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JEE Advanced · Mathematics · 2023

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1
Consider the 6×66\times6 square shown in the figure. Let A1,A2,…,A49A_1,A_2,\ldots,A_{49} be the points of intersection (the dots in the picture), taken in some order. We say AiA_i and AjA_j are friends if they are adjacent along a row or along a column. Assume each point AiA_i has an equal chance of being chosen. Let pip_i be the probability that a randomly chosen point has exactly ii friends, for i=0,1,2,3,4i=0,1,2,3,4. Let XX be a random variable such that P(X=i)=piP(X=i)=p_i for i=0,1,2,3,4i=0,1,2,3,4. Then the value of 7E(X)7E(X) is
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Your answer: —
Correct answer: 24
2
Consider the 6×66\times6 square shown in the figure. Let A1,A2,…,A49A_1,A_2,\ldots,A_{49} be the points of intersection (the dots in the picture), taken in some order. We say AiA_i and AjA_j are friends if they are adjacent along a row or along a column. Assume each point AiA_i has an equal chance of being chosen. Two distinct points are chosen randomly out of the points A1,A2,…,A49A_1,A_2,\ldots,A_{49}. Let pp be the probability that they are friends. Then the value of 7p7p is
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Your answer: —
Correct answer: 0.5
3
Consider an obtuse-angled triangle ABCABC in which the difference between the largest and the smallest angle is π2\dfrac{\pi}{2} and whose sides are in arithmetic progression. Suppose the vertices of this triangle lie on a circle of radius 11. Let aa denote the area of the triangle ABCABC. Then the value of (64a)2(64a)^2 is
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Your answer: —
Correct answer: 1008
4
Consider an obtuse-angled triangle ABCABC in which the difference between the largest and the smallest angle is π2\dfrac{\pi}{2} and whose sides are in arithmetic progression. Suppose the vertices of this triangle lie on a circle of radius 11. Then the inradius of the triangle ABCABC is
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Your answer: —
Correct answer: 0.25
5
Let C1C_1 be the circle of radius 11 with centre at the origin. Let C2C_2 be the circle of radius rr with centre at the point A=(4,1)A=(4,1), where 1<r<31<r<3. Two distinct common tangents PQPQ and STST of C1C_1 and C2C_2 are drawn; PQPQ touches C1C_1 at PP and C2C_2 at QQ, while STST touches C1C_1 at SS and C2C_2 at TT. Midpoints of the segments PQPQ and STST are joined to form a line which meets the xx-axis at a point BB. If AB=5AB=\sqrt5, then the value of r2r^2 is
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Your answer: —
Correct answer: 2
6
Let R={(a3bc2d050):a,b,c,d∈{0,3,5,7,11,13,17,19}}R=\left\{\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}: a,b,c,d\in\{0,3,5,7,11,13,17,19\}\right\}. Then the number of invertible matrices in RR is
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Your answer: —
Correct answer: 3780
7
Let A1,A2,A3,…,A8A_1,A_2,A_3,\ldots,A_8 be the vertices of a regular octagon that lie on a circle of radius 22. Let PP be a point on the circle and let PAiPA_i denote the distance between PP and AiA_i for i=1,2,…,8i=1,2,\ldots,8. If PP varies over the circle, then the maximum value of the product PA1⋅PA2⋯PA8PA_1\cdot PA_2\cdots PA_8 is
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Your answer: —
Correct answer: 512
8
Let XX be the set of all five-digit numbers formed using 1,2,2,2,4,4,01,2,2,2,4,4,0 (each digit used at most as many times as it appears here). For example, 22240∈X22240\in X while 0224402244 and 4442244422 are not in XX. Suppose each element of XX has an equal chance of being chosen. Let pp be the conditional probability that an element chosen at random is a multiple of 2020 given that it is a multiple of 55. Then the value of 38p38p is equal to
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Your answer: —
Correct answer: 31
9
For x∈Rx\in\mathbb{R}, let y(x)y(x) be a solution of the differential equation (x2−5)dydx−2xy=−2x(x2−5)2(x^2-5)\dfrac{dy}{dx}-2xy=-2x(x^2-5)^2 such that y(2)=7y(2)=7. Then the maximum value of the function y(x)y(x) is
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Your answer: —
Correct answer: 16
10
For x∈Rx\in\mathbb{R}, let tan⁡−1(x)∈(−π2,π2)\tan^{-1}(x)\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right). Then the minimum value of the function f:R→Rf:\mathbb{R}\to\mathbb{R} defined by f(x)=∫0xtan⁡−1xe(t−cos⁡t)1+t2023 dtf(x)=\displaystyle\int_0^{x\tan^{-1}x}\dfrac{e^{(t-\cos t)}}{1+t^{2023}}\,dt is
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Your answer: —
Correct answer: 0
11
Let f:(0,1)→Rf:(0,1)\to\mathbb{R} be the function defined as f(x)=[4x](x−14)2(x−12)f(x)=[4x]\left(x-\dfrac14\right)^2\left(x-\dfrac12\right), where [x][x] denotes the greatest integer less than or equal to xx. Then which of the following statements is(are) true?
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A
The function ff is discontinuous exactly at one point in (0,1)(0,1)
B
There is exactly one point in (0,1)(0,1) at which the function ff is continuous but NOT differentiable
C
The function ff is NOT differentiable at more than three points in (0,1)(0,1)
D
The minimum value of the function ff is −1512-\dfrac{1}{512}
12
Let SS be the set of all twice differentiable functions ff from R\mathbb{R} to R\mathbb{R} such that d2fdx2(x)>0\dfrac{d^2f}{dx^2}(x)>0 for all x∈(−1,1)x\in(-1,1). For f∈Sf\in S, let XfX_f be the number of points x∈(−1,1)x\in(-1,1) for which f(x)=xf(x)=x. Then which of the following statements is(are) true?
Not attempted
A
There exists a function f∈Sf\in S such that Xf=0X_f=0
B
For every function f∈Sf\in S, we have Xf≤2X_f\le2
C
There exists a function f∈Sf\in S such that Xf=2X_f=2
D
There does NOT exist any function ff in SS such that Xf=1X_f=1
13
Let M=(aij)M=(a_{ij}), i,j∈{1,2,3}i,j\in\{1,2,3\}, be the 3×33\times3 matrix such that aij=1a_{ij}=1 if j+1j+1 is divisible by ii, and aij=0a_{ij}=0 otherwise. Then which of the following statements is(are) true?
Not attempted
A
MM is invertible
B
There exists a nonzero column matrix (a1a2a3)\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix} such that M(a1a2a3)=(−a1−a2−a3)M\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}=\begin{pmatrix}-a_1\\-a_2\\-a_3\end{pmatrix}
C
The set {X∈R3:MX=0⃗}≠{0⃗}\left\{X\in\mathbb{R}^3:MX=\vec0\right\}\ne\{\vec0\}, where 0⃗=(000)\vec0=\begin{pmatrix}0\\0\\0\end{pmatrix}
D
The matrix (M−2I)(M-2I) is invertible, where II is the 3×33\times3 identity matrix
14
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
15
For any y∈Ry\in\mathbb{R}, let cot⁡−1(y)∈(0,π)\cot^{-1}(y)\in(0,\pi) and tan⁡−1(y)∈(−π2,π2)\tan^{-1}(y)\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right). Then the sum of all the solutions of the equation tan⁡−1 ⁣(6y9−y2)+cot⁡−1 ⁣(9−y26y)=2π3\tan^{-1}\!\left(\dfrac{6y}{9-y^2}\right)+\cot^{-1}\!\left(\dfrac{9-y^2}{6y}\right)=\dfrac{2\pi}{3} for 0<∣y∣<30<|y|<3 is equal to
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A
23−32\sqrt3-3
B
3−233-2\sqrt3
C
43−64\sqrt3-6
D
6−436-4\sqrt3
16
Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is 13\dfrac{1}{3}, then the probability that the experiment stops with a head (i.e. the two matching consecutive tosses are both heads) is
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A
13\dfrac{1}{3}
B
521\dfrac{5}{21}
C
421\dfrac{4}{21}
D
27\dfrac{2}{7}
17
Let f:[1,∞)→Rf:[1,\infty)\to\mathbb{R} be a differentiable function such that f(1)=13f(1)=\dfrac{1}{3} and 3∫1xf(t) dt=xf(x)−x333\int_{1}^{x} f(t)\,dt = xf(x) - \dfrac{x^3}{3} for x∈[1,∞)x\in[1,\infty). Let ee denote the base of the natural logarithm. Then the value of f(e)f(e) is
Not attempted
A
e2+43\dfrac{e^2+4}{3}
B
log⁡e4+e3\dfrac{\log_e 4 + e}{3}
C
4e23\dfrac{4e^2}{3}
D
e2−43\dfrac{e^2-4}{3}
18
Let zz be a complex number satisfying ∣z∣3+2z2+4zˉ−8=0|z|^3+2z^2+4\bar z-8=0, where zˉ\bar z denotes the complex conjugate of zz. Let the imaginary part of zz be nonzero. Match each entry in List-I to the correct entries in List-II. List-I (P) ∣z∣2|z|^2 is equal to (Q) ∣z−zˉ∣2|z-\bar z|^2 is equal to (R) ∣z∣2+∣z+zˉ∣2|z|^2+|z+\bar z|^2 is equal to (S) ∣z+1∣2|z+1|^2 is equal to List-II (1) 1212 (2) 44 (3) 88 (4) 1010 (5) 77 The correct option is:
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A
(P)→(1)(Q)→(3)(R)→(5)(S)→(4)(P)\to(1)\quad(Q)\to(3)\quad(R)\to(5)\quad(S)\to(4)
B
(P)→(2)(Q)→(1)(R)→(3)(S)→(5)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(3)\quad(S)\to(5)
C
(P)→(2)(Q)→(4)(R)→(5)(S)→(1)(P)\to(2)\quad(Q)\to(4)\quad(R)\to(5)\quad(S)\to(1)
D
(P)→(2)(Q)→(3)(R)→(5)(S)→(4)(P)\to(2)\quad(Q)\to(3)\quad(R)\to(5)\quad(S)\to(4)
19
Let ℓ1\ell_1 and ℓ2\ell_2 be the lines r⃗1=λ(i^+j^+k^)\vec r_1=\lambda(\hat i+\hat j+\hat k) and r⃗2=(j^−k^)+μ(i^+k^)\vec r_2=(\hat j-\hat k)+\mu(\hat i+\hat k), respectively. Let XX be the set of all the planes HH that contain the line ℓ1\ell_1. For a plane HH, let d(H)d(H) denote the smallest possible distance between the points of ℓ2\ell_2 and HH. Let H0H_0 be a plane in XX for which d(H0)d(H_0) is the maximum value of d(H)d(H) as HH varies over all planes in XX. Match each entry in List-I to the correct entries in List-II. List-I (P) The value of d(H0)d(H_0) is (Q) The distance of the point (0,1,2)(0,1,2) from H0H_0 is (R) The distance of origin from H0H_0 is (S) The distance of origin from the point of intersection of planes y=zy=z, x=1x=1 and H0H_0 is List-II (1) 3\sqrt3 (2) 13\dfrac1{\sqrt3} (3) 00 (4) 2\sqrt2 (5) 12\dfrac1{\sqrt2} The correct option is:
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A
(P)→(2)(Q)→(4)(R)→(5)(S)→(1)(P)\to(2)\quad(Q)\to(4)\quad(R)\to(5)\quad(S)\to(1)
B
(P)→(5)(Q)→(4)(R)→(3)(S)→(1)(P)\to(5)\quad(Q)\to(4)\quad(R)\to(3)\quad(S)\to(1)
C
(P)→(2)(Q)→(1)(R)→(3)(S)→(2)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(3)\quad(S)\to(2)
D
(P)→(5)(Q)→(1)(R)→(4)(S)→(2)(P)\to(5)\quad(Q)\to(1)\quad(R)\to(4)\quad(S)\to(2)
20
Consider the given data with frequency distribution: xix_i: 3, 8, 11, 10, 5, 43,\ 8,\ 11,\ 10,\ 5,\ 4 fif_i: 5, 2, 3, 2, 4, 45,\ 2,\ 3,\ 2,\ 4,\ 4 Match each entry in List-I to the correct entries in List-II. List-I (P) The mean of the above data is (Q) The median of the above data is (R) The mean deviation about the mean of the above data is (S) The mean deviation about the median of the above data is List-II (1) 2.52.5 (2) 55 (3) 66 (4) 2.72.7 (5) 2.42.4 The correct option is:
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A
(P)→(3)(Q)→(2)(R)→(4)(S)→(5)(P)\to(3)\quad(Q)\to(2)\quad(R)\to(4)\quad(S)\to(5)
B
(P)→(3)(Q)→(2)(R)→(1)(S)→(5)(P)\to(3)\quad(Q)\to(2)\quad(R)\to(1)\quad(S)\to(5)
C
(P)→(2)(Q)→(3)(R)→(4)(S)→(1)(P)\to(2)\quad(Q)\to(3)\quad(R)\to(4)\quad(S)\to(1)
D
(P)→(3)(Q)→(3)(R)→(5)(S)→(5)(P)\to(3)\quad(Q)\to(3)\quad(R)\to(5)\quad(S)\to(5)
21
Let α,β\alpha,\beta and γ\gamma be real numbers. Consider the following system of linear equations x+2y+z=7x+2y+z=7 x+αz=11x+\alpha z=11 2x−3y+βz=γ2x-3y+\beta z=\gamma Match each entry in List-I to the correct entries in List-II. List-I (P) If β=12(7α−3)\beta=\dfrac12(7\alpha-3) and γ=28\gamma=28, then the system has (Q) If β=12(7α−3)\beta=\dfrac12(7\alpha-3) and γ≠28\gamma\ne28, then the system has (R) If β≠12(7α−3)\beta\ne\dfrac12(7\alpha-3) where α=1\alpha=1 and γ≠28\gamma\ne28, then the system has (S) If β≠12(7α−3)\beta\ne\dfrac12(7\alpha-3) where α=1\alpha=1 and γ=28\gamma=28, then the system has List-II (1) a unique solution (2) no solution (3) infinitely many solutions (4) x=11x=11, y=−2y=-2 and z=0z=0 as a solution (5) x=−15x=-15, y=4y=4 and z=0z=0 as a solution The correct option is:
Not attempted
A
(P)→(3)(Q)→(2)(R)→(1)(S)→(4)(P)\to(3)\quad(Q)\to(2)\quad(R)\to(1)\quad(S)\to(4)
B
(P)→(3)(Q)→(2)(R)→(5)(S)→(4)(P)\to(3)\quad(Q)\to(2)\quad(R)\to(5)\quad(S)\to(4)
C
(P)→(2)(Q)→(1)(R)→(4)(S)→(5)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(4)\quad(S)\to(5)
D
(P)→(2)(Q)→(1)(R)→(1)(S)→(3)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(1)\quad(S)\to(3)
22
Let aa and bb be two nonzero real numbers. If the coefficient of x5x^5 in the expansion of (ax2+7027bx)4\left(ax^2+\dfrac{70}{27bx}\right)^4 is equal to the coefficient of x−5x^{-5} in the expansion of (ax−1bx2)7\left(ax-\dfrac{1}{bx^2}\right)^7, then the value of 2b2b is
Not attempted
Your answer: —
Correct answer: 3
23
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
24
Let A={1967+1686 isin⁡θ7−3icos⁡θ:θ∈R}A=\left\{\dfrac{1967+1686\,i\sin\theta}{7-3i\cos\theta} : \theta\in\mathbb{R}\right\}. If AA contains exactly one positive integer nn, then the value of nn is
Not attempted
Your answer: —
Correct answer: 281
25
Let 75⋯5⏞r77\overbrace{5\cdots5}^{r}7 denote the (r+2)(r+2) digit number where the first and the last digits are 77 and the remaining rr digits are 55. Consider the sum S=77+757+7557+⋯+75⋯5⏞987S=77+757+7557+\cdots+7\overbrace{5\cdots5}^{98}7. If S=75⋯5⏞997+mnS=\dfrac{7\overbrace{5\cdots5}^{99}7+m}{n}, where mm and nn are natural numbers less than 30003000, then the value of m+nm+n is
Not attempted
Your answer: —
Correct answer: 1219
26
Let n≥2n\ge2 be a natural number and f:[0,1]→Rf:[0,1]\to\mathbb{R} be the function defined by f(x)={n(1−2nx)if 0≤x≤12n2n(2nx−1)if 12n≤x≤34n4n(1−nx)if 34n≤x≤1nnn−1(nx−1)if 1n≤x≤1f(x)=\begin{cases}n(1-2nx) & \text{if } 0\le x\le\dfrac{1}{2n}\\ 2n(2nx-1) & \text{if } \dfrac{1}{2n}\le x\le\dfrac{3}{4n}\\ 4n(1-nx) & \text{if } \dfrac{3}{4n}\le x\le\dfrac{1}{n}\\ \dfrac{n}{n-1}(nx-1) & \text{if } \dfrac{1}{n}\le x\le1\end{cases}. If nn is such that the area of the region bounded by the curves x=0x=0, x=1x=1, y=0y=0 and y=f(x)y=f(x) is 44, then the maximum value of the function ff is
Not attempted
Your answer: —
Correct answer: 8
27
Let tan⁡−1(x)∈(−π2,π2)\tan^{-1}(x)\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right), for x∈Rx\in\mathbb{R}. Then the number of real solutions of the equation 1+cos⁡(2x)=2 tan⁡−1(tan⁡x)\sqrt{1+\cos(2x)}=\sqrt2\,\tan^{-1}(\tan x) in the set (−3π2,−π2)∪(−π2,π2)∪(π2,3π2)\left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right)\cup\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\cup\left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) is equal to
Not attempted
Your answer: —
Correct answer: 3
28
Let PP be a point on the parabola y2=4axy^2=4ax, where a>0a>0. The normal to the parabola at PP meets the xx-axis at a point QQ. The area of the triangle PFQPFQ, where FF is the focus of the parabola, is 120120. If the slope mm of the normal and aa are both positive integers, then the pair (a,m)(a,m) is
Not attempted
A
(2,3)(2,3)
B
(1,3)(1,3)
C
(2,4)(2,4)
D
(3,4)(3,4)
29
Let X={(x,y)∈Z×Z:x28+y220<1 and y2<5x}X = \left\{(x,y)\in\mathbb{Z}\times\mathbb{Z} : \dfrac{x^2}{8}+\dfrac{y^2}{20}<1 \text{ and } y^2<5x\right\}. Three distinct points PP, QQ and RR are randomly chosen from XX. Then the probability that PP, QQ and RR form a triangle whose area is a positive integer, is
Not attempted
A
71220\dfrac{71}{220}
B
73220\dfrac{73}{220}
C
79220\dfrac{79}{220}
D
83220\dfrac{83}{220}
30
Let QQ be the cube with the set of vertices {(x1,x2,x3)∈R3:x1,x2,x3∈{0,1}}\{(x_1,x_2,x_3)\in\mathbb{R}^3 : x_1,x_2,x_3\in\{0,1\}\}. Let FF be the set of all twelve lines containing the diagonals of the six faces of the cube QQ. Let SS be the set of all four lines containing the main diagonals of the cube QQ; for instance, the line passing through the vertices (0,0,0)(0,0,0) and (1,1,1)(1,1,1) is in SS. For lines ℓ1\ell_1 and ℓ2\ell_2, let d(ℓ1,ℓ2)d(\ell_1,\ell_2) denote the shortest distance between them. Then the maximum value of d(ℓ1,ℓ2)d(\ell_1,\ell_2), as ℓ1\ell_1 varies over FF and ℓ2\ell_2 varies over SS, is
Not attempted
A
16\dfrac{1}{\sqrt6}
B
18\dfrac{1}{\sqrt8}
C
13\dfrac{1}{\sqrt3}
D
112\dfrac{1}{\sqrt{12}}
31
Let f:(0,1)→Rf:(0,1)\to\mathbb{R} be the function defined as f(x)=nf(x)=\sqrt{n} if x∈[1n+1,1n)x\in\left[\dfrac{1}{n+1},\dfrac{1}{n}\right), where n∈Nn\in\mathbb{N}. Let g:(0,1)→Rg:(0,1)\to\mathbb{R} be a function such that ∫x2x1−tt dt<g(x)<2x\displaystyle\int_{x^2}^{x}\sqrt{\dfrac{1-t}{t}}\,dt<g(x)<2\sqrt{x} for all x∈(0,1)x\in(0,1). Then lim⁡x→0f(x)g(x)\displaystyle\lim_{x\to0}f(x)g(x)
Not attempted
A
does NOT exist
B
is equal to 11
C
is equal to 22
D
is equal to 33
32
Let f:[0,1]→[0,1]f:[0,1]\to[0,1] be the function defined by f(x)=x33−x2+5x9+1736f(x)=\dfrac{x^3}{3}-x^2+\dfrac{5x}{9}+\dfrac{17}{36}. Consider the square region S=[0,1]×[0,1]S=[0,1]\times[0,1]. Let G={(x,y)∈S:y>f(x)}G=\{(x,y)\in S: y>f(x)\} be called the green region and R={(x,y)∈S:y<f(x)}R=\{(x,y)\in S: y<f(x)\} be called the red region. Let Lh={(x,h)∈S:x∈[0,1]}L_h=\{(x,h)\in S: x\in[0,1]\} be the horizontal line drawn at a height h∈[0,1]h\in[0,1]. Then which of the following statements is(are) true?
Not attempted
A
There exists an h∈[14,23]h\in\left[\dfrac14,\dfrac23\right] such that the area of the green region above the line LhL_h equals the area of the green region below the line LhL_h
B
There exists an h∈[14,23]h\in\left[\dfrac14,\dfrac23\right] such that the area of the red region above the line LhL_h equals the area of the red region below the line LhL_h
C
There exists an h∈[14,23]h\in\left[\dfrac14,\dfrac23\right] such that the area of the green region above the line LhL_h equals the area of the red region below the line LhL_h
D
There exists an h∈[14,23]h\in\left[\dfrac14,\dfrac23\right] such that the area of the red region above the line LhL_h equals the area of the green region below the line LhL_h
33
Let T1T_1 and T2T_2 be two distinct common tangents to the ellipse E:x26+y23=1E: \dfrac{x^2}{6}+\dfrac{y^2}{3}=1 and the parabola P:y2=12xP: y^2=12x. Suppose that the tangent T1T_1 touches PP and EE at the points A1A_1 and A2A_2, respectively, and the tangent T2T_2 touches PP and EE at the points A4A_4 and A3A_3, respectively. Then which of the following statements is(are) true?
Not attempted
A
The area of the quadrilateral A1A2A3A4A_1A_2A_3A_4 is 3535 square units
B
The area of the quadrilateral A1A2A3A4A_1A_2A_3A_4 is 3636 square units
C
The tangents T1T_1 and T2T_2 meet the xx-axis at the point (−3,0)(-3,0)
D
The tangents T1T_1 and T2T_2 meet the xx-axis at the point (−6,0)(-6,0)
34
Let S=(0,1)∪(1,2)∪(3,4)S = (0,1) \cup (1,2) \cup (3,4) and T={0,1,2,3}T = \{0,1,2,3\}. Then which of the following statements is(are) true?
Not attempted
A
There are infinitely many functions from SS to TT
B
There are infinitely many strictly increasing functions from SS to TT
C
The number of continuous functions from SS to TT is at most 120120
D
Every continuous function from SS to TT is differentiable
Tip\textbf{Tip}: Count the connected components of SS, and recall that a continuous map into a discrete set must be constant on each connected component.
Deconstruction\textbf{Deconstruction}: SS is a union of three disjoint open intervals (three connected components), and TT is a finite discrete set of 44 points. For each option we must decide whether the described class of functions from SS to TT is infinite, bounded, etc. Strategy\textbf{Strategy}: An arbitrary function from SS to TT has no restriction at all. A strictly increasing function must be injective. A continuous function into a discrete space must be constant on each connected component, since the continuous image of a connected set is connected, and the only connected subsets of a discrete space are single points. Execution\textbf{Execution}: (A)\textbf{(A)} Each of the uncountably many points of SS can independently be sent to any of the 44 elements of TT, with no consistency requirement. So there are infinitely (in fact uncountably) many functions from SS to TT. (A) is true. (B)\textbf{(B)} A strictly increasing function is injective. Since SS is uncountable and TT has only 44 elements, no injective function from SS to TT exists at all, let alone infinitely many strictly increasing ones. (B) is false. (C)\textbf{(C)} SS has exactly 33 connected components: (0,1)(0,1), (1,2)(1,2), (3,4)(3,4). A continuous function into TT must be constant on each component, so choosing a continuous function amounts to choosing one of the 44 values of TT independently for each of the 33 components. This gives exactly 43=644^3 = 64 continuous functions, which is indeed at most 120120. (C) is true. (D)\textbf{(D)} Every continuous function from SS to TT is, by the above, locally constant on each open interval component, hence differentiable everywhere with derivative 00. (D) is true. Verification\textbf{Verification}: The multiplication principle directly gives 4×4×4=64≤1204\times4\times4=64\le120, confirming (C). For (B), even a single strictly increasing function would need to be injective, impossible by the pigeonhole principle since ∣S∣>∣T∣|S|>|T|, confirming (B) is false. Final Answer\textbf{Final Answer}: (A), (C), (D).