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JEE Advanced · Mathematics · 2016

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1
Comprehension\textbf{Comprehension}: Let F1(x1,0)F_1(x_1,0) and F2(x2,0)F_2(x_2,0), for x1<0x_1<0 and x2>0x_2>0, be the foci of the ellipse x29+y28=1\dfrac{x^2}{9}+\dfrac{y^2}{8}=1. Suppose a parabola having vertex at the origin and focus at F2F_2 intersects the ellipse at point MM in the first quadrant and at point NN in the fourth quadrant. The orthocentre of the triangle F1MNF_1MN is
Not attempted
A
(−910,0)\left(-\dfrac{9}{10},0\right)
B
(23,0)\left(\dfrac23,0\right)
C
(910,0)\left(\dfrac{9}{10},0\right)
D
(23,6)\left(\dfrac23,\sqrt6\right)
2
Comprehension\textbf{Comprehension}: Let F1(x1,0)F_1(x_1,0) and F2(x2,0)F_2(x_2,0), for x1<0x_1<0 and x2>0x_2>0, be the foci of the ellipse x29+y28=1\dfrac{x^2}{9}+\dfrac{y^2}{8}=1. Suppose a parabola having vertex at the origin and focus at F2F_2 intersects the ellipse at point MM in the first quadrant and at point NN in the fourth quadrant. If the tangents to the ellipse at MM and NN meet at RR, and the normal to the parabola at MM meets the xx-axis at QQ, then the ratio of area of the triangle MQRMQR to area of the quadrilateral MF1NF2MF_1NF_2 is
Not attempted
A
3:43:4
B
4:54:5
C
5:85:8
D
2:32:3
3
Comprehension\textbf{Comprehension}: Football teams T1T_1 and T2T_2 play two games against each other; the outcomes of the two games are independent. The probabilities of T1T_1 winning, drawing and losing a game against T2T_2 are 12,16,13\dfrac12,\dfrac16,\dfrac13 respectively. Each team gets 33 points for a win, 11 point for a draw and 00 points for a loss in a game. Let XX and YY denote the total points scored by T1T_1 and T2T_2 respectively after two games. P(X>Y)P(X>Y) is
Not attempted
A
14\dfrac14
B
512\dfrac{5}{12}
C
12\dfrac12
D
712\dfrac{7}{12}
4
Comprehension\textbf{Comprehension}: Football teams T1T_1 and T2T_2 play two games against each other; the outcomes of the two games are independent. The probabilities of T1T_1 winning, drawing and losing a game against T2T_2 are 12,16,13\dfrac12,\dfrac16,\dfrac13 respectively. Each team gets 33 points for a win, 11 point for a draw and 00 points for a loss in a game. Let XX and YY denote the total points scored by T1T_1 and T2T_2 respectively after two games. P(X=Y)P(X=Y) is
Not attempted
A
1136\dfrac{11}{36}
B
13\dfrac13
C
1336\dfrac{13}{36}
D
12\dfrac12
5
Let u^=u1i^+u2j^+u3k^\hat u=u_1\hat i+u_2\hat j+u_3\hat k be a unit vector in R3\mathbb{R}^3 and w^=16(i^+j^+2k^)\hat w=\dfrac{1}{\sqrt6}(\hat i+\hat j+2\hat k). Given that there exists a vector v⃗\vec v in R3\mathbb{R}^3 such that ∣u^×v⃗∣=1|\hat u\times\vec v|=1 and w^⋅(u^×v⃗)=1\hat w\cdot(\hat u\times\vec v)=1. Which of the following statement(s) is(are) correct?
Not attempted
A
There is exactly one choice for such v⃗\vec v
B
There are infinitely many choices for such v⃗\vec v
C
If u^\hat u lies in the xyxy-plane then ∣u1∣=∣u2∣|u_1|=|u_2|
D
If u^\hat u lies in the xzxz-plane then 2∣u1∣=∣u3∣2|u_1|=|u_3|
6
Let a,λ,μ∈Ra,\lambda,\mu\in\mathbb{R}. Consider the system of linear equations ax+2y=λax+2y=\lambda, 3x−2y=μ3x-2y=\mu. Which of the following statement(s) is(are) correct?
Not attempted
A
If a=−3a=-3, then the system has infinitely many solutions for all values of λ\lambda and μ\mu
B
If a≠−3a\ne-3, then the system has a unique solution for all values of λ\lambda and μ\mu
C
If λ+μ=0\lambda+\mu=0, then the system has infinitely many solutions for a=−3a=-3
D
If λ+μ≠0\lambda+\mu\ne0, then the system has no solution for a=−3a=-3
7
Let PP be the point on the parabola y2=4xy^2=4x which is at the shortest distance from the center SS of the circle x2+y2−4x−16y+64=0x^2+y^2-4x-16y+64=0. Let QQ be the point on the circle dividing the line segment SPSP internally. Then
Not attempted
A
SP=25SP=2\sqrt5
B
SQ:QP=(5+1):2SQ:QP=(\sqrt5+1):2
C
the xx-intercept of the normal to the parabola at PP is 66
D
the slope of the tangent to the circle at QQ is 12\dfrac12
8
Let a,b∈Ra,b\in\mathbb{R} and a2+b2≠0a^2+b^2\ne0. Suppose S={z∈C:z=1a+ibt, t∈R, t≠0}S=\left\{z\in\mathbb{C}: z=\dfrac{1}{a+ibt},\ t\in\mathbb{R},\ t\ne0\right\}, where i=−1i=\sqrt{-1}. If z=x+iyz=x+iy and z∈Sz\in S, then (x,y)(x,y) lies on
Not attempted
A
the circle with radius 12a\dfrac{1}{2a} and centre (12a,0)\left(\dfrac{1}{2a},0\right) for a>0,b≠0a>0,b\ne0
B
the circle with radius −12a-\dfrac{1}{2a} and centre (−12a,0)\left(-\dfrac{1}{2a},0\right) for a<0,b≠0a<0,b\ne0
C
the xx-axis for a≠0,b=0a\ne0,b=0
D
the yy-axis for a=0,b≠0a=0,b\ne0
9
Let f(x)=lim⁡n→∞(nn(x+n)(x+n2)⋯(x+nn)n!(x2+n2)(x2+n24)⋯(x2+n2n2))x/nf(x)=\displaystyle\lim_{n\to\infty}\left(\dfrac{n^n(x+n)\left(x+\dfrac{n}{2}\right)\cdots\left(x+\dfrac{n}{n}\right)}{n!\left(x^2+n^2\right)\left(x^2+\dfrac{n^2}{4}\right)\cdots\left(x^2+\dfrac{n^2}{n^2}\right)}\right)^{x/n}, for all x>0x>0. Then
Not attempted
A
f ⁣(12)≥f(1)f\!\left(\dfrac12\right)\ge f(1)
B
f ⁣(13)≤f ⁣(23)f\!\left(\dfrac13\right)\le f\!\left(\dfrac23\right)
C
f′(2)≤0f'(2)\le0
D
f′(3)f(3)≥f′(2)f(2)\dfrac{f'(3)}{f(3)}\ge\dfrac{f'(2)}{f(2)}
10
Let a,b∈Ra,b\in\mathbb{R} and f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)=acos⁡(∣x3−x∣)+b∣x∣sin⁡(∣x3+x∣)f(x)=a\cos(|x^3-x|)+b|x|\sin(|x^3+x|). Then ff is
Not attempted
A
differentiable at x=0x=0 if a=0a=0 and b=1b=1
B
differentiable at x=1x=1 if a=1a=1 and b=0b=0
C
NOT differentiable at x=0x=0 if a=1a=1 and b=0b=0
D
NOT differentiable at x=1x=1 if a=1a=1 and b=1b=1
11
Let f:R→(0,∞)f:\mathbb{R}\to(0,\infty) and g:R→Rg:\mathbb{R}\to\mathbb{R} be twice differentiable functions such that f′′f'' and g′′g'' are continuous functions on R\mathbb{R}. Suppose f′(2)=g(2)=0f'(2)=g(2)=0, f′′(2)≠0f''(2)\ne0 and g′(2)≠0g'(2)\ne0. If lim⁡x→2f(x)g(x)f′(x)g′(x)=1\displaystyle\lim_{x\to2}\dfrac{f(x)g(x)}{f'(x)g'(x)}=1, then
Not attempted
A
ff has a local minimum at x=2x=2
B
ff has a local maximum at x=2x=2
C
f′′(2)>f(2)f''(2)>f(2)
D
f(x)−f′′(x)=0f(x)-f''(x)=0 for at least one x∈Rx\in\mathbb{R}
12
Let f:[−12,2]→Rf:\left[-\dfrac12,2\right]\to\mathbb{R} and g:[−12,2]→Rg:\left[-\dfrac12,2\right]\to\mathbb{R} be functions defined by f(x)=[x2−3]f(x)=[x^2-3] and g(x)=∣x∣f(x)+∣4x−7∣f(x)g(x)=|x|f(x)+|4x-7|f(x), where [y][y] denotes the greatest integer less than or equal to yy for y∈Ry\in\mathbb{R}. Then
Not attempted
A
ff is discontinuous exactly at three points in [−12,2]\left[-\dfrac12,2\right]
B
ff is discontinuous exactly at four points in [−12,2]\left[-\dfrac12,2\right]
C
gg is NOT differentiable exactly at four points in (−12,2)\left(-\dfrac12,2\right)
D
gg is NOT differentiable exactly at five points in (−12,2)\left(-\dfrac12,2\right)
13
Let PP be the image of the point (3,1,7)(3,1,7) with respect to the plane x−y+z=3x-y+z=3. Then the equation of the plane passing through PP and containing the straight line x1=y2=z1\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{1} is
Not attempted
A
x+y−3z=0x+y-3z=0
B
3x+z=03x+z=0
C
x−4y+7z=0x-4y+7z=0
D
2x−y=02x-y=0
14
The value of ∫−π/2π/2x2cos⁡x1+ex dx\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{x^2\cos x}{1+e^x}\,dx is equal to
Not attempted
A
π24−2\dfrac{\pi^2}{4}-2
B
π24+2\dfrac{\pi^2}{4}+2
C
π2−eπ/2\pi^2-e^{\pi/2}
D
π2+eπ/2\pi^2+e^{\pi/2}
15
Area of the region {(x,y)∈R2:y≥∣x+3∣, 5y≤x+9≤15}\{(x,y)\in\mathbb{R}^2: y\ge\sqrt{|x+3|},\ 5y\le x+9\le15\} is equal to
Not attempted
A
16\dfrac{1}{6}
B
43\dfrac{4}{3}
C
32\dfrac{3}{2}
D
53\dfrac{5}{3}
16
The value of ∑k=1131sin⁡(π4+(k−1)π6)sin⁡(π4+kπ6)\displaystyle\sum_{k=1}^{13}\dfrac{1}{\sin\left(\dfrac{\pi}{4}+\dfrac{(k-1)\pi}{6}\right)\sin\left(\dfrac{\pi}{4}+\dfrac{k\pi}{6}\right)} is equal to
Not attempted
A
3−33-\sqrt3
B
2(3−3)2(3-\sqrt3)
C
2(3−1)2(\sqrt3-1)
D
2(2+3)2(2+\sqrt3)
17
Let bi>1b_i>1 for i=1,2,…,101i=1,2,\ldots,101. Suppose log⁡eb1,log⁡eb2,…,log⁡eb101\log_e b_1,\log_e b_2,\ldots,\log_e b_{101} are in Arithmetic Progression (A.P.) with the common difference log⁡e2\log_e 2. Suppose a1,a2,…,a101a_1,a_2,\ldots,a_{101} are in A.P. such that a1=b1a_1=b_1 and a51=b51a_{51}=b_{51}. If t=b1+b2+⋯+b51t=b_1+b_2+\cdots+b_{51} and s=a1+a2+⋯+a51s=a_1+a_2+\cdots+a_{51}, then
Not attempted
A
s>ts>t and a101>b101a_{101}>b_{101}
B
s>ts>t and a101<b101a_{101}<b_{101}
C
s<ts<t and a101>b101a_{101}>b_{101}
D
s<ts<t and a101<b101a_{101}<b_{101}
18
Let P=(1004101641)P=\begin{pmatrix}1&0&0\\4&1&0\\16&4&1\end{pmatrix} and II be the identity matrix of order 33. If Q=[qij]Q=[q_{ij}] is a matrix such that P50−Q=IP^{50}-Q=I, then q31+q32q21\dfrac{q_{31}+q_{32}}{q_{21}} equals
Not attempted
A
5252
B
103103
C
201201
D
205205
19
Let z=−1+3i2z=\dfrac{-1+\sqrt3i}{2}, where i=−1i=\sqrt{-1}, and r,s∈{1,2,3}r,s\in\{1,2,3\}. Let P=((−z)rz2sz2szr)P=\begin{pmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{pmatrix} and II be the identity matrix of order 2. Then the total number of ordered pairs (r,s)(r,s) for which P2=−IP^2=-I is
Not attempted
Your answer: —
Correct answer: 1
20
Let α,β∈R\alpha,\beta\in\mathbb{R} be such that lim⁡x→0x2sin⁡(βx)αx−sin⁡x=1\displaystyle\lim_{x\to0}\dfrac{x^2\sin(\beta x)}{\alpha x-\sin x}=1. Then 6(α+β)6(\alpha+\beta) equals
Not attempted
Your answer: —
Correct answer: 7
21
The total number of distinct x∈[0,1]x\in[0,1] for which ∫0xt21+t4 dt=2x−1\displaystyle\int_0^x\dfrac{t^2}{1+t^4}\,dt=2x-1 is
Not attempted
Your answer: —
Correct answer: 1
22
Let mm be the smallest positive integer such that the coefficient of x2x^2 in the expansion of (1+x)2+(1+x)3+⋯+(1+x)49+(1+mx)50(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+mx)^{50} is (3n+1) 51C3(3n+1)\,{}^{51}C_3 for some positive integer nn. Then the value of nn is
Not attempted
Your answer: —
Correct answer: 5
23
The total number of distinct x∈Rx\in\mathbb{R} for which ∣xx21+x32x4x21+8x33x9x21+27x3∣=10\begin{vmatrix}x&x^2&1+x^3\\2x&4x^2&1+8x^3\\3x&9x^2&1+27x^3\end{vmatrix}=10 is
Not attempted
Your answer: —
Correct answer: 2
24
The circle C1:x2+y2=3C_1:x^2+y^2=3, with centre at OO, intersects the parabola x2=2yx^2=2y at the point PP in the first quadrant. Let the tangent to the circle C1C_1 at PP touches other two circles C2C_2 and C3C_3 at R2R_2 and R3R_3, respectively. Suppose C2C_2 and C3C_3 have equal radii 232\sqrt3 and centres Q2Q_2 and Q3Q_3, respectively. If Q2Q_2 and Q3Q_3 lie on the yy-axis, then
Not attempted
A
Q2Q3=12Q_2Q_3=12
B
R2R3=46R_2R_3=4\sqrt6
C
area of the triangle OR2R3OR_2R_3 is 626\sqrt2
D
area of the triangle PQ2Q3PQ_2Q_3 is 424\sqrt2
25
Let RSRS be the diameter of the circle x2+y2=1x^2+y^2=1, where SS is the point (1,0)(1,0). Let PP be a variable point (other than RR and SS) on the circle and tangents to the circle at SS and PP meet at the point QQ. The normal to the circle at PP intersects a line drawn through QQ parallel to RSRS at point EE. Then the locus of EE passes through the point(s)
Not attempted
A
(13,13)\left(\dfrac13,\dfrac1{\sqrt3}\right)
B
(14,12)\left(\dfrac14,\dfrac12\right)
C
(13,−13)\left(\dfrac13,-\dfrac1{\sqrt3}\right)
D
(14,−12)\left(\dfrac14,-\dfrac12\right)
26
Let 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} be differentiable functions such that f(x)=x3+3x+2f(x)=x^3+3x+2, g(f(x))=xg(f(x))=x and h(g(g(x)))=xh(g(g(x)))=x for all x∈Rx\in\mathbb{R}. Then
Not attempted
A
g′(2)=115g'(2)=\dfrac1{15}
B
h′(1)=666h'(1)=666
C
h(0)=16h(0)=16
D
h(g(3))=36h(g(3))=36
27
A solution curve of the differential equation (x2+xy+4x+2y+4)dydx−y2=0\left(x^2+xy+4x+2y+4\right)\dfrac{dy}{dx}-y^2=0, x>0x>0, passes through the point (1,3)(1,3). Then the solution curve
Not attempted
A
intersects y=x+2y=x+2 exactly at one point
B
intersects y=x+2y=x+2 exactly at two points
C
intersects y=(x+2)2y=(x+2)^2
D
does NOT intersect y=(x+3)2y=(x+3)^2
28
In a triangle XYZXYZ, let x,y,zx,y,z be the lengths of sides opposite to the angles X,Y,ZX,Y,Z, respectively, and 2s=x+y+z2s=x+y+z. If s−x4=s−y3=s−z2\dfrac{s-x}{4}=\dfrac{s-y}{3}=\dfrac{s-z}{2} and area of incircle of the triangle XYZXYZ is 8π3\dfrac{8\pi}{3}, then
Not attempted
A
area of the triangle XYZXYZ is 666\sqrt6
B
the radius of circumcircle of the triangle XYZXYZ is 3566\dfrac{35}{6}\sqrt6
C
sin⁡X2sin⁡Y2sin⁡Z2=435\sin\dfrac X2\sin\dfrac Y2\sin\dfrac Z2=\dfrac{4}{35}
D
sin⁡2 ⁣(X+Y2)=35\sin^2\!\left(\dfrac{X+Y}{2}\right)=\dfrac35
29
Let P=(3−1−220α3−50)P=\begin{pmatrix}3&-1&-2\\2&0&\alpha\\3&-5&0\end{pmatrix}, where α∈R\alpha\in\mathbb{R}. Suppose Q=[qij]Q=[q_{ij}] is a matrix such that PQ=kIPQ=kI, where k∈Rk\in\mathbb{R}, k≠0k\ne0 and II is the identity matrix of order 3. If q23=−k8q_{23}=-\dfrac{k}{8} and det⁡(Q)=k22\det(Q)=\dfrac{k^2}{2}, then
Not attempted
A
α=0, k=8\alpha=0,\ k=8
B
4α−k+8=04\alpha-k+8=0
C
det⁡(Padj⁡Q)=29\det(P\operatorname{adj}Q)=2^9
D
det⁡(Qadj⁡P)=213\det(Q\operatorname{adj}P)=2^{13}
30
Let f:(0,∞)→Rf:(0,\infty)\to\mathbb{R} be a differentiable function such that f′(x)=2−f(x)xf'(x)=2-\dfrac{f(x)}{x} for all x∈(0,∞)x\in(0,\infty) and f(1)≠1f(1)\ne1. Then
Not attempted
A
lim⁡x→0+f′ ⁣(1x)=1\displaystyle\lim_{x\to0^+}f'\!\left(\dfrac1x\right)=1
B
lim⁡x→0+xf ⁣(1x)=2\displaystyle\lim_{x\to0^+}xf\!\left(\dfrac1x\right)=2
C
lim⁡x→0+x2f′(x)=0\displaystyle\lim_{x\to0^+}x^2f'(x)=0
D
∣f(x)∣≤2|f(x)|\le2 for all x∈(0,2)x\in(0,2)
31
Consider a pyramid OPQRSOPQRS located in the first octant (x≥0,y≥0,z≥0x\ge0,y\ge0,z\ge0) with OO as origin, and OPOP and OROR along the xx-axis and the yy-axis, respectively. The base OPQROPQR of the pyramid is a square with OP=3OP=3. The point SS is directly above the mid-point TT of diagonal OQOQ such that TS=3TS=3. Then
Not attempted
A
the acute angle between OQOQ and OSOS is π3\dfrac{\pi}{3}
B
the equation of the plane containing the triangle OQSOQS is x−y=0x-y=0
C
the length of the perpendicular from PP to the plane containing the triangle OQSOQS is 32\dfrac{3}{\sqrt2}
D
the perpendicular distance from OO to the straight line containing RSRS is 152\sqrt{\dfrac{15}{2}}
32
The least value of α∈R\alpha\in\mathbb{R} for which 4αx2+1x≥14\alpha x^2+\dfrac1x\ge1, for all x>0x>0, is
Not attempted
A
164\dfrac1{64}
B
132\dfrac1{32}
C
127\dfrac1{27}
D
125\dfrac1{25}
33
A computer producing factory has only two plants T1T_1 and T2T_2. Plant T1T_1 produces 20\% and plant T2T_2 produces 80\% of the total computers produced. 7\% of computers produced in the factory turn out to be defective. It is known that P(computer turns out to be defective given that it is produced in plant T1)=10 P(computer turns out to be defective given that it is produced in plant T2)P(\text{computer turns out to be defective given that it is produced in plant } T_1)=10\,P(\text{computer turns out to be defective given that it is produced in plant } T_2), where P(E)P(E) denotes the probability of an event EE. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T2T_2 is
Not attempted
A
3673\dfrac{36}{73}
B
4779\dfrac{47}{79}
C
7893\dfrac{78}{93}
D
7583\dfrac{75}{83}
34
Let S={x∈(−π,π):x≠0,±π2}S=\left\{x\in(-\pi,\pi):x\ne0,\pm\dfrac{\pi}{2}\right\}. The sum of all distinct solutions of the equation 3sec⁡x+cosec⁡x+2(tan⁡x−cot⁡x)=0\sqrt3\sec x+\operatorname{cosec}x+2(\tan x-\cot x)=0 in the set SS is equal to
Not attempted
A
−7π9-\dfrac{7\pi}{9}
B
−2π9-\dfrac{2\pi}{9}
C
0
D
5π9\dfrac{5\pi}{9}
35
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
Not attempted
A
380
B
320
C
260
D
95
36
Let −π6<θ<−π12-\dfrac{\pi}{6} < \theta < -\dfrac{\pi}{12}. Suppose α1\alpha_1 and β1\beta_1 are the roots of the equation x2−2xsec⁡θ+1=0x^2 - 2x\sec\theta + 1 = 0 and α2\alpha_2 and β2\beta_2 are the roots of the equation x2+2xtan⁡θ−1=0x^2 + 2x\tan\theta - 1 = 0. If α1>β1\alpha_1 > \beta_1 and α2>β2\alpha_2 > \beta_2, then α1+β2\alpha_1 + \beta_2 equals
Not attempted
A
2(sec⁡θ−tan⁡θ)2(\sec\theta-\tan\theta)
B
2sec⁡θ2\sec\theta
C
−2tan⁡θ-2\tan\theta
D
00
Solve both quadratics using the quadratic formula, simplify sec⁡2θ−1=∣tan⁡θ∣\sqrt{\sec^2\theta-1}=|\tan\theta| carefully using the sign of tan⁡θ\tan\theta in the given range, then order the roots.
Deconstruction\textbf{Deconstruction}: We are given the range −π/6<θ<−π/12-\pi/6<\theta<-\pi/12, so θ\theta is a small negative angle. This means cos⁡θ>0\cos\theta>0 so sec⁡θ>0\sec\theta>0, and tan⁡θ<0\tan\theta<0. Two quadratics are given and we need α1+β2\alpha_1+\beta_2 where each pair is ordered largest-first. Strategy\textbf{Strategy}: Solve each quadratic by the formula x=−b±b2−4ac2ax=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}, simplifying the discriminant using sec⁡2θ−1=tan⁡2θ\sec^2\theta-1=\tan^2\theta, and fix the sign of tan⁡2θ=∣tan⁡θ∣\sqrt{\tan^2\theta}=|\tan\theta| using tan⁡θ<0\tan\theta<0. Execution\textbf{Execution}: For x2−2xsec⁡θ+1=0x^2-2x\sec\theta+1=0: x=sec⁡θ±sec⁡2θ−1=sec⁡θ±∣tan⁡θ∣=sec⁡θ∓tan⁡θx=\sec\theta\pm\sqrt{\sec^2\theta-1}=\sec\theta\pm|\tan\theta|=\sec\theta\mp\tan\theta (since tan⁡θ<0\tan\theta<0, ∣tan⁡θ∣=−tan⁡θ|\tan\theta|=-\tan\theta). So the two roots are sec⁡θ−tan⁡θ\sec\theta-\tan\theta and sec⁡θ+tan⁡θ\sec\theta+\tan\theta. Since −tan⁡θ>0-\tan\theta>0, the root sec⁡θ−tan⁡θ\sec\theta-\tan\theta is the larger one. So α1=sec⁡θ−tan⁡θ\alpha_1=\sec\theta-\tan\theta, β1=sec⁡θ+tan⁡θ\beta_1=\sec\theta+\tan\theta. For x2+2xtan⁡θ−1=0x^2+2x\tan\theta-1=0: x=−tan⁡θ±tan⁡2θ+1=−tan⁡θ±sec⁡θx=-\tan\theta\pm\sqrt{\tan^2\theta+1}=-\tan\theta\pm\sec\theta. Since sec⁡θ>0\sec\theta>0, the larger root is −tan⁡θ+sec⁡θ-\tan\theta+\sec\theta. So α2=sec⁡θ−tan⁡θ\alpha_2=\sec\theta-\tan\theta, β2=−tan⁡θ−sec⁡θ\beta_2=-\tan\theta-\sec\theta. Therefore α1+β2=(sec⁡θ−tan⁡θ)+(−tan⁡θ−sec⁡θ)=−2tan⁡θ\alpha_1+\beta_2=(\sec\theta-\tan\theta)+(-\tan\theta-\sec\theta)=-2\tan\theta. Verification\textbf{Verification}: Take θ=−π/9\theta=-\pi/9 (inside the given range) numerically: sec⁡θ≈1.0642\sec\theta\approx1.0642, tan⁡θ≈−0.3640\tan\theta\approx-0.3640. Then α1=sec⁡θ−tan⁡θ≈1.4282\alpha_1=\sec\theta-\tan\theta\approx1.4282, β2=−tan⁡θ−sec⁡θ≈−0.7002\beta_2=-\tan\theta-\sec\theta\approx-0.7002. Sum ≈0.7280\approx0.7280, and −2tan⁡θ≈0.7280-2\tan\theta\approx0.7280. Matches, confirming the algebra. Final Answer\textbf{Final Answer}: α1+β2=−2tan⁡θ\alpha_1+\beta_2=-2\tan\theta, option (C).