^{\frac{3}{2}} \right) \right) \]
Then, at \( x = 1 \),

• (1) \( 2y' + 3y = 0 \)
• (2) \( 2y' + 3y' = 0 \)
• (3) \( y' - 3y' = 0 \)
• (4) \( y' + 3y' = 0 \)

If the sum and product of four positive consecutive terms of a G.P. are 126 and 1296, respectively, then the sum of common ratios of all such GPs is:

• (1) 7
• (2) \( \frac{9}{2} \)
• (3) 3
• (4) 14

The number of real roots of the equation \[ \sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6} \]
is:

• (1) 0
• (2) 1
• (3) 3
• (4) 2

Let a differentiable function \(f\) satisfy \[ f(x) + \int_3^x f(t) \, dt = \sqrt{x+1}, \quad x \geq 3. \]
Then \(f(8)\) is equal to:

• (1) 34
• (2) 19
• (3) 17
• (4) 1

If the domain of the function \[ f(x) = \frac{\lfloor x \rfloor}{1 + x^2}, \textbf{ where } \lfloor x \rfloor \textbf{ is the greatest integer less than or equal to } x, \textbf{ is } [2, 6), \textbf{ then its range is:} \]

• (1) \( \left[ \frac{5}{26}, \frac{9}{27} \right] \)
• (2) \( \left[ \frac{5}{26}, \frac{9}{29} \right] \)
• (3) \( \left[ \frac{9}{37}, \frac{29}{109} \right] \)
• (4) \( \left[ \frac{5}{37}, \frac{53}{37} \right] \)

Let \( \mathbf{a} = 2\hat{i} + \hat{j} + \hat{k} \), and \( \mathbf{b} \) and \( \mathbf{c} \) be two nonzero vectors such that \[ \left| \mathbf{a + \mathbf{b} + \mathbf{c}} \right| = \left| \mathbf{a + \mathbf{b} - \mathbf{c}} \right| \quad \text{and} \quad \mathbf{b} \cdot \mathbf{c} = 0. \]

Consider the following two statements:

(B) \( \mathbf{a} \) and \( \mathbf{c} \) are always parallel.

• (1) only (B) is correct
• (2) neither (A) nor (B) is correct
• (3) only (A) is correct
• (4) both (A) and (B) are correct

Let \( \alpha \in (0, 1) \) and \( \beta = \log(1 - \alpha) \). Let \[ P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{x^n}{n}, \quad x \in (0, 1). \]

Then the integral

\[ \int_0^\alpha \frac{1}{1 - t} \, dt \] is equal to:

• (1) \( \beta - P_0(\alpha) \)
• (2) \( -(\beta + P_0(\alpha)) \)
• (3) \( P_0(\alpha) - \beta \)
• (4) \( \beta + P_0(\alpha) \)

If \( \sin^{-1} \left( \frac{\alpha}{17} \right) + \cos^{-1} \left( \frac{4}{5} \right) - \tan^{-1} \left( \frac{77}{36} \right) = 0, \quad 0 < \alpha < 13, \)
then \( \sin^{-1} (\sin \alpha) + \cos^{-1} (\cos \alpha) \) is equal to:

• (1) \( \pi \)
• (2) 16
• (3) 0
• (4) \( 16 - 5\pi \)

Let a circle \( C_1 \) be obtained on rolling the circle \[ x^2 + y^2 - 4x - 6y + 11 = 0 \] upwards 4 units on the tangent \( T \) to it at the point (3, 2). Let \( C_2 \) be the image of \( C_1 \) in \( T \).

Let A and B be the centers of circles \( C_1 \) and \( C_2 \) respectively, and M and N be respectively the feet of perpendiculars drawn from A and B on the x-axis. Then the area of the trapezium AMNB is:

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

(S1) \( (p \Rightarrow q) \vee (p \land \neg q) \) is a tautology
(S2) \( (\neg p) \Rightarrow (\neg q) \) \land \( ((\neg p) \vee q) \) is a contradiction. Then:

• (1) only (S2) is correct
• (2) both (S1) and (S2) are correct
• (3) both (S1) and (S2) are wrong
• (4) only (S1) is correct

The value of \[ \int \frac{(2 + 3 \sin x)}{\sin x (1 + \cos x)} \, dx \]
is equal to:

• (1) \( \frac{7}{2} \sqrt{3} - \log \sqrt{3} \)
• (2) \( 2 + 3\sqrt{3} + \log \sqrt{3} \)
• (3) \( \frac{10}{3} \sqrt{3} - \log \sqrt{3} \)
• (4) \( \sqrt{3} - \log \sqrt{3} \)

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is:

• (1) \( \frac{5}{7} \)
• (2) \( \frac{2}{7} \)
• (3) \( \frac{3}{7} \)
• (4) \( \frac{5}{6} \)

Section – B

Question 21:

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers.
Then the serial number of the number 42923 is:

Let \( a_1, a_2, \dots, a_n \) be in A.P. If \( a_5 = 2a_1 \text{ and } a_1 = 18, \) then

\[ 12 \left( \frac{1}{\sqrt{a_0} + \sqrt{a_1}} + \frac{1}{\sqrt{a_1} + \sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}} \right) \]

is equal to:

Let \( \theta \) be the angle between the planes

\[ P_1: \vec{r} \cdot ( \hat{i} + \hat{j} + 2 \hat{k}) = 9 \quad \text{and} \quad P_2: \vec{r} \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 15. \]

Let \( L \) be the line that meets \( P_2 \) at the point (4, -2, 5) and makes an angle \( \theta \) with the normal of \( P_2 \). If \( \alpha \) is the angle between \( L \) and \( P_2 \), then

\[ (\tan^2 \theta)(\cot^2 \alpha) \] is equal to:

Question 24:

Let α > 0, be the smallest number such that the expansion of (3/x³ + ²/_x)³⁰ has a term βx^-α, β ∈ ℕ. Then α is equal to:

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \[ |\vec{a}| = \sqrt{14}, \quad |\vec{b}| = \sqrt{6}, \quad |\vec{a} \times \vec{b}| = \sqrt{48}. \]
Then \( (\vec{a} \cdot \vec{b})^2 \textbf{ is equal to:} \)

Let the line \( L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-3}{1} \) intersect the plane \[ 2x + y + 3z = 16 \textbf{ at the point } P. \]
Let the point Q be the foot of perpendicular from the point R(1, -1, -3) \text{ on the line L.
If \alpha is the area of triangle PQR, then \alpha^2 is equal to:

The remainder on dividing \( 5^{99} \) by 11 is:

If the variance of the frequency distribution

\[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x_i & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline f_i & 3 & 6 & 16 & \alpha & 9 & 5 & 6 \\ \hline \end{array} \]

is given, find the value of \( \alpha \).

Let for \( x \in \mathbb{R} \) \[ f(x) = \frac{x + |x|}{2} \quad for \quad x \geq 0 \quad and \quad f(x) = \frac{x}{2} \quad for \quad x < 0 \] \[ g(x) = \begin{cases} x^2 & for \quad x \geq 0,
x & for \quad x < 0 \end{cases} \]
\text{Then the area bounded by the curve \( y = f \circ g(x) \) \text{ and the lines \( y = 0, 2y - x = 15 \) \text{ is equal to:

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to: