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JEE Advanced · Mathematics · 2021

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1
For any real number xx, let [x][x] denote the largest integer less than or equal to xx. If I=∫010[10xx+1]dxI = \int_0^{10}\left[\sqrt{\frac{10x}{x+1}}\right]dx, then the value of 9I9I is ___.
Not attempted
Your answer: —
Correct answer: 182
2
Let EE be the ellipse x216+y29=1\frac{x^2}{16}+\frac{y^2}{9}=1. For any three distinct points P,QP,Q and Q′Q' on EE, let M(P,Q)M(P,Q) be the mid-point of the line segment joining PP and QQ, and M(P,Q′)M(P,Q') be the mid-point of the line segment joining PP and Q′Q'. Then the maximum possible value of the distance between M(P,Q)M(P,Q) and M(P,Q′)M(P,Q'), as P,QP,Q and Q′Q' vary on EE, is ___.
Not attempted
Your answer: —
Correct answer: 4
3
A number is chosen at random from the set {1,2,3,…,2000}\{1,2,3,\ldots,2000\}. Let pp be the probability that the chosen number is a multiple of 33 or a multiple of 77. Then the value of 500p500p is ___.
Not attempted
Your answer: —
Correct answer: 214
4
Let ψ1:[0,∞)→R\psi_1:[0,\infty)\to\mathbb{R}, ψ2:[0,∞)→R\psi_2:[0,\infty)\to\mathbb{R}, f:[0,∞)→Rf:[0,\infty)\to\mathbb{R} and g:[0,∞)→Rg:[0,\infty)\to\mathbb{R} be functions such that f(0)=g(0)=0f(0)=g(0)=0, ψ1(x)=e−x+x\psi_1(x)=e^{-x}+x, x≥0x\ge0, ψ2(x)=x2−2x−2e−x+2\psi_2(x)=x^2-2x-2e^{-x}+2, x≥0x\ge0, f(x)=∫−xx(∣t∣−t2)e−t2 dtf(x) = \int_{-x}^{x}(|t|-t^2)e^{-t^2}\,dt, x>0x>0 and g(x)=∫0x2t e−t dtg(x) = \int_0^{x^2}\sqrt{t}\,e^{-t}\,dt, x>0x>0. Which of the following statements is TRUE?
Not attempted
A
f(ln⁡3)+g(ln⁡3)=13f(\sqrt{\ln3})+g(\sqrt{\ln3}) = \frac13
B
For every x>1x>1, there exists an α∈(1,x)\alpha\in(1,x) such that ψ1(x)=1+αx\psi_1(x)=1+\alpha x
C
For every x>0x>0, there exists a β∈(0,x)\beta\in(0,x) such that ψ2(x)=2x(ψ1(β)−1)\psi_2(x)=2x(\psi_1(\beta)-1)
D
ff is an increasing function on the interval [0,32]\left[0,\frac32\right]
5
Let ψ1:[0,∞)→R\psi_1:[0,\infty)\to\mathbb{R}, ψ2:[0,∞)→R\psi_2:[0,\infty)\to\mathbb{R}, f:[0,∞)→Rf:[0,\infty)\to\mathbb{R} and g:[0,∞)→Rg:[0,\infty)\to\mathbb{R} be functions such that f(0)=g(0)=0f(0)=g(0)=0, ψ1(x)=e−x+x\psi_1(x)=e^{-x}+x, x≥0x\ge0, ψ2(x)=x2−2x−2e−x+2\psi_2(x)=x^2-2x-2e^{-x}+2, x≥0x\ge0, f(x)=∫−xx(∣t∣−t2)e−t2 dtf(x) = \int_{-x}^{x}(|t|-t^2)e^{-t^2}\,dt, x>0x>0 and g(x)=∫0x2t e−t dtg(x) = \int_0^{x^2}\sqrt{t}\,e^{-t}\,dt, x>0x>0. Which of the following statements is TRUE?
Not attempted
A
ψ1(x)≤1\psi_1(x)\le1, for all x>0x>0
B
ψ2(x)≤0\psi_2(x)\le0, for all x>0x>0
C
f(x)≥1−e−x2−23x3+25x5f(x) \ge 1-e^{-x^2}-\frac23x^3+\frac25x^5, for all x∈(0,12)x\in\left(0,\frac12\right)
D
g(x)≤23x3−25x5+17x7g(x) \le \frac23x^3-\frac25x^5+\frac17x^7, for all x∈(0,12)x\in\left(0,\frac12\right)
6
Let M={(x,y)∈R×R:x2+y2≤r2}M=\{(x,y)\in\mathbb{R}\times\mathbb{R} : x^2+y^2\le r^2\}, where r>0r>0. Consider the geometric progression an=12n−1a_n=\frac{1}{2^{n-1}}, n=1,2,3,…n=1,2,3,\ldots. Let S0=0S_0=0 and, for n≥1n\ge1, let SnS_n denote the sum of the first nn terms of this progression. For n≥1n\ge1, let CnC_n denote the circle with center (Sn−1,0)(S_{n-1},0) and radius ana_n, and DnD_n denote the circle with center (Sn−1,Sn−1)(S_{n-1},S_{n-1}) and radius ana_n. Consider MM with r=1025513r=\frac{1025}{513}. Let kk be the number of all those circles CnC_n that are inside MM. Let ll be the maximum possible number of circles among these kk circles such that no two circles intersect. Then
Not attempted
A
k+2l=22k+2l=22
B
2k+l=262k+l=26
C
2k+3l=342k+3l=34
D
3k+2l=403k+2l=40
7
Let M={(x,y)∈R×R:x2+y2≤r2}M=\{(x,y)\in\mathbb{R}\times\mathbb{R} : x^2+y^2\le r^2\}, where r>0r>0. Consider the geometric progression an=12n−1a_n=\frac{1}{2^{n-1}}, n=1,2,3,…n=1,2,3,\ldots. Let S0=0S_0=0 and, for n≥1n\ge1, let SnS_n denote the sum of the first nn terms of this progression. For n≥1n\ge1, let CnC_n denote the circle with center (Sn−1,0)(S_{n-1},0) and radius ana_n, and DnD_n denote the circle with center (Sn−1,Sn−1)(S_{n-1},S_{n-1}) and radius ana_n. Consider MM with r=(2199−1)22198r=\frac{(2^{199}-1)\sqrt2}{2^{198}}. The number of all those circles DnD_n that are inside MM is
Not attempted
A
198
B
199
C
200
D
201
8
Let gi:[π8,3π8]→Rg_i:\left[\frac{\pi}{8},\frac{3\pi}{8}\right]\to\mathbb{R}, i=1,2i=1,2, and f:[π8,3π8]→Rf:\left[\frac{\pi}{8},\frac{3\pi}{8}\right]\to\mathbb{R} be functions such that g1(x)=1g_1(x)=1, g2(x)=∣4x−π∣g_2(x)=|4x-\pi| and f(x)=sin⁡2xf(x)=\sin^2x, for all x∈[π8,3π8]x\in\left[\frac{\pi}{8},\frac{3\pi}{8}\right]. Define Si=∫π/83π/8f(x)⋅gi(x) dxS_i = \int_{\pi/8}^{3\pi/8} f(x)\cdot g_i(x)\,dx, i=1,2i=1,2. The value of 16S1π\frac{16S_1}{\pi} is ___.
Not attempted
Your answer: —
Correct answer: 2
9
Let gi:[π8,3π8]→Rg_i:\left[\frac{\pi}{8},\frac{3\pi}{8}\right]\to\mathbb{R}, i=1,2i=1,2, and f:[π8,3π8]→Rf:\left[\frac{\pi}{8},\frac{3\pi}{8}\right]\to\mathbb{R} be functions such that g1(x)=1g_1(x)=1, g2(x)=∣4x−π∣g_2(x)=|4x-\pi| and f(x)=sin⁡2xf(x)=\sin^2x, for all x∈[π8,3π8]x\in\left[\frac{\pi}{8},\frac{3\pi}{8}\right]. Define Si=∫π/83π/8f(x)⋅gi(x) dxS_i = \int_{\pi/8}^{3\pi/8} f(x)\cdot g_i(x)\,dx, i=1,2i=1,2. The value of 48S2π2\frac{48S_2}{\pi^2} is ___.
Not attempted
Your answer: —
Correct answer: 1.5
10
Let f1:(0,∞)→Rf_1:(0,\infty)\to\mathbb{R} and f2:(0,∞)→Rf_2:(0,\infty)\to\mathbb{R} be defined by f1(x)=∫0x∏j=121(t−j)j dtf_1(x) = \int_0^x \prod_{j=1}^{21}(t-j)^j\,dt, x>0x>0 and f2(x)=98(x−1)50−600(x−1)49+2450f_2(x) = 98(x-1)^{50}-600(x-1)^{49}+2450, x>0x>0, where, for any positive integer nn and real numbers a1,a2,…,ana_1,a_2,\ldots,a_n, ∏i=1nai\prod_{i=1}^n a_i denotes the product of a1,a2,…,ana_1,a_2,\ldots,a_n. Let mim_i and nin_i, respectively, denote the number of points of local minima and the number of points of local maxima of fif_i, i=1,2i=1,2, in the interval (0,∞)(0,\infty). The value of 2m1+3n1+m1n12m_1+3n_1+m_1n_1 is ___.
Not attempted
Your answer: —
Correct answer: 57
11
Let f1:(0,∞)→Rf_1:(0,\infty)\to\mathbb{R} and f2:(0,∞)→Rf_2:(0,\infty)\to\mathbb{R} be defined by f1(x)=∫0x∏j=121(t−j)j dtf_1(x) = \int_0^x \prod_{j=1}^{21}(t-j)^j\,dt, x>0x>0 and f2(x)=98(x−1)50−600(x−1)49+2450f_2(x) = 98(x-1)^{50}-600(x-1)^{49}+2450, x>0x>0, where, for any positive integer nn and real numbers a1,a2,…,ana_1,a_2,\ldots,a_n, ∏i=1nai\prod_{i=1}^n a_i denotes the product of a1,a2,…,ana_1,a_2,\ldots,a_n. Let mim_i and nin_i, respectively, denote the number of points of local minima and the number of points of local maxima of fif_i, i=1,2i=1,2, in the interval (0,∞)(0,\infty). The value of 6m2+4n2+8m2n26m_2+4n_2+8m_2n_2 is ___.
Not attempted
Your answer: —
Correct answer: 6
12
Let EE denote the parabola y2=8xy^2=8x. Let P=(−2,4)P=(-2,4), and let QQ and Q′Q' be two distinct points on EE such that the lines PQPQ and PQ′PQ' are tangents to EE. Let FF be the focus of EE. Then which of the following statements is (are) TRUE?
Not attempted
A
The triangle PFQPFQ is a right-angled triangle
B
The triangle QPQ′QPQ' is a right-angled triangle
C
The distance between PP and FF is 525\sqrt2
D
FF lies on the line joining QQ and Q′Q'
13
Consider the region R={(x,y)∈R×R:x≥0 and y2≤4−x}R=\{(x,y)\in\mathbb{R}\times\mathbb{R} : x\ge0 \text{ and } y^2\le 4-x\}. Let F\mathcal F be the family of all circles that are contained in RR and have centers on the xx-axis. Let CC be the circle that has the largest radius among the circles in F\mathcal F. Let (α,β)(\alpha,\beta) be a point where the circle CC meets the curve y2=4−xy^2=4-x. The radius of the circle CC is ___.
Not attempted
Your answer: —
Correct answer: 1.5
14
Consider the region R={(x,y)∈R×R:x≥0 and y2≤4−x}R=\{(x,y)\in\mathbb{R}\times\mathbb{R} : x\ge0 \text{ and } y^2\le 4-x\}. Let F\mathcal F be the family of all circles that are contained in RR and have centers on the xx-axis. Let CC be the circle that has the largest radius among the circles in F\mathcal F. Let (α,β)(\alpha,\beta) be a point where the circle CC meets the curve y2=4−xy^2=4-x. The value of α\alpha is ___.
Not attempted
Your answer: —
Correct answer: 2
15
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}
16
For any real numbers α\alpha and β\beta, let yα,β(x)y_{\alpha,\beta}(x), x∈Rx\in\mathbb{R}, be the solution of the differential equation dydx+αy=xeβx\frac{dy}{dx}+\alpha y = xe^{\beta x}, y(1)=1y(1)=1. Let S={yα,β(x):α,β∈R}S=\{y_{\alpha,\beta}(x) : \alpha,\beta\in\mathbb{R}\}. Then which of the following functions belong(s) to the set SS?
Not attempted
A
f(x)=x22e−x+(e−12)e−xf(x) = \frac{x^2}{2}e^{-x} + \left(e-\frac{1}{2}\right)e^{-x}
B
f(x)=−x22e−x+(e+12)e−xf(x) = -\frac{x^2}{2}e^{-x} + \left(e+\frac{1}{2}\right)e^{-x}
C
f(x)=ex2(x−12)+(e−e24)e−xf(x) = \frac{e^x}{2}\left(x-\frac{1}{2}\right) + \left(e-\frac{e^2}{4}\right)e^{-x}
D
f(x)=ex2(12−x)+(e+e24)e−xf(x) = \frac{e^x}{2}\left(\frac{1}{2}-x\right) + \left(e+\frac{e^2}{4}\right)e^{-x}
17
Let f:[−π2,π2]→Rf:\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\to\mathbb{R} be a continuous function such that f(0)=1f(0)=1 and ∫0π/3f(t) dt=0\int_0^{\pi/3} f(t)\,dt = 0. Then which of the following statements is (are) TRUE?
Not attempted
A
The equation f(x)−3cos⁡3x=0f(x) - 3\cos 3x = 0 has at least one solution in (0,π3)\left(0,\frac{\pi}{3}\right)
B
The equation f(x)−3sin⁡3x=−6πf(x) - 3\sin 3x = -\frac{6}{\pi} has at least one solution in (0,π3)\left(0,\frac{\pi}{3}\right)
C
lim⁡x→0x∫0xf(t) dt1−ex2=−1\lim_{x\to 0} \frac{x\int_0^x f(t)\,dt}{1-e^{x^2}} = -1
D
lim⁡x→0sin⁡x∫0xf(t) dtx2=−1\lim_{x\to 0} \frac{\sin x\int_0^x f(t)\,dt}{x^2} = -1
18
Consider a triangle PQRPQR having sides of lengths p,qp, q and rr opposite to the angles P,QP, Q and RR, respectively. Then which of the following statements is (are) TRUE?
Not attempted
A
cos⁡P≥1−p22qr\cos P \ge 1 - \frac{p^2}{2qr}
B
cos⁡R≥(q−rp+q)cos⁡P+(p−rp+q)cos⁡Q\cos R \ge \left(\frac{q-r}{p+q}\right)\cos P + \left(\frac{p-r}{p+q}\right)\cos Q
C
q+rp<2sin⁡Qsin⁡Rsin⁡P\frac{q+r}{p} < 2\frac{\sqrt{\sin Q \sin R}}{\sin P}
D
If p<qp<q and p<rp<r, then cos⁡Q>pr\cos Q > \frac{p}{r} and cos⁡R>pq\cos R > \frac{p}{q}
19
Let S1={(i,j,k):i,j,k∈{1,2,…,10}}S_1 = \{(i,j,k) : i,j,k \in \{1,2,\ldots,10\}\}, S2={(i,j):1≤i<j+2≤10, i,j∈{1,2,…,10}}S_2 = \{(i,j) : 1 \le i < j+2 \le 10,\ i,j \in \{1,2,\ldots,10\}\}, S3={(i,j,k,l):1≤i<j<k<l, i,j,k,l∈{1,2,…,10}}S_3 = \{(i,j,k,l) : 1 \le i < j < k < l,\ i,j,k,l \in \{1,2,\ldots,10\}\} and S4={(i,j,k,l):i,j,k,l are distinct elements of {1,2,…,10}}S_4 = \{(i,j,k,l) : i,j,k,l \text{ are distinct elements of } \{1,2,\ldots,10\}\}. If the total number of elements in the set SrS_r is nrn_r, r=1,2,3,4r=1,2,3,4, then which of the following statements is (are) TRUE?
Not attempted
A
n1=1000n_1 = 1000
B
n2=44n_2 = 44
C
n3=220n_3 = 220
D
n412=420\frac{n_4}{12} = 420
20
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
21
In a triangle ABCABC, let AB=23AB=\sqrt{23}, BC=3BC=3 and CA=4CA=4. Then the value of cot⁡A+cot⁡Ccot⁡B\frac{\cot A+\cot C}{\cot B} is ____.
Not attempted
Your answer: —
Correct answer: 2
22
For any complex number w=c+idw=c+id, let arg⁡(w)∈(−π,π]\arg(w)\in(-\pi,\pi], where i=−1i=\sqrt{-1}. Let α\alpha and β\beta be real numbers such that for all complex numbers z=x+iyz=x+iy satisfying arg⁡(z+αz+β)=π4\arg\left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}, the ordered pair (x,y)(x,y) lies on the circle x2+y2+5x−3y+4=0x^2+y^2+5x-3y+4=0 Then which of the following statements is (are) TRUE?
Not attempted
A
α=−1\alpha=-1
B
αβ=4\alpha\beta=4
C
αβ=−4\alpha\beta=-4
D
β=4\beta=4
23
For x∈Rx\in\mathbb{R}, the number of real roots of the equation 3x2−4∣x2−1∣+x−1=03x^2-4|x^2-1|+x-1=0 is ____.
Not attempted
Your answer: —
Correct answer: 4
24
For any positive integer nn, let Sn:(0,∞)→RS_n:(0,\infty)\to\mathbb{R} be defined by Sn(x)=∑k=1ncot⁡−1(1+k(k+1)x2x)S_n(x)=\sum_{k=1}^{n}\cot^{-1}\left(\frac{1+k(k+1)x^2}{x}\right), where for any x∈Rx\in\mathbb{R}, cot⁡−1(x)∈(0,π)\cot^{-1}(x)\in(0,\pi) and tan⁡−1(x)∈(−π2,π2)\tan^{-1}(x)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right). Then which of the following statements is (are) TRUE?
Not attempted
A
S10(x)=π2−tan⁡−1(1+11x210x)S_{10}(x)=\frac{\pi}{2}-\tan^{-1}\left(\frac{1+11x^2}{10x}\right), for all x>0x>0
B
lim⁡n→∞cot⁡(Sn(x))=x\lim\limits_{n\to\infty}\cot(S_n(x))=x, for all x>0x>0
C
The equation S3(x)=π4S_3(x)=\frac{\pi}{4} has a root in (0,∞)(0,\infty)
D
tan⁡(Sn(x))≤12\tan(S_n(x))\le\frac{1}{2}, for all n≥1n\ge1 and x>0x>0
25
For any 3×33\times3 matrix MM, let ∣M∣|M| denote the determinant of MM. Let II be the 3×33\times3 identity matrix. Let EE and FF be two 3×33\times3 matrices such that (I−EF)(I-EF) is invertible. If G=(I−EF)−1G=(I-EF)^{-1}, then which of the following statements is (are) TRUE?
Not attempted
A
∣FE∣=∣I−FE∣∣FGE∣|FE|=|I-FE||FGE|
B
(I−FE)(I+FGE)=I(I-FE)(I+FGE)=I
C
EFG=GEFEFG=GEF
D
(I−FE)(I−FGE)=I(I-FE)(I-FGE)=I
26
Let E,FE, F and GG be three events having probabilities P(E)=18P(E)=\frac{1}{8}, P(F)=16P(F)=\frac{1}{6} and P(G)=14P(G)=\frac{1}{4}, and let P(E∩F∩G)=110P(E\cap F\cap G)=\frac{1}{10}. For any event HH, if HcH^c denotes its complement, then which of the following statements is (are) TRUE?
Not attempted
A
P(E∩F∩Gc)≤140P(E\cap F\cap G^c)\le\frac{1}{40}
B
P(Ec∩F∩G)≤115P(E^c\cap F\cap G)\le\frac{1}{15}
C
P(E∪F∪G)≤1324P(E\cup F\cup G)\le\frac{13}{24}
D
P(Ec∩Fc∩Gc)≤512P(E^c\cap F^c\cap G^c)\le\frac{5}{12}
27
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)=x2−3x−6x2+2x+4f(x)=\frac{x^2-3x-6}{x^2+2x+4} Then which of the following statements is (are) TRUE?
Not attempted
A
ff is decreasing in the interval (−2,−1)(-2,-1)
B
ff is increasing in the interval (1,2)(1,2)
C
ff is onto
D
Range of ff is [−32,2]\left[-\frac{3}{2},2\right]
28
For any 3×33\times3 matrix MM, let ∣M∣|M| denote the determinant of MM. Let E=(12323481318)E=\begin{pmatrix}1&2&3\\2&3&4\\8&13&18\end{pmatrix}, P=(100001010)P=\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix} and F=(13281813243)F=\begin{pmatrix}1&3&2\\8&18&13\\2&4&3\end{pmatrix} If QQ is a nonsingular matrix of order 3×33\times3, then which of the following statements is (are) TRUE?
Not attempted
A
F=PEPF=PEP and P2=(100010001)P^2=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}
B
∣EQ+PFQ−1∣=∣EQ∣+∣PFQ−1∣|EQ+PFQ^{-1}|=|EQ|+|PFQ^{-1}|
C
∣(EF)3∣>∣EF∣2|(EF)^3|>|EF|^2
D
Sum of the diagonal entries of P−1EP+FP^{-1}EP+F is equal to the sum of diagonal entries of E+P−1FPE+P^{-1}FP
29
Consider the lines L1L_1 and L2L_2 defined by L1:x2+y−1=0L_1: x\sqrt{2}+y-1=0 and L2:x2−y+1=0L_2: x\sqrt{2}-y+1=0 For a fixed constant λ\lambda, let CC be the locus of a point PP such that the product of the distance of PP from L1L_1 and the distance of PP from L2L_2 is λ2\lambda^2. The line y=2x+1y=2x+1 meets CC at two points RR and SS, where the distance between RR and SS is 270\sqrt{270}. Let the perpendicular bisector of RSRS meet CC at two distinct points R′R' and S′S'. Let DD be the square of the distance between R′R' and S′S'. The value of λ2\lambda^2 is ____.
Not attempted
Your answer: —
Correct answer: 9
30
Consider the lines L1L_1 and L2L_2 defined by L1:x2+y−1=0L_1: x\sqrt{2}+y-1=0 and L2:x2−y+1=0L_2: x\sqrt{2}-y+1=0 For a fixed constant λ\lambda, let CC be the locus of a point PP such that the product of the distance of PP from L1L_1 and the distance of PP from L2L_2 is λ2\lambda^2. The line y=2x+1y=2x+1 meets CC at two points RR and SS, where the distance between RR and SS is 270\sqrt{270}. Let the perpendicular bisector of RSRS meet CC at two distinct points R′R' and S′S'. Let DD be the square of the distance between R′R' and S′S'. The value of DD is ____.
Not attempted
Your answer: —
Correct answer: 77.14
31
Let α,β\alpha, \beta and γ\gamma be real numbers such that the system of linear equations x+2y+3z=αx+2y+3z=\alpha 4x+5y+6z=β4x+5y+6z=\beta 7x+8y+9z=γ−17x+8y+9z=\gamma-1 is consistent. Let ∣M∣|M| represent the determinant of the matrix M=(α2γβ10−101)M=\begin{pmatrix}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{pmatrix} Let PP be the plane containing all those (α,β,γ)(\alpha,\beta,\gamma) for which the above system of linear equations is consistent, and DD be the square of the distance of the point (0,1,0)(0,1,0) from the plane PP. The value of DD is ____.
Not attempted
Your answer: —
Correct answer: 1.5
32
Let α,β\alpha, \beta and γ\gamma be real numbers such that the system of linear equations x+2y+3z=αx+2y+3z=\alpha 4x+5y+6z=β4x+5y+6z=\beta 7x+8y+9z=γ−17x+8y+9z=\gamma-1 is consistent. Let ∣M∣|M| represent the determinant of the matrix M=(α2γβ10−101)M=\begin{pmatrix}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{pmatrix} Let PP be the plane containing all those (α,β,γ)(\alpha,\beta,\gamma) for which the above system of linear equations is consistent, and DD be the square of the distance of the point (0,1,0)(0,1,0) from the plane PP. The value of ∣M∣|M| is ____.
Not attempted
Your answer: —
Correct answer: 1
33
Three numbers are chosen at random, one after another with replacement, from the set S={1,2,3,…,100}S=\{1,2,3,\ldots,100\}. Let p1p_1 be the probability that the maximum of the chosen numbers is at least 8181 and p2p_2 be the probability that the minimum of the chosen numbers is at most 4040. The value of 6254p1\frac{625}{4}p_1 is ____.
Not attempted
Your answer: —
Correct answer: 76.25
34
Three numbers are chosen at random, one after another with replacement, from the set S={1,2,3,…,100}S=\{1,2,3,\ldots,100\}. Let p1p_1 be the probability that the maximum of the chosen numbers is at least 8181 and p2p_2 be the probability that the minimum of the chosen numbers is at most 4040. The value of 1254p2\frac{125}{4}p_2 is ____.
Not attempted
Your answer: —
Correct answer: 24.5
35
Let θ1,θ2,…,θ10\theta_1,\theta_2,\ldots,\theta_{10} be positive valued angles (in radian) such that θ1+θ2+⋯+θ10=2π\theta_1+\theta_2+\cdots+\theta_{10}=2\pi. Define the complex numbers z1=eiθ1z_1=e^{i\theta_1}, zk=zk−1eiθkz_k=z_{k-1}e^{i\theta_k} for k=2,3,…,10k=2,3,\ldots,10, where i=−1i=\sqrt{-1}. Consider the statements PP and QQ given below: P:∣z2−z1∣+∣z3−z2∣+⋯+∣z10−z9∣+∣z1−z10∣≤2πP: |z_2-z_1|+|z_3-z_2|+\cdots+|z_{10}-z_9|+|z_1-z_{10}| \le 2\pi Q:∣z22−z12∣+∣z32−z22∣+⋯+∣z102−z92∣+∣z12−z102∣≤4πQ: |z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2| \le 4\pi Then,
Not attempted
A
PP is TRUE and QQ is FALSE
B
QQ is TRUE and PP is FALSE
C
both PP and QQ are TRUE
D
both PP and QQ are FALSE
36
Consider three sets E1={1,2,3}E_1=\{1,2,3\}, F1={1,3,4}F_1=\{1,3,4\} and G1={2,3,4,5}G_1=\{2,3,4,5\}. Two elements are chosen at random, without replacement, from the set E1E_1, and let S1S_1 denote the set of these chosen elements. Let E2=E1−S1E_2=E_1-S_1 and F2=F1∪S1F_2=F_1\cup S_1. Now two elements are chosen at random, without replacement, from the set F2F_2 and let S2S_2 denote the set of these chosen elements. Let G2=G1∪S2G_2=G_1\cup S_2. Finally, two elements are chosen at random, without replacement, from the set G2G_2 and let S3S_3 denote the set of these chosen elements. Let E3=E2∪S3E_3=E_2\cup S_3. Given that E1=E3E_1=E_3, let pp be the conditional probability of the event S1={1,2}S_1=\{1,2\}. Then the value of pp is
Not attempted
A
15\frac{1}{5}
B
35\frac{3}{5}
C
12\frac{1}{2}
D
25\frac{2}{5}
37
The area of the region {(x,y):0≤x≤94, 0≤y≤1, x≥3y, x+y≥2}\left\{(x,y): 0 \le x \le \frac{9}{4},\ 0 \le y \le 1,\ x \ge 3y,\ x+y \ge 2\right\} is
Not attempted
A
1132\frac{11}{32}
B
3596\frac{35}{96}
C
3796\frac{37}{96}
D
1332\frac{13}{32}
38
Consider a triangle Δ\Delta whose two sides lie on the xx-axis and the line x+y+1=0x+y+1=0. If the orthocenter of Δ\Delta is (1,1)(1,1), then the equation of the circle passing through the vertices of the triangle Δ\Delta is
Not attempted
A
x2+y2−3x+y=0x^2+y^2-3x+y=0
B
x2+y2+x+3y=0x^2+y^2+x+3y=0
C
x2+y2+2y−1=0x^2+y^2+2y-1=0
D
x2+y2+x+y=0x^2+y^2+x+y=0
Hint\textbf{Hint}: One vertex is the intersection of the two given lines. Use the fact that each altitude is perpendicular to the opposite side and passes through the orthocenter to pin down the other two vertices, then fit a general circle x2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0 through all three points.
Deconstruction\textbf{Deconstruction}: Two sides of the triangle lie on y=0y=0 and x+y+1=0x+y+1=0; the orthocenter is (1,1)(1,1). Find the circumcircle. Strategy\textbf{Strategy}: The vertex shared by the two given lines is fixed by their intersection. Use perpendicularity of altitudes through the orthocenter to find the other two vertices, then fit a circle through the three points. Execution\textbf{Execution}: The lines y=0y=0 and x+y+1=0x+y+1=0 meet at A=(−1,0)A=(-1,0). Let B=(b,0)B=(b,0) on the xx-axis and C=(c,−c−1)C=(c,-c-1) on x+y+1=0x+y+1=0. Since ABAB lies along y=0y=0, the altitude from CC is vertical and passes through the orthocenter (1,1)(1,1), so c=1c=1, giving C=(1,−2)C=(1,-2). Since ACAC lies along x+y+1=0x+y+1=0 (slope −1-1), the altitude from BB has slope 11 and passes through B=(b,0)B=(b,0) and (1,1)(1,1): 1−01−b=1⇒b=0\frac{1-0}{1-b}=1 \Rightarrow b=0, giving B=(0,0)B=(0,0). So the vertices are A(−1,0)A(-1,0), B(0,0)B(0,0), C(1,−2)C(1,-2). Let the circle be x2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0. Substituting BB: F=0F=0. Substituting AA: 1−D+F=0⇒D=11-D+F=0 \Rightarrow D=1. Substituting CC: 1+4+D−2E+F=0⇒5+1−2E=0⇒E=31+4+D-2E+F=0 \Rightarrow 5+1-2E=0 \Rightarrow E=3. So the circle is x2+y2+x+3y=0x^2+y^2+x+3y=0. Verification\textbf{Verification}: Side BCBC has slope −2−01−0=−2\frac{-2-0}{1-0}=-2, so the altitude from AA has slope 12\frac{1}{2}: y=12(x+1)y=\frac{1}{2}(x+1), which passes through (1,1)(1,1) exactly, confirming the vertices. Each vertex also satisfies x2+y2+x+3y=0x^2+y^2+x+3y=0: at AA, 1−1+0=01-1+0=0; at BB, 0=00=0; at CC, 1+4+1−6=01+4+1-6=0. Final Answer\textbf{Final Answer}: (B) x2+y2+x+3y=0x^2+y^2+x+3y=0.