CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER … · 24 CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Some...
Transcript of CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER … · 24 CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Some...
CENGAGE/GTEWANIMATHSSOLUTIONS
CHAPTERDETERMINANTS||ALGEBRA
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction
Adeterminantofsecondorderismadewiththeelements0and1.Findthenumberofdeterminantswithnon-negativevalues.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction
Findthevalueof
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction
Provethatthedeterminant isindependentof
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction
If then find the number of real roots of the equation
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction
If provethat
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|124 − 130410|
∣∣∣∣
x sin θ cosθ−sin θ −x 1cos θ 1 x
∣∣∣∣
θ
a, b, c ∈ R,
Δ =∣∣∣∣
x c −b
−c x a
b −a x
∣∣∣∣= 0
x + y + z = 0 |axbyczcyazbxbzcxay| = xyz|abaca|
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction
Iftheentriesina determinantareeither0or1,thenthegreatestvalueoftheirdeterminantsis
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Without expanding at any stage, prove that the value of each of the following
determinants iszero. (1) (2) (3) ,
wherewiscuberootofunity
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If and are all different and nonzero), then
provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Prove that represents a straight line
paralleltothey-axis.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
3 × 3
∣∣∣∣∣
0 p − q p − r
q − p 0 q − r
r − p r − q 0
∣∣∣∣∣
∣∣∣∣
41 1 579 7 929 5 3
∣∣∣∣
∣∣∣∣∣
1 w w2
w w2 1
w2 1 w
∣∣∣∣∣
=∣∣∣∣∣
abc b2c c2b
abc c2a ca2
abc a2b b2a
∣∣∣∣∣
= 0, (a, b, c ∈ R)
a + b + c = 0.
a ≠ 0,⎡⎢⎣x + a x x
x x + a x
x x x + a2
⎤⎥⎦ = 0
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Prove that the value of the determinant
isreal.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If ,thenprovethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Withoutexpandingthedeterminants,provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Findthevalueofdeterminant
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Usingpropertiesofdeterminants,evaluate
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∣∣∣ − 75 + 3i − 4i5 − 3i84 + 5i + 4i4 − fi9∣∣∣23
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ar = (cos2rπ + is ∈ 2rπ)1/ 9 |a1a2a3a4a5a6a7a8a9| = 0.
|103115114111108106104113116|
+ |113116104108106111115114103| = 0
∣∣√(13) + √32√5√5√(15) + √(26)5√(10)3
+ √(65)√(15)5∣∣
|184089408919889198440|.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Solvefor
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
By using properties of determinants, show that:
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
By using properties of determinants, show that:
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Findthevalueofthedeterminant
x|x − 22x − 33 × − 4x− 42x − 93x − 16x − 82x − 273x
− 64| = 0.
∣∣ − a2abacba − b2bacb − c2∣∣ = 4a2b2c2
|x + 42x2x2 × + 42x2x2 × + 4| = (5x + 4)(4 − x)2
|sin α cos α sin(α + δ)sin β cosβ sin(β + δ)sinγ cosγ sin(γ + δ)|= 0
|1111123413610141020|
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are real number, and then prove that either
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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|a − b − c2a2a2 − c − a2b2c2 − a − b| = (a + b + c)3.
∣∣ab + ca2bc + ab2ca + bc2∣∣ = − (a + b + c)
× (a − b)(b − c)(c − a).
a, bandc= |b + + aa + bc + aa + + ca + + + a| − 0,ad + b + c = 0 or a = b = c
∣∣x2x2 − (y − z)2yzy2y2 − (z − x)2zxz2z2 − (x− y)2xy∣∣
= (x− y)(y − z)(z − x)(x + y + z)(x2 + y2 + z2).
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are three-digitpositive integersuch thateach isamultipleof19,thenforsomeintegers provethat isdivisibleby19.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Showthat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Let berealnumberswith Showthattheequation
representsastraightline.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If thenprovethat
xi = aibic + i, i = 1, , 23 x1n, |a1a2a3b1b2b3c1c2c3|
∣∣(b + c)2baacba(c + a)2cbcacb(a + b)2∣∣
= ∣∣(b + c)2a2a2b2(c + a)2b2c2c2(a + b)2∣∣ = 2abc(a + b + c)3
|ab − + ba + cbc − aa − + ac|
= (a + b + c)(a2 + b2 + c2).
a, b, c a2 + b2 + c2 = 1.|ax − by − cbx− aycx + abx + ay − ax+ by − y + bcx
+ acy + b − ax − by + c| = 0
∣∣a2 + 1abaca ^ 2 + 1bcacb ^ 2∣∣ = 1 + a2 + b2 + c2.
a2 + b2 + c2 = 1,
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isindependentof
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Let Show that
iscontant.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
Hence find the value of the determinant if are the roots of the equation
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
Findtheareaofatrianglewhoseverticesare
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant In
∣∣a2 + (b2 + c2)cosφab(1 − cosφ)ac(1 − cosφ)ba(1
− cosφ)b2 + (c2 + a2)cosφbc(1 − cosφ)ca(1 − cosφ)cb(1
− cosφ)c2 + (a2 + b2)cosφ∣∣a, b, ⋅
Deltar = ∣∣r − 1n6(r − 1)22n24n − 2(r − 1)23n33n2 − 3n∣∣.n
∑r=1
Deltar
|1 + a11111 + b11111 + c11111 + d|
= abcd(a + + + + ).1a
1b
1c
1d
a, b, c, dpx4 + qx3 + rx2 + sx + t = 0.
A(3, 2),B(11, 8)andC(8, 12).
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CoordinareGeometry
If aswellas areinG.P.withsamecommonratio,thenprovethatthepoints arecollinear.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
If the lines areconcurrent,showthatthepoint arecollinear.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
Find the value of for which the lines areconcurrent.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
If the lines beingdistinct and different from are concurrent, then prove that
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
If lines areconcurrent,thenprovethat aredistinct
x1, x2, x3 y1, y2, y3(x1, y1), (x2, y2), and(x3, y3)
a1x + b1y + 1 = 0, a2x + b2y + 1 = 0anda3x+ b3y + 1 = 0(a1, b1), (a1, b2)and(a3, b3)
a 2x + y − 1 = 0 2x + y − 1 = 0ax + 3y − 3 = 0 3x + 2y − 2 = 0
ax + y + 1 = 0, x + by + 1 = 0andx + y + c = 0(a, b, c1)
+ + = 1.1
1 − a
11 − b
11 − c
px + qy + r = 0, qx + ry + p = 0andrx+ py + q = 0p + q + r = 0(wherep, q, r ).
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
Find thevalueof represent a pairofstraightlines.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
If arerealnumbers,thenwithoutexpandingatanystage,showthat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
If then find the value of
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λ if 2x2 + 7xy + 3y2 + 8x + 14y + λ = 0
|a1α1 + b1β1a1α2 + b2β2a1α3 + b1β3a2α1 + b2β1a2α2
+ b2β2a2α3 + b2β3a3α1 + b3β1a3α2 + b3β2a3α3 + b3β3| = 0
α, β, γ|1 cos(β − α)cos(γ − α)cos(α − β)1cos(γ − β)cos(α
− γ)cos(β − γ)1| = 0
∣∣1 ×2 ×2 1x21x∣∣ = ,∣∣x3 − 10x − x40x − x4x3 − 1x − x4x3 − 10∣∣
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
Express assquareofadeterminantofhenceevaluateif.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
Showthat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
Provewithoutexpansionthat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If
∣∣(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 + bz)2(1 + cz)2∣∣ = 2(b − c)(c − c)(a − b)
× (y − z)(z − x)(x − y).
= ∣∣2bc − a2c2b2c22ca − b2a2b62a22ab − c2∣∣
∣∣bc − a2ca − b2ab − c2ca − b2ab − c2bc − a2ab − c2bc − a2ca
− b2∣∣ = ∣∣a2c22ac − b22ab − c2b2a2b22bc − a2c2∣∣
|ah + bggab + chbf + bafhb + bcaf + bbg + fc|
= a|ah + bgahbf + bahbaf + bcgf|
y = |sin x cosx sin x cosx − sinx cos × 11| = f ∈ d .dy
dx
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If then find the value of
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If are differentiable functions of
prove that
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
Let bearepeatedrootofaquadraticequation bepolynomials of degrees 3, 4, and 5, respectively, then show that
isdivisibleby ,whereprime denotesthederivatives.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If show thatdenoteshesumofallthecofactorsof
allelementsin dashdenotesthederivativewithrespectof
f(x) =∣∣∣x∩ !2 cos x 4 sin x 8
∣∣∣
cos(nπ)2
sin(nπ)2
([f(x)])x=0
.n ∈ z.
dn
dxn
f , g, andh
xand(x) = ∣∣fgh(xf)' (xg)' (xh)' (xf2f)' ' (x2g)' ' (x2h)' ' ∣∣
^ (' ) = ∣∣fgff' g' h' (x3f ' ' )' (x3g' ' )' (x3h' ' )' ∣∣
α f(x) = 0andA(x), B(x),C(x)
|A(x)B(x)C(x)A(α)B(α)C(α)A' (α)B' (α)C' (α)| f(x)(' )
(x) = |α + xθ + xλ + xβ + xφ + xμ + xγ + xψ + xv + x|Delta^(x) = 0andDelta(0) + Sx,whereS
Delta(0)and x.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Find the value of if the system of equation (i) posses unique solution (ii) infinite
solutions
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If a system of three linear equations is consistent, then
provethat areinH.P.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
SolvebyCramersrule
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Find for which the system of equations is consistent and find the
solutionsforallsuchvaluesof
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
For what values of the following system of equations possesses a nontrivalsolution over the set of rationals:
aandba2x − by = a2 − bandbx = b2y = 2 + 4b
x + 4ay + a = 0, x+ 3by + b = 0, andx+ 2cy + c = 0a, b, c
x+ y + z = 6 x − y + z = 2 3x + 2y − 4z = − 5
λx + y − 2z = 0, 2x − 3y + z = 0, x − 5y + 4z = λ
λ.
k,
+ + 3 = 0, 3 + − 2 = 02 + 3 − 4 = 0.
55 Alsofindthesolutionforthisvalueof
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If have a nontrivialsolution,findthevalueof
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If arenotallzeroandconnectedbytheequations
,showthat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Solve for : =0, a,b
andcbeingdistinctrealnumbers.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
Prove that impliesthat areinA.P.or areinG.P.
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c + ky + 3z = 0, 3c + ky − 2z = 02c + 3y − 4x = 0.k.
2ax − 2y + 3z = 0, x+ ay + 2z = 0, and2 + az = 0a.
x, yandza1x + b1y + c1z = 0, a2x + b2y + c2z = 0, and(p1 + λq1)x
+ (p2 + λq2) + (p3 + λq3z = 0)λ = − |a1b1c1a2b2c2p1p2p3| ÷ |a1b1c1a2b2c2q1q2q3|
x ∈ R
∣∣∣∣∣
(x + a)(x− a) (x + b)(x − b) (x + c)(x − c)
(x− a)3 (x − b)3 (x − c)3
(x+ a)3 (x + b)3 (x + c)3
∣∣∣∣∣
= |a − aa + − cb + ca − − c0a − cxyz1 + x+ y| = 0a, b, c a, c, b
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If isapolynomialofdegree provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Theproductofallvaluesof forwhichthesystemofequations
hasnon-trivialsolution,is (b) (d)
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Let and be the roots of the equation and respectively. If the system of equations and
hasanontrivialsolutionthenprovethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If aretheanglesoftriangle,showthatthesystemofequations
possesnontrivialsolution.Hence,systemhasinfinitesolutions.
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f(x) < 3,|1af(a)/ (x − a)1bf(b)/ (x − b)1cf(c)/ (x − c)|
÷ ∣∣1aa21 ^ 21 ^ 2∣∣ =f(x)
(x − a)(x− b)(x − c)
t,(a − t)x + by + cz = 0, bx + (c − t)y + az = 0, cx + ay
+ (b − t)z = 0|a − c − b − cb − a − b − ac| |abcbcacab| |acaba|
|aa + + c + + a + aa + b|
α1, α2 β1, β2 ax2 + bx + c = 0px2 + qx + r = 0 α1y + α2z = 0
β1yβ _ 2z = 0 =b2
q2ac
pr
A,B, andCx sin 2A + yisnC + z sinB = 0, sinC + y sin 2B + z sinA= 0, andxsinB + y sinA + z sin 2C = 0
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
thenprovethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
Let
.Expressingastheproductoftwodeterminants,showthat Hence,showthatif
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
If thendetA
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
ax12 + by12 + cz12 = ax22 + by22 + cz22 = ax32 + by32+ cz32 = d, ax23 + by2y3 + cz2z3 = ax3x1 + by3y1 + cz3z1
= ax1x2 + by1y2 + cz1z2 = f,
|x1y1z1x2y2z2x3y3z3| = (d − f){ }1/ 2(d + 2f)
abc
= |2a1b1a1b2 + a2b1a1b3 + a3b1a1b2 + a2b12a2b2a2b3
+ a3b2a1b3 + a3b1a3b2 + a2b32a3b3|= 0.
ax2 + 2hxy + by2 + 2gx + 2fy + c
= (lx + my + n)(l ′x + m ′y + n), then|ahghbfgfc| = 0.
2s = a + b + c and A =
∣∣∣∣∣
a2 (s − a)2 (s − a)2
(s − b)2 b2 (s − b)2
(s − c)2 (s − c)2 c2
∣∣∣∣∣
∣ y y y ∣
67 Evaluate
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat:
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If
, show that and that denotes the sumof allthecofactorsofalltheelementin
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are different from 1 and are the roots of
, then prove that
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∣∣ ^ xCx1 C
x2 C
y3C
y1C
y2C
z3C
z1C
z2C3∣∣
| − 2aa + ba + cb + a − 2 + + ac + b − 2c|
= 4(a + b)(b + c)(c + a)
|ax − by − czay + bxcx + azay + bxby − cz − axbz + cycx
+ azbz + cycz − ax − by| = (x2 + y2 + z2)(a2 + b2
+ c2)(ax+ by + cz).
(x) = |a1 + xb1 + xc1 + xa2 + xb2 + xc2 + xa3 + xb3 + xc3
+ x|^ (x) = 0 (x) = (0) + Sx, whereS
(0).
α, β, γ
ax3 + bx2 + cx+ d = 0and(β− γ)(γ − α)(α − β) =252
∣∣∣
αβγα2β2γ 2∣∣∣=
α
1 − α
β
1 − β
γ
1 − γ
25d2(a + b + c + d)
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If then the value of the determnalnt b. c. d.noneofthese
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Ifadeterminantoforder isformedbyusingthenumbers1or-1thenminimumvalueofdeterminantis:
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If is purely realpurelyimaginary d.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are the roots of then the value of the determinantis b. c. d.
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76
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
If and if
p + q + r = 0 = a + b + c,|paqbrcqcrapbrbpcqa|is 0 pa + qb + rc 1
3 × 3
z = | − 53 + 4i5 − 7i3 − 4i68 + 7i5 + 7i8 − 7i9|, thenza + ib, wherea ≠ 0, b ≠ 0 a + ib, whereb = 4
α, β, γ px2 + qx2 + r = 0,|αββγγαβγγααβγααββγ| p q 0 r
a = cosθ + i sin θ, b = cos2θ − i sin 2θ, c = cos3θ + i sin 3θ
then a. b. c.
d.noneofthese
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77
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry
If , then the linepossesthroughthefixedpointwhichis b. c. d.
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78
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
then equals b. c. d.
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79
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are the roots of the equation is equal to
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If denotes the greatest integer less than or equal to the ral number underconsideration, and then the value of the
∣∣∣∣
a b c
b c a
c a b
∣∣∣∣= 0 θ = 2kπ, k ∈ Z θ = (2k+ 1)π, k ∈ Z
θ = (4k + 1)π, k ∈ Z
|ab − + ba + cbc − aa − ba + bc| = 0 ax+ by + c = 0(1, 2) (1, 1) ( − 2, 1) (1, 0)
∣∣xnxn+2xn+3ynyn+ 2yn+3znzn+2zn+3∣∣
= (x− y)(y − z)(z − x)( + + ),1x
1y
1z
n 1 −1 2 −2
f(x) = a = bx + cx2andα, β, γx3 = 1, then|abcbcacab| f(α) + f(β) + f(γ)f(α)f(β) + f(β)f(γ) + f(γ)f(α) f(α)f(β)f(γ) −f(α)f(β)f(γ)
[]−1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ a < 2,
|[ ] + 1[ ][ ][ ][ ] + 1[ ][ ][ ][ ] + 1| [ ] [ ] [ ]
80 determinant is b. c. d. none ofthese
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81
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Let suchthatnotwoofthemareequalandsatisfythenequation hasatlastonerootin atlastonerootin
atlastonerootin atlasttworootsin
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82
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If areinA.P.thenvalueofdeterminant
is0(b)Independentfrom (d)Independentfrom
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83
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If , ,,and
,thenthevalueof is b. c. d.noneofthese
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|[x] + 1[y][z][x][y] + 1[z][x][y][z] + 1| [x] [y] [z]
a, b, c ∈ R |2abcbc2ac2ab| = 0,24ax2 + 4bx + c = 0 [0, 1]
[ − , ]12
12
[ − 1, 0] [0, 2]
p, q, r∣∣a2 + 2n+1 + 2pb2 + 2n+2 + 3qc2 + p2n + p2n+12qa2 + 2n
+ pb2 + 2n+1c2 − r∣∣a, b, c a2b2c2 − 2n n
(x1 − x2)2 + (y1 − y2)
2 = a2 (x2 − x3)2 + (y2 − y3)
2 = b2
(x3 − x1)2 + (y3 − y1)
2 = c2
k
∣∣∣∣∣
x1 y1 1x2 y2 1
x3 y3 1
∣∣∣∣∣
= (a + b + c)(b + c − b)(c + a − b)
× (a + b − c)k 1 2 4
84
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
The value of the determinant is
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85
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
aredifferent,thenthedeterminant
vanisheswhena. b. c. d.
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86
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
The determinant is equal to a.b.
c. d.
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87
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If thenthevalueof is b. c.
d.noneofthese
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88
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Thevalueof isequaltoa.zerob.c. d.noneofthese
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
∣∣kak2 + a21kbk2 + b21kck2 + c21∣∣
k(a + b)(b + c)(c + a) kabc(a2 + bf2 + c2) k(a − b)(b − c)(c − a)
k(a + b − c)(b + c − a)(c + a − b)
∣∣111abca3b2c3∣∣ = (a − b)(b − c)(c − a)(a + b + c),wherea,
b, c
∣∣111(x − a)2(x − b)2(x − c)2(x − b)(x− c)(x − c)(x − a)(x
− a)(x− b)∣∣
a + b + c = 0 x = (a + b + c)13
x = (a + b + c)12
x = a + b + c
∣∣y2 − xyx2abca' b' c' ∣∣∣∣bx + aycx + byb ′x + a' yc ′ x+ b' y∣∣ ∣∣ax + bybx + cya ′x + b'yb'x + c'y∣∣
∣∣bx + cyax+ byb ′x + c' ya ′ x+ b' y∣∣∣∣ax + bybx+ cya ′x + b' yb ′ x+ c' y∣∣
∣∣∣∣∣
b2 + c2 ab ac
ab c2 + a2 bc
ca cb a2 + b2
∣∣∣∣∣
= ka2b2c2, k abc a2b2c2
bc + ca + ab
∣∣ − 1213 + 2√22 + 2√213 − 2√22 − 2√21∣∣−16√2 −8√2
{ , , , }
89
Let be thesetof thirdorderdeterminant thatcanbemadewith
thedistinctnon-zerorealnumbers Then b. c.
d.noneofthese
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90
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If isacomplexcuberootofunity,thenvalueof
isa.0b. c. d.noneofthese
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91
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If areinA.P.,thenthevalueofthedeterminantareinA.P.,thenthevalueofthedeterminant isa.1b. c. d.
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92
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If onerootof isa. b.
c. d.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Intriangle ,if
{D1, D2,D3, Dn}
a1, a2, aq .n
∑i=1
Di = 1n
∑i=1
Di = 0
Di − Dj, ∀ i, j
w= ∣∣a1 + b1wa1w
2 + b1c1 + b1wa2 + b2wa2w2 + b2c2 + b2wa3
+ b3wa3w2 + b3c3 + b3w∣∣
−1 2
x, y, z|a + 2a + 3a + 2xa + 3a + 4a + 2ya + 4a + 5a + 2z|
0 2a a
a + b + c = 0,⎡⎢⎣a − x c b
c b − x a
b a c − x
⎤⎥⎦ = 0 x = 1 x = 2
x = a2 + b2 + c2 x = 0
ABC
cot
93
thenthetrianglemustbea.equilateralb.isoscelesc.obtuseangledd.noneofthese
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94
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are inG.P. then thevalueof dependsonb. c. d.independentof
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95
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If
,then isapolynomialofdegree b. c. d.
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96
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
The value of determinant is equal to b.c. d.noneofthese
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97
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
∣∣∣111 +
+ +∣∣∣
cotA2
cotB2
cotC2
tanB2
tanC2
tanB2
tanA2
tanA2
tanB2
a, b, c, d, e, andf ∣∣a2d2xb2e2yc2f2z∣∣xandy xandz yandz x, y, andz
a2 + b2 + c2 = − 2andf(x) =∣∣a + a2x(1 + b2)x(1 + c2)x(1 + a2)x1
+ b2x(1 + c2)x(1 + a2)x(1 + b2)x1 + c2x∣∣f(x) 0 1 2 3
∣∣111mCm+1
1 Cm+21 Cm
1 Cm+12 Cm+2
2 C2∣∣ 1−1 0
2 1 + 3
If thenthevalueof isa. b. c. d.
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98
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If
,then isequalto b. c. d.noneofthese
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99
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Thevalueof arerespectively,pth, th,and th
termsofanH.P.isa. b. c. d.noneofthese
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100
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are three digit even natural numbers anddivisibleby2butnotnecessarilyby4divisibleby
4butnotnecessarilyby8divisibleby8noneofthese
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101
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
The value of the determinant of order, being given by is b. c. d.
noneofthese
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x ≠ y ≠ zand
∣∣∣∣∣
x x2 1 + x3
y y2 1 + y3
z z2 1 + z3
∣∣∣∣∣
= 0, xyz 1 2 −1
2
x ≠ 0, y ≠ 0, z ≠ 0and|1 + x111 + y1 + 2y11 + z1 + z1
+ 3z| = 0x−1 + y−1 + z −1 1 −1 −3
∣∣∣∣∣
yz zx xy
p 2p 3r1 1 1
∣∣∣∣∣, wherex, y, z (2q) (3r)
−1 0 1
a1b1, c1, a2b2c2anda3b3c3= |c1a1b1c2a2b2c3a3b3|, thenis
nth |x111x111x|(x − 1)n−1(x+ n − 1) (x − 1)n(x + n − 1) (1 − x)−1(x + n − 1)
102
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If arepositiveandarethepth,qth,andrthterms,respectivelyofaG.P.thenis b. c. d.noneofthese
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103
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
,then representsastraightlineparalleltox-axisastraightlineparalleltoy-axisparabolaastraightlinewithnegativeslope
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104
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If then isequalto b.c. d.noneofthese
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105
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
thenvalueof is b. c. d.noneofthese
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
a, b, c= |logap1 log bq1 log cr1| 0 log(abc) = (p + q + r)
f ′ (x) = |mxmx− pmx + p ∩ + pn − pmx + 2nmx + 2n
+ pmx + 2n − p|y = f(x)
|x3636x6x3| = |2x7x7272x| = |45x5x4x45| = 0, x 0 −93
∣∣xnxn+2x2n1xaaxn+5xa+6x2n+5∣∣ = 0, ∀x ∈ R,wheren
∈ N ,a n n − 1 n + 1
106Let then value of is a. none-negative b. none-
positivec.negatived.positive
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107
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Valueof wherex,y,zarenonzerorealnumber,isequal
to b. c. d.
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108
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Whichofthefollowingisnottherootoftheequation a.
b. c. d.
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109
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If then has common roothascommonrootsumofrootsof is noneof
these
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
x < 1,⎡⎢⎣x2 + 2 2x+ 1 12x+ 1 x + 2 1
3 3 1
⎤⎥⎦
⎡⎢⎣x + y z z
x y + z x
y y z + x
⎤⎥⎦,
xyz 2xyz 3xyz 4xyz
∣∣∣∣
x −6 −12 −3x x − 3
−3 2x x + 2
∣∣∣∣= 0? 2
0 1 −3
f(x) = |xaaaxaaax| = 0, f ′ (x) = 0andfx = 0fx = 0andf ′ (x) = 0 f(x) = 0 −3a
110
Roots of the equation are independent ofindependent of depend on independent of
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111
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are different, then the value ofis b. c. d.
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112
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If ,thenvalueof is b.c. d.
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113
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If and then
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114
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Thevalueofthedeterminant isequalto b. c. d.
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|xmn1axn1abx1abc1| = 0 m, andna, b, andc m, n, anda, b, c
m, nanda, b, c
a, b, c∣∣0x2 − ax3 − bx2 + a0x2 + cx4 + bx − c0∣∣ = 0 c c b 0
|b + + aa + ba + + + ac + aa + + c| = k|abaca| k 12 3 4
Δ =∣∣∣∣
a1 b1 c1
a2 b2 c2
a3 b3 c3
∣∣∣∣
Δ1 =∣∣∣∣∣
a1 + pb1 b1 + qc1 c1 + ra1
a2 + pb2 b2 + qc2 c2 + ra2
a3 + pb3 b3 + qc3 c3 + ra3
∣∣∣∣∣
Δ1 =
= 122232422232425 ^ 232425 ^ 26242526272
1 0 2 3
115
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If and discriminant of is negative, then
is b. c. d.
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116
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If form a G.P. and for all , then
isequalto b. c. d.
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117
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Let be three mutually perpendicular unit
vectors,thenthevalueof isequalto b. c. d.noneofthese
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118
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
The number of distinct real roots of in the interval
is b. c. d.
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119
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If then isequalto
b. c. d.
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a > 0 ax2 + 2bx + c∣∣∣∣
a b ax+ b
b c bx+ c
ax+ b bx+ c 0
∣∣∣∣
+ve (ac − b)2(ax2 + 2bx + c) −ve 0
a1, a2, , an ai > 0 i ≥ 1∣∣∣∣∣
log an, logan+1, logan+2
logan+3, logan+4, log an+ 5
logan+6, logan+7, log an+ 8
∣∣∣∣∣
0 1 2 3
→a r = xr i + yr j + zrk, r = 1, 2, 3
∣∣∣∣∣
x1 x2 x3
y1 y2 y3
z1 z2 z3
∣∣∣∣∣
0 ±1 ±2
∣∣∣∣
sin x cos x cos xcos x sin x cos xcos x cos x sin x
∣∣∣∣= 0
π/4 ≤ x ≤ π/4 0 2 1 3
pλ4 + qλ3 + rλ2 + sλ + t =∣∣∣∣∣
λ2 + 3λ λ − 1 λ + 3
λ2 + 1 2 − λ λ − 3
λ2 − 3 λ + 4 3λ
∣∣∣∣∣
p −5
−4 −3 −2
120
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are different from zero and then
thevalueoftheexpression is b. c. d.
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121
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If areanglesofatriangles,thenthevalueof is b.
c. d.
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122
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
For the equation There are exactly two distinct roots There is
onepairofequationrealroots.TherearethreepairsofequalrootsModulusofeachrootis2
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
x, y, z Delta =∣∣∣∣∣
a b − y c − z
a − x b c − z
a − x b − y c
∣∣∣∣∣= 0,
+ +a
x
b
y
c
z0 −1 1 2
A,B, C
∣∣∣∣∣
e2iA e− iC e− iB
e− iC e2iB e− iA
e− iB e− iA e2iC
∣∣∣∣∣
1
−1 −2 −4
∣∣∣∣∣
1 x x2
x2 1 x
x x2 1
∣∣∣∣∣
= 0,
2 − 1 .m 1
123Letmbeapositiveintegerand = .Thenthevalue
of
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124
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
If
then equals b. c. d.noneofthese
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125
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If etc., and , etc. and
then
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126
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Thevalueofdeterminant
isa.alwayspositiveb.alwaysnegativec.alwayszerod.cannotsayanything
Δr∣∣∣∣∣
2r − 1 .m Cr 1
m2 − 1 2m m + 1
sin2(m2) sin2m sin(m2)
∣∣∣∣∣
m
∑r=0
Δr
Dk = 1 ∩ 2kn2 + n + 1n2 + n2k − 1n2n2 + n + 1andn
∑k=1
Dk = 56.
n 4 6 8
l21 + m21 + n2
1 = 1 l1l2 + m1m2 + n1n2 = 0
Δ =∣∣∣∣
l1 m1 n1
l2 m2 n2
l3 m3 n3
∣∣∣∣
∣∣bc − a2ac − b2ab − c2ac − b2ab − c2bc − a2ab − c2bc − a2ac
− b2∣∣
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127
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Valueof
dependsupon b. c. d.noneofthese
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128
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If
,thenhevalueof is b. c. d.
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129
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants
.Thenvalueof isequalto
b. c.zerod.noneofthese
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130
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
Ifthedeterminant
,thenthevalueof s b. c. d.
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131
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Let Then the value ofisequaltoa.zerob. c. d.
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∣∣1 + x11 + x1x1 + x1x21 + x21 + x2x1 + x2x
21 + x31
+ x3x1 + x3x2∣∣
xonly x1only x2only
∣∣a2 + λ2ab + cλca − bλab − cλb2 + λ2bc + aca + bλbc
− aλc2 + λ2∣∣|λc − b − cλab − aλ| = (1 + a2 + b2 + c2)3
λ 8 27 1 −1
f(x) = ∣∣cos × 12sin × 22x tan × 1∣∣ ( lim )x→0
f(x)
x1
−1
∣∣b − − aa − ^ (' ) − c' c ′ − a' a ′ − b' b− c' ' c−a' 'a− b' ' ∣∣
= m|abca' b' c'a' ' b' ' c' ' |m 0 2 1 −1
∣∣x2 ×2 x6 × 6∣∣ = Ax4 + Bx3 + Cx2 + Dx+ E .5A + 4B + 3C + 2D + E −16 11 11
132
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If are the given determinants, then
b. c.
d.
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133
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
If then the value of the determinant
is b. c. d.noneofthese
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134
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Ifthevalueofthedeterminant ispositivethen b.c. d.
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135
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If are respectively, the cofactors of the elements of thedeterminant , then the value of
isequalto b. c. d.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant
Let
Delta1 = |xaxbaax|andDelta2 = |xbax|
Delta1 = 3(Delta2)2 (Delta1) = 3Delta2
d
dx(Delta1) = 3(Delta2)
2d
dxDelta1 = 3Delta23/2
y = sinmx,
|yy1y2y3y4y5y6y7y8|,whereyn =dny
dxn m9 m2 m3
|a111b111c| (a, b, c > 0) abc > 1abc < − 8 abc < − 8 abc ≻ 2
A1,B1, C1, , a1, b1c1, ,Delta = |a1b1c1a1b2c2a3b3c3|, Delta ≠ 0
|B2C2B3C3| a12 a1 a21 a122
136
.Thenthevalueof isa. b. c. d.
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137
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
The number of positive integral solutions of the equation
is
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138
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
are distinct real numbers not equal to one. If have nontrivial solution,
thenthevalueof isequaltoa. b. c. d.noneof
these
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139
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Ifthesystemoflinearequation
hasauniquesolution, then b. c. d. none ofthese
f(x) = ∣∣2 cos2 x sin 2x − sinx sin 2x2 sin2 x cosx sin x
− cosx0∣∣
∫ π /2
0[f(x) + f ′ (x)]dx π π/2 2π 3π/2
∣∣∣∣∣
x3 + 1 x2y x2z
xy2 y3 + 1 y2z
xz2 z2y z3 + 1
∣∣∣∣∣
= 11
a, b, cax + y + z = 0, x + by + z = 0, andx + y + cz = 0
+ +1
1 − a
11 − b 1 − c
1 −1 zero
x + y + z = 6, x + 2y + 3c = 14, and2x + 5y + λz
= μ(λ, μR)λ = 8 λ = 8, μ = 36 λ = 8, μ ≠ 36
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140
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If are the angles of a triangle and system of equations has non-trivial solutions, then
triangleisnecessarilya.equilateralb.isoscelesc.rightangledd.acuteangled
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141
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Given
arenotallzero,thenthevalueof is b. c. d.noneofthese
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142
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If and the system of equation has nontrivial solution, then
valueof is b. c. d.
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143
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Thesystemoflinearequations hasanon-trivialsolutionfor:(1)infinitelymanyvaluesof .(2)exactlyonevalueof.(3)exactlytwovaluesof .(4)exactlythreevaluesof .
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α, β, γcos(α − β)x + cos(β − γ)y + cos(γ − α)z = 0cos(α + β)x + cos(β + γ)y + cos(γ + α)z = 0sin(α + β)x + sin(β + γ)y + sin(γ + α)z = 0
a = x/ (y − z), b = y/(z − x), andc = z/(x − y), wherex,y, zandz
ab + bc + ca 0 1 −1
pqr ≠ 0 (p + a)x + by + cz = 0ax + (q + b)y + cz = 0 ax+ by + (r + c)z = 0
+ +a
p
b
q
c
r−1 0 0 ¬ − 2
x + λy − z = 0 λx − y − z = 0 x + y − λz = 0λ
λ λ λ
144
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If arenon-zero,thenthesystemofequations
hasanon-trivialsolutionif b. c.d.noneofthese
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145
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If and the system of equations is consistent, then the
possiblerealvaluesof are b. c. d.noneof
these
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146
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If areinG.P.withcommonratio areinG.P.withcommonratio and equations have
onlyzerosolution,thenwhichofthefollowingisnottrue?a. b. c. d.noneofthese
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147
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
If , then graphs of issymmetrical about the origin graph of is symmetrical about the y-axis
isanoddfunction
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a, b, c(α + a)x + αy + αz = 0, αx + (α + b)y + αz = 0, αx + αy
+ (α + c)z = 0α−1 = − (a−1 + b−1 + c− 1) α−1 = a + b + c
α + a + b + c = 1
c < 1x + y − 1 = 0, 2x − y − c = 0, andbx+ 3by − c = 0
b b( − 3 )34
b( − , 4)32
b( − , 3)34
a, b, c r1andα, β, γr2 ax + αy + z = 0, bx + βy + z = 0, cx + γy + z = 0
a + b + c abc 1
g(x) = ∣∣a−xex loge ax2a−3xe3x loge ax4a−5xe5x loge a1∣∣ g(x)
g(x)
( ∣ )x=0
= 0d4g(x)
dx4f(x) = g(x) × log( )a − x
a + x
148
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If , then a factor is
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149
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
If
then is independent of is independent of is a constant
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150
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
The determinant is divisible byb. c. d.noneofthese
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151
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If isdivisibleby(a) (b) (c)
(d)noneofthese
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= | − xa − xaab − x| a + b + x
x2 − (a − b) + x+ a2 + b2 + ab x2 − (a + b) + x + a2 + b2 − aba + b − x
=∣∣∣s∫hη cosφ sin θs ∈ φ cosθ cos θ cosφ cosθs ∈ φ − s
∫hη − s∫hηs ∈ φs∫hη cosφ0∣∣∣
θ φ Delta
( ])θ=π/ 2
= 0dDeltadthη
Delta = ∣∣a2 + xabaca ^ 2 + xbcacb ^ 2 + x∣∣ x
x2 x3
f(x) =∣∣∣∣∣
a −1 0ax a −1
ax2 ax a
∣∣∣∣∣, thenf(2x) − f(x) a b c, d, e
152
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
then b. c. d.noneofthese
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153
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If , then (a) (b)
(c) (d)
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154
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If are nonzero real numbers such that
b. c. d.
noneofthese
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155
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
The values of for which the system of equations admits of nontrivial
solutionis b. c. d.
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(x) = ∣∣x2 + 4x − 32x + 4132x2 + 5x − 94x + 5268x2 − 6x
+ 116x − 6104∣∣ = ax3 + bx2 + cx + d,a = 3 b = 0 c = 0
f(θ) =∣∣∣∣∣
sin2A cotA 1
sin2B cosB 1
sin2C cosC 1
∣∣∣∣∣
tanA + tanB + c cotA cotB cotC
sin2 A + sin2B + sin2 C 0
a, b, c |baabcaacaa| = 0, then
+ + = 01a
1bω
1
cω2+ + = 0
1a
1
bω2
1
cω□+ + = 0
1aω
1
bω2
1c
k ∈ Rx + ky + 3z = 0, kx + 2y + 2z = 0, 2x + 3y + 4z = 0
2 5/2 3 5/4
156
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Ifdeterminant
isa.positiveb.independentof c.independentof d.noneofthese
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157
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
The system of equations , , ,has: (a)nosolution if (b)unique solution if (c)infinitenumberofsolutionsif (d)noneofthese
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158
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
isdivisibleby b. c. d.
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159
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Consider the system of the equation Statement 1: System
equations has infinite solutions when Statement 2: If the determinantthen
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160
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Consider the determinantStatement 1: has one root Statement 2: The value of skewsymmetricdeterminantofoddorderisalwayszero.
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161
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Statement1:Ifthesystemofequation
has a non trivial solution, then the value of is 0. Statement 2: the value of skewsymmetricmatrixoforder3isorder.
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∣∣∣cos(θ + φ) − sin(θ + φ)cos 2φs∫hη cos θs ∈ φ − cosθs
∫hη cosφ∣∣∣
q φ
−2x + y + z = a x − 2y + z = b x + y − 2z = ca + b + c ≠ 0 a + b + c = 0a + b + c = 0
= ∣∣aa2012a + b(a + b)012a + 3b∣∣ a + b a + 2b 2a + 3ba2
kx + y + z = 1, x + ky + z = k, andx + y + kz = k2.
k = 1.∣∣111kk1k21k∣∣ = 0, k = − 1.
f(x) = ∣∣0x2 − ax3 − bx2 + a0x2 + cx4 + bx − c0∣∣.f(x) = 0 x = 0.
λx + (b − a)y + (c − a)z = 0, (a − b)x + λy + (c − b)z
= 0, and(a − c)x + (b − c)y + λz = 0λ
162
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Statement1:
isequalto0.Statement2:Thevalueofskewsymmetricmatrixoforder3iszero.
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163
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Consider the system of equation Statement 1: if the
systemhasinfinitenumberofsolutions,then Statement2:Thedeterminant
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164
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Let are the real roots of the equation If the system of equations
given by hasnon-trivialsolutionsthenthevalueof is________.
Delta = |my + nzmq + nrmb + nckz −mxkr − mpkc
− ma − nx− ky − np − kq − na − kb|
x + y + z = 6, x + 2y + 3z = 10, andx + 2y + λz = μ.μ = 10.
|116121012μ| = 0f or μ = 10.
α, β, γx3 + ax2 + bx + c = 0(a, b, c ∈ Randa ≠ 0).( ∈ u, v, andw) αu + βv + γw = 0 βu + γv + αw = 0γu + αv + βw = 0 a2/ b
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165
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If areinH.P.,and ,thenthevalueof representsthegreatestintegerfunction
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166
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Provethat
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167
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Absolutevalueofsumofrootsoftheequation
is_______.
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168
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
The value of for which the system of equation hasnosolutions,is________.
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a1, a2, a3, 54, a6, a7, a8, a9 D = |a1a2a354a6a7a8a9|[D]iswhere[.]
∣∣∣∣∣
(β+ γ − α − δ)4 (γ + γ − α − δ)2 1
(γ + α − β − γ − δ)4 (γ + α − β − δ)2 1
(α + β− γ − δ)4 (α + β − γ − δ)2 1
∣∣∣∣∣
=
− 64(α − β)(α − γ)(α − δ)(βー γ)(β− γ)(γ − δ)
|x + 22x + 33x+ 42x + 33x + 44x + 53x + 55c + 810x
+ 17| = 0
|α| αx + y + z = α − 1x + αy + z = α − 1 x + y + αz = α − 1
169
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Let
thenthevalueof is_____.
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170
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If ,thenthevalueof
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171
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If
thentherealvalueof is__________.
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172
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If
then is__________.
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173
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Given ,thenthevalueof is___________.
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D1 = |aba + bcdc + daba − b|andD2
= |aca + cbdb + daca + b + c|∣∣∣
∣∣∣, whereb ≠ 0andad ≠ bc,
D1
D2
=∣∣∣13 cosθ1s∫hη13 cos θ1s∫hη1∣∣
∣( _ (max)) /2
| × + yx + y + z2x3x + 2y4x + 3y + 2z3x6x + 3y10x + 6y
+ 3z| = 64,x
∣∣xnxn+2xn+4ynyn+ 2yn+4znzn+2zn+4∣∣
= ( − )( − )( − )1y2
1
x2
1
z2
1y2
1
x2
1
z2
n
A = |ab2cde2flm2n|,B = |f2de2n4l2mc2ab| B/A
174
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Thevalueof
is.
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175
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
If
suchthat thenthevalueof
is_______.
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176
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Find the value of for which the homogeneous system of equations: has non-trivial solutions.
Findthesolution.
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177
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Let be positive and not all equal. Show that the value of the determinantisnegative.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
Withoutexpandingadeterminantatanystage,showthat
|2x1y1x1y2 + x2y1x1y3 + x3y1x1y2 + x2y12x2y2x2y3
+ x3y2x1y3 + x3y1x2y3 + x3y22x3y3|
(1 + ax + bx2) = a0 + a1x + a2x2 + + a8x
8, wherea, b, a0,
a1, , a8 ∈ R
a0 + a1 + a2 ≠ 0and|a0a1a2a1a2a0a2a0a1| = 0, 5a
b
λ2x+ 3y − 2z = 0 2x − y + 3z = 0 7x+ λy − z = 0
a, b, c|abcbcacab|
178
aredeterminantoforder3notinvolving
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179
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Show that the system of equations andhasatleastonesolutionforanyrealnumber Findtheset
ofsolutionsof
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180
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Showthat
.
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181
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Considerthesystemlinearequationsin givenby
Findthevalueof forwhichthesystemhasanon-trivialsolution.
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∣∣x2 + × + 1x − 22x2 + 3x − 13x3x − 3x2 + 2x + 32x
− 12x − 1∣∣ = xA + B,whereAandBx.
3x − y + 4z = 3, x+ 2y − 3z = − 26x+ 5y + λz = − 3 λ.
λ = − 5
∣∣ ^ xC xr C
xr+ 1C
y
r+ 2CyrC
y
r+1Czr+2C
zrC
zr+1Cr+1∣∣
= ∣∣ ^ xC x+1r Cx+2
r+1 Cyr+2C
y+1r C
y+2r+1C
zr+2C
z+1r Cz+2
r+1Cr+1∣∣
x, y, andz(s ∈ 3θ)x − y + z = 0, (cos 2θ)x + 4y + 3z = 0, 2x + 7y+ 7z = 0.
θ
182
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If3digitnumbers aredivisiblebyafixedconstant'K'whereA,
B, C are integers lying between 0 and 9, then determinant is always
divisibleby
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183
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If then find the value of
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184
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
Forafixedpositiveinteger if
,thenshowthat isdivisibleby
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185
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
Let be real. Find the set of all values of for which the system of linearequations
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A28, 3B9 and 62C∣∣∣∣
A 3 68 9 C
2 B 2
∣∣∣∣
a ≠ p, b ≠ q, c ≠ rand|pbcaqcabr| = 0,
+ + .p
p − a
q
q − b
r
r − c
n,= |n !(n + 1) !(n + 2)!(n + 1)!(n + 2)!(n + 3) !(n + 2)
!(n + 3) !(n + 4) !|
[/((n !)3) − 4] n.
λandα λ
λx + (s ∈ α)y + (cos α)z = 0, x + (cos α)y + (s ∈ α)z
= 0, − x + (s ∈ α)y − (cos α)z = 0.
186
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Forallvaluesof showthat
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187
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Find thevalueof thedeterminant arerespectively,thepth,qth,andrthtermsofaharmonicprogression.
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188
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Thevalueofthedeterminant
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189
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
ConsiderthesetAofalldeterminantsoforder3withentries0or1only.LetBbethesubsetofAconsistingofalldeterminantswithvalue1.LetCbethesubsetofthesetofalldeterminantswithvalue .Then
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190
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
If is a cube root of unity, then value of the determinantis b. c. d.
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
A,B, CandP ,Q, R|cos(A − P)cos(A −Q)cos(A − R)cos(B − P )cos(B
− Q)cos(B − R)cos(C − P)cos(C −Q)cos(C − R)| = 0
|baabpqr111|,wherea, b, andc
∣∣∣[sin θ, cos θ, sin 2θ], [sin(θ + ), cos(θ + ),
sin(2θ + )], [sin(θ− ), cos(θ − ),
sin(2θ − )∣∣∣
2π3
2π3
4π3
2π3
2π3
4π3
−1
ω( ≠ 1)∣∣11 + i + ω2ω21 − i − 1ω2 − 1 − i − i + ω − 1 − 1∣∣ 0 1 i ω
, , , ,
191
Let betherealnumbers.Thefollowingsystemofequationsin
has a. no solution b. unique solution c. infinitely many solutions d. finitely manysolutions
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192
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Thedeterminant if
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193
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Theparameteronwhichthevalueofthedeterminant
doesnotdependis b. c. d.
a, b, c x, y, andz
+ − = 1, − + = 1, − +
+ = 1
x2
a2
y2
b2
z2
a2
x2
a2
y2
b2
z2
a2
x2
a2
y2
b2
z2
a2
|xp + yxyyp + zyz0xp + yyp + z| = 0 x, y, z
∣∣1aa2cos(p − d)x cospxcos(p + d)xsin(p − d)x sin pxsin(p
+ d)x∣∣a p d x
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194
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If
then isequalto b. c. d.
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195
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
If thesystemofequations hasanonzerosolution,thenthepossiblevalueof are b. c. d.
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196
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Let Then the value of the determinant
is b. c. d.
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197
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
The number of values of for which the system of the equationshasinfinitelymanysolutionsis
b. c. d.infinite
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CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations
f(x) = |1 × + 12 × (x − 1)(x+ 1)x3x(x − 1)x(x − 1)(x
− 2)(x + 1)x(x − 1)|f(5000) 0 1 500 −500
x− ky − z = 0, kx − y − z = 0, x + y − z = 0k −1, 2 1, 2 0, 1 −1, 1
ω = − + i .12
√32
∣∣1111 − 1 − ω2ω21ω2ω4∣∣ 3ω 3ω(ω − 1) 3ω2 3ω(1 − ω)
k(k + 1)x + 8y = 4kandkx + (k + 3)y = 3k − 10 1 2
+ = 0, + = 0, + = 0
198If the system of equations has infinitesolutions,thenthevalueofequationhasnosolutionis b. c. d.
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199
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
If then a. b. c.d.
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200
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants
Thedeterminant if are inA.P.areinG.P. areinH.P. isarootoftheequation isafactorof
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201
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants
Whichofthefollowingvaluesof satisfyingtheequation
b. c. d.
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202
CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Miscellaneous
Let be the complex number . Then the number of
distinctcomplexcosnumberszsatisfying is
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x+ ay = 0, az + y = 0, andax + z = 0−3 1 0 3
|6i − 3i143i − 1203i| = x + iy, x = 3, y = 1 x = 1, y = 3x = 0, y = 3 x = 0, y = 0
|abaα + cbα + caα + α + c0| = 0, a, b, c a, b, ca, b, c α ax2 + bc + c = 0 (x− α)ax2 + 2bx + c
α∣∣(1 + α)2(1 + 2α)2(1 + 3α)2(2 + α)2(2 + 2α)2(2 + 3α)2(3
+ α)2(3 + 2α)2(3 + 3α)2∣∣ = − 648α?−4 9 −9 4
ω cos( ) + i sin( )2π3
2π3
Δ =∣∣∣∣∣
z+ 1 ω ω2
ω z + ω2 1
ω2 1 z + ω
∣∣∣∣∣
= 0