JEE Main 2023 Mathematics Question Paper Feb 1 Shift 2- Download PDF

JEE Main 2023 Feb 1 Shift 2 Mathematics Question Paper with Solution

Question 1:

The sum \( \sum_{n=1}^{\infty} \frac{2n^2 + 3n + 4}{(2n)!} \) is equal to:

• (1) \( \frac{11e}{2} + \frac{7}{2e} \)
• (2) \( \frac{13e}{4} + \frac{5}{4e} - 4 \)
• (3) \( \frac{11e}{2} + \frac{7}{2e} - 4 \)
• (4) \( \frac{13e}{4} + \frac{5}{4e} \)

Let \( S = \{x \in \mathbb{R} : 0 < x < 1 and 2 \tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\} \). If \( n(S) \) denotes the number of elements in \( S \), then:

• (1) \( n(S) = 2 \) and only one element in \( S \) is less than \( \frac{1}{2} \).
• (2) \( n(S) = 1 \) and the element in \( S \) is more than \( \frac{1}{2} \).
• (3) \( n(S) = 1 \) and the element in \( S \) is less than \( \frac{1}{2} \).
• (4) \( n(S) = 0 \).

Let \( \vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k} \), \( \vec{b} = \hat{i} + \hat{k} \), and \( \vec{c} = \hat{i} + 2\hat{j} - 3\hat{k} \) be three given vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = \vec{c} \times \vec{a} \) and \( \vec{r} \cdot \vec{b} = 0 \), then \( |\vec{r}| \) is equal to:

• (1) \( \frac{11}{7} \sqrt{2} \)
• (2) \( \frac{11}{7} \)
• (3) \( \frac{11}{5} \sqrt{2} \)
• (4) \( \frac{\sqrt{914}}{7} \)

If \( A = \frac{1}{2} \begin{bmatrix} 1 & \sqrt{3}
-\sqrt{3} & 1 \end{bmatrix} \), then:

• (1) \( A^{30} - A^{25} = 2I \)
• (2) \( A^{30} + A^{25} + A = I \)
• (3) \( A^{30} + A^{25} - A = I \)
• (4) \( A^{30} = A^{25} \)

Two dice are thrown independently. Let \( A \) be the event that the number appeared on the 1^st die is less than the number appeared on the 2^nd die, \( B \) be the event that the number appeared on the 1^st die is even and that on the 2^nd die is odd, and \( C \) be the event that the number appeared on the 1^st die is odd and that on the 2^nd die is even. Then:

• (1) The number of favourable cases of the event \( (A \cup B) \cap C \) is 6.
• (2) \( A \) and \( B \) are mutually exclusive.
• (3) The number of favourable cases of the events \( A, B, and C \) are 15, 6, and 6 respectively.
• (4) \( B \) and \( C \) are independent.

Which of the following statements is a tautology?

• (1) \( p \to (p \land (p \to q)) \)
• (2) \( (p \land q) \to \sim (p \to q) \)
• (3) \( (p \land (p \to q)) \to \sim q \)
• (4) \( p \lor (p \land q) \)

The number of integral values of \( k \), for which one root of the equation \( 2x^2 - 8x + k = 0 \) lies in the interval \( (1, 2) \) and its other root lies in the interval \( (2, 3) \), is:

• (1) 2
• (2) 0
• (3) 1
• (4) 3

Let \( f : \mathbb{R} - \{0, 1\} \to \mathbb{R} \) be a function such that \( f(x) + f\left(\frac{1}{1-x}\right) = 1 + x \). Then \( f(2) \) is equal to:

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

Let the plane \( P \) pass through the intersection of the planes \( 2x + 3y - z = 2 \) and \( x + 2y + 3z = 6 \), and be perpendicular to the plane \( 2x + y - z + 1 = 0 \). If \( d \) is the distance of \( P \) from the point \((-7, 1, 1)\), then \( d^2 \) is equal to:

• (1) \( \frac{250}{83} \)
• (2) \( \frac{15}{53} \)
• (3) \( \frac{25}{83} \)
• (4) \( \frac{250}{82} \)

Let \( a, b \) be two real numbers such that \( ab < 0 \). If the complex number \( \frac{1 + ai}{b + i} \) is of unit modulus and \( a + ib \) lies on the circle \( |z - 1| = |2z| \), then a possible value of \( \frac{1 + \lfloor a \rfloor}{4b} \), where \( \lfloor t \rfloor \) is the greatest integer function, is:

• (1) \( -\frac{1}{2} \)
• (2) \( -1 \)
• (3) \( 1 \)
• (4) \( \frac{1}{2} \)

The sum of the absolute maximum and minimum values of the function \( f(x) = |x^2 - 5x + 6| - 3x + 2 \) in the interval \([-1, 3]\) is equal to:

• (1) 10
• (2) 12
• (3) 13
• (4) 24

Let \( P(S) \) denote the power set of \( S = \{1, 2, 3, \dots, 10\} \). Define the relations \( R_1 \) and \( R_2 \) on \( P(S) \) as \( A R_1 B \) if \[ (A \cap B^c) \cup (B \cap A^c) = \varnothing, \]
and \( A R_2 B \) if \[ A \cup B^c = B \cup A^c, \]
for all \( A, B \in P(S) \). Then:

• (1) Both \( R_1 \) and \( R_2 \) are equivalence relations.
• (2) Only \( R_1 \) is an equivalence relation.
• (3) Only \( R_2 \) is an equivalence relation.
• (4) Both \( R_1 \) and \( R_2 \) are not equivalence relations.

The area of the region given by \(\{(x, y) : xy \leq 8, \, 1 \leq y \leq x^2\}\) is:

• (1) \( 8 \ln_e 2 - \frac{13}{3} \)
• (2) \( 16 \ln_e 2 - \frac{14}{3} \)
• (3) \( 8 \ln_e 2 + \frac{7}{6} \)
• (4) \( 16 \ln_e 2 + \frac{7}{3} \)

Let \( \alpha x = \exp(x^\beta y^\gamma) \) be the solution of the differential equation \( 2x^2 y \, dy - (1 - xy^2) \, dx = 0 \), \( x > 0 \), \( y(2) = \sqrt{\ln_e 2} \). Then \( \alpha + \beta - \gamma \) equals:

• (1) \( 1 \)
• (2) \( -1 \)
• (3) \( 0 \)
• (4) \( 3 \)

The value of the integral \[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} \, dx is: \]

• (1) \( \frac{\pi^2}{6} \)
• (2) \( \frac{\pi^2}{12 \sqrt{3}} \)
• (3) \( \frac{\pi^2}{3 \sqrt{3}} \)
• (4) \( \frac{\pi^2}{6 \sqrt{3}} \)

Let \( 9 = x_1 < x_2 < \dots < x_7 \) be in an A.P. with common difference \( d \). If the standard deviation of \( x_1, x_2, \dots, x_7 \) is \( 4 \) and the mean is \( \overline{x} \), then \( \overline{x} + x_6 \) is equal to:

• (1) \( 18 \left( 1 + \frac{1}{\sqrt{3}} \right) \)
• (2) \( 34 \)
• (3) \( 2 \left( 9 + \frac{8}{\sqrt{7}} \right) \)
• (4) \( 25 \)

For the system of linear equations \( ax + y + z = 1 \), \( x + ay + z = 1 \), \( x + y + az = \beta \), which one of the following statements is NOT correct?

• (1) It has infinitely many solutions if \( \alpha = 2 \) and \( \beta = -1 \).
• (2) It has no solution if \( \alpha = -2 \) and \( \beta = 1 \).
• (3) \( x + y + z = \frac{3}{4} \) if \( \alpha = 2 \) and \( \beta = 1 \).
• (4) It has infinitely many solutions if \( \alpha = 1 \) and \( \beta = 1 \).

Let \( \vec{a} = 5\hat{i} - \hat{j} - 3\hat{k} \) and \( \vec{b} = \hat{i} + 3\hat{j} + 5\hat{k} \) be two vectors. Then which one of the following statements is TRUE?

• (1) Projection of \( \vec{a} \) on \( \vec{b} \) is \( \frac{17}{\sqrt{35}} \) and the direction of the projection vector is the same as \( \vec{b} \).
• (2) Projection of \( \vec{a} \) on \( \vec{b} \) is \( \frac{-17}{\sqrt{35}} \) and the direction of the projection vector is opposite to \( \vec{b} \).
• (3) Projection of \( \vec{a} \) on \( \vec{b} \) is \( \frac{17}{\sqrt{35}} \) and the direction of the projection vector is opposite to \( \vec{b} \).
• (4) Projection of \( \vec{a} \) on \( \vec{b} \) is \( \frac{-17}{\sqrt{35}} \) and the direction of the projection vector is opposite to \( \vec{b} \).

Let \( P(x_0, y_0) \) be the point on the hyperbola \( 3x^2 - 4y^2 = 36 \), which is nearest to the line \( 3x + 2y = 1 \). Then \( \sqrt{2} \, (y_0 - x_0) \) is equal to:

• (1) \(-3\)
• (2) \(9\)
• (3) \(-9\)
• (4) \(3\)

If \( y(x) = x^x, \, x > 0 \), then \( y''(2) - 2y'(2) \) is equal to:
 

• (1) \( 8 \log_e 2 - 2 \)
• (2) \( 4 \log_e 2 + 2 \)
• (3) \( 4 (\log_e 2)^2 - 2 \)
• (4) \( 4 (\log_e 2)^2 + 2 \)

The total number of six-digit numbers, formed using the digits \(4, 5, 9\) only and divisible by 6, is ____.

Number of integral solutions to the equation \( x + y + z = 21 \), where \( x \geq 1, y \geq 3, z \geq 4 \), is ____.

The line \( x = 8 \) is the directrix of the ellipse \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with the corresponding focus \( (2, 0) \). If the tangent to \( E \) at the point \( P \) in the first quadrant passes through the point \( \left( 0, 4\sqrt{3} \right) \) and intersects the \( x \)-axis at \( Q \), then \( (3PQ)^2 \) is equal to ____.

If the x-intercept of a focal chord of the parabola \( y^2 = 8x + 4y + 4 \) is \( 3 \), then the length of this chord is equal to ____.

If \( \int_0^\pi \frac{5^{\cos x} (1 + \cos x \cos 3x + \cos^2 x + \cos^3 x \cos 3x)}{1 + 5^{\cos x}} dx = \frac{k \pi}{16}, \) then \( k \) is equal to ____.

Let the sixth term in the binomial expansion of \[ \left( \sqrt{2^{\log_2(10 - 3^x)}} + 5 \cdot \sqrt{2^{(x-2)\log_2 3}} \right)^m, \]
in the increasing powers of \( 2^{(x-2)\log_2 3} \), be 21. If the binomial coefficients of the second, third, and fourth terms in the expansion are respectively the first, third, and fifth terms of an A.P., then the sum of the squares of all possible values of \( x \) is ____.

If the term without \(x\) in the expansion of \[ \left( x^{\frac{2}{3}} + \frac{\alpha}{x^3} \right)^{22} \]
is \(7315\), then \(|\alpha|\) is equal to ____.

The sum of the common terms of the following three arithmetic progressions:

- \( 3, 7, 11, 15, \dots, 399 \),

- \( 2, 5, 8, 11, \dots, 359 \),

- \( 2, 7, 12, 17, \dots, 197 \),


is equal to ____.

Let \( \alpha x + \beta y + yz = 1 \) be the equation of a plane passing through the point \((3, -2, 5)\) and perpendicular to the line joining the points \((1, 2, 3)\) and \((-2, 3, 5)\). Then the value of \( \alpha \beta y \) is equal to ____.

The point of intersection \(C\) of the plane \(8x + y + 2z = 0\) and the line joining the points \(A(-3, -6, 1)\) and \(B(2, 4, -3)\) divides the line segment \(AB\) internally in the ratio \(k:1\). If \(a, b, c\) (\(|a|, |b|, |c|\) are coprime) are the direction ratios of the perpendicular from the point \(C\) on the line \(\frac{1 - x}{1} = \frac{y + 4}{2} = \frac{z + 2}{3}\), then \(|a + b + c|\) is equal to ____.