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Question Number 21236 Answers: 1 Comments: 0

If (1/a) + (1/(2a)) + (1/(3a)) = (1/(b^2 − 2b)) a and b are positive integers Find minimum value of a + b

$$\mathrm{If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{a}}\:+\:\frac{\mathrm{1}}{\mathrm{3}{a}}\:=\:\frac{\mathrm{1}}{{b}^{\mathrm{2}} \:−\:\mathrm{2}{b}} \\ $$$${a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{Find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{a}\:+\:{b} \\ $$

Question Number 21235 Answers: 1 Comments: 0

For any integer k, let α_k = cos (((kπ)/7)) + i sin (((kπ)/7)), where i = (√(−1)). The value of the expression ((Σ_(k=1) ^(12) ∣α_(k+1) − α_k ∣)/(Σ_(k=1) ^3 ∣α_(4k−1) − α_(4k−2) ∣)) is

$$\mathrm{For}\:\mathrm{any}\:\mathrm{integer}\:{k},\:\mathrm{let}\:\alpha_{{k}} \:=\:\mathrm{cos}\:\left(\frac{{k}\pi}{\mathrm{7}}\right)\:+ \\ $$$${i}\:\mathrm{sin}\:\left(\frac{{k}\pi}{\mathrm{7}}\right),\:\mathrm{where}\:{i}\:=\:\sqrt{−\mathrm{1}}.\:\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\frac{\underset{{k}=\mathrm{1}} {\overset{\mathrm{12}} {\sum}}\mid\alpha_{{k}+\mathrm{1}} \:−\:\alpha_{{k}} \mid}{\underset{{k}=\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\mid\alpha_{\mathrm{4}{k}−\mathrm{1}} \:−\:\alpha_{\mathrm{4}{k}−\mathrm{2}} \mid}\:\mathrm{is} \\ $$

Question Number 21234 Answers: 1 Comments: 0

Let f(x) = ax^2 + bx + c, where a, b, c are real numbers. If the numbers 2a, a + b, and c are all integers, then the number of integral values between 1 and 5 that f(x) can take is

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c},\:\mathrm{where}\:{a},\:{b},\:{c} \\ $$$$\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{2}{a}, \\ $$$${a}\:+\:{b},\:\mathrm{and}\:{c}\:\mathrm{are}\:\mathrm{all}\:\mathrm{integers},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{integral}\:\mathrm{values}\:\mathrm{between}\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{5}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{can}\:\mathrm{take}\:\mathrm{is} \\ $$

Question Number 21232 Answers: 0 Comments: 0

Question Number 21231 Answers: 0 Comments: 0

Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576, determine the square root of the maximum possible area of the trapezium to the nearest integer.

Question Number 21230 Answers: 1 Comments: 0

For each positive integer n, consider the highest common factor h_n of the two numbers n! + 1 and (n + 1)!. For n < 100, find the largest value of h_n .

$$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:{n},\:\mathrm{consider} \\ $$$$\mathrm{the}\:\mathrm{highest}\:\mathrm{common}\:\mathrm{factor}\:{h}_{{n}} \:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:{n}!\:+\:\mathrm{1}\:\mathrm{and}\:\left({n}\:+\:\mathrm{1}\right)!.\:\mathrm{For}\:{n}\:<\:\mathrm{100}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:{h}_{{n}} . \\ $$

Question Number 21229 Answers: 0 Comments: 0

Let p, q be prime numbers such that n^(3pq) − n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.

$$\mathrm{Let}\:{p},\:{q}\:\mathrm{be}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$${n}^{\mathrm{3}{pq}} \:−\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{3}{pq}\:\mathrm{for}\:\boldsymbol{\mathrm{all}} \\ $$$$\mathrm{positive}\:\mathrm{integers}\:{n}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{p}\:+\:{q}. \\ $$

Question Number 21228 Answers: 0 Comments: 0

Let P be an interior point of a triangle ABC whose sidelengths are 26, 65, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST, are of equal lengths, find this common length.

$$\mathrm{Let}\:{P}\:\mathrm{be}\:\mathrm{an}\:\mathrm{interior}\:\mathrm{point}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle} \\ $$$${ABC}\:\mathrm{whose}\:\mathrm{sidelengths}\:\mathrm{are}\:\mathrm{26},\:\mathrm{65},\:\mathrm{78}. \\ $$$$\mathrm{The}\:\mathrm{line}\:\mathrm{through}\:{P}\:\mathrm{parallel}\:\mathrm{to}\:{BC}\:\mathrm{meets} \\ $$$${AB}\:\mathrm{in}\:{K}\:\mathrm{and}\:{AC}\:\mathrm{in}\:{L}.\:\mathrm{The}\:\mathrm{line}\:\mathrm{through} \\ $$$${P}\:\mathrm{parallel}\:\mathrm{to}\:{CA}\:\mathrm{meets}\:{BC}\:\mathrm{in}\:{M}\:\mathrm{and}\:{BA} \\ $$$$\mathrm{in}\:{N}.\:\mathrm{The}\:\mathrm{line}\:\mathrm{through}\:{P}\:\mathrm{parallel}\:\mathrm{to}\:{AB} \\ $$$$\mathrm{meets}\:{CA}\:\mathrm{in}\:{S}\:\mathrm{and}\:{CB}\:\mathrm{in}\:{T}.\:\mathrm{If}\:{KL},\:{MN}, \\ $$$${ST},\:\mathrm{are}\:\mathrm{of}\:\mathrm{equal}\:\mathrm{lengths},\:\mathrm{find}\:\mathrm{this} \\ $$$$\mathrm{common}\:\mathrm{length}. \\ $$

Question Number 21223 Answers: 1 Comments: 0

lim_(x→π/2) (π−2x)tan (x)

$$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\left(\pi−\mathrm{2}{x}\right)\mathrm{tan}\:\left({x}\right) \\ $$

Question Number 21224 Answers: 0 Comments: 1

One mole of a monoatomic real gas satisfies the equation p(V − b) = RT where b is a constant. The relationship of interatomic potential V(r) and interatomic distance r for the gas is given by

$$\mathrm{One}\:\mathrm{mole}\:\mathrm{of}\:\mathrm{a}\:\mathrm{monoatomic}\:\mathrm{real}\:\mathrm{gas} \\ $$$$\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{p}\left(\mathrm{V}\:−\:\mathrm{b}\right)\:=\:\mathrm{RT} \\ $$$$\mathrm{where}\:\mathrm{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}.\:\mathrm{The}\:\mathrm{relationship} \\ $$$$\mathrm{of}\:\mathrm{interatomic}\:\mathrm{potential}\:\mathrm{V}\left(\mathrm{r}\right)\:\mathrm{and} \\ $$$$\mathrm{interatomic}\:\mathrm{distance}\:\mathrm{r}\:\mathrm{for}\:\mathrm{the}\:\mathrm{gas}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{by} \\ $$

Question Number 21219 Answers: 1 Comments: 0

(A) If ∣w∣ = 2, then the set of points z = w − (1/w) is contained in or equal to (B) If ∣w∣ = 1, then the set of points z = w + (1/w) is contained in or equal to Options for both A and B: (p) An ellipse with eccentricity (4/5) (q) The set of points z satisfying Im z = 0 (r) The set of points z satisfying ∣Im z∣ ≤ 1 (s) The set of points z satisfying ∣Re z∣ ≤ 2 (t) The set of points z satisfying ∣z∣ ≤ 3

$$\left(\mathrm{A}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{2},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:−\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{If}\:\mid{w}\mid\:=\:\mathrm{1},\:\mathrm{then}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points} \\ $$$${z}\:=\:{w}\:+\:\frac{\mathrm{1}}{{w}}\:\mathrm{is}\:\mathrm{contained}\:\mathrm{in}\:\mathrm{or}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{Options}\:\mathrm{for}\:\mathrm{both}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}: \\ $$$$\left(\mathrm{p}\right)\:\mathrm{An}\:\mathrm{ellipse}\:\mathrm{with}\:\mathrm{eccentricity}\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\left(\mathrm{q}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mathrm{Im}\:{z} \\ $$$$=\:\mathrm{0} \\ $$$$\left(\mathrm{r}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid\mathrm{Im}\:{z}\mid \\ $$$$\leqslant\:\mathrm{1} \\ $$$$\left(\mathrm{s}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid\mathrm{Re}\:{z}\mid \\ $$$$\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{t}\right)\:\mathrm{The}\:\mathrm{set}\:\mathrm{of}\:\mathrm{points}\:{z}\:\mathrm{satisfying}\:\mid{z}\mid\:\leqslant\:\mathrm{3} \\ $$

Question Number 21212 Answers: 0 Comments: 0

Question Number 21205 Answers: 0 Comments: 1

Question Number 21210 Answers: 0 Comments: 0

Question Number 21194 Answers: 0 Comments: 4

. lim_(x→2^+ ) {(([x]^3 )/3) −[ (x/3)]^3 } is equal to ...

$$.\:\underset{{x}\rightarrow\mathrm{2}^{+} } {\mathrm{li}{m}}\:\left\{\frac{\left[{x}\right]^{\mathrm{3}} }{\mathrm{3}}\:−\left[\:\frac{{x}}{\mathrm{3}}\right]^{\mathrm{3}} \right\}\:{is}\:{equal}\:{to}\:... \\ $$$$ \\ $$

Question Number 21179 Answers: 1 Comments: 0

lim_(x→∞) (√(16x^2 + 4x)) − (√x^2 ) − (√(9x^2 + 3x))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{16}{x}^{\mathrm{2}} \:+\:\mathrm{4}{x}}\:−\:\sqrt{{x}^{\mathrm{2}} }\:−\:\sqrt{\mathrm{9}{x}^{\mathrm{2}} \:+\:\mathrm{3}{x}} \\ $$

Question Number 21174 Answers: 1 Comments: 0

The factors of determinant ((x,a,b),(a,x,b),(a,b,x))are

$$\mathrm{The}\:\mathrm{factors}\:\mathrm{of}\:\begin{vmatrix}{{x}}&{{a}}&{{b}}\\{{a}}&{{x}}&{{b}}\\{{a}}&{{b}}&{{x}}\end{vmatrix}\mathrm{are} \\ $$

Question Number 21168 Answers: 1 Comments: 0

Let f(x) = ∣x − 1∣ + ∣x − 2∣ + ∣x − 3∣, then find the value of k for which f(x) = k has 1. no solution 2. only one solution 3. two solutions of same sign 4. two solutions of opposite sign

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:−\:\mathrm{2}\mid\:+\:\mid{x}\:−\:\mathrm{3}\mid, \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{for}\:\mathrm{which}\:{f}\left({x}\right) \\ $$$$=\:{k}\:\mathrm{has} \\ $$$$\mathrm{1}.\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{2}.\:\mathrm{only}\:\mathrm{one}\:\mathrm{solution} \\ $$$$\mathrm{3}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{same}\:\mathrm{sign} \\ $$$$\mathrm{4}.\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{opposite}\:\mathrm{sign} \\ $$

Question Number 21154 Answers: 1 Comments: 2

prove: ∀n∈N^∗ ,∀(a,b)∈C^2 , a^(2n+1) +b^(2n+1) = Σ_(k=o) ^(2n) (−1)^k a^k b^(2n−k)

$${prove}:\:\forall{n}\in\mathbb{N}^{\ast} ,\forall\left({a},{b}\right)\in\mathbb{C}^{\mathrm{2}} ,\:{a}^{\mathrm{2}{n}+\mathrm{1}} +{b}^{\mathrm{2}{n}+\mathrm{1}} = \\ $$$$\underset{{k}={o}} {\overset{\mathrm{2}{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {a}^{{k}} {b}^{\mathrm{2}{n}−{k}} \\ $$

Question Number 21150 Answers: 0 Comments: 8

Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance ′a′ from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force F. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x, is

$$\mathrm{Two}\:\mathrm{particles}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{each}\:\mathrm{are}\:\mathrm{tied} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{ends}\:\mathrm{of}\:\mathrm{a}\:\mathrm{light}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length}\:\mathrm{2}{a}. \\ $$$$\mathrm{The}\:\mathrm{whole}\:\mathrm{system}\:\mathrm{is}\:\mathrm{kept}\:\mathrm{on}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}\:\mathrm{with}\:\mathrm{the}\:\mathrm{string}\:\mathrm{held} \\ $$$$\mathrm{tight}\:\mathrm{so}\:\mathrm{that}\:\mathrm{each}\:\mathrm{mass}\:\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance} \\ $$$$'{a}'\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:{P}\:\left(\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\right. \\ $$$$\left.\mathrm{figure}\right).\:\mathrm{Now},\:\mathrm{the}\:\mathrm{mid}-\mathrm{point}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{string}\:\mathrm{is}\:\mathrm{pulled}\:\mathrm{vertically}\:\mathrm{upwards}\:\mathrm{with} \\ $$$$\mathrm{a}\:\mathrm{small}\:\mathrm{but}\:\mathrm{constant}\:\mathrm{force}\:{F}.\:\mathrm{As}\:\mathrm{a}\:\mathrm{result}, \\ $$$$\mathrm{the}\:\mathrm{particles}\:\mathrm{move}\:\mathrm{towards}\:\mathrm{each}\:\mathrm{other} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{surface}.\:\mathrm{The}\:\mathrm{magnitude}\:\mathrm{of} \\ $$$$\mathrm{acceleration},\:\mathrm{when}\:\mathrm{the}\:\mathrm{separation} \\ $$$$\mathrm{between}\:\mathrm{them}\:\mathrm{becomes}\:\mathrm{2}{x},\:\mathrm{is} \\ $$

Question Number 21148 Answers: 0 Comments: 12

Find the compression in the spring if the system shown below is in equilibrium.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{compression}\:\mathrm{in}\:\mathrm{the}\:\mathrm{spring}\:\mathrm{if} \\ $$$$\mathrm{the}\:\mathrm{system}\:\mathrm{shown}\:\mathrm{below}\:\mathrm{is}\:\mathrm{in} \\ $$$$\mathrm{equilibrium}. \\ $$

Question Number 21145 Answers: 0 Comments: 7

Figure shows an arrangement of blocks, pulley and strings. Strings and pulley are massless and frictionless. The relation between acceleration of the blocks as shown in the figure is

$$\mathrm{Figure}\:\mathrm{shows}\:\mathrm{an}\:\mathrm{arrangement}\:\mathrm{of}\:\mathrm{blocks}, \\ $$$$\mathrm{pulley}\:\mathrm{and}\:\mathrm{strings}.\:\mathrm{Strings}\:\mathrm{and}\:\mathrm{pulley} \\ $$$$\mathrm{are}\:\mathrm{massless}\:\mathrm{and}\:\mathrm{frictionless}.\:\mathrm{The} \\ $$$$\mathrm{relation}\:\mathrm{between}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{figure}\:\mathrm{is} \\ $$

Question Number 21142 Answers: 0 Comments: 1

if A+B=(π/4) so proof (1+tan A)(1+tan B)=2

$${if}\:{A}+{B}=\frac{\pi}{\mathrm{4}} \\ $$$${so}\:{proof}\:\left(\mathrm{1}+\mathrm{tan}\:{A}\right)\left(\mathrm{1}+\mathrm{tan}\:{B}\right)=\mathrm{2} \\ $$

Question Number 21141 Answers: 1 Comments: 0

if sin αsin β−cos αcos β+1=0 so proof that 1+cot αtan β=0

$${if}\:\mathrm{sin}\:\alpha\mathrm{sin}\:\beta−\mathrm{cos}\:\alpha\mathrm{cos}\:\beta+\mathrm{1}=\mathrm{0}\: \\ $$$${so}\:{proof}\:{that}\:\mathrm{1}+\mathrm{cot}\:\alpha\mathrm{tan}\:\beta=\mathrm{0} \\ $$

Question Number 21137 Answers: 1 Comments: 0

If the minimum value of ∣z+1+i∣ + ∣z−1−i∣ + ∣2 − z∣ + ∣3 − z∣ is k then (k − 8) equals

$$\mathrm{If}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mid{z}+\mathrm{1}+{i}\mid\:+\:\mid{z}−\mathrm{1}−{i}\mid\:+\:\mid\mathrm{2}\:−\:{z}\mid\:+\:\mid\mathrm{3}\:−\:{z}\mid\:\mathrm{is} \\ $$$${k}\:\mathrm{then}\:\left({k}\:−\:\mathrm{8}\right)\:\mathrm{equals} \\ $$

Question Number 21131 Answers: 0 Comments: 4

Figure shows a small bob of mass m suspended from a point on a thin rod by a light inextensible string of length l. The rod is rigidly fixed on a circular platform. The platform is set into rotation. The minimum angular speed ω, for which the bob loses contact with the vertical rod, is (1) (√(g/l)) (2) (√((2g)/l)) (3) (√(g/(2l))) (4) (√(g/(4l)))

$$\mathrm{Figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{small}\:\mathrm{bob}\:\mathrm{of}\:\mathrm{mass}\:{m} \\ $$$$\mathrm{suspended}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{a}\:\mathrm{thin}\:\mathrm{rod} \\ $$$$\mathrm{by}\:\mathrm{a}\:\mathrm{light}\:\mathrm{inextensible}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length} \\ $$$${l}.\:\mathrm{The}\:\mathrm{rod}\:\mathrm{is}\:\mathrm{rigidly}\:\mathrm{fixed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{platform}.\:\mathrm{The}\:\mathrm{platform}\:\mathrm{is}\:\mathrm{set}\:\mathrm{into} \\ $$$$\mathrm{rotation}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\omega,\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{bob}\:\mathrm{loses}\:\mathrm{contact}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{vertical}\:\mathrm{rod},\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\sqrt{\frac{{g}}{{l}}} \\ $$$$\left(\mathrm{2}\right)\:\sqrt{\frac{\mathrm{2}{g}}{{l}}} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\frac{{g}}{\mathrm{2}{l}}} \\ $$$$\left(\mathrm{4}\right)\:\sqrt{\frac{{g}}{\mathrm{4}{l}}} \\ $$

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