JEE Main 2014 BArch and BPlanning Question Paper with Solutions PDF April 6 Set K

Let \(\mathrm{N}\) be the set of natural numbers and for \(\mathrm{a} \in \mathrm{N}\), \(\mathrm{aN}\) denotes the set \(\{\mathrm{a}x : x \in \mathrm{N}\}\).
If \(\mathrm{bN} \cap \mathrm{cN} = \mathrm{dN}\), where \(\mathrm{b}, \mathrm{c}, \mathrm{d}\) are natural numbers greater than 1 and the greatest common divisor of \(\mathrm{b}\) and \(\mathrm{c}\) is 1, then \(\mathrm{d}\) equals :

• (1) \(\max\{\mathrm{b}, \mathrm{c}\}\)
• (2) \(\min\{\mathrm{b}, \mathrm{c}\}\)
• (3) \(\mathrm{bc}\)
• (4) \(\mathrm{b}+\mathrm{c}\)

If \(z\) is a complex number of unit modulus and argument \(\theta\), then the real part of \(\dfrac{z(1-\bar{z})}{\bar{z}(1+z)}\) is :

• (1) \(1+\cos\dfrac{\theta}{2}\)
• (2) \(1-\sin\dfrac{\theta}{2}\)
• (3) \(-2\sin^{2}\dfrac{\theta}{2}\)
• (4) \(2\cos^{2}\dfrac{\theta}{2}\)

If the roots of the equation
\(\dfrac{1}{x+\mathrm{p}}+\dfrac{1}{x+\mathrm{q}}=\dfrac{1}{\mathrm{r}}\) are equal in magnitude and opposite in sign, then the product of roots is :

• (1) \((\mathrm{p}^{2}+\mathrm{q}^{2})\)
• (2) \(\dfrac{1}{2}(\mathrm{p}^{2}+\mathrm{q}^{2})\)
• (3) \(-\dfrac{1}{2}(\mathrm{p}^{2}+\mathrm{q}^{2})\)
• (4) \(-\dfrac{1}{2}(\mathrm{p}^{2}-\mathrm{q}^{2})\)

If \(S_{k}=\begin{pmatrix}1 & k \\ 0 & 1\end{pmatrix}\), \(k \in \mathrm{N}\), where \(\mathrm{N}\) is the set of natural numbers, then \((\mathrm{S}_{2})^{\mathrm{n}}(\mathrm{S}_{\mathrm{k}})^{-1}\), for \(\mathrm{n} \in \mathrm{N}\), is :

• (1) \(S_{2n+k-1}\)
• (2) \(S_{2^{n}+k-1}\)
• (3) \(S_{2^{n}-k}\)
• (4) \(S_{2n-k}\)

In a \(\Delta \mathrm{ABC}\), if \(\begin{vmatrix}1 & a & b \\ 1 & c & a \\ 1 & b & c\end{vmatrix}=0\), then \(\sin ^{2} \mathrm{A}+\sin ^{2} \mathrm{B}+\sin ^{2} \mathrm{C}\) is :

• (1) \(\dfrac{9}{4}\)
• (2) \(\dfrac{5}{4}\)
• (3) 2
• (4) \(\dfrac{3 \sqrt{3}}{2}\)

Suppose that six students, including Madhu and Puja, are having six beds arranged in a row. Further, suppose that Madhu does not want a bed adjacent to Puja. Then the number of ways, the beds can be allotted to students is :

• (1) 264
• (2) 480
• (3) 600
• (4) 384

Sum of the last 30 coefficients of powers of \(x\) in the binomial expansion of \((1+x)^{59}\) is :

• (1) \(2^{29}\)
• (2) \(2^{28}\)
• (3) \(2^{59}-2^{29}\)
• (4) \(2^{58}\)

If
\(\dfrac{48}{2.3}+\dfrac{47}{3.4}+\dfrac{46}{4.5}+\cdots+\dfrac{2}{48.49}+\dfrac{1}{49.50}\)
\(=\dfrac{51}{2}+\mathrm{K}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{50}\right)\), then
\(\mathrm{K}\) equals :

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

\(\log_{10} 2\), \(\log_{10}\left(2^{x}-1\right)\) and \(\log_{10}\left(2^{x}+3\right)\) are three consecutive terms of an A. P. for :

• (1) no real \(x\)
• (2) exactly one real \(x\)
• (3) exactly two real \(x\)
• (4) more than two real \(x\).

Let \(f(x)=(x+1)^{2}-1\), \(x \geq -1\), then the set \(\left\{x : f(x)=f^{-1}(x)\right\}\) :

• (1) is an empty set
• (2) contains exactly one element
• (3) contains exactly two elements
• (4) contains more than two elements.

\(f(x)=\left|x \log_{e} x\right|\), \(x>0\), is monotonically decreasing in :

• (1) \(\left(0, \frac{1}{e}\right)\)
• (2) \(\left[\frac{1}{e}, 1\right]\)
• (3) \((1, e)\)
• (4) \((e, \infty)\)

Let \(f(x)=\begin{cases}\frac{1}{|x|} & \text{if } |x|>2 \\ a+b x^{2} & \text{if } |x| \leq 2\end{cases}\),
then \(f(x)\) is differentiable at \(x=-2\) for :

• (1) \(a=\frac{3}{4}\) and \(b=-\frac{1}{16}\)
• (2) \(a=-\frac{1}{4}\) and \(b=\frac{1}{16}\)
• (3) \(a=\frac{1}{4}\) and \(b=-\frac{1}{16}\)
• (4) \(a=\frac{3}{4}\) and \(b=\frac{1}{16}\)

If \(f(x)=(x-p)(x-q)(x-r)\), where \(p<q<r\), are real numbers, then the application of Rolle's theorem on \(f\) leads to :

• (1) \((p+q+r)^{2}=3(q r+r p+p q)\)
• (2) \((p+q+r)^{2}>3(q r+r p+p q)\)
• (3) \((p+q+r)^{2}<3(q r+r p+p q)\)
• (4) \((p+q+r)^{2}(q r+r p+p q)=3\)

Let \(f(x)=\left|x-x_{1}\right|+\left|x-x_{2}\right|\), where \(x_{1}\) and \(x_{2}\) are distinct real numbers. Then the number of points at which \(f(x)\) is minimum, is :

• (1) 1
• (2) 2
• (3) 3
• (4) more than 3

If \(\int \frac{1-5 \sin ^{2} x}{\cos ^{5} x \sin ^{2} x}\, d x=\frac{f(x)}{\cos ^{5} x}+C\), then \(f(x)\) is :

• (1) \(-\operatorname{cosec} x\)
• (2) \(\operatorname{cosec} x\)
• (3) \(\cot x\)
• (4) \(-\cot x\)

If \(f(x)=\frac{e^{x}}{1+e^{x}}\),
\(I_{1}=\int_{f(-a)}^{f(a)} x\, g\{x(1-x)\}\, d x\) and
\(I_{2}=\int_{f(-a)}^{f(a)} g\{x(1-x)\}\, d x\),
where \(g\) is not an identity function. Then the value of \(\frac{I_{2}}{I_{1}}\) is :

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

The area bounded by the curves \(y^{2}=12x\) and \(x^{2}=12y\) is divided by the line \(x=3\) in two parts. The area (in square units) of the larger part is :

• (1) \(\frac{147}{4}\)
• (2) \(\frac{45}{4}\)
• (3) \(\frac{137}{4}\)
• (4) \(\frac{245}{4}\)

Consider the differential equation, \(y\,dx-\left(x+y^{2}\right)dy=0\). If for \(y=1\), \(x\) takes value 1, then value of \(x\) when \(y=4\) is :

• (1) \(9\)
• (2) \(16\)
• (3) \(36\)
• (4) \(64\)

The locus of the mid points of the chords of the parabola \(x^{2}=4py\) having slope \(m\) is a :

• (1) line parallel to \(x\)-axis at a distance \(|2pm|\) from it.
• (2) line parallel to \(y\)-axis at a distance \(|2pm|\) from it.
• (3) line parallel to \(y=mx\), \(m\neq 0\) at a distance \(|2pm|\) from it.
• (4) circle with centre at origin and radius \(|2pm|\).

If the point \((p,5)\) lies on the line parallel to the \(y\)-axis and passing through the intersection of the lines \(2\left(a^{2}+1\right)x+by+4\left(a^{3}+a\right)=0\) and \(\left(a^{2}+1\right)x-3by+2\left(a^{3}+a\right)=0\), then \(p\) is equal to :

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

If a circle has two of its diameters along the lines \(x+y=5\) and \(x-y=1\) and has area \(9\pi\), then the equation of the circle is :

• (1) \(x^{2}+y^{2}-6x-4y-3=0\)
• (2) \(x^{2}+y^{2}-6x-4y-4=0\)
• (3) \(x^{2}+y^{2}-6x-4y+3=0\)
• (4) \(x^{2}+y^{2}-6x-4y+4=0\)

Let \(\mathrm{P}\) be a point in the first quadrant lying on the ellipse \(9x^{2}+16y^{2}=144\), such that the tangent at \(\mathrm{P}\) to the ellipse is inclined at an angle \(135^{\circ}\) to the positive direction of \(x\)-axis. Then the coordinates of \(\mathrm{P}\) are :

• (1) \(\left(\frac{\sqrt{143}}{3},\frac{1}{4}\right)\)
• (2) \(\left(\frac{8}{9},\frac{\sqrt{77}}{3}\right)\)
• (3) \(\left(\frac{4}{\sqrt{2}},\frac{3}{\sqrt{2}}\right)\)
• (4) \(\left(\frac{16}{5},\frac{9}{5}\right)\)

A variable plane is at a constant distance \(p\) from the origin \(\mathrm{O}\) and meets the set of rectangular axes \(\mathrm{OX}_{i}\) \((i=1,2,3)\) at points \(\mathrm{A}_{i}\) \((i=1,2,3)\), respectively. If planes are drawn through \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), \(\mathrm{A}_{3}\), which are parallel to the coordinate planes, then the locus of their point of intersection is :

• (1) \(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}=\frac{1}{p}\)
• (2) \(\frac{1}{x_{1}^{2}}+\frac{1}{x_{2}^{2}}+\frac{1}{x_{3}^{2}}=\frac{1}{p^{2}}\)
• (3) \(\frac{1}{x_{1}^{3}}+\frac{1}{x_{2}^{3}}+\frac{1}{x_{3}^{3}}=\frac{1}{p^{3}}\)
• (4) \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=p^{2}\)

If the lines \(\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-\lambda}{3}\) and \(\frac{x}{1}=\frac{y+2}{2}=\frac{z}{4}\) intersect each other, then \(\lambda\) lies in the interval :

• (1) \((-5,-3)\)
• (2) \((13,15)\)
• (3) \((11,13)\)
• (4) \((9,11)\)

Unit vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are coplanar. A unit vector \(\vec{d}\) is perpendicular to them. If \((\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})=\frac{1}{6} \hat{i}-\frac{1}{3} \hat{j}+\frac{1}{3} \hat{k}\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(30^{\circ}\), then \(\vec{c}\) is/are :

• (1) \(\pm\left(\frac{-\hat{i}-2 \hat{j}+2 \hat{k}}{3}\right)\)
• (2) \(\left(\frac{2 \hat{i}+\hat{j}-\hat{k}}{3}\right)\)
• (3) \(\pm\left(\frac{-\hat{i}+2 \hat{j}-2 \hat{k}}{3}\right)\)
• (4) \(\left(\frac{-2 \hat{i}-2 \hat{j}+\hat{k}}{3}\right)\)

A biased coin with probability p, \(0<\mathrm{p}<1\), of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is \(\frac{2}{5}\), then p is equal to :

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

If the mean and the standard deviation of 10 observations \(x_{1}, x_{2}, \ldots \ldots, x_{10}\) are 2 and 3 respectively, then the mean of \(\left(x_{1}+1\right)^{2}\), \(\left(x_{2}+1\right)^{2}, \ldots \ldots,\left(x_{10}+1\right)^{2}\) is equal to :

• (1) 13.5
• (2) 14.4
• (3) 16.0
• (4) 18.0

A vertical pole stands at a point A on the boundary of a circular park of radius \(a\) and subtends an angle \(\alpha\) at another point B on the boundary. If the chord AB subtends an angle \(\alpha\) at the centre of the park, the height of the pole is :

• (1) \(2 a \sin \frac{\alpha}{2} \tan \alpha\)
• (2) \(2 a \cos \frac{\alpha}{2} \tan \alpha\)
• (3) \(2 a \sin \frac{\alpha}{2} \cot \alpha\)
• (4) \(2 a \cos \frac{\alpha}{2} \cot \alpha\)

Let \(\frac{3 \pi}{4}<\theta<\pi\) and
\(\sqrt{2 \cot \theta+\frac{1}{\sin ^{2} \theta}}=\mathrm{K}-\cot \theta\),
then \(\mathrm{K}\) is equal to :

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

Let \(p\) and \(q\) be any two propositions.
Statement 1 : \((p \rightarrow q) \leftrightarrow q \vee \sim p\) is a tautology.
Statement 2 : \(\sim(\sim p \wedge q) \wedge(p \vee q) \leftrightarrow p\) is a fallacy.

• (1) Both statement 1 and statement 2 are true.
• (2) Both statement 1 and statement 2 are false.
• (3) Statement 1 is true and statement 2 is false.
• (4) Statement 1 is false and statement 2 is true.

Directions: Find the odd figure out in the problem figures given below :

• (1)
• (2)
• (3)
• (4)

Directions: Find the odd figure out in the problem figures given below :

• (1)
• (2)
• (3)
• (4)

Directions : Find the odd figure out in the problem figures given below :

• (1)
• (2)
• (3)
• (4)

Directions : Find the odd figure out in the problem figures given below :

• (1)
• (2)
• (3)
• (4)

Directions : The 3 - D problem figure shows the view of an object. Identify, its correct top view, from amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

Directions : The 3 - D problem figure shows the view of an object. Identify, its correct top view, from amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

Directions : The 3 - D problem figure shows the view of an object. Identify, its correct top view, from amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

Directions : The problem figure shows the top view of an object. Identify the correct elevation, from amongst the answer figures, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

Directions : The problem figure shows the top view of an object. Identify the correct elevation, from amongst the answer figures, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

Directions : How many minimum number of straight lines are required to draw the problem figure ?

• (1) 11
• (2) 10
• (3) 9
• (4) 12

Directions: How many minimum number of straight lines are required to draw the problem figure ?

• (1) 18
• (2) 19
• (3) 20
• (4) 21

Directions: The 3 - D problem figure shows a view of an object. Identify the correct front view, from amongst the answer figures, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

Directions: The 3 - D problem figure shows a view of an object. Identify the correct front view, from amongst the answer figures, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

Directions: The 3 - D problem figure shows a view of an object. Identify the correct front view, from amongst the answer figures, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

Directions: The 3 - D problem figure shows a view of an object. Identify the correct front view, from amongst the answer figures, looking in the direction of arrow.

• (1)
• (2)
• (3)
• (4)

Directions: How many total number of triangles are there in the problem figure given below ?

• (1) 11
• (2) 12
• (3) 10
• (4) 9

Directions: How many total number of triangles are there in the problem figure given below ?

• (1) 12
• (2) 13
• (3) 11
• (4) 14

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

• (1)
• (2)
• (3)
• (4)

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

• (1)
• (2)
• (3)
• (4)

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

• (1)
• (2)
• (3)
• (4)

The problem figure shows the elevation of an object. Identify the correct top view from amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

The problem figure shows the elevation of an object. Identify the correct top view from amongst the answer figures.

• (1)
• (2)
• (3)
• (4)

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

• (1)
• (2)
• (3)
• (4)

Which one of the answer figures, shows the correct view of the 3 - D problem figure, after the problem figure is opened up ?

• (1)
• (2)
• (3)
• (4)

Which one of the answer figures, shows the correct view of the 3 - D problem figure, after the problem figure is opened up ?

• (1)
• (2)
• (3)
• (4)

Find out the total number of surfaces of the object, given below in the problem figure.

• (1) 14
• (2) 15
• (3) 17
• (4) 16

Directions : Find out the total number of surfaces of the object, given below in the problem figure.

• (1) 15
• (2) 16
• (3) 14
• (4) 13

Directions : Find out the total number of surfaces of the object, given below in the problem figure.

• (1) 15
• (2) 12
• (3) 13
• (4) 14

Directions : Find out the total number of surfaces of the object, given below in the problem figure.

• (1) 11
• (2) 12
• (3) 14
• (4) 15

Directions : Find out the total number of surfaces of the object, given below in the problem figure.

• (1) 15
• (2) 16
• (3) 14
• (4) 13

Directions : Find out the total number of surfaces of the object, given below in the problem figure.

• (1) 13
• (2) 12
• (3) 11
• (4) 10

Directions : One of the following answer figures is hidden in the problem figure, in the same size and direction. Select, which one is correct ?

• (1)
• (2)
• (3)
• (4)

Directions : One of the following answer figures is hidden in the problem figure, in the same size and direction. Select, which one is correct ?

• (1)
• (2)
• (3)
• (4)

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

• (1)
• (2)
• (3)
• (4)

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

• (1)
• (2)
• (3)
• (4)

Who among the following is not an architect ?

• (1) Zaha Hadid
• (2) M.F. Hussain
• (3) Hafiz Contractor
• (4) Raj Rewal

Fatehpur Sikri was built by :

• (1) Jahangir
• (2) Akbar
• (3) Humayun
• (4) Shah Jahan

Which one of the following is a sound reflecting material ?

• (1) Thermocol
• (2) Jute cloth
• (3) Glass
• (4) Fabric

Buckingham Palace is located in :

• (1) Paris
• (2) London
• (3) Geneva
• (4) Singapore

What secondary colour is obtained by mixing blue and red colours ?

• (1) Pink
• (2) Purple
• (3) Orange
• (4) Brown

Lotus Temple in Delhi was built by :

• (1) Jews
• (2) Jains
• (3) Bahais
• (4) Muslims

Which one of the following material cannot be used in its original form for construction of walls ?

• (1) Fly ash
• (2) Basalt
• (3) Laterite
• (4) Granite

Green architecture is promoted these days because :

• (1) It costs less initially
• (2) It is environment friendly
• (3) It lasts longer
• (4) Green is a good colour

What is texture ?

• (1) A solid colour
• (2) A type of shape
• (3) Lines drawn in one colour
• (4) The way a surface looks and feels

Which one of the following is not an earthquake resistant structure ?

• (1) RCC framed structure
• (2) Load bearing brick walls building
• (3) Steel framed building
• (4) Timber framed building

Which one of the following is an odd match ?

• (1) Deforestation - Climate change
• (2) Ozone layer - UV rays
• (3) Shrinking Polar Caps - Earthquake
• (4) Tsunami - Oceanic Earthquake

Why do large industrial buildings have high located glazing on the North side ?

• (1) To get bright sunlight throughout the day.
• (2) To stop the workers from looking outside.
• (3) To get uniform shadow - less light through the day.
• (4) Because the sun stays on the North side throughout the day.

Which one of the following is an odd match ?

• (1) Hot and Humid - Chennai
• (2) Cold and Dry - Ladakh
• (3) Temperate - Shimla
• (4) Hot and Dry - Jaisalmer

Which one of the following is not a matching set ?

• (1) Sundarbans - Mangroves
• (2) Varanasi - Ghats
• (3) Jaipur - Canals
• (4) Udaipur - Lakes

Which one of the following is not a matching set ?

• (1) San Francisco - Golden Gate Bridge
• (2) Washington - White House
• (3) Egypt - Mississippi River
• (4) England - 10 Downing Street

(a) Draw a rectangular frame of size 140 mm \(\times\) 210 mm in the space provided for answering this question. In this frame, compose an aesthetic 3 - dimensional composition, appropriate to the size of the frame, comprising cubes, cones and cylinders. There is no restriction of numbers, sizes, placement and direction of these solids in this frame. Colour your composition to make it visually exciting.

(b) Draw the graphic given below, such that it is rotated clockwise at an angle of 90 degrees about the point ‘A’. Care should be taken to get the same sizes of all lines in the new rotated position.

Draw from memory a balloon seller, selling balloons to a group of small children.

OR

Draw from memory a scene of a group of village women around a handpump with their pitchers/utensils for filling water.