JEE Main 2020 B.Arch Question Paper with Answer Key (September 1- Forenoon session)

Let a function, \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=1+x|x|\) and let \(\operatorname{sgn}(x)=\begin{cases}1, & \text{if } x\ge 0\\ -1, & \text{if } x<0\end{cases}\)
Then :

• (1) \(f\) is not invertible
• (2) \(f^{-1}(x)=\operatorname{sgn}(x-1)\sqrt{|x-1|}\)
• (3) \(f^{-1}(x)=\sqrt{|x-1|}\)
• (4) \(f^{-1}(x)=\operatorname{sgn}(x)\sqrt{|x-1|}\)

Let \(\alpha\) be a common root of the equations, \(x^{2}-x-12=0\) and \(\lambda x^{2}+10x+3=0\) (where \(\lambda\) is an integer). If \(\beta\) is the other root of the second equation, then the ordered pair \((\lambda,\beta)\) is equal to :

• (1) \(\left(-3,\ \frac{1}{3}\right)\)
• (2) \(\left(-3,\ -\frac{1}{3}\right)\)
• (3) \(\left(3,\ \frac{1}{3}\right)\)
• (4) \(\left(3,\ -\frac{1}{3}\right)\)

If \(z=-1+i\sqrt{2}\), then which one of the following is not true ?

• (1) \(\operatorname{Re}(z^{4})+\operatorname{Re}(z^{3})=-2\)
• (2) \(\operatorname{Im}(z^{4})+\operatorname{Im}(z^{3})=5\sqrt{2}\)
• (3) \(\operatorname{Im}(z^{4})-\operatorname{Re}(z^{3})=4\sqrt{2}-5\)
• (4) \(\operatorname{Re}(z^{4})-\operatorname{Im}(z^{3})=\sqrt{2}-7\)

For \(\theta\in(0,\pi/2)\), \(\theta\in(\pi,2\pi)\) and \(\beta\in\mathbb{R}\), the value of the determinant
\(\begin{vmatrix}\cos 2\alpha & \cos(\alpha+\theta) & -\sin(\alpha+\theta)\\ \sin\alpha & \sin\theta & \cos\theta\\ \beta\cos\alpha & -\cos\theta & \sin\theta\end{vmatrix}\) :

• (1) is independent of \(\alpha\).
• (2) is independent of \(\beta\).
• (3) is independent of \(\theta\).
• (4) depends on all \(\alpha,\ \beta\) and \(\theta\).

Let \(A=\begin{pmatrix}\alpha & 1 & 2\\ 1 & 2 & 3\\ 3 & 1 & 1\end{pmatrix}\), \(\alpha\in\mathbb{R}\) and \(B=[b_{ij}]_{3\times3}\) such that \(AB=rI\), \(r\neq0\) in \(\mathbb{R}\). If \(b_{32}=-\frac{3}{2}r\) and \(|\operatorname{adj}B|=r^{5}\), then \(|\operatorname{adj}(AB)|\) equals :

• (1) \(4^{12}\)
• (2) \(2^{12}\)
• (3) \(2^{6}\)
• (4) \(4^{8}\)

Let \(a_1, a_2, a_3, a_4\) and \(a_5\) be in A.P. with the common difference greater than 2. If \(\sum_{i=1}^{5}a_i=30\) and \(a_1\cdot a_2\cdot a_3\cdot a_4\cdot a_5=3840\), then \(a_1\cdot a_5\) is equal to :

• (1) 82
• (2) \(-128\)
• (3) 128
• (4) \(-82\)

Let the functions \(f\) and \(g\) be defined as \(f(x)=2\cos|x-2|+3|x|\sin|x|\) and \(g(x)=x^{5}+x\). If S be the set of points where the function \((f\circ g)(x)\) is not differentiable, then S is equal to :

• (1) \(\{1\}\)
• (2) \(\{0\}\)
• (3) \(\{0,1\}\)
• (4) \(\phi\) (an empty set)

Let \(f\) be a twice differentiable function on \(\mathbb{R}\) such that \(f(0)=4\) and the function, \(g(x)=\frac{1}{2}f^{2}(x)+3x^{3}-5\) is monotonically increasing in \(\mathbb{R}\). Then \(f(1)\) cannot be :

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

Let \(f(x)=\left(2x^{2}-9x+11\right)\cdot e^{x},\ -\infty<x<\infty\). Then \(f\) has :

• (1) a local maxima at \(x=\frac{1}{2}\) and a local minima at \(x=2\).
• (2) a local minima at \(x=\frac{1}{2}\) and a local maxima at \(x=2\).
• (3) neither a local maxima nor a local minima at \(x=\frac{1}{2}\).
• (4) neither a local maxima nor a local minima at \(x=2\).

If for \(x>0\), \(\int \frac{e^{\sqrt{x}}\sin\left(\frac{\pi}{4}+\sqrt{x}\right)}{\sqrt{x}}\,dx=e^{\sqrt{x}}\cdot f(x)+C\), where \(C\) is a constant of integration, then \(f(x)\) is equal to :

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

If \(y(x)=\int_{\pi^{2}/16}^{x^{2}}\frac{\sin x\,\sin\sqrt{\theta}}{1+\cos^{2}\sqrt{\theta}}\,d\theta\), then \(y'\left(\frac{\pi}{4}\right)\) is equal to :

• (1) \(\frac{\pi}{3\sqrt{2}}\)
• (2) \(\frac{\pi}{12}\)
• (3) \(\frac{\pi}{6}\)
• (4) \(\frac{\pi}{3}\)

The area (in sq. units) of the region \(\{(x,y)\in\mathbb{R}^{2}:2|x|+|y|\le 2\text{ and }x+y\ge 1\}\) is :

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

Let ABC be a triangle such that the vertex A has coordinates \((-3,1)\). If the equation of the median through B be \(2x+y-3=0\) and the equation of the bisector of angle C be \(7x-4y-1=0\), then the equation of the line BC is :

• (1) \(15x+y=50\)
• (2) \(4x-y=7\)
• (3) \(18x-y=49\)
• (4) \(7x-3y=6\)

If the ellipse centered at the origin has a focus at \((0,3)\) and the corresponding directrix is \(y=9\), then which one of the following is not true for this ellipse ?

• (1) Length of its minor axis is \(6\sqrt{2}\)
• (2) Length of its latus rectum is \(4\sqrt{3}\)
• (3) Point \((\sqrt{6},\,2\sqrt{3})\) lies on it
• (4) Point \((\sqrt{6},\,-3\sqrt{2})\) lies on it

If a hyperbola has length of its transverse axis \(=\sqrt{2}\) and it has the same foci as that of the ellipse \(4x^{2}+3y^{2}=24\), then this hyperbola also passes through the point whose coordinates are :

• (1) \(\left(2\sqrt{3},\frac{3}{\sqrt{2}}\right)\)
• (2) \(\left(2\sqrt{3},\frac{3}{2}\right)\)
• (3) \(\left(\frac{1}{\sqrt{6}},1\right)\)
• (4) \(\left(\frac{1}{\sqrt{6}},\frac{\sqrt{2}}{3}\right)\)

Let P be the point of intersection of the plane \(4x-6y+14z=57\) with the line joining the points A\((2,3,5)\) and B\((1,-1,4)\). If \(BP=\lambda\,AP\), then the value of \(\lambda\) is :

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

A six faced biased dice is so loaded that in any of its toss getting any of even scores is equally likely and the probability of getting an even score is three times the probability of getting an odd score. This dice is tossed repeatedly until a six is obtained. If X denotes the number of tosses required, then \(\mathrm{P}(\mathrm{X}<4)\) is :

• (1) \(\frac{19}{3^{3}}\)
• (2) \(\frac{37}{4^{3}}\)
• (3) \(\frac{61}{4^{3}}\)
• (4) \(\frac{397}{(12)^{3}}\)

If \(2 \sin x-\cos 2 x=1\), then :

• (1) \(\cos ^{4} x-\cos ^{2} x=3-\sqrt{5}\)
• (2) \(2\left(\cos ^{4} x-\cos ^{2} x\right)=-1\)
• (3) \(\sin ^{4} x-\sin ^{2} x=2-\sqrt{5}\)
• (4) \(2\left(\sin ^{4} x-\sin ^{2} x\right)=1\)

Let \(\vec{a}, \vec{b}, \vec{c}\) be any three non-coplanar vectors. If the points \(\mathrm{A}(-2 \vec{a}+3 \vec{b}+5 \vec{c})\), \(\mathrm{B}(\vec{a}+2 \vec{b}+3 \vec{c})\) and \(\mathrm{C}(7 \vec{a}-\lambda \vec{c})\) are collinear, then \(\lambda\) is equal to :

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

Which one of the following is a tautology ?

• (1) \(\sim(p \rightarrow q) \rightarrow q\)
• (2) \(\sim(p \rightarrow q) \leftrightarrow \sim q\)
• (3) \(\sim(p \rightarrow q) \leftrightarrow p\)
• (4) \(\sim(p \rightarrow q) \rightarrow p\)

If the sum of all possible 4 digit numbers formed using digits 1, 2, 3, 4 without repetition, is \(1111 \mathrm{k}\), then \(\mathrm{k}\) is equal to ________.

If \(x=\operatorname{cosec} \theta-\sin \theta\) and \(y=\operatorname{cosec}^{5} \theta-\sin ^{5} \theta, 0<\theta<\frac{\pi}{2}\), then \(\left(x^{2}+4\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{2}-25 y^{2}\) is equal to ________.

Let the function \(f: \mathrm{N} \rightarrow \mathrm{R}\) satisfy \(f(x+y)=f(x) \cdot f(y)\) for all \(x, y \in \mathrm{N}\) and \(f(1)=2\). If \(\sum_{\mathrm{k}=1}^{50} f(4+\mathrm{k})=\lambda\left(2^{\beta}-1\right)\), then \(\lambda+\beta\) is equal to ________.

Let the circle, \((x-\mathrm{a})^{2}+(y-\mathrm{b})^{2}=8\) and the parabola, \(y^{2}=4 x\) have a common tangent at the point \(\mathrm{P}(1,2)\). If \(\mathrm{C}\) is the centre of the circle and \(\mathrm{S}\) is the focus of the parabola, then the area (in sq. units) of \(\triangle \mathrm{CPS}\) is equal to ________.

If \(y=y(x)\) is a solution of the differential equation, \(x\frac{dy}{dx}=y(2-3xy),\ x>0\) with \(y(1)=1\), then \(y\left(\frac{1}{4}\right)\) is equal to ________.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Which one of the answer figures is the correct mirror image of the problem figure with respect to X-X ?

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Which one of the answer figures is the correct mirror image of the problem figure with respect to X-X ?

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Which one of the answer figures is the correct mirror image of the problem figure with respect to X-X ?

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Which one of the answer figures is the correct mirror image of the problem figure with respect to X-X ?

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Which one of the answer figures is the correct mirror image of the problem figure with respect to X-X ?

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Find the odd figure out of the problem figures given below.

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Find the odd figure out of the problem figures given below.

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Find the odd figure out of the problem figures given below.

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Find the odd figure out of the problem figures given below.

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Find the odd figure out of the problem figures given below.

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The problem figure shows the top view of objects. Looking in the direction of the arrow, identify the correct elevation, from amongst the answer figures.

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The problem figure shows the top view of objects. Looking in the direction of the arrow, identify the correct elevation, from amongst the answer figures.

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The problem figure shows the top view of objects. Looking in the direction of the arrow, identify the correct elevation, from amongst the answer figures.

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The problem figure shows the top view of objects. Looking in the direction of the arrow, identify the correct elevation, from amongst the answer figures.

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The problem figure shows the top view of objects. Looking in the direction of the arrow, identify the correct elevation, from amongst the answer figures.

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Which one of the answer figure will complete the sequence of the three problem figures ?

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Which one of the answer figure will complete the sequence of the three problem figures ?

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Which one of the answer figure will complete the sequence of the three problem figures ?

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Which one of the answer figure will complete the sequence of the three problem figures ?

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Which one of the answer figure will complete the sequence of the three problem figures ?

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10 Downing Street is the residence of which one of the following ?

• (1) President of America
• (2) Prime Minister of England
• (3) President of France
• (4) President of Sri Lanka

In the Northern Hemisphere the winter sun sets in which one of the following directions ?

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

The famous Badrinath temple is located in which one of the following States ?

• (1) Kerala
• (2) Tamil Nadu
• (3) Assam
• (4) Uttarakhand

What is the thickness of a one brick thick wall ?

• (1) 4.5"
• (2) 9"
• (3) 13.5"
• (4) 12"

Why is the depth of a beam normally more than its width ?

• (1) Horizontal Force
• (2) Momentum
• (3) Moment of Inertia
• (4) Balance

Where is the famous Notre Dame Cathedral which caught fire recently located ?

• (1) Paris
• (2) Hague
• (3) Venice
• (4) Rome

The concept of CharBagh garden found in which one of the following garden concepts ?

• (1) Christian
• (2) Chinese
• (3) Islamic
• (4) Zen

Where is the Bara Imambara located ?

• (1) Kolkata
• (2) Mumbai
• (3) Hyderabad
• (4) Lucknow

Which one of the colors reflects the maximum light ?

• (1) Red
• (2) Green
• (3) White
• (4) Black

Which one of the following colors denotes equality ?

• (1) White
• (2) Blue
• (3) Indigo
• (4) Red

Which one of the following describes the texture of a hand made brick ?

• (1) Smooth
• (2) Glossy
• (3) Rough
• (4) Silky

What are shafts used for in buildings ?

• (1) Beauty
• (2) Concealing pipes
• (3) Structural strength
• (4) Stability

Which one of the following is the most suitable material for toilet floors ?

• (1) Polished granite
• (2) Non skid tiles
• (3) Wood
• (4) Carpet

Who amongst the following designed Indian Institute of Management ?

• (1) Charles Correa
• (2) B.V. Doshi
• (3) Louis Kahn
• (4) A.P. Kanvinde

Who amongst the following designed The Falling Waters ?

• (1) Frank Ghery
• (2) Zaha Hadid
• (3) Norman Foster
• (4) Frank Lloyd Wright

In the space provided for the answer of this question, draw an aesthetic composition appropriate to this space using only rectangles. There is no restriction to numbers, sizes, placement and directions of these shapes. Color this composition so that it becomes visually exciting.

Draw from imagination a highway on which cars and trucks are moving in both directions.
OR
Draw from imagination an interior of cinema hall.
OR
Draw from memory your best teacher's face.