Evaluate |{:(.^(x)C(1),,.^(x)C(2),,.^(x)...

Evaluate `|{:(.^(x)C_(1),,.^(x)C_(2),,.^(x)C_(3)),(.^(y)C_(1),,.^(y)C_(2),,.^(y)C_(3)),(.^(x)C_(1),,.^(z)C_(2),,.^(z)C_(3)):}|`

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`(1)/(12)xyz(x-y)(y-z)(z-x)`

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|{:(.^(x)C_(r),,.^(x)C_(r+1),,.^(x)C_(r+2)),(.^(y)C_(r),,.^(y)C_(r+1),,.^(y)C_(r+2)),(.^(z)C_(r),,.^(z)C_(r+1),,.^(z)C_(r+2)):}| is equal to

The determinant |^(^^)xC_(1),^(x)C_(2),^(x)C_(3)^(^^)yC_(1),^(y)C_(2),^(y)C_(3)sim zC_(1),zC_(2),zC_(3)] is equal to-

Evaluate det[[x_(C_(1)),x_(C_(2)),x_(C_(3))y_(C_(1)),y_(C_(2)),yC_(3)z_(C_(1)),z_(C_(2)),z_(C_(3))]]

prove that |{:((a-x)^(2),,(a-y)^(2),,(a-z)^(2)),((b-x)^(2),,(b-y)^(2),,(b-z)^(2)),((c-x)^(2),,(c-y)^(2),,(c-z)^(2)):}| |{:((1+ax)^(2),,(1+bx)^(2),,(1+cx)^(2)),((1+ay)^(2),,(1+by)^(2),,(1+cy)^(2)),((1+az)^(2),,(1+bx)^(2),,(1+cz)^(2)):}| =2 (b-c)(c-a)(a-b)xx (y-z) (z-x)(x-y)

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0

Show that |^x C_r^x C_(r+1)^x C_(r+2)^y C_r^y C_(r+1)^y C_(r+2)^z C_r^z C_(r+1)^z C_(r+1)|=|^x C_r^(x+1)C_(r+1)^(x+2)C_(r+2)^y C_r^(y+1)C_(r+1)^(y+2)C_(r+2)^z C_r^(z+1)C_(r+1)^(z+2)C_(r+1)| .

Prove that the value of each the following determinants is zero: (a^(x)+a^(-x))^(2),(a^(x)-a^(-x))^(2),1(b^(y)+b^(-y))^(2),(b^(y)-+b^(-y))^(2),1(c^(z)+c^(-z))^(2),(c^(z)-c^(-z))^(2),1]|

ARIHANT MATHS-DETERMINANTS -Exercise (Subjective Type Questions)

Prove that |{:(b+c,c,b),(c,c+a,a),(b,a,a+b):}| =4 abc.
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Prove that: |[a-b-c, 2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3
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Find the value of determinat |{:(sqrt(13)+sqrt(3),2sqrt(5),sqrt(5)),(...
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The value of the determinant |(bc,ca,ab),(p,q,r),(1,1,1)|, where a, b ...
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Whitout expanding the determinat at any stage prove that |{:(-5,3+5i,...
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Prove without expansion that |a h+bgga b+c h bf+b afh b+b c af+b cc bg...
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If A,B and C are the angles of a triangle and |{:(1,1,1),(1+sinA,1+si...
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The value of |{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaal...
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If y=(u)/(v), where u and v are functions of x, show that v^(3)(d^(2...
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Show that the determinant Delta (x) is given by Delta(x) = |{:(sin(...
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Evaluate |{:(.^(x)C(1),,.^(x)C(2),,.^(x)C(3)),(.^(y)C(1),,.^(y)C(2),,....
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(i) Find maximum value of f(x)=|{:(1+sin^(2)x,cos^(2)x,4sin2x),(sin...
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If f(x) = |[x^2-4x+6,2x^2+4x+10,3x^2-2x+16],[x-2,2x+2,3x-1],[1,2,3]| t...
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prove that |(1/(a-a1)^2,1/(a-a1),1/a1),(1/(a-a2)^2,1/(a-a2),1/a2),(1/...
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Show that in general there are three values of t for which the followi...
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Eliminates (i) a,b, and c (ii) x,y,z from the equations -a+(by)...
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If x, y, z are not all zero & if ax + by + cz=0, bx+ cy + az=0 & cx ...
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