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JEE Advanced · Mathematics · 2025

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1
For a non-zero complex number zz, let arg⁡(z)\arg(z) denote the principal argument of zz, with −π<arg⁡(z)≤π-\pi<\arg(z)\le\pi. Let ω\omega be the cube root of unity for which 0<arg⁡(ω)<π0<\arg(\omega)<\pi. Let α=arg⁡(∑n=12025(−ω)n).\alpha = \arg\left(\displaystyle\sum_{n=1}^{2025}(-\omega)^n\right). Then the value of 3απ\dfrac{3\alpha}{\pi} is ________________.
Not attempted
Your answer: —
Correct answer: -2
2
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_____________.
Not attempted
Your answer: —
Correct answer: -2
3
A factory has a total of three manufacturing units, M1M_1, M2M_2, and M3M_3, which produce bulbs independent of each other. The units M1,M2,M3M_1,M_2,M_3 produce bulbs in the proportions 2:2:12:2:1 respectively. It is known that 20%20\% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by M1M_1, 15%15\% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by M2M_2 is 25\dfrac25. If a bulb is chosen randomly from the bulbs produced by M3M_3, then the probability that it is defective is ___________.
Not attempted
Your answer: —
Correct answer: 0.3
4
Let a0,a1,…,a23a_0,a_1,\dots,a_{23} be real numbers such that (1+25x)23=∑i=023aixi\left(1+\dfrac25x\right)^{23} = \displaystyle\sum_{i=0}^{23}a_ix^i for every real number xx. Let ara_r be the largest among the numbers aja_j for 0≤j≤230\le j\le23. Then the value of rr is ___________.
Not attempted
Your answer: —
Correct answer: 6
5
Let y(x)y(x) be the solution of the differential equation x2dydx+xy=x2+y2,x>1e,x^2\dfrac{dy}{dx}+xy = x^2+y^2,\qquad x>\dfrac1e, satisfying y(1)=0y(1)=0. Then the value of 2(y(e))2y(e2)\dfrac{2(y(e))^2}{y(e^2)} is ___________.
Not attempted
Your answer: —
Correct answer: 0.75
6
Let I=(1001)I=\begin{pmatrix}1&0\\0&1\end{pmatrix} and P=(2003)P=\begin{pmatrix}2&0\\0&3\end{pmatrix}. Let Q=(xyz4)Q=\begin{pmatrix}x&y\\z&4\end{pmatrix} for some non-zero real numbers x,y,zx,y,z, for which there is a 2×22\times2 matrix RR with all entries being non-zero real numbers, such that QR=RPQR=RP. Then which of the following statements is (are) TRUE?
Not attempted
A
The determinant of Q−2IQ-2I is zero
B
The determinant of Q−6IQ-6I is 1212
C
The determinant of Q−3IQ-3I is 1515
D
yz=2yz=2
7
Let SS denote the locus of the point of intersection of the pair of lines 4x−3y=12α,4αx+3αy=12,4x-3y=12\alpha,\qquad 4\alpha x+3\alpha y=12, where α\alpha varies over the set of non-zero real numbers. Let TT be the tangent to SS passing through the points (p,0)(p,0) and (0,q)(0,q), q>0q>0, and parallel to the line 4x−32y=04x-\dfrac{3}{\sqrt2}y=0. Then the value of pqpq is
Not attempted
A
−62-6\sqrt2
B
−32-3\sqrt2
C
−92-9\sqrt2
D
−122-12\sqrt2
8
Let SS denote the locus of the mid-points of those chords of the parabola y2=xy^2=x, such that the area of the region enclosed between the parabola and the chord is 43\dfrac43. Let R\mathcal{R} denote the region lying in the first quadrant, enclosed by the parabola y2=xy^2=x, the curve SS, and the lines x=1x=1 and x=4x=4. Then which of the following statements is (are) TRUE?
Not attempted
A
(4,3)∈S(4,\sqrt3)\in S
B
(5,2)∈S(5,\sqrt2)\in S
C
Area of R\mathcal R is 143−23\dfrac{14}{3}-2\sqrt3
D
Area of R\mathcal R is 143−3\dfrac{14}{3}-\sqrt3
9
Let P(x1,y1)P(x_1,y_1) and Q(x2,y2)Q(x_2,y_2) be two distinct points on the ellipse x29+y24=1\dfrac{x^2}{9}+\dfrac{y^2}{4}=1 such that y1>0y_1>0 and y2>0y_2>0. Let CC denote the circle x2+y2=9x^2+y^2=9, and MM be the point (3,0)(3,0). Suppose the line x=x1x=x_1 intersects CC at RR, and the line x=x2x=x_2 intersects CC at SS, such that the yy-coordinates of RR and SS are positive. Let ∠ROM=π6\angle ROM=\dfrac{\pi}{6} and ∠SOM=π3\angle SOM=\dfrac{\pi}{3}, where OO denotes the origin (0,0)(0,0). Let ∣XY∣|XY| denote the length of the line segment XYXY. Then which of the following statements is (are) TRUE?
Not attempted
A
The equation of the line joining PP and QQ is 2x+3y=3(1+3)2x+3y=3(1+\sqrt3)
B
The equation of the line joining PP and QQ is 2x+y=3(1+3)2x+y=3(1+\sqrt3)
C
If N2=(x2,0)N_2=(x_2,0), then 3∣N2Q∣=2∣N2S∣3|N_2Q|=2|N_2S|
D
If N1=(x1,0)N_1=(x_1,0), then 9∣N1P∣=4∣N1R∣9|N_1P|=4|N_1R|
10
The total number of real solutions of the equation θ=tan⁡−1(2tan⁡θ)−12sin⁡−1(6tan⁡θ9+tan⁡2θ)\theta = \tan^{-1}(2\tan\theta) - \dfrac12\sin^{-1}\left(\dfrac{6\tan\theta}{9+\tan^2\theta}\right) is (Here, the inverse trigonometric functions sin⁡−1x\sin^{-1}x and tan⁡−1x\tan^{-1}x assume values in [−π2,π2]\left[-\frac{\pi}{2},\frac{\pi}{2}\right] and (−π2,π2)\left(-\frac{\pi}{2},\frac{\pi}{2}\right), respectively.)
Not attempted
A
1
B
2
C
3
D
5
11
Let α=1sin⁡60∘sin⁡61∘+1sin⁡62∘sin⁡63∘+⋯+1sin⁡118∘sin⁡119∘\alpha = \dfrac{1}{\sin60^\circ\sin61^\circ} + \dfrac{1}{\sin62^\circ\sin63^\circ} + \cdots + \dfrac{1}{\sin118^\circ\sin119^\circ}. Then the value of (cosec 1∘α)2\left(\dfrac{\text{cosec}\,1^\circ}{\alpha}\right)^2 is ______________.
Not attempted
Your answer: —
Correct answer: 3
12
Let R\mathbb{R} denote the set of all real numbers. Then the area of the region {(x,y)∈R×R:x>0, y>1x, 5x−4y−1>0, 4x+4y−17<0}\left\{(x,y)\in\mathbb{R}\times\mathbb{R} : x>0,\ y>\dfrac{1}{x},\ 5x-4y-1>0,\ 4x+4y-17<0\right\} is
Not attempted
A
1716−log⁡e4\dfrac{17}{16}-\log_e4
B
338−log⁡e4\dfrac{33}{8}-\log_e4
C
578−log⁡e4\dfrac{57}{8}-\log_e4
D
172−log⁡e4\dfrac{17}{2}-\log_e4
13
If α=∫1/22tan⁡−1x2x2−3x+2 dx\alpha = \displaystyle\int_{1/2}^{2}\dfrac{\tan^{-1}x}{2x^2-3x+2}\,dx, then the value of 7tan⁡(2α7π)\sqrt7\tan\left(\dfrac{2\alpha\sqrt7}{\pi}\right) is _____________. (Here, the inverse trigonometric function tan⁡−1x\tan^{-1}x assumes values in (−π2,π2)\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right).)
Not attempted
Your answer: —
Correct answer: 21
14
Let x0x_0 be the real number such that ex0+x0=0e^{x_0} + x_0 = 0. For a given real number α\alpha, define g(x)=3xex+3x−αex−αx3(ex+1)g(x) = \dfrac{3xe^{x} + 3x - \alpha e^{x} - \alpha x}{3(e^{x}+1)} for all real numbers xx. Then which one of the following statements is TRUE?
Not attempted
A
For α=2\alpha=2, lim⁡x→x0∣g(x)+ex0x−x0∣=0\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = 0
B
For α=2\alpha=2, lim⁡x→x0∣g(x)+ex0x−x0∣=1\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = 1
C
For α=3\alpha=3, lim⁡x→x0∣g(x)+ex0x−x0∣=0\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = 0
D
For α=3\alpha=3, lim⁡x→x0∣g(x)+ex0x−x0∣=23\displaystyle\lim_{x\to x_0}\left|\dfrac{g(x)+e^{x_0}}{x-x_0}\right| = \dfrac{2}{3}
15
Let R\mathbb{R} denote the set of all real numbers. Let f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)={6x+sin⁡x2x+sin⁡xif x≠0,73if x=0.f(x)=\begin{cases}\dfrac{6x+\sin x}{2x+\sin x} & \text{if } x\ne0,\\[4pt] \dfrac73 & \text{if } x=0.\end{cases} Then which of the following statements is (are) TRUE?
Not attempted
A
The point x=0x=0 is a point of local maxima of ff
B
The point x=0x=0 is a point of local minima of ff
C
Number of points of local maxima of ff in the interval [π,6π][\pi,6\pi] is 33
D
Number of points of local minima of ff in the interval [2π,4π][2\pi,4\pi] is 11
16
Let R\mathbb{R} denote the set of all real numbers. Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→(0,4)g:\mathbb{R}\to(0,4) be functions defined by f(x)=log⁡e(x2+2x+4),g(x)=41+e−2x.f(x)=\log_e(x^2+2x+4),\qquad g(x)=\dfrac{4}{1+e^{-2x}}. Define the composite function f∘g−1f\circ g^{-1} by (f∘g−1)(x)=f(g−1(x))(f\circ g^{-1})(x)=f(g^{-1}(x)), where g−1g^{-1} is the inverse of the function gg. Then the value of the derivative of the composite function f∘g−1f\circ g^{-1} at x=2x=2 is ______________.
Not attempted
Your answer: —
Correct answer: 0.25
17
For all x>0x>0, let y1(x),y2(x)y_1(x), y_2(x), and y3(x)y_3(x) be the functions satisfying dy1dx−(sin⁡x)2 y1=0,y1(1)=5,\dfrac{dy_1}{dx} - (\sin x)^2\,y_1 = 0, \quad y_1(1)=5, dy2dx−(cos⁡x)2 y2=0,y2(1)=13,\dfrac{dy_2}{dx} - (\cos x)^2\,y_2 = 0, \quad y_2(1)=\dfrac13, dy3dx−(2−x3x3)y3=0,y3(1)=35e,\dfrac{dy_3}{dx} - \left(\dfrac{2-x^3}{x^3}\right)y_3 = 0, \quad y_3(1)=\dfrac{3}{5e}, respectively. Then lim⁡x→0+y1(x)y2(x)y3(x)+2xe3xsin⁡x\displaystyle\lim_{x\to0^+}\frac{y_1(x)y_2(x)y_3(x)+2x}{e^{3x}\sin x} is equal to ______________.
Not attempted
Your answer: —
Correct answer: 2
18
Let R\mathbb{R} denote the set of all real numbers. Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function such that f(x)>0f(x)>0 for all x∈Rx\in\mathbb{R}, and f(x+y)=f(x)f(y)f(x+y)=f(x)f(y) for all x,y∈Rx,y\in\mathbb{R}. Let the real numbers a1,a2,…,a50a_1,a_2,\dots,a_{50} be in an arithmetic progression. If f(a31)=64f(a25)f(a_{31})=64f(a_{25}), and ∑i=150f(ai)=3(225+1)\displaystyle\sum_{i=1}^{50} f(a_i) = 3(2^{25}+1), then the value of ∑i=630f(ai)\displaystyle\sum_{i=6}^{30} f(a_i) is _________________.
Not attempted
Your answer: —
Correct answer: 96
19
Let SS be the set of all seven-digit numbers that can be formed using the digits 0,10, 1 and 22. For example, 22102222210222 is in SS, but 02102220210222 is NOT in SS. Then the number of elements xx in SS such that at least one of the digits 00 and 11 appears exactly twice in xx, is equal to ____________.
Not attempted
Your answer: —
Correct answer: 762
20
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
21
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)
22
Let R\mathbb{R} denote the set of all real numbers. Let z1=1+2iz_1 = 1+2i and z2=3iz_2=3i be two complex numbers, where i=−1i=\sqrt{-1}. Let S={(x,y)∈R×R:∣x+iy−z1∣=2∣x+iy−z2∣}S = \{(x,y)\in\mathbb{R}\times\mathbb{R} : |x+iy-z_1| = 2|x+iy-z_2|\}. Then which of the following statements is (are) TRUE?
Not attempted
A
SS is a circle with centre (−13,103)\left(-\dfrac13,\dfrac{10}{3}\right)
B
SS is a circle with centre (13,83)\left(\dfrac13,\dfrac{8}{3}\right)
C
SS is a circle with radius 23\dfrac{\sqrt2}{3}
D
SS is a circle with radius 223\dfrac{2\sqrt2}{3}
23
Let L1L_1 be the line of intersection of the planes given by the equations 2x+3y+z=4andx+2y+z=52x+3y+z=4 \quad \text{and} \quad x+2y+z=5. Let L2L_2 be the line passing through the point P(2,−1,3)P(2,-1,3) and parallel to L1L_1. Let MM denote the plane given by the equation 2x+y−2z=62x+y-2z=6. Suppose that the line L2L_2 meets the plane MM at the point QQ. Let RR be the foot of the perpendicular drawn from PP to the plane MM. Then which of the following statements is (are) TRUE?
Not attempted
A
The length of the line segment PQPQ is 939\sqrt{3}
B
The length of the line segment QRQR is 1515
C
The area of △PQR\triangle PQR is 32234\dfrac{3}{2}\sqrt{234}
D
The acute angle between the line segments PQPQ and PRPR is cos⁡−1 ⁣(123)\cos^{-1}\!\left(\dfrac{1}{2\sqrt{3}}\right)
24
Consider the matrix P=(200020003)P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. Let the transpose of a matrix XX be denoted by XTX^T. Then the number of 3×33\times 3 invertible matrices QQ with integer entries, such that Q−1=QTandPQ=QPQ^{-1} = Q^T \quad \text{and} \quad PQ = QP, is
Not attempted
A
3232
B
88
C
1616
D
2424
25
Let R\mathbb{R} denote the set of all real numbers. Define the function f:R→Rf:\mathbb{R}\to\mathbb{R} by f(x)={2−2x2−x2sin⁡1xif x≠0,2if x=0.f(x) = \begin{cases} 2 - 2x^2 - x^2\sin\dfrac{1}{x} & \text{if } x\neq 0, \\ 2 & \text{if } x = 0. \end{cases} Then which one of the following statements is TRUE?
Not attempted
A
The function ff is NOT differentiable at x=0x=0
B
There is a positive real number δ\delta, such that ff is a decreasing function on the interval (0,δ)(0,\delta)
C
For any positive real number δ\delta, the function ff is NOT an increasing function on the interval (−δ,0)(-\delta,0)
D
x=0x=0 is a point of local minima of ff
26
Let α\alpha and β\beta be real numbers such that lim⁡x→01x3(α2∫0x11−t2 dt  +  βxcos⁡x)=2\displaystyle\lim_{x\to 0}\frac{1}{x^3}\left(\frac{\alpha}{2}\int_0^x \frac{1}{1-t^2}\,dt \;+\; \beta x\cos x\right) = 2. Then the value of α+β\alpha+\beta is __________.
Not attempted
Your answer: —
Correct answer: 2.4
27
Let R\mathbb{R} denote the set of all real numbers. For a real number xx, let [x][x] denote the greatest integer less than or equal to xx. Let nn denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I (P) The minimum value of nn for which the function f(x)=[10x3−45x2+60x+35n]f(x) = \left[\dfrac{10x^3-45x^2+60x+35}{n}\right] is continuous on the interval [1,2][1,2], is (Q) The minimum value of nn for which g(x)=(2n2−13n−15)(x3+3x)g(x)=(2n^2-13n-15)(x^3+3x), x∈Rx\in\mathbb{R}, is an increasing function on R\mathbb{R}, is (R) The smallest natural number nn which is greater than 55, such that x=3x=3 is a point of local minima of h(x)=(x2−9)n(x2+2x+3)h(x)=(x^2-9)^n(x^2+2x+3), is (S) Number of x0∈Rx_0\in\mathbb{R} such that l(x)=∑k=04(sin⁡∣x−k∣+cos⁡∣x−k+12∣)l(x) = \displaystyle\sum_{k=0}^{4}\left(\sin|x-k|+\cos\left|x-k+\dfrac12\right|\right), x∈Rx\in\mathbb{R}, is NOT differentiable at x0x_0, is List-II (1) 88 (2) 99 (3) 55 (4) 66 (5) 1010
Not attempted
A
(P) →\to (1) (Q) →\to (3) (R) →\to (2) (S) →\to (5)
B
(P) →\to (2) (Q) →\to (1) (R) →\to (4) (S) →\to (3)
C
(P) →\to (5) (Q) →\to (1) (R) →\to (4) (S) →\to (3)
D
(P) →\to (2) (Q) →\to (3) (R) →\to (1) (S) →\to (5)
28
Three students S1,S2S_1, S_2, and S3S_3 are given a problem to solve. Consider the following events: UU: At least one of S1,S2,S3S_1, S_2, S_3 can solve the problem, VV: S1S_1 can solve the problem, given that neither S2S_2 nor S3S_3 can solve the problem, WW: S2S_2 can solve the problem and S3S_3 cannot solve the problem, TT: S3S_3 can solve the problem. For any event EE, let P(E)P(E) denote the probability of EE. If P(U)=12P(U) = \dfrac{1}{2}, P(V)=110P(V) = \dfrac{1}{10}, and P(W)=112P(W) = \dfrac{1}{12}, then P(T)P(T) is equal to
Not attempted
A
1336\dfrac{13}{36}
B
13\dfrac{1}{3}
C
1960\dfrac{19}{60}
D
14\dfrac{1}{4}
29
Consider the following frequency distribution: Value458961211Frequency5f1f22113\begin{array}{c|c|c|c|c|c|c|c}\text{Value} & 4 & 5 & 8 & 9 & 6 & 12 & 11 \\ \hline \text{Frequency} & 5 & f_1 & f_2 & 2 & 1 & 1 & 3\end{array} Suppose that the sum of the frequencies is 1919 and the median of this frequency distribution is 66. For the given frequency distribution, let α\alpha denote the mean deviation about the mean, β\beta denote the mean deviation about the median, and σ2\sigma^2 denote the variance. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I (P) 7f1+9f27f_1+9f_2 is equal to (Q) 19α19\alpha is equal to (R) 19β19\beta is equal to (S) 19σ219\sigma^2 is equal to List-II (1) 146146 (2) 4747 (3) 4848 (4) 145145 (5) 5555
Not attempted
A
(P) →\to (5) (Q) →\to (3) (R) →\to (2) (S) →\to (4)
B
(P) →\to (5) (Q) →\to (2) (R) →\to (3) (S) →\to (1)
C
(P) →\to (5) (Q) →\to (3) (R) →\to (2) (S) →\to (1)
D
(P) →\to (3) (Q) →\to (2) (R) →\to (5) (S) →\to (4)
30
Let R\mathbb{R} denote the set of all real numbers. Let ai,bi∈Ra_i, b_i \in \mathbb{R} for i∈{1,2,3}i \in \{1, 2, 3\}. Define the functions f:R→Rf:\mathbb{R}\to\mathbb{R}, g:R→Rg:\mathbb{R}\to\mathbb{R}, and h:R→Rh:\mathbb{R}\to\mathbb{R} by f(x)=a1+10x+a2x2+a3x3+x4f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4, g(x)=b1+3x+b2x2+b3x3+x4g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, h(x)=f(x+1)−g(x+2)h(x) = f(x+1) - g(x+2). If f(x)≠g(x)f(x) \neq g(x) for every x∈Rx \in \mathbb{R}, then the coefficient of x3x^3 in h(x)h(x) is
Not attempted
A
88
B
22
C
−4-4
D
−6-6
31
Let N\mathbb{N} denote the set of all natural numbers, and Z\mathbb{Z} denote the set of all integers. Consider the functions f:N→Zf:\mathbb{N}\to\mathbb{Z} and g:Z→Ng:\mathbb{Z}\to\mathbb{N} defined by f(n)={(n+1)/2if n is odd,(4−n)/2if n is even,g(n)={3+2nif n≥0,−2nif n<0.f(n) = \begin{cases} (n+1)/2 & \text{if } n \text{ is odd}, \\ (4-n)/2 & \text{if } n \text{ is even}, \end{cases} \qquad g(n) = \begin{cases} 3+2n & \text{if } n\ge 0, \\ -2n & \text{if } n<0. \end{cases} Define (g∘f)(n)=g(f(n))(g\circ f)(n) = g(f(n)) for all n∈Nn\in\mathbb{N}, and (f∘g)(n)=f(g(n))(f\circ g)(n)=f(g(n)) for all n∈Zn\in\mathbb{Z}. Then which of the following statements is (are) TRUE?
Not attempted
A
g∘fg\circ f is NOT one-one and g∘fg\circ f is NOT onto
B
f∘gf\circ g is NOT one-one but f∘gf\circ g is onto
C
gg is one-one and gg is onto
D
ff is NOT one-one but ff is onto
32
Let the set of all relations RR on the set {a,b,c,d,e,f}\{a,b,c,d,e,f\}, such that RR is reflexive and symmetric, and RR contains exactly 1010 elements, be denoted by S\mathcal{S}. Then the number of elements in S\mathcal{S} is _________________.
Not attempted
Your answer: —
Correct answer: 105
Reflexivity forces all 66 diagonal pairs into RR already; symmetry means the remaining off-diagonal elements come in pairs {(x,y),(y,x)}\{(x,y),(y,x)\} — so just count how many such pairs are needed to reach 1010 elements, and choose that many from all possible pairs.
Deconstruction\textbf{Deconstruction}: Count reflexive, symmetric relations on a 66-element set with exactly 1010 ordered pairs. Strategy\textbf{Strategy}: Reflexivity is a fixed, mandatory contribution (the 66 pairs (x,x)(x,x)); symmetry forces off-diagonal pairs to appear in twos. Reduce to choosing unordered pairs of distinct elements. Execution\textbf{Execution}: Reflexive ⇒\Rightarrow all 66 pairs (a,a),(b,b),…,(f,f)(a,a),(b,b),\dots,(f,f) must be in RR. This uses up 66 of the required 1010 elements, leaving exactly 44 more elements to place. Symmetric ⇒\Rightarrow off-diagonal elements occur in pairs: if (x,y)∈R(x,y)\in R with x≠yx\neq y, then (y,x)∈R(y,x)\in R too. So the 44 remaining elements must consist of kk symmetric pairs contributing 2k2k elements; we need 2k=4⇒k=22k=4 \Rightarrow k=2. Number of unordered pairs {x,y}\{x,y\}, x≠yx\neq y, from a 66-element set: (62)=15\binom{6}{2}=15. We need to choose 22 of these 1515 unordered pairs to \"activate\" (each activated pair contributes its 22 ordered off-diagonal elements to RR): (152)=15⋅142=105\binom{15}{2} = \dfrac{15\cdot14}{2}=105. Verification\textbf{Verification}: Total elements check: 66 (diagonal, mandatory) +2×2+ 2\times2 (two chosen symmetric pairs) =10=10. ✓\checkmark matches the required count exactly. Also sanity-check the counting method on a smaller case: for a 22-element set with exactly 22 elements required (just the diagonal, k=0k=0), the formula gives (10)=1\binom{1}{0}=1 relation (the identity relation) — correct, since reflexive+symmetric with no extra pairs is uniquely the equality relation. Final Answer\textbf{Final Answer}: 105105.