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JEE Advanced · Mathematics · 2026

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1
For α∈R\alpha\in\mathbb R, let [α][\alpha] denote the greatest integer ≤α\le\alpha. Consider f,g:(−3,3)→(−∞,∞)f,g:(-3,3)\to(-\infty,\infty) defined by f(x)=[x3]log⁡e(1+sin⁡2(π(x−[x])))f(x)=[x^3]\log_e(1+\sin^2(\pi(x-[x]))) and g(x)=x3sin⁡2(πlog⁡e(1+x−[x]))g(x)=x^3\sin^2(\pi\log_e(1+x-[x])). Let A={x∈(−3,3):f discontinuous at x}A=\{x\in(-3,3): f \text{ discontinuous at } x\} and B={x∈(−3,3):g discontinuous at x}B=\{x\in(-3,3): g \text{ discontinuous at }x\}. Find ∣A∣+2∣B∣−∣A∩B∣|A|+2|B|-|A\cap B|.
Not attempted
Your answer: —
Correct answer: 56
2
A bookshelf has 6 distinct Mathematics books and 5 distinct Physics books. From these 11 books, 6 are chosen at random. Let XX be the absolute difference between the number of Mathematics and Physics books chosen. If α\alpha is the mean of XX, find 77α77\alpha.
Not attempted
Your answer: —
Correct answer: 100
3
A data set of 10 observations x1,…,x10x_1,\ldots,x_{10} has mean 5 and variance 7. The first 8 observations have mean 4 and variance 3.5, and x9<x10x_9<x_{10}. Find 3x9+2x103x_9+2x_{10}.
Not attempted
Your answer: —
Correct answer: 44
4
Let A={1,2,3,4,5}A=\{1,2,3,4,5\} and B={1,2,3,4,5,6,7}B=\{1,2,3,4,5,6,7\}. Let SS be the set of functions f:A→Bf:A\to B with f(2)≠2,f(4)≠4f(2)\ne2, f(4)\ne4. Let T={f∈S:there exists g:B→N with g(f(x))=2x for all x∈A}T=\{f\in S: \text{there exists } g:B\to\mathbb N \text{ with } g(f(x))=2^x \text{ for all } x\in A\}. Find ∣T∣|T|.
Not attempted
Your answer: —
Correct answer: 1860
5
Let i=−1i=\sqrt{-1}. Consider S=(0−110)S=\begin{pmatrix}0&-1\\1&0\end{pmatrix} and T=(1101)T=\begin{pmatrix}1&1\\0&1\end{pmatrix}. Let a,b,c,da,b,c,d be real numbers with ST=(abcd)ST=\begin{pmatrix}a&b\\c&d\end{pmatrix}. Let H={x+iy:x,y∈R, y>0}H=\{x+iy: x,y\in\mathbb R,\ y>0\}. Then which of the following statements is (are) TRUE?
Not attempted
A
b+iad+ic=i\dfrac{b+ia}{d+ic}=i
B
If ω=−1+i32\omega=\dfrac{-1+i\sqrt3}2, then aω+bcω+d=ω\dfrac{a\omega+b}{c\omega+d}=\omega
C
If m>2m>2 is an integer such that (ST)2=(ST)m(ST)^2=(ST)^m, then mm is an integer multiple of 88
D
If z∈Hz\in H, then az+bcz+d∈H\dfrac{az+b}{cz+d}\in H
6
Let a,b,ca,b,c be positive integers in arithmetic progression such that ax2+bx+c=0ax^2+bx+c=0 has only integer solutions. Then which of the following statements is (are) TRUE?
Not attempted
A
c−bc-b is an integer multiple of aa
B
Both roots of ax2+bx+c=0ax^2+bx+c=0 are odd integers
C
If c=15c=15, then ab=8ab=8
D
If b=8b=8, then x=3x=3 is a root of ax2+bx+c=0ax^2+bx+c=0
7
Consider f(x)=d10dx10((x2−1)10)f(x)=\dfrac{d^{10}}{dx^{10}}\left((x^2-1)^{10}\right) for all x∈Rx\in\mathbb R. Then which of the following statements is (are) TRUE?
Not attempted
A
The coefficient of x8x^8 in the polynomial f(x)f(x) is (−10)(18!8!)(-10)\left(\dfrac{18!}{8!}\right)
B
The value of f(1)+f(−1)f(1)+f(-1) is equal to 10! 21110!\,2^{11}
C
The degree of the polynomial f(x)f(x) is 1010
D
The constant term of the polynomial f(x)f(x) is −(10!5!)-\left(\dfrac{10!}{5!}\right)
8
Let LL be the line through P(1,2,−1)P(1,2,-1) and Q(2,3,1)Q(2,3,1). Let SS be the foot of the perpendicular from R(4,−1,5)R(4,-1,5) to LL. Another line through RR meets LL at TT such that SS divides PTPT internally in the ratio ∣PS∣:∣ST∣=1:2|PS|:|ST|=1:2. Then which of the following statements is (are) TRUE?
Not attempted
A
The orthocentre of triangle PRTPRT is (235,−4,315)\left(\dfrac{23}5,-4,\dfrac{31}5\right)
B
The orthocentre of triangle PRTPRT is (4,3,5)(4,3,5)
C
The area of triangle PRTPRT is 656\sqrt5
D
The area of triangle PRTPRT is 18518\sqrt5
9
The value of the definite integral ∫0213x+3 dx\displaystyle\int_0^2 \frac{1}{3^x+3}\,dx is
Not attempted
A
12\frac12
B
13\frac13
C
log⁡e33\dfrac{\log_e3}{3}
D
log⁡e32\dfrac{\log_e3}{2}
10
Consider C1:y=e−xC_1: y=e^{-x} for x∈[0,10π]x\in[0,10\pi] and C2:y=e−x(sin⁡x+cos⁡x)C_2: y=e^{-x}(\sin x+\cos x) for x∈[0,10π]x\in[0,10\pi]. Let nn be the total number of points of intersection of C1,C2C_1,C_2. Find nn.
Not attempted
Your answer: —
Correct answer: 11
11
Using the intersection points α1<α2<⋯<α11\alpha_1<\alpha_2<\cdots<\alpha_{11} from the previous question, let β\beta be the area enclosed between C1,C2C_1,C_2 and the lines x=α1, x=α4x=\alpha_1,\ x=\alpha_4. Find −1πlog⁡e(β−2e−π/2)-\frac1\pi\log_e\left(\beta-2e^{-\pi/2}\right).
Not attempted
Your answer: —
Correct answer: 2.5
12
Let y:(−∞,∞)→(0,∞)y:(-\infty,\infty)\to(0,\infty) be the solution of dydx=e5xy3+y3ex+exy4\frac{dy}{dx}=\frac{e^{5x}y^3+y^3}{e^x+e^xy^4} satisfying y(0)=1/2y(0)=1/\sqrt2. Find y(log⁡e2)y(\log_e2).
Not attempted
A
5+352\sqrt{\dfrac{5+\sqrt{35}}{2}}
B
7+532\sqrt{\dfrac{7+\sqrt{53}}{2}}
C
7+532\dfrac{7+\sqrt{53}}{2}
D
5+352\dfrac{5+\sqrt{35}}{2}
13
Let y=f(x)y=f(x) be the real-valued function on (0,∞)(0,\infty) satisfying y(1)=0y(1)=0 and xdydx=y−x3x\frac{dy}{dx}=y-x^3. Then which of the following statements is (are) TRUE?
Not attempted
A
The function ff has a local minimum at x=1/3x=1/\sqrt3
B
The function ff has a local maximum at x=1/3x=1/\sqrt3
C
The function ff is increasing in the interval (1,2)(1,2)
D
If g(x)=4x3−5x2+32xg(x)=4x^3-5x^2+\frac32x for x>0x>0, the number of elements in {x∈(0,∞):f(x)=g(x)}\{x\in(0,\infty): f(x)=g(x)\} is 2
14
Let TT be the tangent to the parabola y2=16xy^2=16x at the point (64,32)(64,32). Let LL be the tangent to the same parabola at another point (x1,y1)(x_1,y_1) on the parabola. If LL and TT are perpendicular, then the distance between (x1,y1)(x_1,y_1) and the focus of the parabola is
Not attempted
A
154\frac{15}{4}
B
44
C
174\frac{17}{4}
D
55
15
Consider the ellipse EE: x218+y212=1\frac{x^2}{18}+\frac{y^2}{12}=1. Let HH be the hyperbola whose eccentricity is the reciprocal of the eccentricity of EE and whose foci are the same as that of EE. Let P,QP,Q be the points of intersection of HH and the parabola 5 y=x2\sqrt5\,y=x^2 in the first quadrant. Let dd be the distance between P,QP,Q. If a,ba,b are integers such that d2=a+b5d^2=a+b\sqrt5, find a−ba-b.
Not attempted
Your answer: —
Correct answer: 18
16
Let θ\theta be the acute angle between the tangents to the ellipses x2+4y2=1x^2+4y^2=1 and 4x2+y2=14x^2+y^2=1 at their first-quadrant intersection point PP. Find 4tan⁡θ4\tan\theta.
Not attempted
Your answer: —
Correct answer: 7.5
17
If α\alpha is the area of the common region that lies inside both the ellipses x2+4y2=1x^2+4y^2=1 and 4x2+y2=14x^2+y^2=1, find cot⁡α\cot\alpha.
Not attempted
Your answer: —
Correct answer: 0.75
18
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
19
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
Not attempted
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
20
Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I\textbf{List-I} (P) If α,β\alpha,\beta are the distinct roots of x2+x+1=0x^2+x+1=0, the quadratic equation with roots 1(α+1)2026\dfrac{1}{(\alpha+1)^{2026}} and 1(β+1)2026\dfrac{1}{(\beta+1)^{2026}} is (Q) If α,β\alpha,\beta are as above, the quadratic equation with roots 1(α+1)2027\dfrac{1}{(\alpha+1)^{2027}} and 1(β+1)2027\dfrac{1}{(\beta+1)^{2027}} is (R) If γ,δ\gamma,\delta are the distinct roots of x2−x+1=0x^2-x+1=0, the value of 1(γ−1)2026+1(δ−1)2026\dfrac{1}{(\gamma-1)^{2026}}+\dfrac{1}{(\delta-1)^{2026}} is (S) If p,rp,r are the distinct roots of x2+x−1=0x^2+x-1=0, the value of 1(p+1)3+1(r+1)3\dfrac{1}{(p+1)^3}+\dfrac{1}{(r+1)^3} is List-II\textbf{List-II} (1) x2+x+1=0x^2+x+1=0 (2) x2−x+1=0x^2-x+1=0 (3) x2+x−1=0x^2+x-1=0 (4) −1-1 (5) −4-4
Not attempted
A
P→1,Q→2,R→5,S→4P\to1,Q\to2,R\to5,S\to4
B
P→3,Q→1,R→4,S→5P\to3,Q\to1,R\to4,S\to5
C
P→1,Q→2,R→4,S→5P\to1,Q\to2,R\to4,S\to5
D
P→2,Q→3,R→5,S→4P\to2,Q\to3,R\to5,S\to4
21
The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets exactly 6 pens, is
Not attempted
Your answer: —
Correct answer: 206
22
Considering only the principal values of the inverse trigonometric functions, the value of cot⁡−1(cot⁡(−11))+10sin⁡ ⁣(2cos⁡−1 ⁣(12))+10sin⁡(2tan⁡−1(2))\cot^{-1}(\cot(-11)) + 10\sin\!\left(2\cos^{-1}\!\left(\dfrac{1}{\sqrt2}\right)\right) + 10\sin(2\tan^{-1}(2)) is
Not attempted
A
3π+73\pi+7
B
77
C
4π+74\pi+7
D
3π−53\pi-5
23
α=(1−2cos⁡π11)(1−2cos⁡3π11)(1−2cos⁡9π11)(1−2cos⁡27π11)(1−2cos⁡81π11).\alpha = \left(1-2\cos\dfrac{\pi}{11}\right)\left(1-2\cos\dfrac{3\pi}{11}\right)\left(1-2\cos\dfrac{9\pi}{11}\right)\left(1-2\cos\dfrac{27\pi}{11}\right)\left(1-2\cos\dfrac{81\pi}{11}\right). Then the value of 5−α25-\alpha^2 is
Not attempted
Your answer: —
Correct answer: 4
24
Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I\textbf{List-I} (P) The number of elements in {x∈[−π,π]:sin⁡6x+cos⁡4x=1}\{x\in[-\pi,\pi]: \sin^6x+\cos^4x=1\} (Q) The number of elements in {x∈[−π/2,π/2]:sin⁡2x+cos⁡6x=1}\{x\in[-\pi/2,\pi/2]: \sin^2x+\cos^6x=1\} (R) The number of elements in {x∈[−π,π]:cos⁡2(x/2)−sin⁡2x=1/2}\{x\in[-\pi,\pi]: \cos^2(x/2)-\sin^2x=1/2\} (S) The number of elements in {x∈[−2π,2π]:6sin⁡2(x/2)−cos⁡3x=3}\{x\in[-2\pi,2\pi]: 6\sin^2(x/2)-\cos3x=3\} List-II\textbf{List-II} (1) is 1 (2) is 2 (3) is 3 (4) is 4 (5) is 5
Not attempted
A
P→2,Q→5,R→3,S→4P\to2,Q\to5,R\to3,S\to4
B
P→5,Q→3,R→2,S→4P\to5,Q\to3,R\to2,S\to4
C
P→5,Q→4,R→1,S→3P\to5,Q\to4,R\to1,S\to3
D
P→4,Q→3,R→2,S→5P\to4,Q\to3,R\to2,S\to5
25
Let PP be the point on the parabola y=x2y=x^2 such that the slope of the tangent at PP is 44. Let QQ be the point in the first quadrant on the circle x2+y2=2x^2+y^2=2 such that the slope of the tangent at QQ is −1-1. Let RR be the point in the first quadrant on the ellipse x2+4y2=8x^2+4y^2=8 such that the slope of the tangent at RR is −12-\dfrac{1}{2}. Then the radius of the circle passing through P,Q,RP, Q, R is
Not attempted
A
10\sqrt{10}
B
5\sqrt{5}
C
5/2\sqrt{5/2}
D
252\sqrt{5}
26
Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I\textbf{List-I} (P) The circle with centre (1,2)(1,2) and touching the line 3x+4y=13x+4y=1, passes through (Q) The common tangent to the circle x2+y2=2x^2+y^2=2 and the parabola y2=8xy^2=8x with positive slope, passes through (R) Let MM be the end point of the latus rectum of the ellipse 3x2+4y2=483x^2+4y^2=48 in the first quadrant. The normal to the ellipse at MM passes through (S) Let HH be the hyperbola centred at the origin with one focus at (5,0)(5,0) and one directrix 5x+16=05x+16=0. Then HH passes through List-II\textbf{List-II} (1) the point (1,1)(1,1) (2) the point (7,9)(7,9) (3) the point (3,2)(3,2) (4) the point (2,5)(2,5) (5) the point (8,33)(8, 3\sqrt{3})
Not attempted
A
P→3,Q→4,R→1,S→2P\to3, Q\to4, R\to1, S\to2
B
P→3,Q→2,R→1,S→5P\to3, Q\to2, R\to1, S\to5
C
P→3,Q→2,R→4,S→5P\to3, Q\to2, R\to4, S\to5
D
P→4,Q→1,R→2,S→3P\to4, Q\to1, R\to2, S\to3
27
Consider the function f:(0,∞)→(−∞,∞)f: (0, \infty) \to (-\infty, \infty) given by f(x)=x log⁡e(x)−x+1f(x) = \sqrt{x}\,\log_e(x) - x + 1. Then which one of the following statements is TRUE?
Not attempted
A
The derivative of the function ff is decreasing in the interval (0,1)(0,1)
B
The function ff has a local maximum at some point a∈(0,∞)a \in (0,\infty)
C
The function ff has a local minimum at some point b∈(0,∞)b \in (0,\infty)
D
The function ff has NEITHER a point of local maximum NOR a point of local minimum in (0,∞)(0,\infty)
28
Let f:R→Rf:\mathbb R\to\mathbb R be an arbitrary function and let g:R→Rg:\mathbb R\to\mathbb R be defined by g(x)=xf(x)g(x)=xf(x) for all x∈Rx\in\mathbb R. Then which of the following statements is (are) TRUE?
Not attempted
A
The function gg is always continuous at x=0x=0
B
If ff is continuous at x=0x=0, then gg is differentiable at x=0x=0
C
If gg is differentiable at x=0x=0, then ff is continuous at x=0x=0
D
If gg is differentiable at x=0x=0, then lim⁡x→0f(x)\lim_{x\to0}f(x) exists
29
Consider f:(−π2,π2)→(−∞,∞)f:\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\to(-\infty,\infty) defined by f(x)=(∣x∣+∣x−1∣)sin⁡x+[xsin⁡x]f(x)=(|x|+|x-1|)\sin x + [x\sin x], where [⋅][\cdot] is the greatest integer function. Let α\alpha be the number of points in (−π2,π2)\left(-\dfrac\pi2,\dfrac\pi2\right) where ff is NOT continuous, and β\beta the number of points where ff is NOT differentiable. Find α+β\alpha+\beta.
Not attempted
Your answer: —
Correct answer: 5
30
Let S={1,2,…,10}S=\{1,2,\dots,10\}. Consider X={R:R is an equivalence relation on S with exactly 42 elements}X=\{R : R\text{ is an equivalence relation on }S\text{ with exactly }42\text{ elements}\} (as a set of ordered pairs). Find the number of elements in XX.
Not attempted
Your answer: —
Correct answer: 2520
31
Let PP be the plane containing the line x−12=y−33=z+21\dfrac{x-1}{2}=\dfrac{y-3}{3}=\dfrac{z+2}{1} and perpendicular to the plane x+2y+3z=4x+2y+3z=4. Let P1P_1 be the plane through (4,2,2)(4,2,2) parallel to PP. Then which of the following statements is (are) TRUE?
Not attempted
A
The equation of the plane PP is 7x−5y+z=−107x-5y+z=-10
B
The distance between the planes PP and P1P_1 is 3030
C
The distance of the plane PP from the origin is 232\sqrt3
D
The acute angle between PP and 2x+2y+z=32x+2y+z=3 is cos⁡−1 ⁣(133)\cos^{-1}\!\left(\dfrac{1}{3\sqrt3}\right)
32
Suppose Box I contains 6 red and 9 green balls, and Box II contains 8 red and 12 green balls. All balls are mixed and one is chosen at random. Let E1,E2E_1,E_2 be the events that the chosen ball belonged to Box I, Box II respectively, and let F1,F2F_1,F_2 be the events that the chosen ball is red, green respectively. Then which of the following statements is(are) TRUE?
Not attempted
A
The events E1E_1 and F1F_1 are independent
B
The events E2E_2 and F2F_2 are dependent
C
P(F1∣E1)=P(F1∣E2)P(F_1|E_1) = P(F_1|E_2)
D
P(F1∣E1)>P(F2∣E2)P(F_1|E_1) > P(F_2|E_2)
33
Which one of the following matrices can be obtained by performing elementary row transformations on the 3×33\times3 identity matrix? (A) (111111111)\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix} (B) (111234121)\begin{pmatrix}1&1&1\\2&3&4\\1&2&1\end{pmatrix} (C) (111234258)\begin{pmatrix}1&1&1\\2&3&4\\2&5&8\end{pmatrix} (D) (111−112023)\begin{pmatrix}1&1&1\\-1&1&2\\0&2&3\end{pmatrix}
Not attempted
A
(111111111)\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}
B
(111234121)\begin{pmatrix}1&1&1\\2&3&4\\1&2&1\end{pmatrix}
C
(111234258)\begin{pmatrix}1&1&1\\2&3&4\\2&5&8\end{pmatrix}
D
(111−112023)\begin{pmatrix}1&1&1\\-1&1&2\\0&2&3\end{pmatrix}
34
Consider the matrix M=(2−110)M=\begin{pmatrix}2&-1\\1&0\end{pmatrix}. Let p,q,r,s,a,b,c,dp,q,r,s,a,b,c,d be integers such that M26=(pqrs)M^{26}=\begin{pmatrix}p&q\\r&s\end{pmatrix} and ∑k=126Mk=(abcd)\displaystyle\sum_{k=1}^{26}M^k=\begin{pmatrix}a&b\\c&d\end{pmatrix}. Then which of the following statements is (are) TRUE?
Not attempted
A
There exists a 2×22\times2 invertible matrix NN with real entries such that MN=N(1101)MN=N\begin{pmatrix}1&1\\0&1\end{pmatrix}
B
The value of aa is 378378
C
For any two given integers m,nm,n, there exist unique integers x,yx,y such that px+qy=mpx+qy=m and rx+sy=nrx+sy=n
D
For each positive real number tt, the system (a+t)x+by=1, cx+(d+t)y=−1(a+t)x+by=1,\ cx+(d+t)y=-1 has a unique solution
Find the eigenvalues of MM (they are repeated), write M=I+NM=I+N with NN nilpotent, and use Mk=I+kNM^k=I+kN.
tr(M)=2, det⁡(M)=1\text{tr}(M)=2,\ \det(M)=1, so the characteristic equation is (λ−1)2=0(\lambda-1)^2=0, a repeated eigenvalue λ=1\lambda=1. Let N=M−I=(1−11−1)N=M-I=\begin{pmatrix}1&-1\\1&-1\end{pmatrix}; direct computation gives N2=0N^2=0. So Mk=(I+N)k=I+kNM^k=(I+N)^k=I+kN. M26=I+26N=(27−2626−25)M^{26}=I+26N=\begin{pmatrix}27&-26\\26&-25\end{pmatrix}, so (p,q,r,s)=(27,−26,26,−25)(p,q,r,s)=(27,-26,26,-25). ∑k=126Mk=26I+(∑k=126k)N=26I+351N=(377−351351−325)\sum_{k=1}^{26}M^k = 26I+\left(\sum_{k=1}^{26}k\right)N = 26I+351N = \begin{pmatrix}377&-351\\351&-325\end{pmatrix}, so a=377,b=−351,c=351,d=−325a=377,b=-351,c=351,d=-325. (A)\textbf{(A)}: Since M≠IM\ne I but (M−I)2=0(M-I)^2=0, MM has a single non-trivial Jordan block for eigenvalue 11, exactly the Jordan form (1101)\begin{pmatrix}1&1\\0&1\end{pmatrix}. So an invertible change-of-basis NN with MN=N(1101)MN=N\begin{pmatrix}1&1\\0&1\end{pmatrix} exists. TRUE\textbf{TRUE}. (B)\textbf{(B)}: We computed a=377a=377, not 378378. FALSE\textbf{FALSE}. (C)\textbf{(C)}: det⁡(M26)=det⁡(M)26=1\det(M^{26})=\det(M)^{26}=1, a unit in Z\mathbb{Z}, so M26M^{26}'s inverse has integer entries, a unique integer solution (x,y)(x,y) exists for every integer (m,n)(m,n). TRUE\textbf{TRUE}. (D)\textbf{(D)}: The matrix is (a+tbcd+t)=(26+t)I+351N\begin{pmatrix}a+t&b\\c&d+t\end{pmatrix}=(26+t)I+351N. Its determinant works out to (t+26)2>0(t+26)^2>0 for all t>0t>0, so it is always invertible, a unique solution exists for every t>0t>0. TRUE\textbf{TRUE}. Final Answer\textbf{Final Answer}: (A), (C), (D)