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JEE Advanced · Mathematics · 2022

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1
For positive integer nn, define f(n)=n+16+5n−3n24n+3n2+32+n−3n28n+3n2+48−3n−3n212n+3n2+⋯+25n−7n27n2f(n)=n+\dfrac{16+5n-3n^2}{4n+3n^2}+\dfrac{32+n-3n^2}{8n+3n^2}+\dfrac{48-3n-3n^2}{12n+3n^2}+\cdots+\dfrac{25n-7n^2}{7n^2}. Then, the value of lim⁡n→∞f(n)\displaystyle\lim_{n\to\infty}f(n) is equal to
Not attempted
A
3+43log⁡e73+\dfrac43\log_e7
B
4−34log⁡e(73)4-\dfrac34\log_e\left(\dfrac73\right)
C
4−43log⁡e(73)4-\dfrac43\log_e\left(\dfrac73\right)
D
3+34log⁡e73+\dfrac34\log_e7
2
Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls. A ball is chosen randomly from Box-I; call this ball bb. If bb is red then a ball is chosen randomly from Box-II, if bb is blue then a ball is chosen randomly from Box-III, and if bb is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen balls is green' has happened, is equal to
Not attempted
A
15256\dfrac{15}{256}
B
316\dfrac{3}{16}
C
552\dfrac{5}{52}
D
18\dfrac{1}{8}
3
If M=(5232−32−12)M=\begin{pmatrix}\dfrac52 & \dfrac32\\[4pt] -\dfrac32 & -\dfrac12\end{pmatrix}, then which of the following matrices is equal to M2022M^{2022}?
Not attempted
A
(30343033−3033−3032)\begin{pmatrix}3034 & 3033\\ -3033 & -3032\end{pmatrix}
B
(3034−30333033−3032)\begin{pmatrix}3034 & -3033\\ 3033 & -3032\end{pmatrix}
C
(30333032−3032−3031)\begin{pmatrix}3033 & 3032\\ -3032 & -3031\end{pmatrix}
D
(30323031−3031−3030)\begin{pmatrix}3032 & 3031\\ -3031 & -3030\end{pmatrix}
4
Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?
Not attempted
A
2181621816
B
8553685536
C
1209612096
D
156816156816
5
For x∈Rx\in\mathbb{R}, let the function y(x)y(x) be the solution of the differential equation dydx+12y=cos⁡(π12x),y(0)=0\dfrac{dy}{dx}+12y=\cos\left(\dfrac{\pi}{12}x\right),\qquad y(0)=0. Then, which of the following statements is/are TRUE?
Not attempted
A
y(x)y(x) is an increasing function
B
y(x)y(x) is a decreasing function
C
There exists a real number β\beta such that the line y=βy=\beta intersects the curve y=y(x)y=y(x) at infinitely many points
D
y(x)y(x) is a periodic function
6
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}
7
Let GG be a circle of radius R>0R>0. Let G1,G2,…,GnG_1,G_2,\ldots,G_n be nn circles of equal radius r>0r>0. Suppose each of the nn circles G1,G2,…,GnG_1,G_2,\ldots,G_n touches the circle GG externally. Also, for i=1,2,…,n−1i=1,2,\ldots,n-1, the circle GiG_i touches Gi+1G_{i+1} externally, and GnG_n touches G1G_1 externally. Then, which of the following statements is/are TRUE?
Not attempted
A
If n=4n=4, then (2−1)r<R(\sqrt2-1)r<R
B
If n=5n=5, then r<Rr<R
C
If n=8n=8, then (2−1)r<R(\sqrt2-1)r<R
D
If n=12n=12, then 2(3+1)r>R\sqrt2(\sqrt3+1)r>R
8
Let zˉ\bar z denote the complex conjugate of a complex number zz. If zz is a non-zero complex number for which both real and imaginary parts of (zˉ)2+1z2(\bar z)^2+\dfrac{1}{z^2} are integers, then which of the following is/are possible value(s) of ∣z∣|z|?
Not attempted
A
(43+32052)1/4\left(\dfrac{43+3\sqrt{205}}{2}\right)^{1/4}
B
(7+334)1/4\left(\dfrac{7+\sqrt{33}}{4}\right)^{1/4}
C
(9+654)1/4\left(\dfrac{9+\sqrt{65}}{4}\right)^{1/4}
D
(7+136)1/4\left(\dfrac{7+\sqrt{13}}{6}\right)^{1/4}
9
Let α=∑k=1∞sin⁡2k(π6)\alpha=\displaystyle\sum_{k=1}^{\infty}\sin^{2k}\left(\dfrac{\pi}{6}\right). Let g:[0,1]→Rg:[0,1]\to\mathbb{R} be the function defined by g(x)=2αx+2α(1−x)g(x)=2^{\alpha x}+2^{\alpha(1-x)}. Then, which of the following statements is/are TRUE?
Not attempted
A
The minimum value of g(x)g(x) is 27/62^{7/6}
B
The maximum value of g(x)g(x) is 1+21/31+2^{1/3}
C
The function g(x)g(x) attains its maximum at more than one point
D
The function g(x)g(x) attains its minimum at more than one point
10
Let PQRSPQRS be a quadrilateral in a plane, where QR=1QR=1, ∠PQR=∠QRS=70∘\angle PQR=\angle QRS=70^\circ, ∠PQS=15∘\angle PQS=15^\circ and ∠PRS=40∘\angle PRS=40^\circ. If ∠RPS=θ∘\angle RPS=\theta^\circ, PQ=αPQ=\alpha and PS=βPS=\beta, then the interval(s) that contain(s) the value of 4αβsin⁡θ∘4\alpha\beta\sin\theta^\circ is/are
Not attempted
A
(0,2)(0,\sqrt2)
B
(1,2)(1,2)
C
(2,3)(\sqrt2,3)
D
(22,32)(2\sqrt2,3\sqrt2)
11
Consider the functions f,g:R→Rf,g:\mathbb{R}\to\mathbb{R} defined by f(x)=x2+512f(x)=x^2+\dfrac{5}{12} and g(x)={2(1−4∣x∣3),∣x∣≤34,0,∣x∣>34.g(x)=\begin{cases}2\left(1-\dfrac{4|x|}{3}\right), & |x|\le\dfrac34,\\[4pt] 0, & |x|>\dfrac34.\end{cases} If α\alpha is the area of the region {(x,y)∈R×R:∣x∣≤34, 0≤y≤min⁡{f(x),g(x)}}\left\{(x,y)\in\mathbb{R}\times\mathbb{R} : |x|\le\dfrac34,\ 0\le y\le\min\{f(x),g(x)\}\right\}, then the value of 9α9\alpha is _______.
Not attempted
Your answer: —
Correct answer: 6
12
Consider the hyperbola x2100−y264=1\dfrac{x^2}{100}-\dfrac{y^2}{64}=1 with foci at SS and S1S_1, where SS lies on the positive xx-axis. Let PP be a point on the hyperbola, in the first quadrant. Let ∠SPS1=α\angle SPS_1=\alpha, with α<π2\alpha<\dfrac{\pi}{2}. The straight line passing through the point SS and having the same slope as that of the tangent at PP to the hyperbola, intersects the straight line S1PS_1P at P1P_1. Let δ\delta be the distance of PP from the straight line SP1SP_1, and β=S1P\beta=S_1P. Then the greatest integer less than or equal to βδ9sin⁡α2\dfrac{\beta\delta}{9}\sin\dfrac{\alpha}{2} is _______.
Not attempted
Your answer: —
Correct answer: 7
13
Let β\beta be a real number. Consider the matrix A=(β0121−231−2)A=\begin{pmatrix}\beta & 0 & 1\\ 2 & 1 & -2\\ 3 & 1 & -2\end{pmatrix}. If A7−(β−1)A6−βA5A^7-(\beta-1)A^6-\beta A^5 is a singular matrix, then the value of 9β9\beta is _______.
Not attempted
Your answer: —
Correct answer: 3
14
If β=lim⁡x→0ex3−(1−x3)1/3+((1−x2)1/2−1)sin⁡xxsin⁡2x\beta = \displaystyle\lim_{x\to0} \dfrac{e^{x^3}-(1-x^3)^{1/3}+\left((1-x^2)^{1/2}-1\right)\sin x}{x\sin^2 x}, then the value of 6β6\beta is _______.
Not attempted
Your answer: —
Correct answer: 5
15
The product of all positive real values of xx satisfying the equation x(16(log⁡5x)3−68log⁡5x)=5−16x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16} is _______.
Not attempted
Your answer: —
Correct answer: 1
16
The greatest integer less than or equal to ∫12log⁡2(x3+1) dx  +  ∫1log⁡29(2x−1)1/3 dx\displaystyle\int_1^2 \log_2(x^3+1)\,dx \;+\; \int_1^{\log_2 9} (2^x-1)^{1/3}\,dx is _______.
Not attempted
Your answer: —
Correct answer: 5
17
If y(x)y(x) is the solution of the differential equation x dy−(y2−4y) dx=0x\,dy - (y^2-4y)\,dx = 0 for x>0x>0, y(1)=2\quad y(1)=2, and the slope of the curve y=y(x)y=y(x) is never zero, then the value of 10 y(2)10\,y(\sqrt2) is _______.
Not attempted
Your answer: —
Correct answer: 8
18
Let α\alpha and β\beta be real numbers such that −π4<β<0<α<π4-\dfrac{\pi}{4} < \beta < 0 < \alpha < \dfrac{\pi}{4}. If sin⁡(α+β)=13\sin(\alpha+\beta) = \dfrac{1}{3} and cos⁡(α−β)=23\cos(\alpha-\beta) = \dfrac{2}{3}, then the greatest integer less than or equal to (sin⁡αcos⁡β+cos⁡βsin⁡α+cos⁡αsin⁡β+sin⁡βcos⁡α)2\left(\dfrac{\sin\alpha}{\cos\beta} + \dfrac{\cos\beta}{\sin\alpha} + \dfrac{\cos\alpha}{\sin\beta} + \dfrac{\sin\beta}{\cos\alpha}\right)^2 is _______.
Not attempted
Your answer: —
Correct answer: 1
19
Consider the ellipse x24+y23=1\dfrac{x^2}4+\dfrac{y^2}3=1. Let H(α,0)H(\alpha,0), 0<α<20<\alpha<2, be a point. A straight line drawn through HH parallel to the yy-axis crosses the ellipse and its auxiliary circle at points EE and FF respectively, in the first quadrant. The tangent to the ellipse at the point EE intersects the positive xx-axis at a point GG. Suppose the straight line joining FF and the origin makes an angle ϕ\phi with the positive xx-axis. List-I (I) If ϕ=π4\phi=\dfrac\pi4, then the area of the triangle FGHFGH is (II) If ϕ=π3\phi=\dfrac\pi3, then the area of the triangle FGHFGH is (III) If ϕ=π6\phi=\dfrac\pi6, then the area of the triangle FGHFGH is (IV) If ϕ=π12\phi=\dfrac\pi{12}, then the area of the triangle FGHFGH is List-II (P) (3−1)48\dfrac{(\sqrt3-1)^4}8 (Q) 11 (R) 34\dfrac34 (S) 123\dfrac1{2\sqrt3} (T) 332\dfrac{3\sqrt3}2 The correct option is:
Not attempted
A
(I)→(R); (II)→(S); (III)→(Q); (IV)→(P)(I)\to(R);\ (II)\to(S);\ (III)\to(Q);\ (IV)\to(P)
B
(I)→(R); (II)→(T); (III)→(S); (IV)→(P)(I)\to(R);\ (II)\to(T);\ (III)\to(S);\ (IV)\to(P)
C
(I)→(Q); (II)→(T); (III)→(S); (IV)→(P)(I)\to(Q);\ (II)\to(T);\ (III)\to(S);\ (IV)\to(P)
D
(I)→(Q); (II)→(S); (III)→(Q); (IV)→(P)(I)\to(Q);\ (II)\to(S);\ (III)\to(Q);\ (IV)\to(P)
20
Let p,q,rp,q,r be nonzero real numbers that are, respectively, the 10th10^{th}, 100th100^{th} and 1000th1000^{th} terms of a harmonic progression. Consider the system of linear equations x+y+z=1x+y+z=1 10x+100y+1000z=010x+100y+1000z=0 qr x+pr y+pq z=0qr\,x+pr\,y+pq\,z=0 List-I (I) If qr=10\dfrac qr=10, then the system of linear equations has (II) If pr≠100\dfrac pr\ne100, then the system of linear equations has (III) If pq≠10\dfrac pq\ne10, then the system of linear equations has (IV) If pq=10\dfrac pq=10, then the system of linear equations has List-II (P) x=0, y=109, z=−19x=0,\ y=\dfrac{10}9,\ z=-\dfrac19 as a solution (Q) x=109, y=−19, z=0x=\dfrac{10}9,\ y=-\dfrac19,\ z=0 as a solution (R) infinitely many solutions (S) no solution (T) at least one solution The correct option is:
Not attempted
A
(I)→(T); (II)→(R); (III)→(S); (IV)→(T)(I)\to(T);\ (II)\to(R);\ (III)\to(S);\ (IV)\to(T)
B
(I)→(Q); (II)→(S); (III)→(S); (IV)→(R)(I)\to(Q);\ (II)\to(S);\ (III)\to(S);\ (IV)\to(R)
C
(I)→(Q); (II)→(R); (III)→(P); (IV)→(R)(I)\to(Q);\ (II)\to(R);\ (III)\to(P);\ (IV)\to(R)
D
(I)→(T); (II)→(S); (III)→(P); (IV)→(T)(I)\to(T);\ (II)\to(S);\ (III)\to(P);\ (IV)\to(T)
21
Two players, P1P_1 and P2P_2, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let xx and yy denote the readings on the die rolled by P1P_1 and P2P_2, respectively. If x>yx>y, then P1P_1 scores 55 points and P2P_2 scores 00 point. If x=yx=y, then each player scores 22 points. If x<yx<y, then P1P_1 scores 00 point and P2P_2 scores 55 points. Let XiX_i and YiY_i be the total scores of P1P_1 and P2P_2, respectively, after playing the ithi^{th} round. List-I (I) Probability of (X2≥Y2)(X_2\ge Y_2) is (II) Probability of (X2>Y2)(X_2>Y_2) is (III) Probability of (X3=Y3)(X_3=Y_3) is (IV) Probability of (X3>Y3)(X_3>Y_3) is List-II (P) 38\dfrac38 (Q) 1116\dfrac{11}{16} (R) 516\dfrac5{16} (S) 355864\dfrac{355}{864} (T) 77432\dfrac{77}{432} The correct option is:
Not attempted
A
(I)→(Q); (II)→(R); (III)→(T); (IV)→(S)(I)\to(Q);\ (II)\to(R);\ (III)\to(T);\ (IV)\to(S)
B
(I)→(Q); (II)→(R); (III)→(T); (IV)→(T)(I)\to(Q);\ (II)\to(R);\ (III)\to(T);\ (IV)\to(T)
C
(I)→(P); (II)→(R); (III)→(Q); (IV)→(S)(I)\to(P);\ (II)\to(R);\ (III)\to(Q);\ (IV)\to(S)
D
(I)→(P); (II)→(R); (III)→(Q); (IV)→(T)(I)\to(P);\ (II)\to(R);\ (III)\to(Q);\ (IV)\to(T)
22
Consider the following lists: List-I (I) {x∈[−2π3,2π3]:cos⁡x+sin⁡x=1}\left\{x\in\left[-\dfrac{2\pi}{3},\dfrac{2\pi}{3}\right]: \cos x+\sin x=1\right\} (II) {x∈[−5π18,5π18]:3tan⁡3x=1}\left\{x\in\left[-\dfrac{5\pi}{18},\dfrac{5\pi}{18}\right]: \sqrt3\tan3x=1\right\} (III) {x∈[−6π5,6π5]:2cos⁡(2x)=3}\left\{x\in\left[-\dfrac{6\pi}{5},\dfrac{6\pi}{5}\right]: 2\cos(2x)=\sqrt3\right\} (IV) {x∈[−7π4,7π4]:sin⁡x−cos⁡x=1}\left\{x\in\left[-\dfrac{7\pi}{4},\dfrac{7\pi}{4}\right]: \sin x-\cos x=1\right\} List-II (P) has two elements (Q) has three elements (R) has four elements (S) has five elements (T) has six elements The correct option is:
Not attempted
A
(I)→(P); (II)→(S); (III)→(P); (IV)→(S)(I)\to(P);\ (II)\to(S);\ (III)\to(P);\ (IV)\to(S)
B
(I)→(P); (II)→(P); (III)→(T); (IV)→(R)(I)\to(P);\ (II)\to(P);\ (III)\to(T);\ (IV)\to(R)
C
(I)→(Q); (II)→(P); (III)→(T); (IV)→(S)(I)\to(Q);\ (II)\to(P);\ (III)\to(T);\ (IV)\to(S)
D
(I)→(Q); (II)→(S); (III)→(P); (IV)→(R)(I)\to(Q);\ (II)\to(S);\ (III)\to(P);\ (IV)\to(R)
23
Let ∣M∣|M| denote the determinant of a square matrix MM. Let g:[0,π2]→Rg:\left[0,\dfrac{\pi}{2}\right]\to\mathbb{R} be the function defined by g(θ)=f(θ)−1+f(π2−θ)−1g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\dfrac{\pi}{2}-\theta\right)-1} where f(θ)=12∣1sin⁡θ1−sin⁡θ1sin⁡θ−1−sin⁡θ1∣+∣sin⁡πcos⁡(θ+π4)tan⁡(θ−π4)sin⁡(θ−π4)−cos⁡π2log⁡e(4π)cot⁡(θ+π4)log⁡e(π4)tan⁡π∣f(\theta)=\dfrac12\begin{vmatrix}1 & \sin\theta & 1\\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1\end{vmatrix} + \begin{vmatrix}\sin\pi & \cos\left(\theta+\dfrac{\pi}{4}\right) & \tan\left(\theta-\dfrac{\pi}{4}\right)\\ \sin\left(\theta-\dfrac{\pi}{4}\right) & -\cos\dfrac{\pi}{2} & \log_e\left(\dfrac{4}{\pi}\right)\\ \cot\left(\theta+\dfrac{\pi}{4}\right) & \log_e\left(\dfrac{\pi}{4}\right) & \tan\pi\end{vmatrix}. Let p(x)p(x) be a quadratic polynomial whose roots are the maximum and minimum values of the function g(θ)g(\theta), and p(2)=2−2p(2)=2-\sqrt2. Then, which of the following is/are TRUE?
Not attempted
A
p(3+24)<0p\left(\dfrac{3+\sqrt2}{4}\right)<0
B
p(1+324)>0p\left(\dfrac{1+3\sqrt2}{4}\right)>0
C
p(52−14)>0p\left(\dfrac{5\sqrt2-1}{4}\right)>0
D
p(5−24)<0p\left(\dfrac{5-\sqrt2}{4}\right)<0
24
Consider the parabola y2=4xy^2=4x. Let SS be the focus of the parabola. A pair of tangents drawn to the parabola from the point P=(−2,1)P=(-2,1) meet the parabola at P1P_1 and P2P_2. Let Q1Q_1 and Q2Q_2 be points on the lines SP1SP_1 and SP2SP_2 respectively such that PQ1PQ_1 is perpendicular to SP1SP_1 and PQ2PQ_2 is perpendicular to SP2SP_2. Then, which of the following is/are TRUE?
Not attempted
A
SQ1=2SQ_1=2
B
Q1Q2=3105Q_1Q_2=\dfrac{3\sqrt{10}}{5}
C
PQ1=3PQ_1=3
D
SQ2=1SQ_2=1
25
Let P1P_1 and P2P_2 be two planes given by P1:10x+15y+12z−60=0P_1: 10x+15y+12z-60=0, P2:−2x+5y+4z−20=0P_2: -2x+5y+4z-20=0. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1P_1 and P2P_2?
Not attempted
A
x−10=y−10=z−15\dfrac{x-1}{0}=\dfrac{y-1}{0}=\dfrac{z-1}{5}
B
x−6−5=y2=z3\dfrac{x-6}{-5}=\dfrac{y}{2}=\dfrac{z}{3}
C
x−2=y−45=z4\dfrac{x}{-2}=\dfrac{y-4}{5}=\dfrac{z}{4}
D
x1=y−4−2=z3\dfrac{x}{1}=\dfrac{y-4}{-2}=\dfrac{z}{3}
26
Let SS be the reflection of a point QQ with respect to the plane given by r⃗=−(t+p)i^+tj^+(1+p)k^\vec r = -(t+p)\hat i+t\hat j+(1+p)\hat k where t,pt,p are real parameters and i^,j^,k^\hat i,\hat j,\hat k are the unit vectors along the three positive coordinate axes. If the position vectors of QQ and SS are 10i^+15j^+20k^10\hat i+15\hat j+20\hat k and αi^+βj^+γk^\alpha\hat i+\beta\hat j+\gamma\hat k respectively, then which of the following is/are TRUE?
Not attempted
A
3(α+β)=−1013(\alpha+\beta)=-101
B
3(β+γ)=−713(\beta+\gamma)=-71
C
3(γ+α)=−863(\gamma+\alpha)=-86
D
3(α+β+γ)=−1213(\alpha+\beta+\gamma)=-121
27
Let a1,a2,a3,…a_1,a_2,a_3,\ldots be an arithmetic progression with a1=7a_1=7 and common difference 88. Let T1,T2,T3,…T_1,T_2,T_3,\ldots be such that T1=3T_1=3 and Tn+1−Tn=anT_{n+1}-T_n=a_n for n≥1n\ge1. Then, which of the following is/are TRUE?
Not attempted
A
T20=1604T_{20}=1604
B
∑k=120Tk=10510\displaystyle\sum_{k=1}^{20}T_k=10510
C
T30=3454T_{30}=3454
D
∑k=130Tk=35610\displaystyle\sum_{k=1}^{30}T_k=35610
28
Consider the equation ∫1e(log⁡ex)1/2x(a−(log⁡ex)3/2)2 dx=1,a∈(−∞,0)∪(1,∞)\displaystyle\int_1^e \dfrac{(\log_e x)^{1/2}}{x\left(a-(\log_e x)^{3/2}\right)^2}\,dx = 1, \quad a\in(-\infty,0)\cup(1,\infty). Which of the following statements is/are TRUE?
Not attempted
A
No aa satisfies the above equation
B
An integer aa satisfies the above equation
C
An irrational number aa satisfies the above equation
D
More than one aa satisfy the above equation
29
Let ABCABC be the triangle with AB=1AB=1, AC=3AC=3 and ∠BAC=π2\angle BAC=\dfrac{\pi}{2}. If a circle of radius r>0r>0 touches the sides ABAB, ACAC and also touches internally the circumcircle of the triangle ABCABC, then the value of rr is __________.
Not attempted
Your answer: —
Correct answer: 0.84
30
The number of 44-digit integers in the closed interval [2022,4482][2022,4482] formed by using the digits 0,2,3,4,6,70,2,3,4,6,7 is __________.
Not attempted
Your answer: —
Correct answer: 569
31
Let l1,l2,…,l100l_1,l_2,\ldots,l_{100} be consecutive terms of an arithmetic progression with common difference d1d_1, and let w1,w2,…,w100w_1,w_2,\ldots,w_{100} be consecutive terms of another arithmetic progression with common difference d2d_2, where d1d2=10d_1 d_2=10. For each i=1,2,…,100i=1,2,\ldots,100, let RiR_i be a rectangle with length lil_i, width wiw_i and area AiA_i. If A51−A50=1000A_{51}-A_{50}=1000, then the value of A100−A90A_{100}-A_{90} is __________.
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Your answer: —
Correct answer: 18900
32
Let zz be a complex number with non-zero imaginary part. If 2+3z+4z22−3z+4z2\dfrac{2+3z+4z^2}{2-3z+4z^2} is a real number, then the value of ∣z∣2|z|^2 is __________.
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Correct answer: 0.5
33
Let zˉ\bar z denote the complex conjugate of a complex number zz and let i=−1i=\sqrt{-1}. In the set of complex numbers, the number of distinct roots of the equation zˉ−z2=i(zˉ+z2)\bar z - z^2 = i(\bar z+z^2) is __________.
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Correct answer: 4
34
In a study about a pandemic, data of 900900 persons was collected. It was found that 190190 persons had symptom of fever, 220220 persons had symptom of cough, 220220 persons had symptom of breathing problem, 330330 persons had symptom of fever or cough or both, 350350 persons had symptom of cough or breathing problem or both, 340340 persons had symptom of fever or breathing problem or both, 3030 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900900 persons, then the probability that the person has at most one symptom is __________.
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Correct answer: 0.8
35
Let α\alpha be a positive real number. Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:(α,∞)→Rg:(\alpha,\infty)\to\mathbb{R} be the functions defined by f(x)=sin⁡(πx12)f(x)=\sin\left(\dfrac{\pi x}{12}\right) and g(x)=2log⁡e(x−α)log⁡e(ex−eα)g(x)=\dfrac{2\log_e(\sqrt{x}-\sqrt{\alpha})}{\log_e(e^{\sqrt{x}}-e^{\sqrt{\alpha}})}. Then the value of lim⁡x→α+f(g(x))\displaystyle\lim_{x\to\alpha^+} f(g(x)) is __________.
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Correct answer: 0.5
36
Considering only the principal values of the inverse trigonometric functions, the value of 32cos⁡−122+π2+14sin⁡−122 π2+π2+tan⁡−12π\dfrac{3}{2}\cos^{-1}\sqrt{\dfrac{2}{2+\pi^2}} + \dfrac{1}{4}\sin^{-1}\dfrac{2\sqrt{2}\,\pi}{2+\pi^2} + \tan^{-1}\dfrac{\sqrt{2}}{\pi} is __________.
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Your answer: —
Correct answer: 2.36
Substitute tan⁡φ=2π\tan\varphi = \dfrac{\sqrt2}{\pi} for some φ∈(0,π/2)\varphi \in (0,\pi/2), then express 22+π2\sqrt{\dfrac{2}{2+\pi^2}} as sin⁡φ\sin\varphi and 22π2+π2\dfrac{2\sqrt2\pi}{2+\pi^2} as sin⁡(2φ)\sin(2\varphi) using the right triangle with legs 2\sqrt2, π\pi and hypotenuse 2+π2\sqrt{2+\pi^2}.
Deconstruction\textbf{Deconstruction}: This is a numerical-value question combining cos⁡−1\cos^{-1}, sin⁡−1\sin^{-1}, and tan⁡−1\tan^{-1} of related expressions built from 2\sqrt2 and π\pi. The key is to recognize a common angle hidden inside all three terms. Strategy\textbf{Strategy}: Let tan⁡φ=2π\tan\varphi = \dfrac{\sqrt2}{\pi} with φ∈(0,π/2)\varphi \in (0,\pi/2). Build the right triangle with opposite side 2\sqrt2, adjacent side π\pi, hypotenuse 2+π2\sqrt{2+\pi^2}. Then sin⁡φ=22+π2\sin\varphi = \dfrac{\sqrt2}{\sqrt{2+\pi^2}} and cos⁡φ=π2+π2\cos\varphi = \dfrac{\pi}{\sqrt{2+\pi^2}}, so each term in the expression reduces to a simple multiple of φ\varphi. Execution\textbf{Execution}: Since 22+π2=sin⁡φ\sqrt{\dfrac{2}{2+\pi^2}} = \sin\varphi, we get cos⁡−122+π2=cos⁡−1(sin⁡φ)=π2−φ\cos^{-1}\sqrt{\dfrac{2}{2+\pi^2}} = \cos^{-1}(\sin\varphi) = \dfrac{\pi}{2}-\varphi. Since 2sin⁡φcos⁡φ=22 π2+π2=sin⁡(2φ)2\sin\varphi\cos\varphi = \dfrac{2\sqrt2\,\pi}{2+\pi^2} = \sin(2\varphi), and 2φ∈(0,π/2)2\varphi \in (0,\pi/2) (because φ≈24.2∘\varphi \approx 24.2^\circ), we get sin⁡−122 π2+π2=sin⁡−1(sin⁡2φ)=2φ\sin^{-1}\dfrac{2\sqrt2\,\pi}{2+\pi^2} = \sin^{-1}(\sin 2\varphi) = 2\varphi. Also tan⁡−12π=φ\tan^{-1}\dfrac{\sqrt2}{\pi} = \varphi. So the expression becomes 32(π2−φ)+14(2φ)+φ=3π4−32φ+12φ+φ=3π4\dfrac{3}{2}\left(\dfrac{\pi}{2}-\varphi\right) + \dfrac{1}{4}(2\varphi) + \varphi = \dfrac{3\pi}{4} - \dfrac{3}{2}\varphi + \dfrac{1}{2}\varphi + \varphi = \dfrac{3\pi}{4}, since the φ\varphi terms cancel exactly. Verification\textbf{Verification}: Numerically, φ=tan⁡−1(0.4502)≈0.4234\varphi = \tan^{-1}(0.4502) \approx 0.4234 rad. Then 32(π/2−0.4234)+14(0.8468)+0.4234≈1.7211+0.2117+0.4234=2.3562=3π4\dfrac{3}{2}(\pi/2-0.4234) + \dfrac{1}{4}(0.8468) + 0.4234 \approx 1.7211+0.2117+0.4234 = 2.3562 = \dfrac{3\pi}{4}, confirming the algebraic cancellation. Final Answer\textbf{Final Answer}: 3π4≈2.36\dfrac{3\pi}{4} \approx 2.36.