JEE Main 2023 Mathematics April 15 Shift 1 Question Paper with Solution PDF- Download Here

Let \( S \) be the set of all values of \( \lambda \), for which the shortest distance between the lines \[ \frac{x - 0}{1} = \frac{y - 4}{3} = \frac{z + \lambda}{6} \quad and \quad \frac{x - 3}{1} = \frac{y + \lambda}{-4} = \frac{z}{0} \]
is 13. Then, \( \sum \lambda \in S \) is equal to:

• (1) 302
• (2) 306
• (3) 304
• (4) 308

Let \( S \) be the set of all \( (\lambda, \mu) \) for which the vectors \( \lambda \hat{i} - \hat{j} + \hat{k}, \, \hat{i} + 2\hat{j} + \hat{k} \) and \( 3\hat{i} - 4\hat{j} + 5\hat{k} \), where \( \lambda - \mu = 5 \), are coplanar, then \[ \sum_{(\lambda, \mu) \in S} 80(\lambda^2 + \mu^2) \]
is equal to:

• (1) 2130
• (2) 2210
• (3) 2290
• (4) 2370

Let the foot of perpendicular of the point \( P(3, -2, -9) \) on the plane passing through the points \( (1, -2, -3), (9, 3, 4), (9, -2, 1) \) be \( Q(\alpha, \beta, \gamma) \). Then the distance of \( Q \) from the origin is:

• (1) \( \sqrt{29} \)
• (2) \( \sqrt{38} \)
• (3) \( \sqrt{42} \)
• (4) \( \sqrt{35} \)

If the set \( \left\{ Re \left( \frac{z - \bar{z} + z^2}{2 - 3z + 5z^2} \right): z \in \mathbb{C}, Re(z) = 3 \right\} \) is equal to the interval \( (\alpha, \beta) \), then \( 24(\beta - \alpha) \) is equal to:

• (1) 36
• (2) 27
• (3) 30
• (4) 42

Let \( x = y \) be the solution of the differential equation \[ 2(y + 2) \log(y + 2) \, dx + (x + 4) - 2 \log(x + 2) \, dy = 0, \quad with \quad x(1) = -2. \]
Then, \( x'(-2) \) is equal to:

• (1) \( \frac{4}{9} \)
• (2) \( \frac{32}{9} \)
• (3) \( \frac{10}{3} \)
• (4) 3

If \[ \int_{0}^{2} \frac{1}{(5 + 2x - 2x^2) \left( 1 + \left( e^{2 - 4x} \right) \right)} \, dx = \frac{1}{\alpha} \log \left( \frac{\alpha + 1}{\beta} \right), \quad \alpha, \beta > 0, \]
then \( \alpha^4 - \beta^4 \) is equal to:

• (1) 19
• (2) -21
• (3) 21
• (4) 0

The number of common tangents, to the circles \( x^2 + y^2 - 18x - 15y + 131 = 0 \) and \( x^2 + y^2 - 6x - 6y - 7 = 0 \), is

• (1) 4
• (2) 1
• (3) 3
• (4) 2

Let ABCD be a quadrilateral. If E and F are the midpoints of the diagonals AC and BD respectively and \[ (AB - BC) + (AD - DC) = k \, FE \quad then \quad k \, is equal to: \]

• (1) 4
• (2) 2
• (3) -2
• (4) -4

Let \( (a + bx + cx^2)^{10} = \sum_{i=0}^{20} P_i x^i \), where \( a, b, c \in \mathbb{N} \). If \( p_1 = 20 \) and \( p_2 = 210 \), then \( 2(a + b + c) \) is equal to:

• (1) 8
• (2) 12
• (3) 6
• (4) 15

Let \( [x] \) denote the greatest integer function and \( f(x) = \max \{ 1 + x + [x], 2 + x, x + 2[x] \} \), where \( 0 \leq x \leq 2 \). Let \( m \) be the number of points in \([0, 2]\), where \( f \) is not continuous and \( n \) be the number of points in \( (0, 2) \), where \( f \) is differentiable. Then \( (m + n)^2 + 2 \) is equal to:

• (1) 6
• (2) 2
• (3) 3
• (4) 11

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:

• (1) \( \frac{1}{4} \)
• (2) \( \frac{9}{50} \)
• (3) \( \frac{11}{50} \)
• (4) \( \frac{1}{5} \)

If the domain of the function \[ f(x) = \log_e \left( 4x^2 + 11x + 6 \right) + \sin^{-1} \left( 4x + 3 \right) + \cos^{-1} \left( \frac{10x + 6}{3} \right), \]
then \( 36|\alpha + \beta| \) is equal to:

• (1) 72
• (2) 63
• (3) 45
• (4) 54

Let the determinant of a square matrix A of order \( m \) be \( m - n \), where \( m \) and \( n \) satisfy \( 4m + n = 22 \) and \( 17m + 4n = 93 \). If \( det (n \, adj(adj(mA))) = 3^a 5^b 6^c \), then \( a + b + c \) is equal to:

• (1) 101
• (2) 84
• (3) 109
• (4) 96

The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is:

• (1) 14
• (2) 11
• (3) 12
• (4) 13

If \( (\alpha, \beta) \) is the orthocenter of the triangle ABC with vertices \( A(3, -7), B(-1, 2), C(4, 5) \), then \( 9\alpha - 6\beta + 60 \) is equal to:

• (1) 30
• (2) 35
• (3) 40
• (4) 25

The number of real roots of the equation \[ x |x| - 5 |x + 2| + 6 = 0, \]
is:

• (1) 5
• (2) 6
• (3) 4
• (4) 3

Let the system of linear equations \[ -x + 2y - 9z = 7
-x + 3y + 7z = 9
-2x + y + 5z = 8
-3x + y + 13z = \lambda \]
has a unique solution \( x = \alpha, y = \beta, z = \gamma \). Then the distance of the point \( (\alpha, \beta, \gamma) \) from the plane \( 2x - 2y + z = \lambda \) is:

• (1) 7
• (2) 9
• (3) 13
• (4) 11

Let \( A_1 \) and \( A_2 \) be two arithmetic means and \( G_1, G_2, G_3 \) be three geometric means of two distinct positive numbers. Then \[ G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2 is equal to: \]

• (1) \( 2(A_1 + A_2) G_1 G_3 \)
• (2) \( (A_1 + A_2)^2 G_1 G_3 \)
• (3) \( 2(A_1 + A_2)^2 G_1^2 G_3^2 \)
• (4) \( (A_1 + A_2) G_1^2 G_2^2 G_3^2 \)

Negation of \( p \land (q \land \neg (p \land q)) \) is:

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

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is:

• (1) 21
• (2) 18
• (3) 20
• (4) 22

Let \( A = \{ 1, 2, 3, 4 \} \) and \( R \) be a relation on the set \( A \times A \) defined by \[ R = \{(a, b), (c, d): 2a + 3b = 4c + 5d \}. \]
Then the number of elements in \( R \) is:

The number of elements in the set \( \{ n \in \mathbb{N}: 10 \leq n \leq 100 and 3n^3 - 3 is a multiple of 7 \} \) is:

Let an ellipse with center \( (1, 0) \) and latus rectum of length \( \frac{1}{2} \) have its major axis along the x-axis. If its minor axis subtends an angle of \( 60^\circ \) at the foci, then the square of the sum of the lengths of its minor and major axes is equal to:

If the area bounded by the curve \( 2y^2 = 3x \), lines \( x + y = 3 \), \( y = 0 \), and outside the circle \( (x - 3)^2 + y^2 = 2 \) is \( A \), then \( 4(\pi + 4A) \) is equal to:

Consider the triangles with vertices \( A(2,1) \), \( B(0,0) \) and \( C(t,4) \), \( t \in [0,4] \). If the maximum and the minimum perimeters of such triangles are obtained at \( t = \alpha \) and \( t = \beta \) respectively, then \( 6\alpha + 21\beta \) is equal to:

Let the plane \( P \) contain the line \( 2x + y - z = 3 = 0 \), \( 5x - 3y + 4z + 9 = 0 \) and be parallel to the line \( \frac{x + 2}{2} = \frac{3 - y}{4} = \frac{z - 7}{5} \). Then the distance of the point \( A(8, -1, -19) \) from the plane \( P \), measured parallel to the line is equal to:

If the sum of the series \[ \left( \frac{1}{2} + \frac{1}{3} \right) + \left( \frac{1}{2^2} + \frac{1}{2 \cdot 3^2} \right) + \left( \frac{1}{3^2} + \frac{1}{2^2 \cdot 3^2} \right) + \left( \frac{1}{2^3} + \frac{1}{2^2 \cdot 3^3} \right) + \ldots \]
is \( \frac{\alpha}{\beta} \), where \( \alpha \) and \( \beta \) are co-prime, then \( \alpha + 3\beta \) is equal to:

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is:

If the line \( x = y = z \) intersects the line \( x \sin A + y \sin B + z \sin C - 18 = 0 \) and \( x \sin 2A + y \sin 2B + z \sin 2C - 9 = 0 \), where A, B, C are the angles of a triangle ABC, then \( 80 \left( \frac{\sin A}{\sin B} \frac{\sin C}{\sin B} \right) \) is equal to:

Let \( f(x) = \frac{dx}{(3 + 4x^2) \sqrt{4 - 3x^2}} \), \( |x| < \frac{2}{\sqrt{3}} \), and \( f(0) = 0 \). If \( f(0) = 0 \) and \( f(1) = 1 \), then \( \alpha \beta > 0 \), then \( \alpha^2 + \beta^2 \) is equal to: