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Check whether (l+m+n) is a factor of the...

Check whether (l+m+n) is a factor of the determinant `|{:(1+m,m+n,n+1),(" "n," "1," "m),(" "2," "2," "2):}|` or not. Give reason.

Text Solution

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`"Apply"R_1toR_1+R_2|{:(l+m+n,,m+n+l,,n+l+m),(n,,1,,m),(2,,2,,2):}|=2(l+m+n)|{:(1,,1,,1),(n,,1,,m),(1,,1,,1):}|`, "yes (l+m+n)is a factor".
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The quantum numbers of six elements are given below. Arrange them in order of increasing energies. If any of these combinations has/have the same energies, list them. {:((1) n = 4 ",", l = 2",", m_l=-2",", m_s = - 1/2),((2) n = 3 ",", l = 2",", m_l=0",", m_s = + 1/2),((3) n = 4 ",", l = 1",", m_l= 0",", m_s = + 1/2),((4) n = 3 ",", l = 2",", m_l=-2",", m_s = - 1/2),((5) n = 3 ",", l = 1",", m_l=-1",", m_s = + 1/2),((6) n = 4 ",", l = 1",", m_l=0",", m_s = + 1/2):}

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 are proportional to l_1+l_2,m_1+m_2, n_1+n_2dot Statement 2: The angle between the two intersection lines having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 is given by costheta=l_1l_2+m_1m_2+n_1n_2dot

Knowledge Check

  • Let I_(m","n)= int sin^(n)x cos^(m)x dx . Then , we can relate I_(n ","m) with each of the following : (i) I_(n-2","m) " " (ii) I_(n+2","m) (iii) I_(n","m-2) " " (iv) I_(n","m+2) (v) I_(n-2","m+2)" " I_(n+2","m-2) Suppose we want to establish a relation between I_(n","m) and I_(n","m-2) , then we get P(x)=sin^(n+1)x cos^(m-1)x ...(i) In I_(n","m) and I_(n","m-2) the exponent of cos x in m and m-2 respectively, the minimum of the two is m - 2, adding 1 to the minimum we get m-2+1=m-1 . Now, choose the exponent of sin x for m - 1 of cos x in P(x). Similarly, choose the exponent of sin x for P(x)=(nH)sin^(n)x cos^(m)x-(m-1)sin^(n+2) x cos^(m-2)x . Now, differentiating both the sides of Eq. (i), we get =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x(1-cos^(2)x)cos^(m-2)x =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x+(m-1)sin^(n)x cos^(n)x =(n+m)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x Now, integrating both the sides, we get sin^(n+1)x cos^(m-1)x=(n+m)I_(n","m)-(m-1)I_(n","m-2) Similarly, we can establish the other relations. The relation between I_(4","2) and I_(2","2) is

    A
    `I_(4","2)=(1)/(6)(-sin^(3)x cos^(3)x+3I_(2","2))`
    B
    `I_(4","2)=(1)/(6)(sin^(3)x cos^(3)x+3I_(2","2))`
    C
    `I_(4","2)=(1)/(6)(sin^(3)x cos^(3)x-3I_(2","2))`
    D
    `I_(4","2)=(1)/(4)(-sin^(3)x cos^(3)x+2I_(2","2))`

  • Two lines with direction cosines l_(1),m_(1),n_(1) and l_(2), m_(2), n_(2) are at right angle of

    A
    `l_(1)l_(2)-m_(1)m_(2)-n_(1)n_(2)=0`
    B
    `l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)=0`
    C
    `l_(2)=l_(2), m_(1)=m_(2),n_(1)=n_(2)`
    D
    `(l_(1))/(l_(2))=(m_(1))/(m_(2))=(n_(1))/(n_(2))`

  • The number of ways of factoring 91,000 into two factors, m and n, such that m gt 1,n gt 1 and gcd (m, n) = 1 is

    A
    7
    B
    15
    C
    32
    D
    37

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