CENGAGE/GTEWANIMATHSSOLUTIONS

CHAPTERDETERMINANTS||ALGEBRA

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1

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction

Adeterminantofsecondorderismadewiththeelements0and1.Findthenumberofdeterminantswithnon-negativevalues.

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2

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction

Findthevalueof

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3

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction

Provethatthedeterminant isindependentof

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4

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction

If then find the number of real roots of the equation

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5

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|

6

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Introduction

Iftheentriesina determinantareeither0or1,thenthegreatestvalueoftheirdeterminantsis

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7

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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8

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

If and are all different and nonzero), then

provethat

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9

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

10

Prove that the value of the determinant

isreal.

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11

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

If ,thenprovethat

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12

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Withoutexpandingthedeterminants,provethat

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13

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Findthevalueofdeterminant

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14

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Usingpropertiesofdeterminants,evaluate

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∣∣∣ − 75 + 3i − 4i5 − 3i84 + 5i + 4i4 − fi9∣∣∣23

23

ar = (cos2rπ + is ∈ 2rπ)1/ 9 |a1a2a3a4a5a6a7a8a9| = 0.

|103115114111108106104113116|

+ |113116104108106111115114103| = 0

∣∣√(13) + √32√5√5√(15) + √(26)5√(10)3

+ √(65)√(15)5∣∣

|184089408919889198440|.

15

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Solvefor

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16

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

By using properties of determinants, show that:

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17

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

By using properties of determinants, show that:

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18

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat

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19

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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20

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat

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21

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat

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22

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

If are real number, and then prove that either

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23

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).

24

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

If are three-digitpositive integersuch thateach isamultipleof19,thenforsomeintegers provethat isdivisibleby19.

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25

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat

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26

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Showthat

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27

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Let berealnumberswith Showthattheequation

representsastraightline.

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28

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,

29

isindependentof

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30

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Let Show that

iscontant.

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31

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat

Hence find the value of the determinant if are the roots of the equation

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32

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).

33

CoordinareGeometry

If aswellas areinG.P.withsamecommonratio,thenprovethatthepoints arecollinear.

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34

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry

If the lines areconcurrent,showthatthepoint arecollinear.

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35

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry

Find the value of for which the lines areconcurrent.

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36

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry

If the lines beingdistinct and different from are concurrent, then prove that

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37

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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38

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Use Of Determinant InCoordinareGeometry

Find thevalueof represent a pairofstraightlines.

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39

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

Provethat

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40

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

If arerealnumbers,thenwithoutexpandingatanystage,showthat

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41

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∣∣

42

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

Provethat

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43

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

Express assquareofadeterminantofhenceevaluateif.

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44

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

Showthat

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45

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

Provewithoutexpansionthat

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46

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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47

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant

If then find the value of

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48

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant

If are differentiable functions of

prove that

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49

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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50

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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51

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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52

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations

If a system of three linear equations is consistent, then

provethat areinH.P.

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53

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations

SolvebyCramersrule

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54

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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56

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations

If have a nontrivialsolution,findthevalueof

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57

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations

If arenotallzeroandconnectedbytheequations

,showthat

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58

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Solve for : =0, a,b

andcbeingdistinctrealnumbers.

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59

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

60

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

If isapolynomialofdegree provethat

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61

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SystemOfLinearEquations

Theproductofallvaluesof forwhichthesystemofequations

hasnon-trivialsolution,is (b) (d)

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62

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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63

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

64

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants

If

thenprovethat

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65

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_ProductOfTwoDeterminants

Let

.Expressingastheproductoftwodeterminants,showthat Hence,showthatif

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66

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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68

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat:

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69

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

Provethat

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70

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_Differentiation Of ADeterminant

If

, show that and that denotes the sumof allthecofactorsofalltheelementin

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71

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)

72

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants

If then the value of the determnalnt b. c. d.noneofthese

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73

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_PropertiesOfDeterminants

Ifadeterminantoforder isformedbyusingthenumbers1or-1thenminimumvalueofdeterminantis:

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74

CENGAGE_MATHS_ALGEBRA_DETERMINANTS_SomeImportantDeterminants

If is purely realpurelyimaginary d.

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75

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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Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage(G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana,VidyaMandir

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

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