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JEE Advanced · Mathematics · 2020

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1
Let the function f:(0,π)→Rf:(0,\pi) \to \mathbb{R} be defined by f(θ)=(sin⁡θ+cos⁡θ)2+(sin⁡θ−cos⁡θ)4f(\theta) = (\sin\theta+\cos\theta)^2 + (\sin\theta-\cos\theta)^4. Suppose the function ff has a local minimum at θ\theta precisely when θ∈{λ1π,…,λrπ}\theta \in \{\lambda_1\pi,\ldots,\lambda_r\pi\}, where 0<λ1<⋯<λr<10<\lambda_1<\cdots<\lambda_r<1. Then the value of λ1+⋯+λr\lambda_1+\cdots+\lambda_r is ____
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Your answer: —
Correct answer: 0.5
2
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function such that its derivative f′f' is continuous and f(π)=−6f(\pi)=-6. If F:[0,π]→RF:[0,\pi]\to\mathbb{R} is defined by F(x)=∫0xf(t) dtF(x)=\displaystyle\int_0^x f(t)\,dt, and if ∫0π(f′(x)+F(x))cos⁡x dx=2\displaystyle\int_0^\pi \left(f'(x)+F(x)\right)\cos x\,dx = 2, then the value of f(0)f(0) is ____
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Your answer: —
Correct answer: 4
3
Let the function f:[0,1]→Rf:[0,1] \to \mathbb{R} be defined by f(x)=4x4x+2f(x) = \dfrac{4^x}{4^x+2}. Then the value of f(140)+f(240)+f(340)+⋯+f(3940)−f(12)f\left(\dfrac1{40}\right)+f\left(\dfrac2{40}\right)+f\left(\dfrac3{40}\right)+\cdots+f\left(\dfrac{39}{40}\right)-f\left(\dfrac12\right) is ____
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Your answer: —
Correct answer: 19
4
Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If pp is the probability that this perfect square is an odd number, then the value of 14p14p is ____
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Your answer: —
Correct answer: 8
5
An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is ____
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Your answer: —
Correct answer: 495
6
In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is ____
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Your answer: —
Correct answer: 1080
7
For nonnegative integers ss and rr, let (sr)={s!r! (s−r)!if r≤s,0if r>s.\dbinom{s}{r} = \begin{cases} \dfrac{s!}{r!\,(s-r)!} & \text{if } r \le s, \\ 0 & \text{if } r>s. \end{cases} For positive integers mm and nn, let g(m,n)=∑p=0m+nf(m,n,p)(n+pp)g(m,n) = \displaystyle\sum_{p=0}^{m+n} \dfrac{f(m,n,p)}{\dbinom{n+p}{p}}, where for any nonnegative integer pp, f(m,n,p)=∑i=0p(mi)(n+ip)(p+np−i)f(m,n,p) = \displaystyle\sum_{i=0}^{p} \dbinom{m}{i}\dbinom{n+i}{p}\dbinom{p+n}{p-i}. Then which of the following statements is/are TRUE?
Not attempted
A
g(m,n)=g(n,m)g(m,n)=g(n,m) for all positive integers m,nm,n
B
g(m,n+1)=g(m+1,n)g(m,n+1)=g(m+1,n) for all positive integers m,nm,n
C
g(2m,2n)=2 g(m,n)g(2m,2n)=2\,g(m,n) for all positive integers m,nm,n
D
g(2m,2n)=(g(m,n))2g(2m,2n)=(g(m,n))^2 for all positive integers m,nm,n
8
Let aa and bb be positive real numbers. Suppose PQ→=ai^+bj^\overrightarrow{PQ}=a\hat i+b\hat j and PS→=ai^−bj^\overrightarrow{PS}=a\hat i - b\hat j are adjacent sides of a parallelogram PQRSPQRS. Let u⃗\vec u and v⃗\vec v be the projection vectors of w⃗=i^+j^\vec w = \hat i+\hat j along PQ→\overrightarrow{PQ} and PS→\overrightarrow{PS}, respectively. If ∣u⃗∣+∣v⃗∣=∣w⃗∣|\vec u|+|\vec v|=|\vec w| and if the area of the parallelogram PQRSPQRS is 88, then which of the following statements is/are TRUE?
Not attempted
A
a+b=4a+b=4
B
a−b=2a-b=2
C
The length of the diagonal PRPR of the parallelogram PQRSPQRS is 44
D
w⃗\vec w is an angle bisector of the vectors PQ→\overrightarrow{PQ} and PS→\overrightarrow{PS}
9
Let α,β,γ,δ\alpha,\beta,\gamma,\delta be real numbers such that α2+β2+γ2≠0\alpha^2+\beta^2+\gamma^2 \ne 0 and α+γ=1\alpha+\gamma=1. Suppose the point (3,2,−1)(3,2,-1) is the mirror image of the point (1,0,−1)(1,0,-1) with respect to the plane αx+βy+γz=δ\alpha x+\beta y+\gamma z=\delta. Then which of the following statements is/are TRUE?
Not attempted
A
α+β=2\alpha+\beta=2
B
δ−γ=3\delta-\gamma=3
C
δ+β=4\delta+\beta=4
D
α+β+γ=δ\alpha+\beta+\gamma=\delta
10
Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→Rg:\mathbb{R}\to\mathbb{R} be functions satisfying f(x+y)=f(x)+f(y)+f(x)f(y)f(x+y)=f(x)+f(y)+f(x)f(y) and f(x)=xg(x)f(x)=xg(x) for all x,y∈Rx,y \in \mathbb{R}. If lim⁡x→0g(x)=1\displaystyle\lim_{x\to0} g(x)=1, then which of the following statements is/are TRUE?
Not attempted
A
ff is differentiable at every x∈Rx \in \mathbb{R}
B
If g(0)=1g(0)=1, then gg is differentiable at every x∈Rx \in \mathbb{R}
C
The derivative f′(1)f'(1) is equal to 11
D
The derivative f′(0)f'(0) is equal to 11
11
Let aa and bb be positive real numbers such that a>1a>1 and b<ab<a. Let PP be a point in the first quadrant that lies on the hyperbola x2a2−y2b2=1\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1. Suppose the tangent to the hyperbola at PP passes through the point (1,0)(1,0), and suppose the normal to the hyperbola at PP cuts off equal intercepts on the coordinate axes. Let Δ\Delta denote the area of the triangle formed by the tangent at PP, the normal at PP and the xx-axis. If ee denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
Not attempted
A
1<e<21 < e < \sqrt2
B
2<e<2\sqrt2 < e < 2
C
Δ=a4\Delta = a^4
D
Δ=b4\Delta = b^4
12
Let bb be a nonzero real number. Suppose f:R→Rf:\mathbb{R}\to\mathbb{R} is a differentiable function such that f(0)=1f(0)=1. If the derivative f′f' of ff satisfies the equation f′(x)=f(x)b2+x2f'(x) = \dfrac{f(x)}{b^2+x^2} for all x∈Rx \in \mathbb{R}, then which of the following statements is/are TRUE?
Not attempted
A
If b>0b>0, then ff is an increasing function
B
If b<0b<0, then ff is a decreasing function
C
f(x)f(−x)=1f(x)f(-x)=1 for all x∈Rx \in \mathbb{R}
D
f(x)−f(−x)=0f(x)-f(-x)=0 for all x∈Rx \in \mathbb{R}
13
Let the functions f:(−1,1)→Rf:(-1,1) \to \mathbb{R} and g:(−1,1)→(−1,1)g:(-1,1) \to (-1,1) be defined by f(x)=∣2x−1∣+∣2x+1∣f(x)=|2x-1|+|2x+1| and g(x)=x−[x]g(x)=x-[x], where [x][x] denotes the greatest integer less than or equal to xx. Let f∘g:(−1,1)→Rf \circ g : (-1,1) \to \mathbb{R} be the composite function defined by (f∘g)(x)=f(g(x))(f\circ g)(x) = f(g(x)). Suppose cc is the number of points in the interval (−1,1)(-1,1) at which f∘gf\circ g is NOT continuous, and suppose dd is the number of points in the interval (−1,1)(-1,1) at which f∘gf\circ g is NOT differentiable. Then the value of c+dc+d is ____
Not attempted
Your answer: —
Correct answer: 4
14
The value of the limit lim⁡x→π242(sin⁡3x+sin⁡x)(2sin⁡2xsin⁡3x2+cos⁡5x2)−(2+2cos⁡2x+cos⁡3x2)\displaystyle\lim_{x \to \frac{\pi}{2}} \dfrac{4\sqrt2(\sin3x+\sin x)}{\left(2\sin2x\sin\frac{3x}{2}+\cos\frac{5x}{2}\right) - \left(\sqrt2+\sqrt2\cos2x+\cos\frac{3x}{2}\right)} is ____
Not attempted
Your answer: —
Correct answer: 8
15
The trace of a square matrix is defined to be the sum of its diagonal entries. If AA is a 2×22\times2 matrix such that the trace of AA is 33 and the trace of A3A^3 is −18-18, then the value of the determinant of AA is ____
Not attempted
Your answer: —
Correct answer: 5
16
Let OO be the centre of the circle x2+y2=r2x^2+y^2=r^2, where r>52r > \dfrac{\sqrt5}{2}. Suppose PQPQ is a chord of this circle and the equation of the line passing through PP and QQ is 2x+4y=52x+4y=5. If the centre of the circumcircle of the triangle OPQOPQ lies on the line x+2y=4x+2y=4, then the value of rr is ____
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Your answer: —
Correct answer: 2
17
The probability that a missile hits a target successfully is 0.750.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.950.95, is ____
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Your answer: —
Correct answer: 6
18
For a complex number zz, let Re(z)\text{Re}(z) denote the real part of zz. Let SS be the set of all complex numbers zz satisfying z4−∣z∣4=4iz2z^4 - |z|^4 = 4iz^2, where i=−1i=\sqrt{-1}. Then the minimum possible value of ∣z1−z2∣2|z_1-z_2|^2, where z1,z2∈Sz_1, z_2 \in S with Re(z1)>0\text{Re}(z_1) > 0 and Re(z2)<0\text{Re}(z_2) < 0, is ____
Not attempted
Your answer: —
Correct answer: 8
19
For a polynomial g(x)g(x) with real coefficients, let mgm_g denote the number of distinct real roots of g(x)g(x). Suppose SS is the set of polynomials with real coefficients defined by S={(x2−1)2(a0+a1x+a2x2+a3x3):a0,a1,a2,a3∈R}S = \{(x^2-1)^2(a_0+a_1x+a_2x^2+a_3x^3) : a_0,a_1,a_2,a_3 \in \mathbb{R}\}. For a polynomial ff, let f′f' and f′′f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (mf′+mf′′)(m_{f'} + m_{f''}), where f∈Sf \in S, is
Not attempted
Your answer: —
Correct answer: 5
20
Let ee denote the base of the natural logarithm. The value of the real number aa for which the right hand limit lim⁡x→0+(1−x)1/x−e−1xa\displaystyle\lim_{x\to0^+} \frac{(1-x)^{1/x} - e^{-1}}{x^a} is equal to a nonzero real number, is
Not attempted
Your answer: —
Correct answer: 1
21
In a triangle PQRPQR, let a⃗=QR⃗\vec{a} = \vec{QR}, b⃗=RP⃗\vec{b} = \vec{RP} and c⃗=PQ⃗\vec{c} = \vec{PQ}. If ∣a⃗∣=3,∣b⃗∣=4|\vec{a}| = 3, |\vec{b}| = 4 and a⃗⋅(c⃗−b⃗)c⃗⋅(a⃗−b⃗)=∣a⃗∣∣a⃗∣+∣b⃗∣\dfrac{\vec{a}\cdot(\vec{c}-\vec{b})}{\vec{c}\cdot(\vec{a}-\vec{b})} = \dfrac{|\vec{a}|}{|\vec{a}|+|\vec{b}|}, then the value of ∣a⃗×b⃗∣2|\vec{a}\times\vec{b}|^2 is
Not attempted
Your answer: —
Correct answer: 108
22
Let f:[0,2]→Rf: [0, 2] \to \mathbb{R} be the function defined by f(x)=(3−sin⁡(2πx))sin⁡(πx−π4)−sin⁡(3πx+π4)f(x) = (3 - \sin(2\pi x))\sin\left(\pi x - \frac{\pi}{4}\right) - \sin\left(3\pi x + \frac{\pi}{4}\right). If α,β∈[0,2]\alpha, \beta \in [0, 2] are such that {x∈[0,2]:f(x)≥0}=[α,β]\{x \in [0,2] : f(x) \geq 0\} = [\alpha, \beta], then the value of β−α\beta - \alpha is
Not attempted
Your answer: —
Correct answer: 1
23
Let mm be the minimum possible value of log⁡3(3y1+3y2+3y3)\log_3(3^{y_1}+3^{y_2}+3^{y_3}), where y1,y2,y3y_1, y_2, y_3 are real numbers for which y1+y2+y3=9y_1+y_2+y_3=9. Let MM be the maximum possible value of (log⁡3x1+log⁡3x2+log⁡3x3)(\log_3 x_1 + \log_3 x_2 + \log_3 x_3), where x1,x2,x3x_1, x_2, x_3 are positive real numbers for which x1+x2+x3=9x_1+x_2+x_3=9. Then the value of log⁡2(m3)+log⁡3(M2)\log_2(m^3) + \log_3(M^2) is
Not attempted
Your answer: —
Correct answer: 8
24
Let a1,a2,a3,…a_1, a_2, a_3, \ldots be a sequence of positive integers in arithmetic progression with common difference 22. Also, let b1,b2,b3,…b_1, b_2, b_3, \ldots be a sequence of positive integers in geometric progression with common ratio 22. If a1=b1=ca_1 = b_1 = c, then the number of all possible values of cc, for which the equality 2(a1+a2+⋯+an)=b1+b2+⋯+bn2(a_1+a_2+\cdots+a_n) = b_1+b_2+\cdots+b_n holds for some positive integer nn, is
Not attempted
Your answer: —
Correct answer: 1
25
Which of the following inequalities is/are TRUE?
Not attempted
A
∫01xcos⁡x dx≥38\int_0^1 x\cos x \, dx \geq \frac{3}{8}
B
∫01xsin⁡x dx≥310\int_0^1 x\sin x \, dx \geq \frac{3}{10}
C
∫01x2cos⁡x dx≥12\int_0^1 x^2\cos x \, dx \geq \frac{1}{2}
D
∫01x2sin⁡x dx≥29\int_0^1 x^2\sin x \, dx \geq \frac{2}{9}
26
Let L1L_1 and L2L_2 be the following straight lines. L1:x−11=y−1=z−13L_1: \frac{x-1}{1} = \frac{y}{-1} = \frac{z-1}{3} and L2:x−1−3=y−1=z−11L_2: \frac{x-1}{-3} = \frac{y}{-1} = \frac{z-1}{1}. Suppose the straight line L:x−αl=y−1m=z−γ−2L: \frac{x-\alpha}{l} = \frac{y-1}{m} = \frac{z-\gamma}{-2} lies in the plane containing L1L_1 and L2L_2, and passes through the point of intersection of L1L_1 and L2L_2. If the line LL bisects the acute angle between the lines L1L_1 and L2L_2, then which of the following statements is/are TRUE?
Not attempted
A
α−γ=3\alpha - \gamma = 3
B
l+m=2l+m = 2
C
α−γ=1\alpha - \gamma = 1
D
l+m=0l+m = 0
27
Let x,yx, y and zz be positive real numbers. Suppose x,yx, y and zz are the lengths of the sides of a triangle opposite to its angles X,YX, Y and ZZ, respectively. If tan⁡X2+tan⁡Z2=2yx+y+z\tan\frac{X}{2} + \tan\frac{Z}{2} = \frac{2y}{x+y+z}, then which of the following statements is/are TRUE?
Not attempted
A
2Y=X+Z2Y = X+Z
B
Y=X+ZY = X+Z
C
tan⁡X2=xy+z\tan\frac{X}{2} = \frac{x}{y+z}
D
x2+z2−y2=xzx^2+z^2-y^2 = xz
28
Let SS be the set of all complex numbers zz satisfying ∣z2+z+1∣=1|z^2+z+1|=1. Then which of the following statements is/are TRUE?
Not attempted
A
∣z+12∣≤12|z+\frac{1}{2}| \leq \frac{1}{2} for all z∈Sz \in S
B
∣z∣≤2|z| \leq 2 for all z∈Sz \in S
C
∣z+12∣≥12|z+\frac{1}{2}| \geq \frac{1}{2} for all z∈Sz \in S
D
The set SS has exactly four elements
29
Let MM be a 3×33 \times 3 invertible matrix with real entries and let II denote the 3×33 \times 3 identity matrix. If M−1=adj(adj M)M^{-1} = \text{adj}(\text{adj}\, M), then which of the following statements is/are ALWAYS TRUE?
Not attempted
A
M=IM = I
B
det⁡M=1\det M = 1
C
M2=IM^2 = I
D
(adj M)2=I(\text{adj}\, M)^2 = I
30
Consider all rectangles lying in the region {(x,y)∈R×R:0≤x≤π2 and 0≤y≤2sin⁡(2x)}\{(x, y) \in \mathbb{R} \times \mathbb{R} : 0 \leq x \leq \frac{\pi}{2} \text{ and } 0 \leq y \leq 2\sin(2x)\} and having one side on the xx-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
Not attempted
A
3π2\frac{3\pi}{2}
B
π\pi
C
π23\frac{\pi}{2\sqrt{3}}
D
π32\frac{\pi\sqrt{3}}{2}
31
Let the function f:R→Rf: \mathbb{R} \to \mathbb{R} be defined by f(x)=x3−x2+(x−1)sin⁡xf(x) = x^3 - x^2 + (x-1)\sin x and let g:R→Rg: \mathbb{R} \to \mathbb{R} be an arbitrary function. Let fg:R→Rfg: \mathbb{R} \to \mathbb{R} be the product function defined by (fg)(x)=f(x)g(x)(fg)(x) = f(x)g(x). Then which of the following statements is/are TRUE?
Not attempted
A
If gg is continuous at x=1x=1, then fgfg is differentiable at x=1x=1
B
If fgfg is differentiable at x=1x=1, then gg is continuous at x=1x=1
C
If gg is differentiable at x=1x=1, then fgfg is differentiable at x=1x=1
D
If fgfg is differentiable at x=1x=1, then gg is differentiable at x=1x=1
32
Let C1C_1 and C2C_2 be two biased coins such that the probabilities of getting head in a single toss are 23\frac{2}{3} and 13\frac{1}{3}, respectively. Suppose α\alpha is the number of heads that appear when C1C_1 is tossed twice, independently, and suppose β\beta is the number of heads that appear when C2C_2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x2−αx+βx^2 - \alpha x + \beta are real and equal, is
Not attempted
A
4081\frac{40}{81}
B
2081\frac{20}{81}
C
12\frac{1}{2}
D
14\frac{1}{4}
33
Let a,ba, b and λ\lambda be positive real numbers. Suppose PP is an end point of the latus rectum of the parabola y2=4λxy^2 = 4\lambda x, and suppose the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 passes through the point PP. If the tangents to the parabola and the ellipse at the point PP are perpendicular to each other, then the eccentricity of the ellipse is
Not attempted
A
12\frac{1}{\sqrt{2}}
B
12\frac{1}{2}
C
13\frac{1}{3}
D
25\frac{2}{5}
34
Let the functions f:R→Rf: \mathbb{R} \to \mathbb{R} and g:R→Rg: \mathbb{R} \to \mathbb{R} be defined by f(x)=ex−1−e−∣x−1∣f(x) = e^{x-1} - e^{-|x-1|} and g(x)=12(ex−1+e1−x)g(x) = \frac{1}{2}(e^{x-1} + e^{1-x}). Then the area of the region in the first quadrant bounded by the curves y=f(x)y=f(x), y=g(x)y=g(x) and x=0x=0 is
Not attempted
A
(2−3)+12(e−e−1)(2-\sqrt{3}) + \frac{1}{2}(e-e^{-1})
B
(2+3)+12(e−e−1)(2+\sqrt{3}) + \frac{1}{2}(e-e^{-1})
C
(2−3)+12(e+e−1)(2-\sqrt{3}) + \frac{1}{2}(e+e^{-1})
D
(2+3)+12(e+e−1)(2+\sqrt{3}) + \frac{1}{2}(e+e^{-1})
35
If the function f:R→Rf: \mathbb{R} \to \mathbb{R} is defined by f(x)=∣x∣(x−sin⁡x)f(x) = |x|(x - \sin x), then which of the following statements is TRUE?
Not attempted
A
ff is one-one, but NOT onto
B
ff is onto, but NOT one-one
C
ff is BOTH one-one and onto
D
ff is NEITHER one-one NOR onto
36
Suppose a,ba, b denote the distinct real roots of the quadratic polynomial x2+20x−2020x^2 + 20x - 2020, and suppose c,dc, d denote the distinct complex roots of the quadratic polynomial x2−20x+2020x^2 - 20x + 2020. Then the value of ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d) is
Not attempted
A
00
B
80008000
C
80808080
D
1600016000
Hint\textbf{Hint}: Regroup the expression as (a2+b2)(c+d)−(c2+d2)(a+b)(a^2+b^2)(c+d) - (c^2+d^2)(a+b), then use a+ba+b, abab, c+dc+d, cdcd from Vieta's formulas.
Deconstruction\textbf{Deconstruction}: We need the value of ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d) where a,ba,b are the real roots of x2+20x−2020=0x^2+20x-2020=0 and c,dc,d are the complex roots of x2−20x+2020=0x^2-20x+2020=0. Strategy\textbf{Strategy}: Expand and regroup the expression in terms of the symmetric functions a+ba+b, abab, c+dc+d, cdcd from Vieta's formulas, avoiding the need to find a,b,c,da,b,c,d individually. Execution\textbf{Execution}: By Vieta's formulas: a+b=−20a+b=-20, ab=−2020ab=-2020, and c+d=20c+d=20, cd=2020cd=2020. Expanding: ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)=a2c−ac2+a2d−ad2+b2c−bc2+b2d−bd2ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d) = a^2c-ac^2+a^2d-ad^2+b^2c-bc^2+b^2d-bd^2 Grouping: =a2(c+d)+b2(c+d)−c2(a+b)−d2(a+b)=(a2+b2)(c+d)−(c2+d2)(a+b)= a^2(c+d) + b^2(c+d) - c^2(a+b) - d^2(a+b) = (a^2+b^2)(c+d) - (c^2+d^2)(a+b) a2+b2=(a+b)2−2ab=400−2(−2020)=4440a^2+b^2 = (a+b)^2 - 2ab = 400 - 2(-2020) = 4440 c2+d2=(c+d)2−2cd=400−2(2020)=−3640c^2+d^2 = (c+d)^2 - 2cd = 400 - 2(2020) = -3640 Substituting: 4440(20)−(−3640)(−20)=88800−72800=160004440(20) - (-3640)(-20) = 88800 - 72800 = 16000 Verification\textbf{Verification}: The identity used is an exact algebraic regrouping of the expanded expression, valid regardless of whether c,dc,d are real or complex, since only the real symmetric functions c+dc+d, cdcd, c2+d2c^2+d^2 are used. Final Answer\textbf{Final Answer}: 1600016000, option (D).