JEE Main 2023 April 12 Shift 2 B.Arch Question Paper with Solution PDF- Download Here

If \( f(x)=\frac{x^{3}}{x-1}+\frac{x^{3}}{x+1} \) and \( g(x)=\sqrt{x} \), then \( f \circ g:[0,2]-\{1\} \rightarrow \mathbb{R} \) is :

• (1) one-one but not onto
• (2) one-one and onto
• (3) onto but not one-one
• (4) neither one-one nor onto

The sum of squares of the values of \( \alpha \in \mathbb{R} \) such that the argument of \( \frac{\alpha+i}{\alpha-i} \) is \( 60^{\circ} \), \( i=\sqrt{-1} \), is :

• (1) 3
• (2) \( \frac{10}{3} \)
• (3) \( \frac{11}{3} \)
• (4) 4

For the system of equations
\( x+\lambda y-z=1 \)
\( x+2y+\lambda z=2 \)
\( x+2y+z=2 \),
which one of the following is NOT correct ?

• (1) it has unique solution if \( \lambda \) is not a root of the equation \( t^{2}-3t+2=0 \)
• (2) it has unique solution if \( \lambda \) is not a root of the equation \( t^{2}-t-2=0 \)
• (3) it has infinitely many solutions if \( \lambda=1 \)
• (4) it has no solution if \( \lambda=2 \)

If \[ a_n=(2n^2-n+2)(n!) , \] then \[ \sum_{n=1}^{20} a_n \] is equal to:

• (1) \(37(20!)-1\)
  
• (2) \(37(20!)+1\)
  
• (3) \(39(21!)+1\)
  
• (4) \(39(21!)-1\)

\( \lim_{x\rightarrow0^{+}}\frac{1}{\sqrt{x}}\left(\frac{1}{\sin x}-\frac{1}{x}\right) \)

• (1) is equal to 0
• (2) is equal to \( \frac{1}{5} \)
• (3) is equal to 1
• (4) does not exist

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

The slope of the tangent to the curve \( y=y(x)=\int_{\sin^{-1}(x)}^{\cos^{-1}(x)}\sqrt{1+4\sin^{2}t}\,dt\ ;\ 0\le x\le1 \) at the point \( \left(\frac{1}{\sqrt{2}},0\right) \) on the curve is :

• (1) \( -2\sqrt{6} \)
• (2) \( 2\sqrt{6} \)
• (3) \( -4\sqrt{3} \)
• (4) \( 4\sqrt{3} \)

Let \( \alpha,\beta,\gamma\left(0<\alpha,\beta,\gamma<\frac{\pi}{2}\right) \) be the angles between non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \vec{b} \) and \( \vec{c} \), \( \vec{c} \) and \( \vec{a} \) respectively. If \( \theta \) is the angle that the vector \( \vec{a} \) makes with the plane containing \( \vec{b} \) and \( \vec{c} \), then

• (1) \( \cos^{2}\theta=\operatorname{cosec}^{2}\beta\left(\cos^{2}\alpha+\cos^{2}\gamma-2\cos\alpha\cos\beta\cos\gamma\right) \)
• (2) \( \cos^{2}\theta=\sec^{2}\beta\left(\cos^{2}\alpha+\cos^{2}\gamma+2\cos\alpha\cos\beta\cos\gamma\right) \)
• (3) \( \sin^{2}\theta=\operatorname{cosec}^{2}\beta\left(\cos^{2}\alpha+\cos^{2}\gamma-2\cos\alpha\cos\beta\cos\gamma\right) \)
• (4) \( \sin^{2}\theta=\sec^{2}\beta\left(\cos^{2}\alpha+\cos^{2}\gamma+2\cos\alpha\cos\beta\cos\gamma\right) \)

The domain of the function \(f(x)=\sin ^{-1}\left(\log _{2}((x-1)(x-2))\right)\) is :

• (1) \([0,3]\)
• (2) \(\left[0, \frac{3-\sqrt{3}}{2}\right] \cup\left[\frac{3+\sqrt{3}}{2}, 3\right]\)
• (3) \(\left(\frac{3-\sqrt{3}}{2}, 1\right) \cup\left(2, \frac{3+\sqrt{3}}{2}\right)\)
• (4) \(\left[0, \frac{3-\sqrt{6}}{2}\right] \cup\left[\frac{3+\sqrt{6}}{2}, 3\right]\)

Let \(\alpha\) and \(\beta\) be the roots of the equation \[ 2x^2-5x-1=0. \] For \(n\in\mathbb{N}\), let \[ P_n=\alpha^n+\beta^n. \] Then the value of \[ \frac{2P_{11}\,(2P_{10}-5P_9)}{P_8\,(5P_{10}+P_9)} \] is equal to:

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

If the image of the point \((1,1,2)\) in the plane \(2 x-y+z+3=0\) is the point P, then the distance of P from origin is

• (1) \(2 \sqrt{3}\)
• (2) \(3 \sqrt{2}\)
• (3) 4
• (4) 6

For three non–coplanar vectors \(\vec a,\vec b,\vec c\), if \[ (\vec b+\vec c)\cdot\big((\vec c+\vec a)\times(\vec a+\vec b)\big) = \alpha \,[\vec a\,\vec b\,\vec c] \] and \[ (\vec a+\vec b)\cdot\big((\vec b+\vec c)\times(\vec a+\vec b+\vec c)\big) = \beta \,[\vec a\,\vec b\,\vec c], \] then \(\alpha+\beta\) is equal to:

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

If \(I(x)=\int \frac{d x}{1-2 \sin ^{2} x \cos ^{2} x}\), then \(\tan \left(\sqrt{2}\left(I\left(\frac{\pi}{8}\right)-I(0)\right)\right)\) is equal to :

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

Let \(X\) have a binomial distribution \(B(6,p)\). If the sum of the mean and the variance of \(X\) is \(\dfrac{21}{8}\), then \[ \frac{P(2\le X<4)}{P(4<X<6)} \] is equal to:

• (1) \(65\)
  
• (2) \(195\)
  
• (3) \(\dfrac{195}{2}\)
  
• (4) \(\dfrac{225}{2}\)

If the statement
\(((\mathrm{p} * (\sim \mathrm{q})) \wedge(\mathrm{p} \vee \mathrm{q})) \Leftrightarrow \mathrm{p}\)
is a tautology, then \(*\) is :

• (1) \(\wedge\)
• (2) \(\vee\)
• (3) \(\Rightarrow\)
• (4) \(\Leftrightarrow\)

If the circles \(x^{2}+y^{2}-2 x-4 y+4=0\) and \(x^{2}+y^{2}-6 x-10 y+20+2 \sqrt{13}=0\) touch each other at the point \((a, \mathrm{b})\), then \((3 a-2 \mathrm{b})^{2}\) is equal to :

• (1) 1
• (2) 4
• (3) 9
• (4) 13

If the angle between the line \(l: \frac{x-1}{2}=\frac{y+1}{1}=\frac{z-2}{2}\) and the plane \(\mathrm{P}: \lambda x+4y-7=0,\ \lambda \neq 0\), is \(\operatorname{cosec}^{-1}\left(\frac{3}{2}\right)\), then the sum of co-ordinates of the point where line \(l\) crosses the plane P is :

• (1) \(-33\)
• (2) \(-2\)
• (3) 3
• (4) 6

Let three distinct normal be drawn to the parabola \(y^{2}+4y-6x-8=0\) from a point \((a, \mathrm{b})\) on the axis of the parabola. Then :

• (1) \(a \in (1, \infty)\) and \(b=-2\)
• (2) \(a \in (0, \infty)\) and \(b=-2\)
• (3) \(a \in (1, \infty)\) and \(b=2\)
• (4) \(a \in (2, \infty)\) and \(b=2\)

Given \({}^{n}C_{r}=\frac{n!}{r!(n-r)!}\), \(0 \leq r \leq n\) and n is a non-negative integer. Then a possible value of k for which the equality
\({}^{50}C_{k-1}+\sum_{r=1}^{50}{}^{100-r}C_{k-2}={}^{100}C_{49}\) holds, is

• (1) 40
• (2) 49
• (3) 50
• (4) 25

If an unbiased die, marked with \(-3,-2,-1,0,1,2\) on its faces, is thrown four times, then the probability of getting \(-1\) as the sum of outputs is :

• (1) \(\frac{7}{81}\)
• (2) \(\frac{35}{324}\)
• (3) \(\frac{8}{81}\)
• (4) \(\frac{26}{243}\)

\(\left(\sin 5^{\circ} \sin 55^{\circ} \sin 65^{\circ} \sin 75^{\circ}\right)^{-1}\) is equal to ________.

If the shortest distance between the lines \(3x+2y-4z-5=0=5x-7y-17z+2\) and \(\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+1}{2}\) is \(\frac{10}{\sqrt{k}}\), then \(k\) is equal to ________.

For \(p \in \mathbb{N}\), if the angle between pair of tangents drawn to the ellipse \(3x^{2}+2y^{2}=5\) from the point \((1, p)\) is \(\tan^{-1}\left(\frac{12}{\sqrt{5}}\right)\), then the distance of the vertex of the parabola \(y=x^{2}-px+p+1\) from the point \((-7,8)\) is equal to ________.

Let P be a polygon with \(n\) vertices such that the line segment joining any two points of P remains entirely in P. If the number of diagonals of P is \(n+25\), then \(n\) is equal to ________.

Let \(f(x)\) be a polynomial of degree 5 such that \(\lim_{x \to 0} \frac{f(x)}{x^{2}}=1\), \(f(-1)=-1\), \(f(x)-14x\) has an extrema at \(x=1\) and \(f(x)-10x\) has an extrema at \(x=-1\). Then \(f(2)\) is equal to ________.

The number of 7 digits integers formed by using the digits 2, 3, 4, 5 only and having the sum of digits equal to 18 is ________.

The remainder when \((556)^{40}\) is divided by 7 is ________.

Let \([\mathrm{t}]\) denote the greatest integer less than or equal to t. Then the value of \(10-10 \int_{-2}^{2}\left[x+x^{3}\right] dx\) is ________.

Let \(A=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]\). If \(k, l \in \mathbb{N}\) be such that \(A^{k} B^{l}=I\), then the minimum value of \(k+l\) is ________.

Let \(\mathrm{A}_{1}=[1]\), \(\mathrm{A}_{2}=\left[\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right]\), \(\mathrm{A}_{3}=\left[\begin{array}{ccc}6 & 7 & 8 \\ 9 & 10 & 11 \\ 12 & 13 & 14\end{array}\right], \ldots\) Then the sum of the diagonal elements of \(\mathrm{A}_{20}\) is ________.

Which of the following is not a sustainable development goal targeted to be achieved by 2030?

• (1) gender equality
• (2) zero hunger
• (3) good health & well being
• (4) space Research

KANDARIYA MAHADEV TEMPLE is situated in which state of India?

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

Vertical elements in a building are:-

• (1) Column, Lintel, Wall
• (2) Lintel, Beam, Arch
• (3) Column, Pier, Post
• (4) Floor, Column, Arch

Cavity Walls are used for

• (1) Waterproofing
• (2) Heat Insulation
• (3) Aesthetics
• (4) Cost saving

Lintel is related to:

• (1) Window
• (2) Roof
• (3) Foundation
• (4) Pipe

Choose the correct option among the following by looking at the image given above:

• (1) Lodhi Garden
• (2) Jantar Mantar
• (3) Rajghat
• (4) Bahai Temple

ARCHITECT ACT 1972 was enforced in which month

• (1) 31^st Dec
• (2) 31^st May
• (3) 31^st July
• (4) 31^st March

The dome of Rashtrapati Bhawan is influenced by which of the following:

• (1) Gurudwara
• (2) Mosque
• (3) Stupa
• (4) Temple

Dominion tower is designed by

• (1) Bjarke Ingels
• (2) Zaha Hadid
• (3) Frei otto
• (4) F.L. Wright

The largest river island of the world is present in which state of India?

• (1) Maharastra
• (2) Assam
• (3) Uttar Pradesh
• (4) West Bengal

Who prepared the first layout of the town of Jamshedpur?

• (1) Charles Correa
• (2) Otto Koenigsberger
• (3) Le Corbusier
• (4) Messrs Julian Kennedy Sahlin

Identify the historical monument which is given in the image.

• (1) Sanchi Stupa
• (2) Victory Tower
• (3) Sarnath Stupa
• (4) Nalanda

Who was known as “Father of modern concrete”?

• (1) John Paul
• (2) John Nash
• (3) John Smeaton
• (4) Joseph Stein

Who is the architect of TWA Airport, New York?

• (1) Eero Saarinen
• (2) Mies Vander Rohe
• (3) Joseph Allen Stein
• (4) Alvar Aalto

Starting from a point, Maria walked 12 meters towards North, she turned right and walked 10 meters. She again turned right and walked 12 meters, then she turned left and walked 5 meters. How far is she now and in which direction from the starting point? Consider the starting point is in south direction.

• (1) 27 meters towards East
• (2) 5 meters towards East
• (3) 10 meters towards West
• (4) 15 meters towards East

Who among the following is not Pritzker prize winner?

• (1) Wang Shu
• (2) B.V. Doshi
• (3) Arata Isozaki
• (4) Denise Scott Brown

Which of the following is not Antonio Gaudi’s project?

• (1) Casa Mila
• (2) Sagrada Familia
• (3) Casa Batllo
• (4) Casa Milano

Which one of the following is most earthquake resistant structure?

• (1) RCC framed structure
• (2) Load bearing structure
• (3) Mud walls structure
• (4) Random stone masonry structure

"CHALK" belongs to which category?

• (1) Rock
• (2) Mineral
• (3) Wood
• (4) Polymer

What are the three parts of colour theory?

• (1) hyper, value and chrome
• (2) honey, violet and chroma
• (3) hue, value and chroma
• (4) hue, chrome and violet

Identify the three primary colours?

• (1) Red, Blue and Green
• (2) Red, Yellow and Blue
• (3) Black, White and Yellow
• (4) Blue, White and Yellow

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

A square has an area of 100 sq.mt. How much will be its area on a scale of 1 : 20?

• (1) 5 sq.mt.
• (2) 50 sq.mt.
• (3) 0.25 sq.mt.
• (4) 25 sq.mt.

Given below are two statements:
Statement I: Shakti Sthal is a memorial in Raj Ghat marking the cremation site of Indira Gandhi.
Statement II: A.P. Kanvinde in the landscape designer known for designing Shakti Sthal.
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

What is the full form of "UDAY"?

• (1) Ujjwala Discom Assitance Yojana
• (2) Unnati Development Ayojan Yojana
• (3) Ujwal Distribution Assurance Yojana
• (4) Ujwal Discom Assurance Yojana

What is the acronym for "AYUSH"?

• (1) Ayurveda, Yoga and Naturopathy, Unani, Siddha and Homeopathy
• (2) Ayurveda, Yogic and Unani Siddha and Homeopathy
• (3) Acroveda, Yoga, Unani, Siddha and Homeopathy
• (4) Ayurveda, Yoga, Naturopathy, Unani Surgery and Homeopathy

Select the correct option from answer figure which will be best suitable to continue the same pattern as established by the problem figure.

• (1)
• (2)
• (3)
• (4)

Choose the correct option from the answer figures which completes the question series.

• (1)
• (2)
• (3)
• (4)

Identify the number of surfaces in the given 3D object.

• (1) 12
• (2) 13
• (3) 10
• (4) 9

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

• (1)
• (2)
• (3)
• (4)

The 3D figure shows the view of an object. Identify the correct view looking in the direction of arrow, amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

Choose the correct option from the answer figures which completes the sequence.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3D view of an object, identify the correct view looking in the direction of the arrow?

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3D view of an object. Identify the top view of the object in the direction of the arrow.

• (1)
• (2)
• (3)
• (4)

Identify the total number of surfaces in the given 3D object.

• (1) 12
• (2) 13
• (3) 15
• (4) 16

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

• (1)
• (2)
• (3)
• (4)

In the question figure it is a 3D view of an object. Identify the correct top view among the given options.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3D view of an object. Identify the most appropriate top view of the object from the given direction of the arrow.

• (1)
• (2)
• (3)
• (4)

Identify the total number of surfaces in the given object.

• (1) 13
• (2) 14
• (3) 11
• (4) 10

Identify the mirror image of the question figure, amongst the answer figures with respect to P-P.

• (1)
• (2)
• (3)
• (4)

Identify the correct mirror image of the question image with respect to P-P.

• (1)
• (2)
• (3)
• (4)

Which of the answer figures fit best to the left side of the given question figure?

• (1)
• (2)
• (3)
• (4)

In the question figure, the top, front and right side view is given. Identify the most appropriate 3D view of the object from the answer figures.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3D view of an object. Identify the most appropriate elevation (front view) looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

The question figure shows the 3D view of an object. Identify the most appropriate view of the object looking in the direction of the arrow.

• (1)
• (2)
• (3)
• (4)

The question figure is the 3D view of an object. Identify the most appropriate front view of the object looking in the direction of the arrow.

• (1)
• (2)
• (3)
• (4)

Question figure shows 3D view of an object. Identify the most appropriate top view of the given 3D object looking in the direction of an arrow, from the given answers figures.

• (1)
• (2)
• (3)
• (4)

Identify the number of surfaces in the problem figure.

• (1) 24
• (2) 25
• (3) 23
• (4) 20

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

• (1)
• (2)
• (3)
• (4)

Find out the total number of surfaces of the 3D object.

• (1) 12
• (2) 13
• (3) 14
• (4) 15

Draw the Sketch of the given image in detail.

Create an interesting sculpture, using basic 3D shapes of sports elements eg. different Balls, Bats etc. You can scale each object to any size of your choice, and use them as per your choice. Draw the effect of light and shadow on your composition. Use colours of your choice.