JEE Main 2023 BArch Question Paper with Solutions PDF January 28

Let \(\mathrm{R}_1\) and \(\mathrm{R}_2\) be two relations on \(\mathbb{R}^2\) defined as
\((a, b)\ \mathrm{R}_1\ (c, d)\) if \(ad - bc \geq 0\)
\((a, b)\ \mathrm{R}_2\ (c, d)\) if \(a + d \geq b + c\). Then :

• (1) \(\mathrm{R}_1\) is transitive but \(\mathrm{R}_2\) is not transitive
• (2) \(\mathrm{R}_2\) is transitive but \(\mathrm{R}_1\) is not transitive
• (3) Both \(\mathrm{R}_1\) and \(\mathrm{R}_2\) are transitive
• (4) Neither \(\mathrm{R}_1\) nor \(\mathrm{R}_2\) is transitive

Let \(\alpha\) and \(\beta\) be the roots of \(x^2 - 3x + 9 = 0\). Then \(\left(\frac{\beta^{30}}{(9\alpha)^{10}} + \frac{\alpha^{30}}{(9\beta)^{10}}\right)^2\) is equal to

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

For \(z = 2 + 5i\), the modulus of \(2z^3 + 21z^2 - 58z + 4\) is :

• (1) 1153
• (2) 947
• (3) 537
• (4) 837

If the system of equations
\(Kx - \sqrt{2}y + \sqrt{5}z = \sqrt{7}\)
\(\sqrt{5}x + \sqrt{3}y - \sqrt{2}z = \sqrt{11}\)
\(30x + (3\sqrt{15} - 5\sqrt{6})y + (5\sqrt{15} - 3\sqrt{10})z = 5\sqrt{21} + 3\sqrt{55}\)
has infinitely many solutions, then \(\mathrm{K}^2\) is

• (1) 27
• (2) \(\frac{1}{3}\)
• (3) 9
• (4) 3

For \(\alpha, \beta \in \mathbb{R}\), if the matrices \(\mathrm{A} = \begin{pmatrix} \alpha & 0 \\ 0 & \beta \end{pmatrix}\), \(\mathrm{B} = \begin{pmatrix} \alpha & 0 \\ 0 & \alpha \end{pmatrix}\) and \(\mathrm{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) satisfy the equation \((\mathrm{A} * \mathrm{B}) * 2\mathrm{I} = 20\mathrm{I}\), where \(*\) is defined as \(\mathrm{A} * \mathrm{B} = \mathrm{A}^2 + \mathrm{B}^2\), then \(|\alpha\beta|\) is equal to :

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

The sum of the first eleven terms of the series is \(\frac{1}{1 + 1^2 + 1^4} + \frac{2}{1 + 2^2 + 2^4} + \frac{3}{1 + 3^2 + 3^4} + \ldots\)

• (1) \(\frac{61}{133}\)
• (2) \(\frac{66}{133}\)
• (3) \(\frac{16}{33}\)
• (4) \(\frac{33}{67}\)

\(\lim_{x \to 0} (1 + 3x)^{\frac{x + 2}{x}}\) is equal to

• (1) e
• (2) \(e^3\)
• (3) \(e^6\)
• (4) \(e^9\)

Let \(P(\alpha, \beta, \lambda)\) be the image of the point \(\mathrm{Q}(1, 2, 0)\) in the line \(\frac{x - 5}{3} = \frac{y - 12}{1} = \frac{z - 10}{2}\), then \((\mathrm{PQ})^2\) is equal to

• (1) 90
• (2) 180
• (3) 360
• (4) 270

A box contains 7 red and 9 white balls. The number of ways of drawing 8 balls such that there are at least three balls of each colour, is :

• (1) 8820
• (2) 10584
• (3) 1764
• (4) 3515

Let \([t]\) denote the greatest integer function. If \(\int_0^1 [1 + x^2 + x^4]\,\mathrm{d}x = a\), then \(36a - 25a^2 + 8a^3 - a^4\) is equal to

• (1) 19
• (2) \(-19\)
• (3) 18
• (4) \(-21\)

Let \(\mathrm{PL} = 8\) units and \(\mathrm{QM} = 2\) units be two parallel line segments such that the line segments \(\mathrm{PM}\) and \(\mathrm{QL}\) intersect at the point \(\mathrm{R}\). If \(\mathrm{PL}\) and \(\mathrm{QM}\) are tangents to a circle passing through points \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) then radius of this circle is

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

For some \(\alpha \in \mathbb{N}\), let \(\mathrm{PQR}\) be a triangle with two fixed vertices \(\mathrm{P}(2, 5)\) and \(\mathrm{Q}(\alpha, -11)\). If the point \(\mathrm{R}\) moves on the line \(l_1 : 9x + 7y + \alpha = 0\), then the centroid of \(\Delta \mathrm{PQR}\) moves on the line \(l_2\), which is parallel to \(l_1\) at a distance \(\frac{20}{3\sqrt{130}}\) units from it. If the distance of \(\mathrm{Q}\) from \(l_2\) is \(\frac{k}{3\sqrt{130}}\), then k is equal to :

• (1) 117
• (2) 129
• (3) 131
• (4) 133

Let \(\mathrm{A}_i(x_i, y_i)\), \(i = 1, 2, 3\) be points on the circle \(x^2 + y^2 = 10\) such that \(\mathrm{A}_1\) lies in the \(1^{\mathrm{st}}\) quadrant and it is the image of point \(\mathrm{A}_2\) with respect to \(y\)-axis. If the distance of point \(\mathrm{A}_1\) from each of the points \(\mathrm{A}_2\) and \(\mathrm{A}_3\) is 2, then twenty times the area of the \(\Delta\,\mathrm{A}_1 \mathrm{A}_2 \mathrm{A}_3\) is

• (1) 12
• (2) 30
• (3) 24
• (4) 48

The remainder when \(7^{89}\) is divided by 15 is

• (1) 5
• (2) 7
• (3) 9
• (4) 11

If the plane \(y = \alpha x - \beta z + \gamma\) passing through the point \((1, -1, 3)\) is perpendicular to each of the planes \(2x + y + z = 1\) and \(3x - 2y + 2z = 0\), then \(\alpha + \beta + \gamma\) is equal to :

• (1) 5
• (2) 13
• (3) 19
• (4) 27

Let \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = \hat{i} - \hat{j} + 2\hat{k}\), \(\vec{c} = 2\hat{i} + \hat{j} - 4\hat{k}\) be three vectors. If \(\vec{r}\) is the vector such that \(\vec{r} \times \vec{a} = (\vec{b} + \vec{c}) \times \vec{a}\) and \(\vec{r} \cdot (\vec{b} - \vec{c}) = 0\), then \(\vec{r} \cdot (\hat{i} + \hat{j} - \hat{k})\) is equal to :

• (1) 3
• (2) 4
• (3) 5
• (4) 6

The probability that a randomly selected root of the equation \( 1 + x + x^2 + \ldots + x^{118} = 0 \) satisfies the equation \( x^7 = 1 \), is

• (1) 0
• (2) \( \frac{1}{59} \)
• (3) \( \frac{3}{59} \)
• (4) \( \frac{7}{118} \)

Let \( X \) have the binomial distribution \( \mathrm{B}(n, p) \). If its mean is 3 and variance is 2, then \( \mathrm{P}\left(X < \frac{n}{4}\right) \) is equal to :

• (1) \( \frac{29 \times 2^8}{3^9} \)
• (2) \( \frac{25 \times 2^9}{3^9} \)
• (3) \( \frac{163}{3^9} \)
• (4) \( \frac{835}{3^9} \)

The domain of the function \( f(x) = \cos^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x - 1}\right) \) is :

• (1) \( \left(\sqrt{2} - 1, \frac{3}{5}\right] \)
• (2) \( \left[\frac{3}{5}, \infty\right) \)
• (3) \( \mathbb{R} - \{-\sqrt{2} - 1, \sqrt{2} - 1\} \)
• (4) \( (-\infty, -1 - \sqrt{2}) \cup (\sqrt{2} - 1, \infty) \)

Which of the following statements is a tautology ?

• (1) \( ((p \Rightarrow q) \vee p) \Rightarrow q \)
• (2) \( ((p \wedge q) \Rightarrow p) \Rightarrow q \)
• (3) \( ((p \wedge q) \wedge (\sim q)) \Rightarrow p \)
• (4) \( ((p \Rightarrow q) \vee p) \Rightarrow p \)

The curve \( y = x^2 + 1 \) divides the area enclosed by the curves \( y + |x| = 3 \) and \( y = |x - 1| \) in the ratio m : n, where m and n are coprime, then m + n is equal to ________.

The number of ways in which 30 identical pens can be distributed among 12 students so that each student gets at least one pen and exactly two students get at least two pens each, is ________.

Let \( (1 + x^2 - x^4)^{12} = \sum_{n=0}^{48} a_n x^n \). Then \( a_0 + a_2 + a_4 + \ldots + a_{44} \) is equal to ________.

If \( \mathrm{S}_n \) denotes the sum of first n terms of the series \( 7 + 10 + 16 + 25 + 37 + \ldots \), then \( \mathrm{S}_{30} - \mathrm{S}_{20} \) is equal to ________.

If \([t]\) denotes the greatest integer \(\leq t\), then the number of points, at which the function \(f(x) = [x + x^3] + |x - x^3| + \left|x + \frac{1}{2}\right|\) is not differentiable in the open interval \((-10, 10)\), is ________.

If \(\int \frac{dx}{(3x^2 + 5)\sqrt{10x^2 + 7}} = -\frac{1}{\sqrt{580}} \log_e |f(x)| + C\) where C is an arbitrary constant, then \(f(0)\) is equal to ________.

Let the equation of the hyperbola with foci \((1, 5), (1, -1)\) and eccentricity \(\sqrt{3}\) be \(x^2 - 2y^2 + ax + by + c = 0\). Then \(|a + b + c|\) is equal to ________.

Let \(\alpha_1, \alpha_2\) be the values of \(\alpha\) such that the distance between the point \((2, 4, 3)\) and the plane \(3x + y + \alpha z + 10 = 0\) is \(\sqrt{35}\) units. Then the area of the triangle with vertices \((\alpha_1, \alpha_2, 0)\), \((\alpha_2, \alpha_1, 0)\) and \(\left(\frac{164}{13}, 5, 0\right)\) is ________ unit\(^2\).

Let O be the origin and let the vectors \(\overrightarrow{OA} = -3\hat{i} + 7\hat{j} + 5\hat{k}\), \(\overrightarrow{OB} = -5\hat{i} + 7\hat{j} - 3\hat{k}\) and \(\overrightarrow{OC} = \hat{u}\) represent three sides of a parallelopiped, where \(\hat{u}\) is a unit vector in the \(xy\)- plane. If the maximum volume of the parallelopiped is \(2\sqrt{\alpha}\), then \(\alpha\) is equal to ________.

If the solution curve of the differential equation \(\frac{x + y - 2}{x + y - 1}\,\frac{dy}{dx} = \frac{x + y + 2}{x + y + 1}\), \(x + y > 2\), passes through the points \((\sqrt{2}, \sqrt{2})\) and \((2, \alpha)\), then \(2\alpha - \log_e\left(\frac{\alpha^2 + 4\alpha + 2}{6}\right)\) is equal to ________.

Adobe is a

• (1) Type of cement
• (2) Type of floor finish
• (3) Type of paint
• (4) Type of Brick

Given below are two statements:
Statement I : Chandigarh is the first planned city of Independent India
Statement II : Chandigarh city was designed by Swiss French architect Le Corbusier
In the light of above statements, choose the most appropriate answer from the options given below

• (1) Both Statement I and Statement II are correct
• (2) Both Statement I and Statement II are incorrect
• (3) Statement I is correct but statement II is incorrect
• (4) Statement I is incorrect but statement II is correct

Given below are two statements:
Statement I : Glass has low thermal conductivity
Statement II : Glass can absorb, refract and transmit light.
In the light of above statements, choose the most appropriate answer from the options given below

• (1) Both Statement I and Statement II are correct
• (2) Both Statement I and Statement II are incorrect
• (3) Statement I is correct but statement II is incorrect
• (4) Statement I is incorrect but statement II is correct

• (1) A-I, B-III, C-II, D-IV
• (2) A-III, B-IV, C-I, D-II
• (3) A-IV, B-II, C-III, D-I
• (4) A-IV, B-III, C-I, D-II

• (1) A-IV, B-II, C-I, D-III
• (2) A-III, B-I, C-II, D-IV
• (3) A-II, B-I, C-IV, D-III
• (4) A-I, B-IV, C-III, D-II

Which one of these is not a complimentary colour?

• (1) Blue-Orange
• (2) Red-Green
• (3) Blue-Green
• (4) Violet-Yellow

In which State of India, Robbers cave is situated:

• (1) Himachal Pradesh
• (2) Uttarakhand
• (3) Uttar Pradesh
• (4) Madhya Pradesh

Choose the correct option among the following:
Petronas Tower is situated in:

• (1) Paris
• (2) Dubai
• (3) Kuala Lumpur
• (4) New York

The Konark Temple is located in which state?

• (1) Madhya Pradesh
• (2) Odisha
• (3) Karnataka
• (4) Rajasthan

Who is the architect of the Lotus Temple?

• (1) Louis I Kahn
• (2) Moshe Safdie
• (3) Fariborz Sahba
• (4) Richard Meier

A small lift for carrying only a small load is known as:

• (1) A dead Bearer
• (2) A Dumb Waiter
• (3) A Jockey Boy
• (4) A push upper

Which are often referred as 'twin cities' of Odisha?

• (1) Bhubaneshwar-Puri
• (2) Puri-Cuttack
• (3) Bhubaneshwar-Cuttack
• (4) Bhubaneshwar-Rourkela

Which is the correct chronology of Human Civilizations in terms of their existence?

• (1) Mesopotamia-Harappa-Egyptian-Chinese
• (2) Mesopotamia-Egyptian-Harappa-Chinese
• (3) Mesopotamia-Chinese-Harappa-Egyptian
• (4) Egyptian-Mesopotamia-Harappa-Sumerian

Qutub-Minar in Delhi was built by:

• (1) Shah Jahan
• (2) Jahangir
• (3) Akbar
• (4) Qutub ud-din Aibak

Which direction in the southern hemisphere would you get glare free (diffused) light throughout the year?

• (1) North
• (2) South
• (3) East
• (4) West

Who is the architect of the famous "Jawaharlal Kala Complex" in Jaipur?

• (1) Raj Rewal
• (2) Charles Correa
• (3) Achyut Kanvinde
• (4) Hafeez Contractor

Dhajji-Dewari is a construction style popular predominantly in ___________.

• (1) Coastal Areas
• (2) Plains
• (3) Mountainous Region
• (4) Desert Areas

Which stone is used for roofing in mountainous regions?

• (1) Marble
• (2) Granite
• (3) Shale
• (4) Sand Stone

Chandigarh is an example of which type of city planning.

• (1) Radio Centric
• (2) Grid-iron
• (3) Linear
• (4) Organic

If you have to build on the seashore in Goa, which rooms would have the best view of the sea?

• (1) Those facing North
• (2) Those facing South
• (3) Those facing East
• (4) Those facing West

"NIFT" National Institute of Fashion Technology, Delhi is designed by

• (1) B.V Doshi
• (2) C.P Kukreja
• (3) Raj Rewal
• (4) Bimal Patel

• (1) A-II, B-I, C-IV, D-III
• (2) A-I, B-III, C-IV, D-II
• (3) A-I, B-II, C-III, D-IV
• (4) A-IV, B-III, C-II, D-I

• (1) A-IV, B-I, C-II, D-III
• (2) A-II, B-III, C-I, D-IV
• (3) A-III, B-II, C-I, D-IV
• (4) A-IV, B-III, C-II, D-I

Given below are two statements:
Statement I : Red, Blue and Yellow are the primary colours of a colour wheel.
Statement II : The colours which are positioned opposite to each other in a colour wheel are known as complementary colours.
In the light of above statements, choose the correct answer form the options given below

• (1) Both Statement I and Statement II are correct
• (2) Both Statement I and Statement II are incorrect
• (3) Statement I is correct but statement II is incorrect
• (4) Statement I is incorrect but statement II is correct

Following question consists of problem figures followed by answer figures. Select a figure from amongst the answer figures which will continue the same series or pattern as established by the problem figures.
PROBLEM FIGURES

• (1)
• (2)
• (3)
• (4)

How many triangles are there in given figure:-

• (1) 26
• (2) 25
• (3) 27
• (4) 24

Find the odd one out:-
7, 9, 25, 32, 43, 59

• (1) 59
• (2) 32
• (3) 25
• (4) 9

A residential building has 15 floors. The height of ground floor is 4.2 meter (including length and slab thickness). Rest all other floors are of 3.3 meter high (including slab thickness). What is the total height of the building (from ground to terrace) in meters?

• (1) 45.6 meter
• (2) 50 meter
• (3) 50.4 meter
• (4) 51.6 meter

Identify the mirror image of the given word:-

• (1)
• (2)
• (3)
• (4)

The scale of a map is 1:1000. If a car travels 7 cm from point 'A' to point 'B' on the Map. Then how much the car has travelled in original:-

• (1) 0.7 km
• (2) 7000 mm
• (3) 7 km
• (4) 70 meter

A land size of 80 meter × 40 meter for a house design is drawn on paper at a scale of 1:100, then what size is drawn on paper to represent the land?

• (1) 8 meter × 4 meter
• (2) 80 meter × 40 meter
• (3) 8 centimeter × 4 centimeter
• (4) 4 meter × 2 meter

Find the missing number in given series.
16, 33, 65, 131, 261, (.....)

• (1) 523
• (2) 521
• (3) 524
• (4) 520

In a code language if ROMAN is written as TQOCP: then ITALY is.......

• (1) KVCMA
• (2) KWCNB
• (3) KUCLA
• (4) KVCNA

Select a suitable figure from the four alternatives which will come in the empty box.

• (1) D
• (2) C
• (3) B
• (4) A

The 3D figure shows the view of an object. Looking in the direction of arrow, identify the most appropriate elevation from the given answer figures.

• (1)
• (2)
• (3)
• (4)

Question figure shows top view/ plan of an object. Identify the INCORRECT 3D view from the given answers figure.

• (1)
• (2)
• (3)
• (4)

How many surfaces does the object have?

• (1) 9
• (2) 10
• (3) 8
• (4) 11

If the question figure is cut into two parts, which of the answer figures complete the question figure without any overlappings?

• (1)
• (2)
• (3)
• (4)

The 3D problem figure shows the view of an object. Identify its appropriate top view from the answer figures.

• (1)
• (2)
• (3)
• (4)

Identify the correct elevation when you look into the object from the marked arrow side of the plan of the object.

• (1) D
• (2) A
• (3) B
• (4) C

A paper is folded in a given pattern and it is cut at the end. Identify which pattern is formed when the paper is unfold.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3-D view of an object. Identify the correct view, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3-D view of an object. Identify the correct view, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

From the given options below, choose the correct plan of the 3-D object, when viewed from the top.

• (1)
• (2)
• (3)
• (4)

The second figure in the first part of the problem figures bears certain relationship to the first figure. Similarly, one of the figures of answer figures bears the same relationship to the first figure of the second part. Identify the correct option from the given answer figures.

• (1)
• (2)
• (3)
• (4)

Identify the true mirror image of the figure amongst the answer figures with respect to X-X

• (1)
• (2)
• (3)
• (4)

Choose the correct option amongst the answer figures which complete the series.

• (1)
• (2)
• (3)
• (4)

Sheet is folded in marked format and cut led at last as shown. Identify from the options below, how the pattern will be made when it is fully unfolded?

• (1) B
• (2) A
• (3) D
• (4) C

In the figure mentioned below find the missing series:-

• (1)
• (2)
• (3)
• (4)

The diagram shows the supply and demand of different users for different items. Which of the following is/are correct?

A. Sanitizer only meet 50% of the demand
B. Cosmetics has the least supply among all.
C. Among all, two items have equal demand but difference in supply
D. Among all, two items have equal supply and two items have equal demand.
Choose the correct answer from the options given below:-

• (1) B and D only
• (2) A and B only
• (3) A and C only
• (4) B, C and D only

Draw a proportionate sketch of given reference image. Use black and white rendering techniques of your choice.

Use the basic 2D shapes found in a motor cycle and create an interesting 2D composition of your choice, colour with any three colours of your choice.