JEE Main 2024 BArch Question Paper with Solutions PDF January 24

The domain of the function \(f(x)=\sqrt{4-x^{2}}+\sqrt{2+x}\) is \([a, b]\), then \(a^{2}+3 b^{2}\) is equal to

• (1) 4
• (2) 8
• (3) 12
• (4) 16

Let \(\mathbb{R}\) denote the set of real numbers, \(\mathbb{Z}\) the set of integers and W the set of Non-negative integers, and define a map
(i) \(f_{1}: \mathbb{Z} \rightarrow \mathbb{Z}\) by \(f_{1}(x)=x^{3}\). Then \(f_{1}\) is one-one and not onto.
(ii) \(f_{2}: \mathbb{R} \rightarrow \mathbb{R}\) by \(f_{2}(x)=x^{3}\). Then \(f_{2}\) is one-one and onto.
(iii) \(f_{3}: \mathbb{Z} \rightarrow\) W by \(f_{3}(x)=|x|\). Then \(f_{3}\) is not one-one but is onto.
(iv) \(f_{4}: \mathbb{Z} \rightarrow \mathbb{Z}\) by \(f_{4}(x)=x^{2}\). Then \(f_{4}\) is neither one-one nor onto.
Then which of the following options is correct?

• (1) (i), (iii), (iv) are true but (ii) is false
• (2) All are true
• (3) (ii), (iii) and (iv) are true but (i) is false
• (4) (i), (ii) and (iv) are true but (iii) is false

If \(\frac{\sqrt{5+12 i}+\sqrt{5-12 i}}{\sqrt{5+12 i}-\sqrt{5-12 i}}=a+b i\), then \(a-b\) is :

• (1) \(\frac{5}{6}\)
• (2) \(-\frac{5}{12}\)
• (3) \(\frac{3}{2}\)
• (4) \(\frac{5}{12}\)

Let \(A\) be a \(3 \times 3\) matrix such that \(\operatorname{adj}(A)=\left[\begin{array}{ccc}25 & 0 & 100 \\ 0 & 4 & 3 \\ 0 & 100 & 100\end{array}\right]\), then determinant of \(A\) is

• (1) 25
• (2) 50
• (3) 100
• (4) 1000

The total number of times the digit 3 will be written when listing integers from 1 to 1000 is

• (1) 297
• (2) 243
• (3) 273
• (4) 300

The coefficient of \(x^{24}\) in \(\left(1+x^{2}\right)^{12}\left(1+x^{12}\right)\left(1+x^{24}\right)\) is

• (1) \({ }^{12} C_{6}+3\)
• (2) \({ }^{12} C_{6}\)
• (3) \({ }^{12} C_{6}+1\)
• (4) \({ }^{12} C_{6}+2\)

Sum to \(n\) terms of the series \(\log _{e} a+\log _{e}\left(\frac{a^{2}}{b}\right)+\log _{e}\left(\frac{a^{3}}{b^{2}}\right)+\log _{e}\left(\frac{a^{4}}{b^{3}}\right)+\ldots \ldots \ldots \ldots\) is

• (1) \(\frac{n}{2}\left[(n+1) \log _{e}\left(\frac{a}{b}\right)-\log _{e} a b\right]\)
• (2) \(\frac{n}{2}\left[n \log _{e}\left(\frac{a}{b}\right)-\log _{e} a b\right]\)
• (3) \(\frac{n}{2}\left[\log _{e}\left(\frac{a}{b}\right)+n \log _{e} a b\right]\)
• (4) \(\frac{n}{2}\left[n \log _{e}\left(\frac{a}{b}\right)+\log _{e} a b\right]\)

If the sum of first 10 terms and first 26 terms of an A.P. are 190 and 1326 respectively, then find the sum of its first 50 terms.

• (1) 4900
• (2) 4924
• (3) 4950
• (4) 4980

If \(\lim_{x \to 1} \frac{x^{5}-1}{x-1}=\lim_{x \to k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}\), then \(k\) is equal to

• (1) \(\frac{5}{3}\)
• (2) \(\frac{7}{3}\)
• (3) \(\frac{8}{3}\)
• (4) \(\frac{10}{3}\)

Let \(f:[0, \pi] \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\begin{cases}x+\sqrt{2}\, a \sin x, & 0 \leq x<\frac{\pi}{4} \\ 2 x \cot x+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a \cos 2 x-b \sin x, & \frac{\pi}{2}<x \leq \pi\end{cases}\)
If \(f\) is continuous for \(0 \leq x \leq \pi\), then \(a^{2}+140 b^{2}\) is equal to

• (1) \(0\)
• (2) \(\pi^{2}\)
• (3) \(2 \pi^{2}\)
• (4) \(\frac{\pi^{2}}{2}\)

\(\int \frac{\mathrm{d} x}{\sin ^{2} x \cos ^{2} x}\) is equal to

• (1) \(\tan x+\cot x+c\)
• (2) \((\tan x+\cot x)^{2}+c\)
• (3) \(\tan x-\cot x+c\)
• (4) \(2(\sec x \tan x-\operatorname{cosec} x \cot x)+c\)

The solution of differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y\) is

• (1) \(x=y^{2}+c\)
• (2) \(y=x^{2}+c\)
• (3) \(x=y\left(y^{2}+c\right)\)
• (4) \(y=x\left(x^{2}+c\right)\)

The point diametrically opposite to the point \((-1,1)\) on the circle \(x^{2}+y^{2}-x-4 y+1=0\) is

• (1) \((-2,5)\)
• (2) \((2,-5)\)
• (3) \((0,3)\)
• (4) \((2,3)\)

The vertices of a triangle \(OAB\) are \((0,0)\), \((a, 0)\) and \((0, b)\) respectively. The distance between its orthocentre and circumcentre is

• (1) \((a+b)\)
• (2) \((a-b)\)
• (3) \(\frac{1}{2} \sqrt{a^{2}+b^{2}}\)
• (4) \(\frac{1}{2} \sqrt{a^{2}-b^{2}}\)

The shortest distance between the lines \(\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-3 \hat{j}+2 \hat{k})\) and \(\vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+\hat{k})\) is :

• (1) \(\frac{1}{19}\) units
• (2) \(\sqrt{19}\) units
• (3) \(\frac{3}{\sqrt{19}}\) units
• (4) \(\frac{1}{\sqrt{19}}\) units

The projection vector of \(\vec{a}=\hat{i}-3 \hat{j}-3 \hat{k}\) on \(\vec{b}=3 \hat{i}-4 \hat{j}\) is

• (1) \(\frac{9}{5} \hat{i}-\frac{12}{5} \hat{j}\)
• (2) \(\frac{3}{\sqrt{19}} \hat{i}-\frac{4}{\sqrt{19}} \hat{j}\)
• (3) \(\frac{12}{5} \hat{i}-\frac{36}{5} \hat{j}-\frac{36}{5} \hat{k}\)
• (4) \(9 \hat{i}-12 \hat{j}\)

Given \( |\vec{a}|=|\vec{b}|=1 \) and \( |\vec{a}+\vec{b}|=\sqrt{3} \). If \( \vec{c} \) is a vector such that \( \vec{c}-\vec{a}-2\vec{b}=3(\vec{a}\times\vec{b}) \), then \( \vec{c}\cdot\vec{b} \) is equal to

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

If \( A \) and \( B \) are two events such that \( P(A\cup B)=\frac{5}{8} \) and \( P(A'\cup B')=\frac{3}{4} \), then \( P(A')+P(B')= \)

• (1) \(\frac{1}{8}\)
• (2) \(\frac{3}{8}\)
• (3) \(\frac{9}{8}\)
• (4) \(\frac{11}{8}\)

The value of \( \sec 15^{\circ}-\sqrt{3}\,\operatorname{cosec} 15^{\circ} \) is

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

\( \cot^{-1}7+\cot^{-1}8+\cot^{-1}5= \)

• (1) \(\cot^{-1}2\)
• (2) \(\cot^{-1}10\)
• (3) \(\cot^{-1}6\)
• (4) \(\cot^{-1}4\)

Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^{2}-6x-2=0 \), with \( \alpha>\beta \). If \( a_{n}=\alpha^{n}-\beta^{n} \) for \( n\geq1 \), then the value of \( \frac{a_{10}-2a_{8}}{3a_{9}} \) is equal to ________

If \( \alpha,\beta,\gamma \) are the angles of a triangle, then the value of the determinant \( \begin{vmatrix}\cot\alpha&\cot\beta&\cot\gamma\\1&1&1\\\sin^{2}\alpha&\sin^{2}\beta&\sin^{2}\gamma\end{vmatrix} \) is equal to ________

If \( r^{\text{th}} \) term in the expansion of \( \left(\sqrt[3]{\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{a}}}\right)^{21} \) contains equal powers of \( a \) and \( b \), then value of \( r \) is ________

If \( y=\tan^{-1}\left(\frac{3x}{1+54x^{2}}\right)+\tan^{-1}\left(\frac{3x}{1+18x^{2}}\right) \), then \( \frac{dy}{dx} \) at \( x=0 \) is ________

The value of \( \int_{1}^{5} f(x)\, dx \), where \( f(x)=|x-1|+|x-2|+|x-3| \) is ________

Let \( y=y(x) \) be the solution of differential equation \( \sin x \frac{dy}{dx}+y\cos x=4x \). If \( y\left(\frac{\pi}{2}\right)=0 \) and \( y\left(-\frac{\pi}{6}\right)=\frac{\alpha}{\beta}\pi^{2} \), where \( \alpha \) and \( \beta \) are co-prime numbers, then value of \( (\alpha+\beta) \) is ________

Let \( L \) denote the line in the \( xy \)-plane with \( x \) and \( y \) intercepts as 2 and 1 respectively. If the image of the point \( (-1,-3) \) in this line is \( \left(\frac{\alpha}{5}, \frac{\beta}{5}\right) \), then \( \alpha+\beta \) is equal to ________

The length of projection of the line segment joining the points \( (1,0,-1) \) and \( (-1,2,2) \) on the plane \( x+3y-5z=6 \) is \( d \), then find \( 70d^{2} \) ________

There are two bags. The first bag contains 3 white and 4 red balls and the second bag contains 4 white and 3 red balls. Two balls are drawn at random from the first bag and are put into the second bag without noticing their colours. Then two balls are drawn from the second bag. If the probability that one ball is white and the other is red is \( \frac{x}{y} \), where \( \frac{x}{y} \) is in the simplest form, then value of \( (2x-y) \) is ________

In a \( \Delta ABC \), \( \angle A=\frac{\pi}{2} \). If \( \tan\frac{B}{2} \) and \( \tan\frac{C}{2} \) are the roots of \( 2x^{2}-bx-c=0 \), then what is the value of \( (b-c)^{3} \) ________ ?

Saturation level of colors represents

• (1) Distribution
• (2) Darkness
• (3) Brilliance
• (4) Warmth

Which of the following is a shell structure?

• (1) Millenium dome
• (2) Stansted airport
• (3) Bank of China tower
• (4) Louvre, Museum

Identify the building material represented in the given figure

• (1) Tile
• (2) Paving Block
• (3) Wooden Plank
• (4) Granite

Before plastering, the surface has to be :

• (1) Cemented
• (2) Polished
• (3) Smooth
• (4) Rough

Who designed the 'Sydney Opera House'?

• (1) Jorn Utzon
• (2) Jan Gehl
• (3) Bjarke Ingels
• (4) Henning Lorsen

The design elements which are associated with the sense of touch

• (1) Pattern, Shape, Form
• (2) Texture, Shape, Form
• (3) Shape, Pattern, Texture
• (4) Form, Pattern, Texture

The First World Environment Day was held in which year?

• (1) 1973
• (2) 1976
• (3) 1978
• (4) 1970

Elephanta Caves are dedicated to which God?

• (1) Budha
• (2) Tirthankar Mahavir
• (3) Shiva
• (4) Vishnu

Which one of the following is not a city type?

• (1) Garden City
• (2) Linear City
• (3) Radiant City
• (4) Stack City

A gigantic chariot with elaborately carved stone wheels, pillars and walls are the characteristics of :

• (1) Konark Sun temple
• (2) Golden temple
• (3) Meenakshi temple
• (4) Dilwara temple

Who built the Moti Masjid in Red Fort at Delhi?

• (1) Shah Jahan
• (2) Aurangzeb
• (3) Qutub-Ud-Din Aibak
• (4) Akbar

Spray painting is used to :

• (1) Get a textured pain
• (2) Reach high areas
• (3) Apply paint without touching surface
• (4) Apply large amount of paint

An escalator looks like which one of the following?

• (1) Ladder
• (2) Stair case
• (3) Ramp
• (4) Lift

Concrete is a mixture of

• (1) Cement + Sand + Water
• (2) Cement + Coarse Aggregate + Fine Aggregate + Water
• (3) Cement + Water + Brick Bat
• (4) Cement + Sand + Water + Brick Bat

What type of tile are suitable for air - conditioned rooms and gymnasium?

• (1) Cork Tile
• (2) Ceramic Tile
• (3) Concrete Tile
• (4) Wooden Tile

Famous Tassel House was designed by

• (1) Victor Horta
• (2) Gerrit Rietvelt
• (3) Mies Van der Rohe
• (4) Frank Lloyd Wright

Who is the Author of "Toward an Architecture"?

• (1) Frank Ching
• (2) Le Corbusier
• (3) Kevin A. Lynch
• (4) Frank Wright

Hiroshima Peace Memorial Museum in Japan was designed by :

• (1) I. M. Pie
• (2) Tadao Ando
• (3) Kenzo Tange
• (4) Kisho Kurokawa

Which of the following is not a principle of Design?

• (1) Emphasis
• (2) Focal Point
• (3) Unity
• (4) Rhythm

Which of the following software is not used by architects?

• (1) Staad. Pro
• (2) 3D Max
• (3) Revit
• (4) Sketch Up

Who built the Shalimar Bagh, Srinagar?

• (1) Bega Begum
• (2) Akbar
• (3) Shah Jahan
• (4) Jahangir

Kanpur, Uttar Pradesh is also known as :

• (1) The Leather City
• (2) Wine Capital of India
• (3) The Orange City
• (4) City of Nawabs

How the load is transfered from the super structure to sub structure?
A. Beam
B. Slab
C. Foundation
D. Earth
E. column
Choose the correct answer from the options given below:

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

The Question figure shows the 3D view of an object. Identify the correct view, looking in the direction of Arrow, identify the correct view of the object.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3D view of an object. Identify the correct top view amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

Identify the mirror image of the given word with respect to X-X

• (1)
• (2)
• (3)
• (4)

When the 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)

Identify the figure which is odd from rest of the figures.

• (1)
• (2)
• (3)
• (4)

Given below are two statements :
Statement I : Stone masonary is classified into two parts - Random rubble and ashlar masonary.
Statement II : Random Rubble masonary, each stone is cut to uniform size and shape with all sides rectangular so that stones gives perfect joints.
In the light of the above statements, choose the correct answer from the options given below :

• (1) Both Statement I and Statement II are true
• (2) Both Statement I and Statement II are false
• (3) Statement I is true but Statement II is false
• (4) Statement I is false but Statement II is true

Identify the type of Garden by understanding the elements used in the given picture:

• (1) Roman Garden
• (2) Japanese Garden
• (3) French Garden
• (4) Chinese Garden

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

Identify the name of the monument :

• (1) The Pantheon
• (2) The Cathedral of Pisa
• (3) The Parthenon
• (4) Great Temple of Karnak

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

• (1)
• (2)
• (3)
• (4)

Identify the correct mirror image of the given figure along X-axis.

• (1)
• (2)
• (3)
• (4)

Identify the Mirror image of given word

• (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)

Identify the correct figure to complete the sequence :

• (1)
• (2)
• (3)
• (4)

Complete the number series
10, 30, 68, 130, ________, 350

• (1) 222
• (2) 192
• (3) 220
• (4) 216

In a clock, the time is 11:35 AM. The clock is so placed that its hour hand is facing North-West. In which direction will be minute hand facing?

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

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

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

Identify the famous building in the given picture :

• (1) Mysore Palace
• (2) Khalsa Heritage Complex
• (3) Hotel Taj Mahal Palace
• (4) Victoria Memorial

'LEED' stands for :

• (1) Leadership in Energy and Environmental Design
• (2) Leadership in Environmental and Energy Design
• (3) Leadership in Energy Efficient Design
• (4) Leadership in Environmental Energy Design

From the given 3D view of an object, identify the most appropriate elevation looking in the direction of the given arrow :

• (1)
• (2)
• (3)
• (4)

From the given 3D view of an object, identify the most appropriate elevation looking in the direction of the given arrow.

• (1)
• (2)
• (3)
• (4)

Choose the correct top view of the given 3D image from the options.

• (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)

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Green buildings have a lower life cycle cost than conventional buildings.
Reason R : Green buildings require less maintenance and have lower energy and water consumption.
In the light of the above statements, choose the most appropriate answer from the options given below :

• (1) Both A and R are correct and R is the correct explanation of A
• (2) Both A and R are correct but R is NOT the correct explanation of A
• (3) A is correct but R is not correct
• (4) A is not correct but R is correct

Match List I with List II

Choose the correct answer from the options given below:

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

Identify the building material primarily used for cladding in Lotus Temple :

• (1) Dolomite
• (2) Lime
• (3) Marble
• (4) Granite

Draw a proportionate sketch of the image given below in detail. Sketch this using Black and White medium only.

Combine Musical Instruments like Guitar, Microphone (mic), Drums, Tabla, Keyboard, etc. into a 3D Abstract Sculpture, using different colors to represent various Texture and Instruments. You can scale each object to any size of your choice. Apply the effect of light and shadow on your Sculpture.
OR
Considering yourself as one of the spectators, draw a beautiful sketch of a Football stadium where a match is going on.