JEE Main 2023 Mathematics April 6 Shift 1 Question Paper PDF with Solutions

Download the official JEE Main 2023 Mathematics April 6 Shift 1 Question Paper PDF with Solutions and Answer Key using the link below, and kickstart your preparation for JEE Main 2025! This JEE Main 2023 Mathematics Question Paper is divided into two sections: Section A with 20 Multiple Choice Questions (MCQs) and Section B with 10 numerical-type questions, where candidates must answer all from Section A and any 5 from Section B. Perfect for students aiming to master the JEE Main 2025 Mathematics exam, this free resource offers a comprehensive way to understand the exam pattern, practice with real questions, and enhance your skills using the JEE Main 2023 Question Paper, Answer Key, and Solutions for April 6 Shift 1.

JEE Main 2023 Mathematics Question Paper PDF With Answer Key (April 6 Shift 1)

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JEE Mains 2023 Mathematics Question Paper With Solutions

Question 1:

The straight lines l₁ and l₂ pass through the origin and trisect the line segment of the line L : 9x + 5y = 45 between the axes. If m₁ and m₂ are the slopes of the lines l₁ and l₂, then the point of intersection of the line y = (m₁ + m₂)x with L lies on:

6x + y = 10
6x - y = 15
y - 2x = 5
y − x = 5

Correct Answer: (4) y − x = 5
View Solution

Question 2:

Let the position vectors of the points A, B, C and D be 5i + 5j + 2λk, i + 2j + 3k, −2i + xj + 4k and -i + 5j + 6k. Let the set S = {λ ∈ R: the points A, B, C and D are coplanar}. Then Σ_λ∈S(λ + 2)² is equal to:

Correct Answer: (4) 41
View Solution

Question 3:

Let I(x) = ∫ ^{(x²(xsec²x + tanx))}⁄_{(xtanx + 1)²} dx. If I(0) = 0, then I(π/4) is equal to:

loge^((π+4)²)⁄_{(16 + 4(π+4)/π²)}
loge^((π+4)²)⁄_{π²/(4(π+4))}
loge^((π+4)²)⁄_{(16 - 4(π+4)/π²)}
loge^((π+4)²)⁄_{(32 + 4(π+4)/π²)}

Correct Answer: (2) loge^((π+4)²)⁄_{π²/(4(π+4))}
View Solution

Question 4:

The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + ... is:

3450
3420
3520
3250

Correct Answer: (3) 3520
View Solution

Question 5:

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If probability of at least 4 successes is ^k⁄₃₁₁, then k is equal to:

Correct Answer: (2) 123
View Solution

Question 6:

Let A = [a_ij]_2x2, where a_ij ≠ 0 for all i, j, and A² = I. Let a be the sum of all diagonal elements of A and b = |A|. Then 3a² + 4b² is equal to:

Correct Answer: (2) 4 View Solution

Question 7:

Let a₁, a₂, a₃, ..., a_n be n positive consecutive terms of an arithmetic progression. If d > 0 is its common difference, then: lim_n→∞ (¹⁄_√n)(¹⁄_{√a₁+√a₂} + ¹⁄_{√a₂+√a₃}+ ... + ¹⁄_{√a_n-1+√a_n}) is:

¹⁄₁₆
1
√d
0

Correct Answer: (2) 1 <View Solution.

Question 8:

If ²ⁿC₃ : ⁿC₃ = 10:1, then the ratio (n² + 3n) : (n² - 3n + 4) is:

27:11
35:16
2:1
65:37

Correct Answer: (3) 2:1 View Solution

Question 9:

Let A = {x ∈ R : |x+3| + |x+4| ≤ 3}, B = {x ∈ R: 3 · Σ^3x-3_r=110^-r < 3^-[x]} where [x] denotes the greatest integer function. Then,

A ⊂ B, A ≠ B
A ∩ B = ∅
A = B
B ⊂ A, A ≠ B

Correct Answer: (3) A = B
View Solution

Question 10:

One vertex of a rectangular parallelepiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to the z axis, not passing through O or P is:

¹²⁄_5√5
^12√5⁄₅
¹²⁄₅
12

Correct Answer: (3) ¹²⁄₅
View Solution

Question 11:

If the equation of the plane passing through the line of intersection of the planes 2x - y + z = 3 and 4x – 3y + 5z + 9 = 0 and parallel to the line ^(x+1)⁄_-2 = ^(y+3)⁄₄ = ^z⁄₅ is ax + by + cz + 6 = 0, then a + b + c is equal to :

Correct Answer: (2) 14
View Solution

Question 12:

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of (√2 + ¹⁄_√3)ⁿ is √6 : 1, then the third term from the beginning is :

30√2
60√2
30√3
60√3

Correct Answer: (4) 60√3
View Solution

Question 13:

The sum of all the roots of the equation |x² - 8x + 15| - 2x + 7 = 0 is:

11 - √3
9 - √3
9 + √3
11 + √3

Correct Answer: (3) 9 + √3
View Solution

Question 14:

From the top A of a vertical wall AB of height 30 m, the angles of depression of the top P and bottom Q of a vertical tower PQ are 15° and 60° respectively. B and Q are on the same horizontal level. If C is a point on AB such that CB = 15 m, then the area (in m²) of the quadrilateral BCPQ is equal to:

300(√3-√5)
300(√3+1)
300(3-1)
600(√3-1)

Correct Answer: (4) 600(√3-1)
View Solution

Question 15:

Let a = 2i + 3j + 4k, b = i – 2j – 2k and c = −i + 4j + 3k. If d is a vector perpendicular to both b and c, and a · d = 18, then |a x d|² is equal to :

Correct Answer: (3) 720
View Solution

Question 16:

If 2^xy + 3^{y^x} = 20, then ^dy⁄_dx at (2,2) is equal to:

^{(3+loge 8)}⁄_{(2+loge 4)}
^{(2+loge 8)}⁄_{(3+loge 4)}
^{(3+loge 4)}⁄_{(2+loge 8)}
^{(3+loge 16)}⁄_{(4+loge 8)}

Correct Answer: (2) ^{(2+loge 8)}⁄_{(3+loge 4)} View Solution

Question 17:

If the system of equations x+y+az = b, 2x + 5y + 2z = 6, x+2y+3z = 3 has infinitely many solutions, then 2a+3b is equal to:

Correct Answer: (4) 23
View Solution

Question 18:

Statement (P⇒Q)∧(R⇒Q) is logically equivalent to:

(P∨R)⇒Q
(P⇒R)∨(Q⇒R)
(P⇒R)∧(Q⇒R)
(P∧R)⇒Q

Correct Answer: (1) (P∨R)⇒Q
View Solution

Question 19:

Let 5f(x) + 4f(¹⁄_x) = ¹⁄_x + 3, x>0. Then 18∫²₁f(x)dx is equal to:

10loge2 - 6
10loge2 + 6
5loge2 - 3
5loge2 + 3

Correct Answer: (1) 10loge2 - 6
View Solution

Question 20:

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and σ² respectively. If the variance of all the 30 numbers in the two sets is 13, then σ² is equal to:

Correct Answer: (2) 10
View Solution

Question 21:

Let the tangents to the curve x² + 2x – 4y + 9 = 0 at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line x - 3y = 6 meet the parabola y² = 4x at B. If B lies on the line 2x – 3y = 8, then (AB)² is equal to:

Correct Answer: 292
View Solution

Question 22:

Let the point (p, p + 1) lie inside the region E = {(x,y): 3 - x ≤ y ≤ √(9 - x²), 0 < x < 3}. If the set of all values of p is the interval (a, b), then b² + b − a² is equal to:

Correct Answer: 3
View Solution

Question 23:

Let y = y(x) be a solution of the differential equation (xcosx)dy + (xysinx + ycosx - 1)dx = 0, 0 < x < π/2. If y(π/3) = √3, then |y"(π/6) + 2y'(π/6)| is equal to:

Correct Answer: 2
View Solution

Question 24:

Let a ∈ Z and [t] be the greatest integer ≤ t. Then the number of points, where the function f(x) = [a + 13sinx], x ∈ (0,π) is not differentiable, is:

Correct Answer: 25
View Solution

Question 25:

If the area of the region S = {(x,y): 2y - y² ≤ x ≤ 2y, x ≥ y} is equal to ^{(1 - ¹⁄_n)π}⁄₄, then the natural number n is equal to:

Correct Answer: 5
View Solution

Question 26:

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is:

Correct Answer: 3483638676
View Solution

Question 27:

Let the image of the point P(1, 2, 3) in the plane 2x - y + z = 9 be Q. If the coordinates of the point R are (6, 10, 7), then the square of the area of the triangle PQR is:

Correct Answer: 594
View Solution

Question 28:

A circle passing through the point P(α,β) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of the perpendicular from P on AB. If PQ is equal to 11 units, then the value of αβ is:

Correct Answer: 121
View Solution

Question 29:

The coefficient of x¹⁸ in the expansion of (1+x²+2x)¹⁵ is:

Correct Answer: 5005 View Solution

Question 30:

Let A = {1,2,3,4,...,10} and B = {0,1,2,3,4}. The number of elements in the relation R = {(a,b) ∈ A x A : 2(a-b)² + 3(a-b) ∈ B} is:

Correct Answer: 18
View Solution

JEE Main 2023 Mathematics Paper Analysis April 6 Shift 1

JEE Main 2023 Mathematics Paper Analysis for the exam scheduled on April 6 Shift 1 is available here. Candidates can check subject-wise paper analysis for the exam scheduled on April 6 Shift 1 here along with the topics with the highest weightage.

JEE Main 2023 Mathematics Paper Analysis April 6 Shift 1

JEE Main 2023 Mathematics Question Paper Pattern

Feature	Question Paper Pattern
Examination Mode	Computer-based Test
Exam Language	13 languages (English, Hindi, Assamese, Bengali, Gujarati, Kannada, Malayalam, Marathi, Odia, Punjabi, Tamil, Telugu, and Urdu)
Exam Duration	3 hours
Sectional Time Limit	None
Mathematics Marks	100 marks
Total Number of Questions Asked	20 MCQs + 10 Numerical Type Questions
Total Number of Questions to be Answered	20 MCQs + 5 Numerical Type Questions
Marking Scheme	+4 for each correct answer
Negative Marking	-1 for each incorrect answer

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JEE Main 2023 Mathematics April 6 Shift 1 Question Paper PDF: Download with Solutions for JEE 2025 Preparation

JEE Main 2023 Mathematics Question Paper PDF With Answer Key (April 6 Shift 1)

JEE Mains 2023 Mathematics Question Paper With Solutions

JEE Main 2023 Mathematics Paper Analysis April 6 Shift 1

JEE Main 2023 Mathematics Question Paper Pattern

JEE Main Previous Year Question Paper

JEE Main Questions

1.
Find sum up to 8 terms of the series
\[ \frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots \]

2.
$2.8 \times 10^{-3}$ mol of CO$_2$ is left after removing $10^{21}$ molecules from its $x$ mg sample. The mass of CO$_2$ taken initially is: Given: $N_A = 6.02 \times 10^{23}\,\text{mol}^{-1}$

3.
In a microscope of tube length $10\,\text{cm}$ two convex lenses are arranged with focal lengths $2\,\text{cm}$ and $5\,\text{cm}$. Total magnification obtained with this system for normal adjustment is $(5)^k$. The value of $k$ is ___.

5.
What is the mole fraction of water in \(10%\) by weight (w/w) of aqueous urea solution? [Given: Molar mass of H, O, C and N are 1, 16, 12 and \(14 \text{ g mol}^{-1}\) respectively.]

6.
A molecule \( (X) \) with the following structure under mild acidic conditions is hydrolyzed to produce \( (Y) \) and \( (Z) \). Identify the correct statements about \( (Y) \) and \( (Z) \).}

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JEE Main 2023 Mathematics Question Paper PDF With Answer Key (April 6 Shift 1)

JEE Mains 2023 Mathematics Question Paper With Solutions

JEE Main 2023 Mathematics Paper Analysis April 6 Shift 1

JEE Main 2023 Mathematics Question Paper Pattern

JEE Main Previous Year Question Paper

JEE Main Questions

1.Find sum up to 8 terms of the series \[ \frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots \]

2.$2.8 \times 10^{-3}$ mol of CO$_2$ is left after removing $10^{21}$ molecules from its $x$ mg sample. The mass of CO$_2$ taken initially is: Given: $N_A = 6.02 \times 10^{23}\,\text{mol}^{-1}$

3.In a microscope of tube length $10\,\text{cm}$ two convex lenses are arranged with focal lengths $2\,\text{cm}$ and $5\,\text{cm}$. Total magnification obtained with this system for normal adjustment is $(5)^k$. The value of $k$ is ___.

5.What is the mole fraction of water in \(10%\) by weight (w/w) of aqueous urea solution? [Given: Molar mass of H, O, C and N are 1, 16, 12 and \(14 \text{ g mol}^{-1}\) respectively.]

6.A molecule \( (X) \) with the following structure under mild acidic conditions is hydrolyzed to produce \( (Y) \) and \( (Z) \). Identify the correct statements about \( (Y) \) and \( (Z) \).}

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1.
Find sum up to 8 terms of the series
\[ \frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots \]

2.
$2.8 \times 10^{-3}$ mol of CO$_2$ is left after removing $10^{21}$ molecules from its $x$ mg sample. The mass of CO$_2$ taken initially is: Given: $N_A = 6.02 \times 10^{23}\,\text{mol}^{-1}$

3.
In a microscope of tube length $10\,\text{cm}$ two convex lenses are arranged with focal lengths $2\,\text{cm}$ and $5\,\text{cm}$. Total magnification obtained with this system for normal adjustment is $(5)^k$. The value of $k$ is ___.

5.
What is the mole fraction of water in \(10%\) by weight (w/w) of aqueous urea solution? [Given: Molar mass of H, O, C and N are 1, 16, 12 and \(14 \text{ g mol}^{-1}\) respectively.]

6.
A molecule \( (X) \) with the following structure under mild acidic conditions is hydrolyzed to produce \( (Y) \) and \( (Z) \). Identify the correct statements about \( (Y) \) and \( (Z) \).}