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JEE Advanced · Mathematics · Vector Algebra

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1
Let a⃗,b⃗\vec a, \vec b be two vectors, and let P,Q,RP,Q,R be the points with position vectors a⃗,b⃗,a⃗+b⃗\vec a, \vec b, \vec a+\vec b respectively, with respect to origin OO. If ∣a⃗+b⃗∣=21|\vec a+\vec b|=\sqrt{21}, ∣a⃗−b⃗∣=3|\vec a-\vec b|=3, and a⃗\vec a and (a⃗−b⃗)(\vec a-\vec b) are perpendicular, then the area of triangle OPROPR is
Not attempted
A
3\sqrt3
B
32\frac{\sqrt3}{2}
C
332\frac{3\sqrt3}{2}
D
32\frac32
2
For real numbers α,β,γ,δ,μ\alpha,\beta,\gamma,\delta,\mu, consider M=(α12−1213β13γδμ)M=\begin{pmatrix}\alpha & \frac{1}{\sqrt2} & -\frac1{\sqrt2}\\ \frac1{\sqrt3} & \beta & \frac1{\sqrt3}\\ \gamma & \delta & \mu\end{pmatrix} with MMT=IMM^T=I. Let u⃗=αi^+13j^+γk^\vec u=\alpha\hat i+\frac1{\sqrt3}\hat j+\gamma\hat k, v⃗=12i^+βj^+δk^\vec v=\frac1{\sqrt2}\hat i+\beta\hat j+\delta\hat k, w⃗=−12i^+13j^+μk^\vec w=-\frac1{\sqrt2}\hat i+\frac1{\sqrt3}\hat j+\mu\hat k. Match each entry in List-I to the correct entry in List-II. List-I\textbf{List-I} (P) The value of γ2+δ2\gamma^2+\delta^2 (Q) If xu⃗+yv⃗+zw⃗=j^x\vec u+y\vec v+z\vec w=\hat j, the value of xx (R) The value of ∣u⃗⋅(v⃗×w⃗)∣|\vec u\cdot(\vec v\times\vec w)| (S) The value of ∣u⃗×(v⃗×w⃗)∣|\vec u\times(\vec v\times\vec w)| List-II\textbf{List-II} (1) 00 (2) 11 (3) 12\frac1{\sqrt2} (4) 13\frac1{\sqrt3} (5) 56\frac56
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A
P→5,Q→4,R→2,S→1P\to5,Q\to4,R\to2,S\to1
B
P→4,Q→5,R→1,S→2P\to4,Q\to5,R\to1,S\to2
C
P→5,Q→3,R→2,S→1P\to5,Q\to3,R\to2,S\to1
D
P→5,Q→4,R→1,S→2P\to5,Q\to4,R\to1,S\to2
3
Consider the vectors x⃗=ı^+2ȷ^+3k^,y⃗=2ı^+3ȷ^+k^,z⃗=3ı^+ȷ^+2k^\vec x=\hat\imath+2\hat\jmath+3\hat k,\quad \vec y=2\hat\imath+3\hat\jmath+\hat k,\quad \vec z=3\hat\imath+\hat\jmath+2\hat k. For two distinct positive real numbers α\alpha and β\beta, define X⃗=αx⃗+βy⃗−z⃗,Y⃗=αy⃗+βz⃗−x⃗,Z⃗=αz⃗+βx⃗−y⃗.\vec X=\alpha\vec x+\beta\vec y-\vec z,\qquad \vec Y=\alpha\vec y+\beta\vec z-\vec x,\qquad \vec Z=\alpha\vec z+\beta\vec x-\vec y. If the vectors X⃗,Y⃗,Z⃗\vec X,\vec Y,\vec Z lie in a plane, then the value of α+β−3\alpha+\beta-3 is_____________.
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Your answer: —
Correct answer: -2
4
For any two points MM and NN in the XYXY-plane, let MN→\overrightarrow{MN} denote the vector from MM to NN, and 0⃗\vec{0} denote the zero vector. Let P,QP, Q and RR be three distinct points in the XYXY-plane. Let SS be a point inside the triangle △PQR\triangle PQR such that SP→+5SQ→+6SR→=0⃗\overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \vec{0}. Let EE and FF be the mid-points of the sides PRPR and QRQR, respectively. Then the value of length of the line segment EFlength of the line segment ES\dfrac{\text{length of the line segment } EF}{\text{length of the line segment } ES} is _______________.
Not attempted
Your answer: —
Correct answer: 1.2
5
Let w⃗=i^+j^−2k^\vec{w} = \hat{i}+\hat{j}-2\hat{k}, and u⃗\vec{u} and v⃗\vec{v} be two vectors, such that u⃗×v⃗=w⃗\vec u\times\vec v = \vec w and v⃗×w⃗=u⃗\vec v\times\vec w = \vec u. Let α,β,γ\alpha,\beta,\gamma, and tt be real numbers such that u⃗=αi^+βj^+γk^,−tα+β+γ=0,α−tβ+γ=0,andα+β−tγ=0.\vec u = \alpha\hat i+\beta\hat j+\gamma\hat k, \quad -t\alpha+\beta+\gamma=0, \quad \alpha-t\beta+\gamma=0, \quad \text{and} \quad \alpha+\beta-t\gamma=0. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I (P) ∣v⃗∣2|\vec v|^2 is equal to (Q) If α=3\alpha=\sqrt3, then γ2\gamma^2 is equal to (R) If α=3\alpha=\sqrt3, then (β+γ)2(\beta+\gamma)^2 is equal to (S) If α=2\alpha=\sqrt2, then t+3t+3 is equal to List-II (1) 00 (2) 11 (3) 22 (4) 33 (5) 55
Not attempted
A
(P) →\to (2) (Q) →\to (1) (R) →\to (4) (S) →\to (5)
B
(P) →\to (2) (Q) →\to (4) (R) →\to (3) (S) →\to (5)
C
(P) →\to (2) (Q) →\to (1) (R) →\to (4) (S) →\to (3)
D
(P) →\to (5) (Q) →\to (4) (R) →\to (1) (S) →\to (3)
6
Let p⃗=2i^+j^+3k^\vec p=2\hat i+\hat j+3\hat k and q⃗=i^−j^+k^\vec q=\hat i-\hat j+\hat k. If for some real numbers α,β,\alpha,\beta, and γ\gamma, we have 15i^+10j^+6k^=α(2p⃗+q⃗)+β(p⃗−2q⃗)+γ(p⃗×q⃗)15\hat i+10\hat j+6\hat k=\alpha(2\vec p+\vec q)+\beta(\vec p-2\vec q)+\gamma(\vec p\times\vec q), then the value of γ\gamma is _______.
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Your answer: —
Correct answer: 2
7
Let OP→=α−1αi^+j^+k^\overrightarrow{OP}=\dfrac{\alpha-1}{\alpha}\hat i+\hat j+\hat k, OQ→=i^+β−1βj^+k^\overrightarrow{OQ}=\hat i+\dfrac{\beta-1}{\beta}\hat j+\hat k and OR→=i^+j^+12k^\overrightarrow{OR}=\hat i+\hat j+\dfrac12\hat k be three vectors, where α,β∈R−{0}\alpha,\beta\in\mathbb R-\{0\} and OO denotes the origin. If (OP→×OQ→)⋅OR→=0\left(\overrightarrow{OP}\times\overrightarrow{OQ}\right)\cdot\overrightarrow{OR}=0 and the point (α,β,2)(\alpha,\beta,2) lies on the plane 3x+3y−z+l=03x+3y-z+l=0, then the value of ll is ________.
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Your answer: —
Correct answer: 5
8
Let the position vectors of the points P,Q,RP,Q,R and SS be a⃗=i^+2j^−5k^\vec a=\hat i+2\hat j-5\hat k, b⃗=3i^+6j^+3k^\vec b=3\hat i+6\hat j+3\hat k, c⃗=175i^+165j^+7k^\vec c=\dfrac{17}{5}\hat i+\dfrac{16}{5}\hat j+7\hat k and d⃗=2i^+j^+k^\vec d=2\hat i+\hat j+\hat k, respectively. Then which of the following statements is true?
Not attempted
A
The points P,Q,RP,Q,R and SS are NOT coplanar
B
b⃗+2d⃗3\dfrac{\vec b+2\vec d}{3} is the position vector of a point which divides PRPR internally in the ratio 5:45:4
C
b⃗+2d⃗3\dfrac{\vec b+2\vec d}{3} is the position vector of a point which divides PRPR externally in the ratio 5:45:4
D
The square of the magnitude of the vector b⃗×d⃗\vec b\times\vec d is 9595
9
Let PP be the plane 3x+2y+3z=16\sqrt3x+2y+3z=16 and let S={αi^+βj^+γk^:α2+β2+γ2=1 and the distance of (α,β,γ) from the plane P is 72}S=\left\{\alpha\hat i+\beta\hat j+\gamma\hat k : \alpha^2+\beta^2+\gamma^2=1 \text{ and the distance of } (\alpha,\beta,\gamma) \text{ from the plane } P \text{ is } \dfrac72\right\}. Let u⃗,v⃗\vec u,\vec v and w⃗\vec w be three distinct vectors in SS such that ∣u⃗−v⃗∣=∣v⃗−w⃗∣=∣w⃗−u⃗∣|\vec u-\vec v|=|\vec v-\vec w|=|\vec w-\vec u|. Let VV be the volume of the parallelepiped determined by vectors u⃗,v⃗\vec u,\vec v and w⃗\vec w. Then the value of 803V\dfrac{80}{\sqrt3}V is
Not attempted
Your answer: —
Correct answer: 45
10
Let ı^,ȷ^\hat\imath,\hat\jmath and k^\hat k be the unit vectors along the three positive coordinate axes. Let a⃗=3ı^+ȷ^−k^\vec a=3\hat\imath+\hat\jmath-\hat k, b⃗=ı^+b2ȷ^+b3k^,b2,b3∈R\vec b=\hat\imath+b_2\hat\jmath+b_3\hat k,\quad b_2,b_3\in\mathbb{R}, c⃗=c1ı^+c2ȷ^+c3k^,c1,c2,c3∈R\vec c=c_1\hat\imath+c_2\hat\jmath+c_3\hat k,\quad c_1,c_2,c_3\in\mathbb{R} be three vectors such that b2b3>0b_2b_3>0, a⃗⋅b⃗=0\vec a\cdot\vec b=0 and (0−c3c2c30−c1−c2c10)(1b2b3)=(3−c11−c2−1−c3)\begin{pmatrix}0 & -c_3 & c_2\\ c_3 & 0 & -c_1\\ -c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix}1\\ b_2\\ b_3\end{pmatrix}=\begin{pmatrix}3-c_1\\ 1-c_2\\ -1-c_3\end{pmatrix}. Then, which of the following is/are TRUE?
Not attempted
A
a⃗⋅c⃗=0\vec a\cdot\vec c=0
B
b⃗⋅c⃗=0\vec b\cdot\vec c=0
C
∣b⃗∣>10|\vec b|>\sqrt{10}
D
∣c⃗∣≤11|\vec c|\le\sqrt{11}
11
Let OO be the origin and OA⃗=2i^+2j^+k^\vec{OA} = 2\hat{i}+2\hat{j}+\hat{k}, OB⃗=i^−2j^+2k^\vec{OB} = \hat{i}-2\hat{j}+2\hat{k} and OC⃗=12(OB⃗−λOA⃗)\vec{OC} = \frac{1}{2}\left(\vec{OB}-\lambda\vec{OA}\right) for some λ>0\lambda>0. If ∣OB⃗×OC⃗∣=92|\vec{OB}\times\vec{OC}| = \frac{9}{2}, then which of the following statements is (are) TRUE?
Not attempted
A
Projection of OC⃗\vec{OC} on OA⃗\vec{OA} is −32-\frac{3}{2}
B
Area of the triangle OABOAB is 92\frac{9}{2}
C
Area of the triangle ABCABC is 92\frac{9}{2}
D
The acute angle between the diagonals of the parallelogram with adjacent sides OA⃗\vec{OA} and OC⃗\vec{OC} is π3\frac{\pi}{3}
12
Let u⃗,v⃗\vec{u}, \vec{v} and w⃗\vec{w} be vectors in three-dimensional space, where u⃗\vec{u} and v⃗\vec{v} are unit vectors which are not perpendicular to each other and u⃗⋅w⃗=1,v⃗⋅w⃗=1,w⃗⋅w⃗=4\vec{u}\cdot\vec{w}=1,\quad \vec{v}\cdot\vec{w}=1,\quad \vec{w}\cdot\vec{w}=4 If the volume of the parallelepiped, whose adjacent sides are represented by the vectors u⃗,v⃗\vec{u},\vec{v} and w⃗\vec{w}, is 2\sqrt2, then the value of ∣3u⃗+5v⃗∣|3\vec{u}+5\vec{v}| is ____.
Not attempted
Your answer: —
Correct answer: 7
13
Let aa and bb be positive real numbers. Suppose PQ→=ai^+bj^\overrightarrow{PQ}=a\hat i+b\hat j and PS→=ai^−bj^\overrightarrow{PS}=a\hat i - b\hat j are adjacent sides of a parallelogram PQRSPQRS. Let u⃗\vec u and v⃗\vec v be the projection vectors of w⃗=i^+j^\vec w = \hat i+\hat j along PQ→\overrightarrow{PQ} and PS→\overrightarrow{PS}, respectively. If ∣u⃗∣+∣v⃗∣=∣w⃗∣|\vec u|+|\vec v|=|\vec w| and if the area of the parallelogram PQRSPQRS is 88, then which of the following statements is/are TRUE?
Not attempted
A
a+b=4a+b=4
B
a−b=2a-b=2
C
The length of the diagonal PRPR of the parallelogram PQRSPQRS is 44
D
w⃗\vec w is an angle bisector of the vectors PQ→\overrightarrow{PQ} and PS→\overrightarrow{PS}
14
In a triangle PQRPQR, let a⃗=QR⃗\vec{a} = \vec{QR}, b⃗=RP⃗\vec{b} = \vec{RP} and c⃗=PQ⃗\vec{c} = \vec{PQ}. If ∣a⃗∣=3,∣b⃗∣=4|\vec{a}| = 3, |\vec{b}| = 4 and a⃗⋅(c⃗−b⃗)c⃗⋅(a⃗−b⃗)=∣a⃗∣∣a⃗∣+∣b⃗∣\dfrac{\vec{a}\cdot(\vec{c}-\vec{b})}{\vec{c}\cdot(\vec{a}-\vec{b})} = \dfrac{|\vec{a}|}{|\vec{a}|+|\vec{b}|}, then the value of ∣a⃗×b⃗∣2|\vec{a}\times\vec{b}|^2 is
Not attempted
Your answer: —
Correct answer: 108
15
Let a⃗=2i^+j^−k^\vec a=2\hat i+\hat j-\hat k and b⃗=i^+2j^+k^\vec b=\hat i+2\hat j+\hat k be two vectors. Consider a vector c⃗=αa⃗+βb⃗\vec c=\alpha\vec a+\beta\vec b, α,β∈R\alpha,\beta\in\mathbb{R}. If the projection of c⃗\vec c on the vector (a⃗+b⃗)(\vec a+\vec b) is 323\sqrt2, then the minimum value of (c⃗−(a⃗×b⃗))⋅c⃗(\vec c-(\vec a\times\vec b))\cdot\vec c equals ____
Not attempted
Your answer: —
Correct answer: 18
16
Consider the cube in the first octant with sides OPOP, OQOQ and OROR of length 11, along the xx-axis, yy-axis and zz-axis respectively, where O(0,0,0)O(0,0,0) is the origin. Let S(12,12,12)S\left(\frac12,\frac12,\frac12\right) be the centre of the cube and TT be the vertex of the cube opposite to the origin OO such that SS lies on the diagonal OTOT. If p⃗=SP⃗\vec p=\vec{SP}, q⃗=SQ⃗\vec q=\vec{SQ}, r⃗=RS⃗\vec r=\vec{RS} and t⃗=ST⃗\vec t=\vec{ST}, then the value of ∣(p⃗×q⃗)×(r⃗×t⃗)∣\left|(\vec p\times\vec q)\times(\vec r\times\vec t)\right| is _____.
Not attempted
Your answer: —
Correct answer: 0.5
17
Let a⃗\vec a and b⃗\vec b be two unit vectors such that a⃗⋅b⃗=0\vec a\cdot\vec b=0. For some x,y∈Rx,y\in\mathbb{R}, let c⃗=xa⃗+yb⃗+(a⃗×b⃗)\vec c = x\vec a+y\vec b+(\vec a\times\vec b). If ∣c⃗∣=2|\vec c|=2 and the vector c⃗\vec c is inclined at the same angle α\alpha to both a⃗\vec a and b⃗\vec b, then the value of 8cos⁡2α8\cos^2\alpha is ______.
Not attempted
Your answer: —
Correct answer: 3
18
Let OO be the origin and let PQRPQR be an arbitrary triangle. The point SS is such that OP⃗⋅OQ⃗+OR⃗⋅OS⃗=OR⃗⋅OP⃗+OQ⃗⋅OS⃗=OQ⃗⋅OR⃗+OP⃗⋅OS⃗\vec{OP} \cdot \vec{OQ} + \vec{OR} \cdot \vec{OS} = \vec{OR} \cdot \vec{OP} + \vec{OQ} \cdot \vec{OS} = \vec{OQ} \cdot \vec{OR} + \vec{OP} \cdot \vec{OS} Then the triangle PQRPQR has SS as its
Not attempted
A
centroid
B
circumcentre
C
incentre
D
orthocenter
19
Paragraph\textbf{Paragraph}: Let OO be the origin, and OX⃗\vec{OX}, OY⃗\vec{OY}, OZ⃗\vec{OZ} be three unit vectors in the directions of the sides QR⃗\vec{QR}, RP⃗\vec{RP}, PQ⃗\vec{PQ}, respectively, of a triangle PQRPQR. ∣OX⃗×OY⃗∣=|\vec{OX} \times \vec{OY}| =
Not attempted
A
sin⁡(P+Q)\sin(P+Q)
B
sin⁡2R\sin 2R
C
sin⁡(P+R)\sin(P+R)
D
sin⁡(Q+R)\sin(Q+R)
20
Paragraph\textbf{Paragraph}: Let OO be the origin, and OX⃗\vec{OX}, OY⃗\vec{OY}, OZ⃗\vec{OZ} be three unit vectors in the directions of the sides QR⃗\vec{QR}, RP⃗\vec{RP}, PQ⃗\vec{PQ}, respectively, of a triangle PQRPQR. If the triangle PQRPQR varies, then the minimum value of cos⁡(P+Q)+cos⁡(Q+R)+cos⁡(R+P)\cos(P+Q) + \cos(Q+R) + \cos(R+P) is
Not attempted
A
−53-\frac{5}{3}
B
−32-\frac{3}{2}
C
32\frac{3}{2}
D
53\frac{5}{3}
21
Let u^=u1i^+u2j^+u3k^\hat u=u_1\hat i+u_2\hat j+u_3\hat k be a unit vector in R3\mathbb{R}^3 and w^=16(i^+j^+2k^)\hat w=\dfrac{1}{\sqrt6}(\hat i+\hat j+2\hat k). Given that there exists a vector v⃗\vec v in R3\mathbb{R}^3 such that ∣u^×v⃗∣=1|\hat u\times\vec v|=1 and w^⋅(u^×v⃗)=1\hat w\cdot(\hat u\times\vec v)=1. Which of the following statement(s) is(are) correct?
Not attempted
A
There is exactly one choice for such v⃗\vec v
B
There are infinitely many choices for such v⃗\vec v
C
If u^\hat u lies in the xyxy-plane then ∣u1∣=∣u2∣|u_1|=|u_2|
D
If u^\hat u lies in the xzxz-plane then 2∣u1∣=∣u3∣2|u_1|=|u_3|
22
Suppose that p⃗,q⃗\vec p,\vec q and r⃗\vec r are three non-coplanar vectors in R3\mathbb R^3. Let the components of a vector s⃗\vec s along p⃗,q⃗\vec p,\vec q and r⃗\vec r be 4,34,3 and 55, respectively. If the components of this vector s⃗\vec s along (−p⃗+q⃗+r⃗)(-\vec p+\vec q+\vec r), (p⃗−q⃗+r⃗)(\vec p-\vec q+\vec r) and (−p⃗−q⃗+r⃗)(-\vec p-\vec q+\vec r) are x,yx,y and zz, respectively, then the value of 2x+y+z2x+y+z is
Not attempted
Your answer: —
Correct answer: 9
23
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.60). Column II lists the values 1,2,3,5,61,2,3,5,6; select every value that correctly matches this Column-I entry. Entry (C)\textbf{Entry (C)}: In R2\mathbb{R}^2, let 3i^+j^, i^+3j^\sqrt3\hat i+\hat j,\ \hat i+\sqrt3\hat j and βi^+(1−β)j^\beta\hat i+(1-\beta)\hat j be the position vectors of X,YX,Y and ZZ with respect to the origin OO, respectively. If the distance of ZZ from the bisector of the acute angle of OX⃗\vec{OX} with OY⃗\vec{OY} is 32\dfrac{3}{\sqrt2}, then possible value(s) of ∣β∣|\beta| is (are)
Not attempted
A
11
B
22
C
33
D
55
E
66
24
Let △PQR\triangle PQR be a triangle. Let a⃗=QR⃗, b⃗=RP⃗\vec a=\vec{QR},\ \vec b=\vec{RP} and c⃗=PQ⃗\vec c=\vec{PQ}. If ∣a⃗∣=12, ∣b⃗∣=43|\vec a|=12,\ |\vec b|=4\sqrt3 and b⃗⋅c⃗=24\vec b\cdot\vec c=24, then which of the following is (are) true?
Not attempted
A
∣c⃗∣22−∣a⃗∣=12\dfrac{|\vec c|^2}{2}-|\vec a|=12
B
∣c⃗∣22+∣a⃗∣=30\dfrac{|\vec c|^2}{2}+|\vec a|=30
C
∣a⃗×b⃗+c⃗×a⃗∣=483|\vec a\times\vec b+\vec c\times\vec a|=48\sqrt3
D
a⃗⋅b⃗=−72\vec a\cdot\vec b=-72
25
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (A)\textbf{Entry (A)}: In R2\mathbb{R}^2, if the magnitude of the projection vector of the vector αi^+βj^\alpha\hat i+\beta\hat j on 3i^+j^\sqrt3\hat i+\hat j is 3\sqrt3, and if α=2+3β\alpha=2+\sqrt3\beta, then the possible value(s) of ∣α∣|\alpha| is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
26
Match each entry of List I\textbf{List I} to its value in List II\textbf{List II}. List I\textbf{List I} P.\textbf{P.} Let y(x)=cos⁡(3cos⁡−1x)y(x)=\cos(3\cos^{-1}x), x∈[−1,1]x\in[-1,1], x≠±32x\ne\pm\dfrac{\sqrt3}{2}. Then 1y(x){(x2−1)d2y(x)dx2+xdy(x)dx}\dfrac{1}{y(x)}\left\{(x^2-1)\dfrac{d^2y(x)}{dx^2}+x\dfrac{dy(x)}{dx}\right\} equals Q.\textbf{Q.} Let A1,A2,…,AnA_1,A_2,\dots,A_n (n>2n>2) be the vertices of a regular polygon of nn sides with centre at the origin. Let a⃗k\vec{a}_k be the position vector of AkA_k. If ∣∑k=1n−1(a⃗k×a⃗k+1)∣=∣∑k=1n−1(a⃗k⋅a⃗k+1)∣\left|\displaystyle\sum_{k=1}^{n-1}(\vec a_k\times\vec a_{k+1})\right|=\left|\displaystyle\sum_{k=1}^{n-1}(\vec a_k\cdot\vec a_{k+1})\right|, the minimum value of nn is R.\textbf{R.} If the normal from the point P(h,1)P(h,1) on the ellipse x26+y23=1\dfrac{x^2}{6}+\dfrac{y^2}{3}=1 is perpendicular to the line x+y=8x+y=8, then the value of hh is S.\textbf{S.} Number of positive solutions satisfying the equation tan⁡−1 ⁣(12x+1)+tan⁡−1 ⁣(14x+1)=tan⁡−1 ⁣(2x2)\tan^{-1}\!\left(\dfrac{1}{2x+1}\right)+\tan^{-1}\!\left(\dfrac{1}{4x+1}\right)=\tan^{-1}\!\left(\dfrac{2}{x^2}\right) is List II\textbf{List II} 1.\textbf{1.} 11 2.\textbf{2.} 22 3.\textbf{3.} 88 4.\textbf{4.} 99 Which of the following is the correct combination?
Not attempted
A
P→4, Q→3, R→2, S→1P{\to}4,\ Q{\to}3,\ R{\to}2,\ S{\to}1
B
P→2, Q→4, R→3, S→1P{\to}2,\ Q{\to}4,\ R{\to}3,\ S{\to}1
C
P→4, Q→3, R→1, S→2P{\to}4,\ Q{\to}3,\ R{\to}1,\ S{\to}2
D
P→2, Q→4, R→1, S→3P{\to}2,\ Q{\to}4,\ R{\to}1,\ S{\to}3
27
Let x⃗,y⃗\vec x,\vec y and z⃗\vec z be three vectors each of magnitude 2\sqrt 2 and the angle between each pair of them is π3\frac{\pi}{3}. If a⃗\vec a is a nonzero vector perpendicular to x⃗\vec x and y⃗×z⃗\vec y\times\vec z, and b⃗\vec b is a nonzero vector perpendicular to y⃗\vec y and z⃗×x⃗\vec z\times\vec x, then:
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A
b⃗=(b⃗⋅z⃗)(z⃗−x⃗)\vec b = (\vec b\cdot\vec z)(\vec z-\vec x)
B
a⃗=(a⃗⋅y⃗)(y⃗−z⃗)\vec a = (\vec a\cdot\vec y)(\vec y-\vec z)
C
a⃗⋅b⃗=−(a⃗⋅y⃗)(b⃗⋅z⃗)\vec a\cdot\vec b = -(\vec a\cdot\vec y)(\vec b\cdot\vec z)
D
a⃗=(a⃗⋅y⃗)(z⃗−y⃗)\vec a = (\vec a\cdot\vec y)(\vec z-\vec y)
28
Let a⃗,b⃗\vec a,\vec b, and c⃗\vec c be three non-coplanar unit vectors such that the angle between every pair of them is π3\frac{\pi}{3}. If a⃗×b⃗+b⃗×c⃗=pa⃗+qb⃗+rc⃗\vec a\times\vec b+\vec b\times\vec c=p\vec a+q\vec b+r\vec c, where p,qp,q and rr are scalars, then the value of p2+2q2+r2q2\dfrac{p^2+2q^2+r^2}{q^2} is
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Your answer: —
Correct answer: 4
29
Match List I with List II and select the correct answer using the code given below the lists: List I: P. Volume of the parallelepiped determined by vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c is 22. Then the volume of the parallelepiped determined by vectors 2(a⃗×b⃗), 3(b⃗×c⃗), (c⃗×a⃗)2(\vec a\times\vec b),\ 3(\vec b\times\vec c),\ (\vec c\times\vec a) is Q. Volume of the parallelepiped determined by vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c is 55. Then the volume of the parallelepiped determined by vectors 3(a⃗+b⃗), (b⃗+c⃗), 2(c⃗+a⃗)3(\vec a+\vec b),\ (\vec b+\vec c),\ 2(\vec c+\vec a) is R. Area of a triangle with adjacent sides determined by vectors a⃗\vec a and b⃗\vec b is 2020. Then the area of the triangle with adjacent sides determined by vectors (2a⃗+3b⃗)(2\vec a+3\vec b) and (a⃗−b⃗)(\vec a-\vec b) is S. Area of a parallelogram with adjacent sides determined by vectors a⃗\vec a and b⃗\vec b is 3030. Then the area of the parallelogram with adjacent sides determined by vectors (a⃗+b⃗)(\vec a+\vec b) and a⃗\vec a is List II: 1. 100100 2. 3030 3. 2424 4. 6060 Codes:
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A
(A) P→4, Q→2, R→3, S→1P\to4,\ Q\to2,\ R\to3,\ S\to1
B
(B) P→2, Q→3, R→1, S→4P\to2,\ Q\to3,\ R\to1,\ S\to4
C
(C) P→3, Q→4, R→1, S→2P\to3,\ Q\to4,\ R\to1,\ S\to2
D
(D) P→1, Q→4, R→3, S→2P\to1,\ Q\to4,\ R\to3,\ S\to2
30
Consider the set of eight vectors V={ai^+bj^+ck^:a,b,c∈{−1,1}}V=\{a\hat i+b\hat j+c\hat k : a,b,c\in\{-1,1\}\}. Three non-coplanar vectors can be chosen from VV in 2p2^p ways. Then pp is
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Your answer: —
Correct answer: 5
31
Let PR⃗=3i^+j^−2k^\vec{PR}=3\hat{i}+\hat{j}-2\hat{k} and SQ⃗=i^−3j^−4k^\vec{SQ}=\hat{i}-3\hat{j}-4\hat{k} determine diagonals of a parallelogram PQRSPQRS and PT⃗=i^+2j^+3k^\vec{PT}=\hat{i}+2\hat{j}+3\hat{k} be another vector. Then the volume of the parallelepiped determined by the vectors PT⃗,PQ⃗\vec{PT},\vec{PQ} and PS⃗\vec{PS} is
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A
5
B
20
C
10
D
30
In a parallelogram PQRSPQRS with PQ⃗=a⃗\vec{PQ}=\vec a, PS⃗=b⃗\vec{PS}=\vec b, the diagonals satisfy PR⃗=a⃗+b⃗\vec{PR}=\vec a+\vec b and SQ⃗=a⃗−b⃗\vec{SQ}=\vec a-\vec b; solve this 2×22\times2 linear system for a⃗,b⃗\vec a,\vec b before computing the scalar triple product.
Deconstruction\textbf{Deconstruction}: Given both diagonals of a parallelogram (as vectors) and a third vector, find the volume of the parallelepiped they span with the two sides. Strategy\textbf{Strategy}: Express the parallelogram's sides PQ⃗\vec{PQ} and PS⃗\vec{PS} in terms of the given diagonals, then compute the absolute value of the scalar triple product PT⃗⋅(PQ⃗×PS⃗)\vec{PT}\cdot(\vec{PQ}\times\vec{PS}). Execution\textbf{Execution}: Let a⃗=PQ⃗\vec a=\vec{PQ}, b⃗=PS⃗\vec b=\vec{PS}. Since PQRSPQRS is a parallelogram, PR⃗=a⃗+b⃗\vec{PR}=\vec a+\vec b and SQ⃗=a⃗−b⃗\vec{SQ}=\vec a-\vec b. - a⃗=PR⃗+SQ⃗2=(3+1, 1−3, −2−4)2=(2,−1,−3)\vec a=\dfrac{\vec{PR}+\vec{SQ}}{2}=\dfrac{(3+1,\,1-3,\,-2-4)}{2}=(2,-1,-3) - b⃗=PR⃗−SQ⃗2=(3−1, 1+3, −2+4)2=(1,2,1)\vec b=\dfrac{\vec{PR}-\vec{SQ}}{2}=\dfrac{(3-1,\,1+3,\,-2+4)}{2}=(1,2,1) - PQ⃗×PS⃗=∣i^j^k^2−1−3121∣=i^(−1−(−6))−j^(2−(−3))+k^(4−(−1))=(5,−5,5)\vec{PQ}\times\vec{PS}=\begin{vmatrix}\hat i&\hat j&\hat k\\2&-1&-3\\1&2&1\end{vmatrix}=\hat i(-1-(-6))-\hat j(2-(-3))+\hat k(4-(-1))=(5,-5,5) - PT⃗⋅(5,−5,5)=1(5)+2(−5)+3(5)=5−10+15=10\vec{PT}\cdot(5,-5,5)=1(5)+2(-5)+3(5)=5-10+15=10 - Volume =∣10∣=10=|10|=10. Verification\textbf{Verification}: A widely circulated third-party solution reaches PS⃗=i^−2j^+k^\vec{PS}=\hat i-2\hat j+\hat k instead of the correct i^+2j^+k^\hat i+2\hat j+\hat k — an arithmetic sign slip in computing 1−(−3)2\dfrac{1-(-3)}{2} (which equals +2+2, not −2-2) — and this single sign error propagates to a determinant of −30-30 instead of the correct 1010. Recomputing PR⃗−SQ⃗2\dfrac{\vec{PR}-\vec{SQ}}{2} carefully confirms the jj-component is 1−(−3)2=2\dfrac{1-(-3)}{2}=2 (positive), so PS⃗=(1,2,1)\vec{PS}=(1,2,1) is correct, and the resulting volume is 1010, matching the exam's own official answer key. Final Answer\textbf{Final Answer}: (C) 10