mastersheet: binomial theorem exercise # 1 [vector]

IIT JEE MAINS & ADVANCED MATHEMATICS BY OM SIR

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MASTERSHEET: Binomial Theorem

EXERCISE # 1 [VECTOR]

Q.1 If ABCDE is a pentagon then the resultant of

forces AB , AE , BC , DC , ED and AC in

terms of AC is-

(A) 2 AC (B) 3 AC

(C)5 AC (D) None of these

Q.2 Points a + b + c , 4 a +3 b , 10 a + 7 b – 2 c are

(A) collinear (B) coplanar

(C) non-collinear (D) None of these

Q.3 If Five forces AB , AC , AD , AE , AF act at

the vertex A of a regular hexagon ABCDEF. then

their resultant is (where O is the centroid of the

hexagon)-

(A) 2 AO (B) 3 AO (C) 5 AO (D) 6 AO

Q.4 If D, E, F are the mid points of the sides BC, CA

and AB respectively of a triangle ABC and 'O' is

any point then AD + BE + CF is-

(A) 1 (B) 0 (C) 2 (D) None

Q.5 If the vector b

is collinear with the vector

a

= (2 2 ,–1, 4) and | b

| = 10, then

(A) a

± b

= 0 (B) a

± 2 b

= 0

(C) 2 a

± b

= 0 (D) None of these

Q.6 If points A(1, 2, 3), B(3, 4, 7), C(– 3, – 2, – 5) are

collinear then the ratio in which B divides AC is-

(A) – 1 : 3 (B) 1 : 3

(C) 3 : 1 (D) None of these

Q.7 The position vectors of points A, B, C are

respectively a

, b

, c

. If L divides AB in 3 : 4 & M

divides BC in 2 : 1 both externally, then LM is-

(A) 4 a

–2 b

+2 c

(B) 4 a

+2 b

+2 c

(C) –4 a

+2 b

+2 c

(D) 4 a

–2 b

–2 c

Q.8 If A(4, 7, 8), B(2, 3, 4), C(2, 5, 7) are the position

vectors of the vertices of ABC. Then length of

angle bisector of angle A is -

(A) 342

3 (B) 34

3

2

(C) 342

1 (D) 34

3

1

Q.9 If 1e

& 2e

are non collinear unit vectors, such

that | 1e

+ 2e

| = 3 then (2 1e

– 5 2e

).(3 1e

+ 2e

)

is equal to

(A) –2

11 (B)

2

13 (C)

11

2 (D)

2

11

Q.10 The vector p

perpendicular to the vectors

a

= 2 i + 3 j – k and b

= i – 2 j + 3 k and

satisfying the condition p

.(2 i – j + k ) = – 6 is

(A) – i + j + k (B) 3(– i + j + k )

(C) 2(– i + j + k ) (D) i – j + k

Q.11 If | a

| = 5, | a

– b

| = 8 and | a

+ b

| = 10, then | b

| is

equal to

(A) 1 (B) 57 (C) 3 (D) None

Q.12 Angle between diagonals of a parallelogram

whose side are represented by a

= 2 i + j + k

and b

= i – j – k

(A) cos–1

3

1 (B) cos–1

2

1

(C) cos–1

9

4 (D) cos–1

9

5

Q.13 Vectors a

and b

make an angle = 3

2.

If | a

| = 1, | b

| = 2, then {( a

+ 3 b

) × (3 a

– b

)}2 is

equal to

(A) 225 (B) 250 (C) 275 (D) 300

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Q.14 Unit vector perpendicular to the plane of the

triangle ABC with position vectors a

, b

, c

of

the vertices A, B, C is

(A)

)accbba(

(B)

2

)accbba(

(C)

4

)accbba(

(D) None of these

Q.15 Given the three vectors a

= –2 i + j + k ,

b

= i + 5 j and c

= 4 i + 4 j – 2 k . The projection

of the vector 3 a

– 2 b

on the vector c

is

(A) 11 (B) –11 (C) 13 (D) None

Q.16 For three vectors u

, v

, w

which of the

following expressions is not equal to any of the

remaining three ?

(A) u

. ( v

× w

) (B) ( v

× w

) . u

(C) v

. ( u

× w

) (D) ( u

× v

) . w

Q.17 Which of the following expression is meaningful ?

(A) u

. ( v

× w

) (B) ( u

. v

) . w

(C) ( u

. v

) w

(D) u

× ( v

. w

)

Q.18 For any three vectors a

, b

and c

,

( a

– b

). ( b

– c

) × ( c

– a

) =

(A) 0 (B) a

. b

× c

(C) 2 a

. b

× c

(D) None of these

Q.19 A , B and C are three non coplanar vectors, then

( A + B + C ) . (( A + B ) × ( A + C )) = (A) 0

(B) [ A , B , C ]

(C) 2 [ A , B , C ] (D) – [ A , B , C ]

Q.20 If A

, B

, C

are three non-coplanar vectors, then

B.AC

CB.A

+

BA.C

CA.B

=

(A) 0 (B) 1 (C) 2 (D) None

Q.21 The value of [( a

+ 2 b

– c

), ( a

– b

), ( a

– b

– c

)] is

equal to the box product:

(A) [ a

b

c

] (B) 2[ a

b

c

]

(C) 3 [ a

b

c

] (D) 4 [ a

b

c

]

Q.22 If a

= i + j + k , b

= i – j + k , c

= i + 2 j – k ,

then the value of

c.cb.ca.c

c.bb.ba.b

c.ab.aa.a

is equal to

(A) 2 (B) 4 (C) 16 (D) 64

Q.23 If b

and c

are two non- collinear vectors such

that a

|| ( b

× c

), then ( a

× b

). ( a

× c

) is equal to

(A) 2a

( b

. c

) (B) 2b

( a

. c

)

(C) 2c

( a

. b

) (D) None of these

Q.24 Let a

= x i + 12 j – k , b

= 2 i + 2x j + k and

c

= i + k . If the ordered set [ b

c

a

] is left

handed, then:

(A) x (2, ) (B) x (–, –3)

(C) x (–3, 2) (D) x {–3, 2}

Q.25 If a

, b

, c

be the unit vectors such that b

is not

parallel to c

and a

× (2 b

× c

) = b

, then the

angle that a

makes with b

and c

are

respectively

(A) 3

&

4

(B)

3

&

3

2

(C) 2

&

3

2 (D)

2

&

3

Q.26 Vector of length 3 unit which is perpendicular to

i + j + k and lies in the plane of i + j + k and

2 i –3 j , is-

(A) 6

3( i –2 j + k )

(B) 6

3 (2 i – j – k )

(C) 114

3(8 i –7 j – k )

(D)114

3(–7 i +8 j – k )



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Q.27 Given unit vectors m

, n

& p

such that angle

between m

& n

= angle between p

and

( m

× n

) = 6

then [ n

p

m

] =

(A)2

3 (B)

4

3 (C)

5

3 (D) None

Q.28 Let u

, v

, w

be vectors such that u

+ v

+ w

= 0.

If | u

| = 3, | v

| = 5, | w

| = 4. Then the value of

the u

. v

+ v

. w

+ w

. u

is-

(A) 47 (B) – 25 (C) 0 (D) 25

Q.29 Let a

= i + j and b

= 2 i – k . The point of

intersection of the lines r

× a

= b

× a

and

r

× b

= a

× b

is

(A) – i + j + 2 k (B) 3 i – j + k

(C) 3 i + j – k (D) i – j – k

Q.30 If a line has a vector equation

r

= 2 i + 6 j + ( i –3 j ), then which of the

following statements hold good?

(A) the line is parallel to 2 i + 6 j

(B) the line passes through the point 3 i + 3 j

(C) the line passes through the point i + 9 j

(D) the line is parallel to XY- plane

Q.31 A line passes through a point A with position

vector 3 i + j – k and is parallel to the vector

2 i – j +2 k . If P is a point on this line such that

AP = 15 units, then the position vector of the

point P is/are

(A) 13 i +4 j –9 k (B) 13 i – 4 j +9 k

(C) 7 i – 6 j +11 k (D) –7 i +6 j –11 k

Q.32 The perpendicular distance of

A

(1, 4, – 2) from the segment BC where

B

= (2, 1, – 2) and C

= (0, – 5, 1) is-

(A) 267

3 (B) 26

7

6

(C) 267

4 (D) 26

7

2

Q.33 If line r

= ( i –2 j – k ) + (2 i + j +2 k ) is parallel

to the plane r

. (3 i –2 j –m k ) = 14, then the

value of m is

(A) 2

(B) –2

(C) 0

(D) can not be predicted with these information

Q.34 Shortest distance between the lines:

r

= (4 i – j ) + ( i +2 j – 3 k ) and

r

= ( i – j +2 k ) + (2 i + 4 j – 5 k ) is

(A) 6/ 5 (B) 12/ 5

(C) 18/ 5 (D) None of these

Q.35 The distance between the line

r

= 2 i –2 j +3 k + ( i – j +4 k ) and the plane

r

.( i +5 j + k ) = 5 is

(A) 10/3 (B) 3/10

(C) 33

10 (D) 10/9

Q.36 Equation of a line which passes through a point

with position vector c

, parallel to the plane

r

. n

= 1 & perpendicular to the line r

= a

+ t b

is-

(A) r

= c

+ ( c

– a

) × n

(B) r

= c

+ ( a

× n

)

(C) r

= c

+ ( b

× n

)

(D) r

= c

+ ( b

. n

) a

Q.37 The vectors a

= –4 i + 3 k , b

= 14 i + 2 j – 5 k are

co-initial. The vector d

which is bisecting the

angle between the vectors a

and b

, is having the

magnitude 6 , is

(A) i + j +2 k (B) i – j +2 k

(C) i + j – 2 k (D) None of these

Q.38 The set of values of 'm' for which the vectors

a

= m i + (m + 1) j + (m + 8) k ,

b

= (m + 3) i + (m + 4) j + (m + 5) k and

c

= (m + 6) i + (m + 7) j + (m + 8) k are

non-coplanar is

(A) R (B) R – {1}

(C) R – {1, 2} (D)



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Q.39 If r

= i + j + i + j + 4 k ) & r

. ( i +2 j – k ) = 3

are the equations of a line and a plane respectively,

then which of the following is false?

(A) line is perpendicular to the plane

(B) line lies in the plane

(C) line is parallel to the plane but does not lie in

the plane

(D) line cuts the plane is one point only

Q.40 Which of the following system is linearly

dependent-

(A) a

= i + j , b

= i + k , c

= 3 i + 3 j + 2 k

(B) a

= –2 i – 4 k , b

= i –2 j – k , c

= i – 4 j + 3 k

(C) a

= i – 2 j + 3 k , b

= 3 i – 6 j + 9 k

(D) a

=–2 i – 4 k , b

= i –2 j – k , c

= i –4 j + 3 k

Q.41 If a

, b

, c

are linearly independent vectors, then

which one of the following set of vectors is

linearly dependent?

(A) a

+ b

, b

+ c

, c

+ a

(B) a

– b

, b

– c

, c

– a

(C) a

× b

, b

× c

, c

× a

(D) None of these

Q.42 Points 4 i +8 j +12 k , 2 i +4 j +6 k , 3 i + 5 j + 4 k ,

5 i + 8 j +5 k are-

(A) Linearly independent (B) coplanar

(C) Linearly dependent (D) None of these

Q.43 If a

, b

, c

and d

are linearly independent set of

vectors and K1 a

+ K2 b

+ K3 c

+ K4 d

= 0, then K1,

K2, K3, K4 satisfies

(A) K1 + K2 + K3 + K4 = 0

(B) K1 + K3 = K2 + K4 = 0

(C) K1 + K4 = K2 + K3 = 0

(D) None of these

Q.44 Vector x

satisfying the relation A

. x

= c and

A

× x

= B

is

(A) |A|

)BA(Ac

(B) 2|A|

)BA(Ac

(C) 2|A|

)BA(Ac

(D) 2|A|

)BA(2Ac

Q.45 For a non- zero vector A

if the equation

A

. B

= A

. C

and A

× B

= A

× C

hold

simultaneously, then:

(A) A

is perpendicular to B

– C

(B) A

= B

(C) B

= C

(D) C

= A

Q.46 Points a – 2 b + 2 c , 2 a + 3 b – 4 c ,

– 7 b + 10 c are collinear.

Q.47 If D, E, F are the mid points of the sides BC, CA

and AB respectively of a triangle ABC and 'O' is

any point,

(i) OA + OB + OC = OD + OE + OF

(iii) AD + 3

2 BE +

3

1CF =

2

1AC

Q.48 Let r =

a +

and r =

b +

m be two lines in

space wherea = 5 i + j + 2 k &

b = – i +7 j + 8 k ,

= – 4 i – j + k and m = 2 i – 5 j – 7 k then the

position vector of a point which lies on both of

these lines is 2 i + j + k .

Q.49 Given system of points is coplanar

3 a

+2 b

–5 c

, 3 a

+ 8 b

+ 5 c

,

–3 a

+ 2 b

+ c

, a

+ 4 b

–3 c

Fill in the blanks type questions

Q.50 The vectors

AB= 3 i – 2 j + 2 k and

BC = – i + 2 k are the adjacent sides of a

parallelogram ABCD, then the angle between the

diagonals is ………………

Q.51 Let A ,

B ,

C be vectors of length 3, 4, 5

respectively. Let A be perpendicular to

B +

C ,

B to

C +

A and

C to

A +

B . Then the

length of vector A +

B +

C is...........



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Q.52 Let r ,

a ,

b and

c be four non zero vectors,

such that

r ·

a = 0 =

r ·

b , |

r ×

c | = |

r | |

c | then

[a

b

c ] = ............ .

53 A, B, C and D are four points in a plane with

position vectors a , b , c and d

respectively

such that ( a – d

).( b – c ) = ( b – d

) . ( c – a ) = 0.

The point D, then, is the ................ of the triangle

ABC.

Q.54 Let b

= 4 i + 3 j & c

be two vectors perpendicular

to each other in the xy-plane. All vectors in the

same plane having projections 1 and 2 along b

and c

, respectively, are given by ................

Q.55 A non zero vector a

is parallel to the line of

intersection of the plane determined by the

vectors i , i + j and the plane determined by the

i – j , i + k . The angle between a

and the vector

i – 2 j + 2 k is.......

Q.56 If b

and c

are any two non-collinear unit

vectors and a

is any vector, then

( a

. b

) b

+ ( a

. c

) c

+2|cb|

)cb.(a

( b

× c

) = .......



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EXERCISE # 2

Q.1 The points A(5, –1, 1); B(7, –4, 7); C(1, –6, 10)

and D(–1, –3, 4) are vertices of a -

(A) square (B) rhombus

(C) rectangle (D) none of these

Q.2 Points (1, 2, 3); (3, 5, 7) and (–1, –1, –1) are-

(A) Vertices of a equilateral triangle

(B) Vertices of a right angle triangle

(C) Vertices of a isosceles triangle

(D) Collinear

Q.3 The ratio in which the segment joining the points

(2, 4, 5), (3, 5, –4) is divided by the

yz-plane is-

(A) – 2 : 3 (B) 2 : 3 (C) 3 : 2 (D) – 3 : 2

Q.4 The ratio in which the segment joining

(1, 2, –1) and (4, –5, 2) is divided by the plane

2x – 3y + z = 4 is-

(A) 7 : 3 (B) 3 : 7

(C) 3 : 5 (D) None of these

Q.5 If distance of any point from z-axis is thrice its

distance from xy-plane, then its locus is-

(A) x2 + y2 – 9z2 = 0 (B) y2 + z2 – 9x2 = 0

(C) x2 – 9y2 + z2 = 0 (D) x2 + y2 + z2 = 0

Q.6 The co-ordinates of the point where the line

joining the points (2, –3, 1), (3, –4, –5) cuts the

plane 2x + y + z = 7 are-

(A) (2, 1, 0) (B) (3, 2, 5)

(C) (1, –2, 7) (D) None of these

Q.7 A point moves in such a way that sum of square

of its distances from the co-ordinate axis are 36,

then distance of these given point from origin are-

(A) 6 (B) 2 3 (C) 3 2 (D) None

Q.8 The d.c's of a line whose direction ratios are

2, 3, –6, are-

(A) 7

6,

7

3,

7

2 (B)

7

6,

7

3,

7

2

(C) 7

6,

7

3,

7

2 (D) None of these

Q.9 The projections of a line segment on x, y and z

axes are respectively 3, 4 and 5, then the length

and direction cosines of the line segment is

(A) 5 2 ; 25

3,

25

4,

2

1

(B) 3 2 ; 23

3,

25

4,

2

1

(C) 5 2 ; 25

3,

23

4,

2

1

(D) 3 2 ; 25

3,

25

4,–

2

1

Q.10 The direction cosines of a line equally inclined

with the coordinate axes are -

(A) (1, 1, 1) or (–1, –1, –1)

(B)

3

1,

3

1,

3

1or

3

1,

3

1,

3

1

(C)

2

1,

2

1,

2

1or

2

1,

2

1,

2

1

(D) none of these

Q.11 Direction ratios of two lines are a, b, c and

bc

1,

ca

1,

ab

1. The lines are -

(A) Mutually perpendicular

(B) Parallel

(C) Coincident

(D) None of these

Q.12 If , m, n and , m, n be the direction cosines of

two lines which include an angle , then -

(A) cos = + mm + nn

(B) sin = + mm + nn

(C) cos = mm + mn + n +n+ m + m

(D) sin = mn + mn + n + n + m + m



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Q.13 P (x1, y1, z1) and Q (x2, y2, z2) are two points

if direction cosines of a line AB are , m, n then

projection of PQ on AB are -

(A)

1 (x2 – x1) +

m

1 (y2 – y1) +

n

1 (z2 – z1)

(B) (x2 – x1) + m (y2 – y1) + n (z2 – z1)

(C) mn

1

[(x2 – x1) + m(y2 – y1) + n(z2 – z1)]

(D) None of these

Q.14 If the line OP of length r makes an angle with

x-axis and lies in the xz plane, then the coordinate

of P are -

(A) (r cos , 0, r sin ) (B) (0, 0, r sin )

(C) (0, 0, r cos ) (D) (r cos , 0, 0)

Q.15 If coordinates of point P, Q, R, S are respectively

(6, 3, 2) ; (5, 1, 4) ; (3, 4, –7) and

(0, 2, 5) then the projection of PQ on RS are-

(A) 157

31 (B)

157

131

(C) 7

13 (D)

7

13

Q.16 The angle between two lines

1

4z

2

3y

2

1x

and

2

1z

2

4y

1

4x

is-

(A) cos–1 (2/9) (B) cos–1 (4/9)

(C) cos–1 (5/9) (D) cos–1 (7/9)

Q.17 The equation of a line passing through the origin

and parallel to the line whose direction ratios are

1, –1, 2 is -

(A) 1

x =

1

y

=

2

z

(B) 1

1x =

1

1y

=

2

2z

(C) 6/1

x=

6/1

y

=

6/2

z

(D) 2/1

7x =

2/1

7y

=

1

14z

Q.18 3

2x =

4

3y =

0

4z is -

(A) parallel to yz plane

(B) parallel to zx plane

(C) perpendicular to z axis

(D) parallel to z axis

Q.19 The point in which the join of (–9, 4, 5) and

(11, 0, –1) is met by the perpendicular from the

origin is-

(A) (2, 1, 2) (B) (2, 2, 1)

(C) (1, 2, 2) (D) None of these

Q.20 If the lines 2

3z

k2

2y

3

1x

and

5

6z

1

5y

k3

1x

are at right angles, then the

value of k will be –

(A) 7

10 (B)

10

7

(C) –10 (D) –7

Q.21 The equation of straight line passing through the

points (a, b, c) and (a – b, b – c, c – a), is -

(A) ba

ax

=

cb

by

=

ac

cz

(B) b

ax =

c

by =

a

cz

(C) ba

ax

=

b

by =

c

cz

(D) ba2

ax

=

cb2

by

=

ac2

cz

Q.22 The normal form of the plane 2x + 6y + 3z = 1, is-

(A) 7

1z

7

3y

7

6x

7

2

(B) 7

1z

7

3y

7

6x

7

2

(C) 7

1z

7

3y

7

6x

7

2

(D) 7

1z

7

3y

7

6x

7

2



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Q.23 A point which lie in yz plane, the sum of

co-ordinate is 3, if distance of point from xz plane

is twice the distance of point from xy plane, then

co-ordinates are -

(A) (1, 2, 0) (B) (0, 1, 2)

(C) (0, 2, 1) (D) (2, 0, 1)

Q.24 A point located in space is moves in such a way

that sum of algebraic distance from xy and yz

plane is equal to distance from zx plane the locus

of the point are -

(A) x – y + z = 2 (B) x + y – z = 0

(C) x + y – z = 2 (D) x – y + z = 0

Q.25 A plane meets the co-ordinate axes in A, B, C

such that the centroid of the triangle is the point

(1, r, r2), the equation of the plane is -

(A) x + ry + r2z = 3r2 (B) r2x + ry + z = 3r2

(C) x + ry + r2z = 3 (D) r2x + ry + z = 3

Q.26 The plane x – 2y + 7z + 21 = 0

(A) contains the line 3

1x

=

2

3y =

1

2z

(B) contains the point (0, 7, –1)

(C) is perpendicular to the line 1

x=

2

y

=

7

z

(D) is parallel to the plane x – 2y + 7z = 0

Q.27 If the line 2

3x =

3

4y =

4

5z lies in the plane

4x + 4y – kz – d = 0, then the value of k and d,

are-

(A) 3, 5 (B) 5, 3 (C) 2, 5 (D) 5, 2

True or false type questions

Q.28 The foot of the perpendicular from (a, b, c) on the

line x = y = z is the point (r, r, r) where

3r = a + b + c.

Q.29 The line x – 2y + 4z + 4 = 0, x + y + z – 8 = 0

intersects the plane x – y + 2z + 1 = 0 at the point

(2, 5, 1).

Fill in the blanks type questions

Q.30 The planes bx – ay = n, cy – bz = , az – cx = m

intersect in a line if ..............

Q.31 If a plane cuts off intercepts OA = a, OB = b,

OC = c from the coordinate axes, then the area of

the triangle ABC is .....



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EXERCISE # 3

Only single correct answer type

questions Part-A

Q.1 Let

A and

B be two non-parallel unit vectors in

a plane. If (

A +

B ) bisects the internal angle

between

A and

B , then is equal to-

(A) 1/2 (B) 1 (C) 2 (D) 4

Q.2 Given a parallelogram OACB. The length of the

vectors

OA ,

OB and

AB are a, b and c

respectively. The scalar product of the vectors

OC and

OB is -

(A) (a2 – 3b2 + c2)/2 (B) (3a2 + b2 – c2)/2

(C) (3a2 – b2 + c2)/2 (D) (a2 + 3b2 – c2)/2

Q.3 The vertices of triangle have the position vectors

a ,

b ,

c and P(

r ) is a point in the plane of

such that : a .

b +

c .

r =

a .

c +

b .

r =

b .

c +

a .

r then for the , P is the

(A) circumcentre (B) centroid

(C) orthocentre (D) incentre

Q.4 If A, B, C, D are four points in space satisfying

AB . CD =K [ | AD |2 + | BC |2 – | AC |2– | BD |2]

then the value of K is –

(A) 2 (B) 1/3

(C) 1/2 (D) 1

Q.5 If in a right angled triangle ABC, the hypotenuse AB

= p, then

CB.CABA.BCAC.AB is -

(A) 2p2 (B) 2

p2

(C) p2 (D) None

Q.6 Let a ,

b and

c be three non–zero and non

coplanar vectors and p ,

q and

r be three

vectors given by p =

a +

b – 2

c ,

q = 3

a – 2

b +

c ,

r =

a – 4

b + 2

c .

If the volume of the parallelopiped determined by

a ,

b and

c is v1 and that of the parallelopiped

determined by p ,

q and

r is v2, then v2 : v1 =

(A) 3 : 1 (B) 7 : 1

(C) 11 : 1 (D) 15 :1

Q.7 A vector x is coplanar with vectors

a = – i + j + k and

b = 2 i + k and is

orthogonal to the vector b . If

x .

a = 7 then the

vector x is equal to-

(A) (– 3 i + 5 j + 6 k ) (B) 2

1(–3 i + 5 j + 6 k )

(C) (3 i – 5 j – 6 k ) (D) none of these

Q.8 If b is a vector whose initial point divides the

join of 5 i and 5 j in the ratio k : 1 and terminal

point is origin and | b | 37 , then k lies in the

interval -

(A)

6

1,6 (B) (–, – 6]

,

6

1

(C) [0, 6] (D) None of these

Q.9 If a and

b are mutually perpendicular vectors,

then the projection of the vector

|ba|

)ba(n

|b|

bm

|a|

a

along the angle

bisector of the vectorsa &

b may be given as-

(A) 222

22

nm

m

(B)

222 nm

(C) 222

22

nm

m

(D)

2

m

Q.10 Let co-ordinates of a point 'p' with respect to the

system non-coplanar vectors

a ,

b and c is (3,

2, 1). Then, co-ordinates of 'p' with respect to the

system of vectors

a +

b +c ,

a –

b +c and

a +

b –c is -

(A)

1,

2

1,

2

3 (B)

2

1,1,

2

3

(C)

1,

2

3,

2

1 (D) none of these



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Q.11 The position vectors of the points P and Q are

p and

q respectively. If O is the origin and R is

a point in the interior of POQ such that OR

bisects the POQ then unit vector along OR is

(A)

|q||p|

qp

(B)

|q|

q

|p|

p

(C)

|q|

q

|p|

p

|q|

q

|p|

p

(D) none of these

Q.12 If

DA =a ,

AB =

b and

CB =

ka where

k > 0 and X, Y are the mid-points of DB & AC

respectively, such that |a | = 17 & |

XY | = 4, then

k equal to -

(A) 17

8 (B)

17

13

(C) 17

25 (D)

17

4

Q.13 Let a ,

b and c be three non-zero vectors, no

two of which are collinear. If the vector

3a + 7

b is collinear with c and 3

b + 2c is

collinear with a , then 9

a + 21

b + 14c is equal

to -

(A) a (B)

c

(C) 0 (D) none of these

Q.14 Let a

be a unit vector and b

a non-zero vector

not parallel to a

. The angles of the triangle, two

of whose sides are represented by

3 ( a

× b

) and b

– ( a

. b

) a

are -

(A) /4, /4, /2 (B) /4, /3, 5/12

(C) /6, /3, /2 (D) None of these

Q.15 Let a

= a1 i + a2 j + a3 k , b

= b1 i + b2 j + b3 k

and c

= c1 i + c2 j + c3 k be three non-zero

vectors such that c

is a unit vector perpendicular

to both the vectors a

and b

. If the angle between

a

and b

is 6

, then

2

321

321

321

ccc

bbb

aaa

is equal to-

(A) 0

(B) 1

(C) 4

1 (a1

2 + a22 + a3

2) (b12 + b2

2 + b32)

(D) 4

3 (a1

2 + a22 + a3

2) (b12 + b2

2 + b32)

× (c12 + c2

2 + c32)

Q.16 Let a

= i – j , b

= j – k , c

= k – i . If d

is

a unit vector such that a

. d

= 0 = [ b

, c

, d

], then

d

equals

(A) ±6

k2ji (B) ±

3

kji

(C) 3

kji (D) ± k

Q.17 If p

, q

, r

be three mutually perpendicular vectors

of the same magnitude. If a vector

x

satisfies the equation p

× p)qx(

+

q

× q)rx(

+ r

× r)px(

= 0

then x

is given by-

(A) 2

1 ( p

+ q

– 2 r

) (B) 2

1 ( p

+ q

+ r

)

(C) 3

1 ( p

+ q

+ r

) (D) 3

1 (2 p

+ q

– r

)

Q.18 If a

= i + j + k , b

= 4 i + 3 j + 4 k and

c

= i + j + k are linearly dependent

vectors and | c

| = 3 , then-

(A) = 1, = –1

(B) = 1, = ±1

(C) = –1, = ±1

(D) = ±1, = 1

Q.19 Let a

= 2 i + j + k , b

= i + 2 j – k and a

unit vector c

be coplanar. If c

is perpendicular

to a

, then c

=

(A) 2

1 (– j + k ) (B)

3

1 (– i – j – k )

(C) 5

1 ( i – 2 j ) (D)

3

1 ( i – j – k )



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One or more than one correct

answer type questions Part-B

Q.20 Let ABC be a triangle, the position vector of

whose vertices are respectively 7 j + 10 k ,

– i + 6 j + 6 k & – 4 i + 9 j + 6 k . Then ABC

is -

(A) Isosceles (B) Equilateral

(C) Right angled (D) none of these

Q.21 Let a = 2 i – j + k ;

b = i + 2 j + k ,

c = i +2 j – 2 k be three vectors. A vector in the

plane of b and

c whose projection on

a is

magnitude 3

2 is -

(A) 3 i + 6 j – 2 k (B) 2 i + 3 j + 3 k

(C) – 2 i – j + 2 k (D) 2 i + j + 5 k

Q.22 Unit vectors a and b are inclined at an angle 2

and | a – b | 1, if 0 . Then may belong

to -

(A) [0, /6] (B) (5/6, )

(C) [/6, /2] (D) [/6, 5/6]

Q.23 If | a

. b

| = | a

× b

|, then the angle between a

and

b

is -

(A) 0º (B) 180º

(C) 135º (D) 45º

Q.24 If a ×

b =

c ×

d and

a ×

c =

b ×

d then -

(A) (a –

d ) = (

b –

c )

(B) (a +

d ) = (

b +

c )

(C) (a –

b ) = (

c +

d )

(D) none of these

Q.25 The scalar A .(

B +

C ) × (

A +

B +

C ) equals

(A) 0 (B)[A

B

C ]+[

B

C

A ]

(C) [A

B

C ] (D) None of these

Q.26 The adjacent sides of a parallelogram are

represented by the vectors 2 i + 4 j – 5 k and

i + 2 j + 3 k respectively. The unit vectors

parallel to the diagonals of the parallelogram are

(A) 69

)k8j2i(

(B) 7

)k2j6i3(

(C) 7

)k2j6i3(

(D) 69

)k8j2i(

Q.27 If a ,

b and

c are non-coplanar vectors, then

the following vectors are coplanar -

(A)a + 2

b + 3

c ,–2

a + 3

b – 4

c ,

a –3

b +5

c

(B) 3a – 7

b –4

c ,3

a –2

b +

c ,

a +

b +2

c

(C)a –2

b +3

c ,–2

a +3

b –4

c ,–

b +2

c

(D) 7a – 8

b + 9

c , 3

a + 20

b + 5

c ,

5a + 6

b +7

c

Q.28 If a vector r of magnitude 3 6 is directed

along the bisector of the angle between the

vectorsa = 7 i – 4 j – 4 k and

b = – 2 i – j + 2 k

then r =

(A) i – 7 j + 2 k (B) i + 7 j – 2 k

(C) – i + 7 j – 2 k (D) i – 7 j – 2 k

Q.29 The vector 3

1 (2 i –2 j + k )

(A) is a unit vector

(B) makes an angle3

with vector

(2 i – 4 j + 3 k )

(C) is parallel to the vector

k

2

1ji

(D) is perpendicular to the vector 3 i +2 j –2 k



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Assertion-Reason type questions Part-C

The following questions 30 to 33 consists of two

statements each, printed as (Assertion) Statement-1

and Reason (Statement-2). While answering these

questions you are to choose any one of the

following four responses.

(A) If both Statement-1 and Statement-2 are true

and the Statement-2 is correct explanation of

the Statement-1.

(B) If both Statement-1 and Statement-2 are true

but Statement-2 is not correct explanation of

the Statement-1.

(C) If Statement-1 is true but the Statement-2 is

false.

(D) If Statement-1 is false but Statement-2 is true

Q.30 Statement-1 : A vector c

, directed along the

internal bisector of the angle between the vector

a

= 7 i – 4 j – 4 k & b

= –2 i – j + 2 k , with | c

| =

5 6 is 3

5 ( i –7 j + 2 k ).

Statement-2 : The vector bisecting the angle of

a

& b

is given by c

=

|b|

b

|a|

a

< 0.

Q.31 Statement-1 : The components of a vector b

along and perpendicular to a non-zero vectora

are

2|a|

a)ab(

.

&

2|a|

)ab(a

respectively.

Statement-2 : IfA .

B and

C are three non-

coplanar vectors then

B.AC

)CB(.A

+

BAC

CAB

.

. = 2

Q.32 Statement-1: Let a

, b

, c

be unit vectors such

that a

+ 5 b

+ 3 c

= 0,

then a

.( b

× c

) = b

.( a

+ c

).

Statement-2 : Box product of three coplanar

vectors is 0.

Q.33 Statement-1: For non-coplanar vectors A ,

B

and C , | [

A

B

C ] | = |

A | |

B | |

C | holds iff

A .

B =

B .

C =

C .

A = 0.

Statement-2 : |A ×

B .

C | = | A | | B | | C | sin .

cos where be angle between A and

B and

the angle between C and

A ×

B .

Column Matching type questions Part-D

Q.34 Match the following :

Column-I Column-II

(A) If a

= i + 2 j + 3 k , (P) –1

b

= – i + 2 j + k and c

= 3 i + j ,

then t, such that a

+ t b

is

perpendicular to c

, will be

(B) If | a

| = 2, | b

| = 5 and (Q) 4

| a

× b

| = 8, then a

. b

=

(C) If four point A (1, 0, 3), (R) 5

B(–1, 3, 4), C(1, 2, 1) and

D(k, 2, 5) are coplanar, then k =

(D) If A, B, C, and D are four (S) 6

points and

|

AB ×

CD +

BC ×

AD +

CA ×

BD |

= (area of the BAC), then =



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Q.35 For any three given vectors a

, b

and c

, match the

following column :

Column-I Column-II

(A) If a

= i + 2 j + 3 k , (P) 0

b

= – 2 i + j + k ,

c

= 10 j – k and

a

× ( b

× c

) = u a

+ v b

+ w c

,

then u =

(B) Volume of the tetrahedron (Q) 1

whose vertices are the

points with position vectors

i – 6 j + 10 k , – i – 3 j + 7 k ,

5 i – j + k and 7 i – 4 j + 7 k

is 11 (units)3 then =

(C) Given two vectors (R) 7

a

= 2 i –3 j + 6 k ,

b

= –2 i + 2 j – k and

=aonb of projection the

bona of projection the

,

then the value of 3 is

(D) Let a

, b

, c

be three non- (S)| a

|| b

|| c

|

zero vectors such that

a

+ b

+ c

= 0

, then (T) 2

b

× a

+ b

× c

+ c

× a

= 0,

where is equal to



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EXERCISE # 44

Only single correct answer type

questions Part-A

Q.1 OABC is a tetrahedron whose vertices are

O (0, 0, 0); A (a, 2, 3); B (1, b, 2) and

C (2, 1, c) if its centroid is (1, 2, –1) then distance

of point (a, b, c) from origin are -

(A) 14 (B) 107

(C) 14/107 (D) None of these

Q.2 A point P (x, y, z) moves parallel to z-axis. Which

of the three variable x, y, z remain fixed ?

(A) x and y (B) y and z

(C) x and z (D) none of these

Q.3 A line segment (vector) has length 21 and

direction ratios as 2, –3, 6. If the line makes

obtuse angle with x - axis, the components of the

line (vector) are -

(A) –6, 9, –18 (B) 2, –3, 6

(C) 6, – 9, 18 (D) –18, 27, – 54

Q.4 A line passes through the points (6, – 7, –1) and

(2, –3, 1). The direction cosines of the line so

directed that the angle made by it with positive

direction of x-axis is acute, are -

(A) 3

2,

3

2 ,

3

1 (B)

3

2,

3

2,

3

1

(C) 3

2,

3

2 ,

3

1 (D) –

3

2 ,

3

2,

3

1

Q.5 If the coordinates of the vertices of a

triangle ABC be A(–1, 3, 2), B(2, 3, 5) and C(3, 5,

–2), then A is equal to-

(A) 45º (B) 60º (C) 90º (D) 30º

Q.6 The shortest distance between the lines

3

3x =

1

8y

=

1

3z and

3

3x

=

2

7y =

4

6z

is-

(A) 30 (B) 2 30 (C) 5 30 (D) 3 30

Q.7 The plane passing through the point

(–2, –2, 2) and containing the line joining the

points (1, 1, 1) and (1, –1, 2) makes intercepts on

the co-ordinates axes, the sum of whose length is-

(A) 3 (B) 4 (C) 6 (D) 12

Q.8 If the plane x – 3y + 5z = d passes through the

point (1, 2, 4), then the intercepts cut by it on the

axes of x, y, z are respectively-

(A) 15, –5, 3 (B) 1, –5, 3

(C) –15, 5, –3 (D) 1, –6, 20

Q.9 In three dimensional space, the equation

3y + 4z = 0 represents-

(A) A plane containing x-axis

(B) A plane containing y-axis

(C) A plane containing z-axis

(D) A line with direction ratios 0, 3, 4

One or more than one correct

answer type questions Part-B

Q.10 The coordinates of a point, square of whose

distance from the origin is 90 is -

(A) (5, 4, 7) (B) (–1, 8, 5)

(C) (4, – 5, –7) (D) (0, 9, 3)

Q.11 If 1, m1, n1 and 2, m2, n2 are D.C.'s of the two

lines inclined to each other at an angle , then the

D.C.'s of the internal and external bisectors of the

angle between these lines are-

(A) )2/sin(2

21

,

)2/sin(2

mm 21

,

)2/sin(2

nn 21

(B) )2/cos(2

21

,

)2/cos(2

mm 21

,

)2/cos(2

nn 21

(C) )2/sin(2

21

,

)2/sin(2

mm 21

,

)2/sin(2

nn 21

(D) )2/cos(2

21

,

)2/cos(2

mm 21

,

)2/cos(2

nn 21

Q.12 The equation of a line passing through the point

with position vector 2 i – 3 j + 4 k and in the

direction of the vector 3 i + 4 – 5 k is -

(A)

r = 2 i – 3 j + 4 k + (3 i + 4 j – 5 k ),

where is a parameter

(B) 3

2x =

4

3y =

5

4z

(C) 4x – 3y = 17, 5y + 4z = 1

(D)

r = 3 i + 4 j – 5 k + (2 i – 3 j + 4 k ),

where is a parameter



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Q.13 The lines 1

2x =

1

3y =

k

4z

and

k

1x =

2

4y =

1

5z are coplanar if -

(A) k = 0 (B) k = – 1(C) k = – 3 (D) k = 3

Q.14 The lines 3

1x =

1

1y

=

0

1z and

2

4x =

0

0y =

3

1z

(A) do not intersect

(B) intersect

(C) intersect at (4, 0, –1)

(D) intersect at (1, 1, –1)

Q.15 The equation of a line passing through the origin

and parallel to the line whose direction ratio are 1,

– 1, 2 is -

(A) 1

x =

1

y

=

2

z

(B) 1

1x =

1

1y

=

2

2z

(C) 6/1

x =

6/1

y

=

6/2

z

(D) 2/1

7x =

2/1

7y

=

1

14z

Q.16 The equation of a plane L is given by

x + 2y – 2z = 9, then -

(A) Intercept made by L on x-axis is 9 units in length

(B) Intercept made by L on y-axis = 9/2 units in

length

(C) Intercept made by L on z-axis is 9/2 units in

length

(D) direction cosines of the normal to the plane

are 1/3, 2/3, –2/3

Q.17 The equation of the line passing through

3 i – 5 j + 7 k and perpendicular to the plane 3x

– 4y + 5z = 8 is -

(A) 3

3x =

4

5y

=

5

7z

(B) 3

3x =

5

4y

=

7

5z

(C) r = 3 i – 5 j + 7 k + (3 i – 4 j + 5 k )

(D) r = 3 i – 4 j + 5 k + (3 i – 5 j + 7 k )

(, are parameter)

Column Matching type questions Part-C

Q.18 Column-I Column-II

(A) 4

2x =

7

3y =

6

4z (P) lies in

3x + 2y + 6z – 12=0

(B) 2

2x =

3

3y =

2

4z

(Q) is parallel to

2x + 6y – 2z = 3

(C) 2

x

=

3

y

=

4

z (R) is perpendicular

to 4x+7y+6z = 0

(D) 4

1x =

7

y =

6

1z (S) passes through

(–2, –3, 4).

Q.19 Equation of a plane

Column-I Column-II

(A) through the origin (P) 2x + 3y – 4z = 5

and (1, 1, 1)

(B) perpendicular to (Q)3x – 2y + 4z = 7

the plane

2x + 3y + 4z = 5

(C) parallel to the plane (R) 4x + 4y – 5z = 3

3x – 2y + 4z = 5

(D) containing the line (S) x – 2y + z = 0

2

3x =

3

4y =

4

5z

Q.20 The equation of a plane passing through the line

of intersection of the planes

2x + 3y – 4z = 1, 3x – y + z + 2 = 0 is

2x + 3y – 4z – 1 + (3x – y + z + 2) = 0

Column-I Column-II

(A) = 1/2 (P) it is parallel to 14x – y = 0

(B) = 29 (Q) it makes an intercept of 4

on the positive x-axis

(C) = 4 (R) is passes through the origin

(D) = – 1/2 (S) it is perpendicular to

2x + 3y – 4z = 0



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EXERCISE # 5 [JEE MAINS PYQ]

1. If a and b are unit vectors, then the greatest value of 3 | a b | + | a – b | is ____. [IIT JEE MAINS 2020]

2. If x and y be two non-zero vectors such that | x + y | = | x | and 2x + y is perpendicular to y , then the value

of is _____.

[IIT JEE MAINS 2020]

3. Let the vectors a , b , c be such that | a | = 2| b | = 4 and | c | = 4. If the projection of b on a is equal to the

projection of c on a and b is perpendicular to c , then the value of | a + b – c | is ______.


4. If the equation of a plane P, passing through the intersection of the planes, x + 4y – z + 7 = 0 and

3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b R, then the distance of the point (3, 2, – 1) from the plane P is

_____.


5. If a = ˆ2i + j + ˆ2k , then the value of 2

ˆ î (a i) + 2

ˆ ˆj (a j) + 2

ˆ ˆk (a k) is equal to ….


6. Let a plane P contain two lines r = i + ˆ î j , R and r = – j + ˆ ˆj k , R

If Q(, , ) is the foot of the perpendicular drawn from the point M(1, 0, 1) to P, then 3( + + ) equals …….


7. Let ABC be a triangle with vertices at points A(2, 3, 5), B(–1, 3, 2) and C(, 5, ) in three dimensional

space. If the median through A is equally inclined with the axes, then (, ) is equal to : [JEE- MAIN - 2013]

(1) (10, 7) (2) (7.5) (3) (7, 10) (4) (5, 7)

8. Let a , b and c be three unit vectors such that | a – b |2 + | a – c |2 = 8.

then | a + 2 b |2 + | a + 2 c |2 is equal to …. [IIT JEE MAINS 2020]

9. The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points

(–1, 2, 3) and (3, –2, 10) is ________.


10. If the vectors, p

= (a + 1) i + aj + âk , q

= âi + (a + 1) j + âk and r

= âi + aj + (a + 1) k (a R) are

coplanar and 3

2

p q

–

2

r q

= 0, then the value of is ________.


11. Let a

, b

and c

be three vectors such that | a

| = 3 , | b

| = 5, b

c

= 10 and the angle between b

and c

is

3

. If a

is perpendicular to the vector b

× c

, then a b c

is equal to ________.




https://t.me/omsir





12. If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane containing the lines

x 1

2

=

y – 3

4 =

z 1

3

and

x 3

2

=

y 2

6

=

z –1

(R) is equal to

k

633, then k is equal to _______.


13. If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (, 7, 1) is5 7 17

, ,3 3 3

,

then is equal to ______.


14. Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the

image of R in the plane P is :

[IIT JEE MAINS 2020] (1) (6, 5, 2) (2) (4, 3, 2) (3) (6, 5, – 2) (4) (3, 4, – 2)

15. A vector a

= i + ˆ2 j + k (, R) lies in the plane of the vectors, b

= i + j and c

= i – j + ˆ4k . If a

bisects the angle between b

and c

then


(1) a

i + 1 = 0 (2) a

i + 3 = 0 (3) a

k + 2 = 0 (4) a

k + 4 = 0

16. Let a

, b

and c

be three unit vectors such that a

+ b

+ c

= 0

.If = a

b

+ b

c

+ c

a

and d

= a

× b

+ b

× c

+ c

× a

, then the ordered pair, , d

is equal to :


(1) 3

– , 3a b2

(2)

3– , 3 c b

2

(3)

3, 3b c

2

(4)

3, 3a c

2

17. The shortest distance between the lines x – 3

3 =

y – 8

–1 =

z – 3

1 and

x 3

–3

=

y 7

2

=

z – 6

4 is :


(1) 7

230 (2) 3 30 (3) 3 (4) 2 30

18. Let the volume of a parallelepiped whose coterminous edges are given by u

= i + j + k , v

= i + j + ˆ3k and

w

= ˆ2i + j + k be 1 cubic unit. If be the angle between the edges u

and w

, then cos can be :


(1) 7

6 3 (2)

5

7 (3)

7

6 6 (4)

5

3 3

19. Let a

= i – ˆ2 j + k and b

= i – j + k be two vectors. If c

is a vector such that b

× c

= b

× a

and

c

a

= 0, then c

b

is equal to :


(1) – 1

2 (2) – 1 (3)

1

2 (4) –

3

2



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20. The mirror image of the point (1, 2, 3) in a plane is 7 4 1

– , – , –3 3 3

. Which of the following points lies on this

plane?

[IIT JEE MAINS 2020] (1) (1, –1, 1) (2) (1, 1, 1) (3) (–1, –1, 1) (4) (–1, –1, –1)

21. The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line 2x = 3y, z = 1 also passes through

the point :


(A) (0, 6, – 2) (B) (–2, 0, 1) (C) (0, –6, 2) (D) (2, 0, –1)

22. A plane passing through the point (3,1,1) contains two lines whose direction ratios are 1, –2, 2 and 2, 3 – 1

respectively. If this plane also passes through the point (, –3, 5), then is equal to :

[IIT JEE MAINS 2020] (A) – 10 (B) 5 (C) 10 (D) – 5

23. The plane which bisects the line joining the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the

point :

[IIT JEE MAINS 2020] (A) (4, 0, –1) (B) (4, 0, 1) (C) (0, 1, –1) (D) (0, –1, 1)

24. Let a, b, c R be such that a2 + b2 + c2 = 1. If a cos = b cos2

3

= c cos4

3

, where = 9

, then

the angle between the vectors ˆai + bj + ˆck and ˆbi + cj + ˆak is :


(A) 2

(B) 0 (C)

9

(D)

2

3

25. The lines r = ( i – j ) + (2 i + k ) and r = (2 i – j ) + m( i + j – k ) [IIT JEE MAINS 2020]

(A) Intersect when = 1 and m = 2 (B) Intersect when = 2 and m = 1

2

(C) Do not intersect for any values of and m (D) Intersect for all values of and m

26. The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, –2, 3) and (1, 1, 0) lies

on the plane :

[IIT JEE MAINS 2020] (A) x + 2y – z = 1 (B) x – 2y + z = 1 (C) x –y – 2z = 1 (D) 2x + y – z = 1

27. The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line x

2=

y

3=

y

–6 is


(A) 7 (B) 1 (C) 1

7 (D)

7

5

28. Let x0 be the point of local maxima of f(x) = a .( b × c ), where a = ˆxi – ˆ2 j + ˆ3k , b = – ˆ2i + xj – k and c =

ˆ7i – ˆ2 j + ˆxk . Then the value of a b + b c + c a at x = x0 is :

[IIT JEE MAINS 2020] (A) – 30 (B) 14 (C) – 4 (D) – 22



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29. If for some R, the lines [IIT JEE MAINS 2020]

L1 : x 1

2

=

y – 2

–1 =

z –1

1 and L2 :

x 2

=

y 1

5 –

=

z 1

1

are coplanar, then the line L2 passes through the

point :

(A) (–2, 10, 2) (B) (10, 2, 2) (C) (10, –2, –2) (D) (2, –10, –2)

30. If the volume of a parallelepiped, whose conterminous edges are given by the vectors

a = i + j + ˆnk . b = ˆ2i + ˆ4 j – ˆnk and c = i + nj + ˆ3k (n 0), is 158 cu. Units, then :


(A) a c = 17 (B) b c = 10 (C) n = 7 (D) n = 9

31. If (a, b, c) is the image of the point (1, 2, –3) in the line, x 1

2

=

y – 3

–2 =

z

–1, then a + b + c is equal to

[IIT JEE MAINS 2020] (A) –1 (B) 2 (C) 3 (D) 1

32. A plane P meets the coordinate axes at A, B and C respectively. The centroid of ABC is given to be (1, 1, 2).

Then the equation of the line through this centroid and perpendicular to the plane P is :


(A) x –1

1 =

y –1

2 =

z – 2

2 (B)

x –1

2 =

y –1

2 =

z – 2

1

(C) x –1

2 =

y –1

1 =

z – 2

1 (D)

x –1

1 =

y –1

1 =

z – 2

2

33. The shortest distance between the lines x –1

0 =

y 1

–1

=

z

1 and x + y + z + 1 = 0, 2x – y + z + 3 = 0 is :


(A) 1

2 (B) 1 (C)

1

2 (D)

1

3

34. Let a

= i – j , b

= i + j + k

and c

be a vector such that a

× c

+ b

= 0

and a

. c

= 4, then | c

|2 is

[2019]

(1) 19

2 (2) 8 (3)

17

2 (4) 9

35. The equation of the line passing through (–4, 3, 1), parallel to the plane x + 2y – z – 5 = 0 and intersecting the line

x 1

–3

=

y – 3

2 =

z – 2

–1 is :

[2019]

(1) x 4

3

=

y – 3

–1 =

z –1

1 (2)

x 4

1

=

y – 3

1 =

z –1

3

(3) x 4

–1

=

y – 3

1 =

z –1

1 (4)

x – 4

2 =

y 3

1

=

z 1

4

36. The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis also

passes through the point :



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[2019] (1) (–3, 1, 1) (2) (–3, 0, –1) (3) (3, 3, –1) (4) (3, 2, 1)

37. Let a = i + j + ˆ2k , b = 1ˆb i + 2

ˆb j + ˆ2k and c = ˆ5i + j + ˆ2k be three vectors such that the projection

vector of b on a is a . If a + b is perpendicular to c , then | b | is equal to :

[2019]

(1) 4 (2) 22 (3) 6 (4) 32

38. If the lines x = ay + b, z = cy + d and x = az + by = cz + d are perpendicular then :

[2019]

(1) cc + a + a = 0 (2) aa + c + c = 0 (3) bb + cc + 1 = 0 (4) ab + bc + 1 = 0

39. The equation of the plane containing the straight line x

2 =

y

3 =

z

4 and perpendicular to the plane containing the

straight lines x

3 =

y

4 =

z

2 and

x

4 =

y

2 =

z

3 is :

[2019]

(1) 3x + 2y – 3z = 0 (2) x + 2y – 2z = 0 (3) x – 2y + z = 0 (4) 5x + 2y – 4z = 0

40. Let a = ˆ2i + 1 j + 3 k , b = ˆ4i + (3 – 2) j + ˆ6k , and c = ˆ3i + ˆ6 j + (3 – 1) k be three vectors such that

b = 2a and a is perpendicular to c . Then a possible value of (1, 2, 3) is -

[2019]

(1) (1, 5, 1) (2) (1, 3, 1) (3) 1

– ,4,02

(4) 1

, 4, – 22

41. The plane passing through the point (4, –1, 2) and parallel to the lines x 2

3

=

y – 2

–1 =

z 1

2

and

x – 2

1 =

y – 3

2 =

y – 4

3 also passes through the point -

[2019]

(1) (1, 1, –1) (2) (1, 1, 1) (3) (–1, –1, –1) (4) (–1, –1, 1)

42. Let A be a point on the line r = (1 – 3) i + ( – 1) j + (2 + 5) k and B(3, 2, 6) be a point in the space. Then the

value of for which the vector –

AB

is parallel to the plane x – 4y + 3z = 1 is –

[2019]

(1) 1

8 (2)

1

2 (3)

1

4 (4) –

1

4

43. On which of the following lines lies the point of intersect in of the line, x – 4

2 =

y – 5

2 =

z – 3

1 and the plane, x +

y + z = 2?

[2019]

(1) x – 4

1 =

y – 5

1 =

z – 5

–1 (2)

x – 2

2 =

y – 3

2 =

z – 3

3

(3) x –1

1 =

y – 3

2 =

z 4

–5

(4)

x – 3

3 =

4 – y

3 =

z 1

–2



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44. If = ( – 2) a + b and = (4 – 2) a + 3 b be two given vectors where vectors a and b are non-collinear.

The value of for which vectors and are collinear, is –

[2019]

(1) 4 (2) 3 (3) – 3 (4) – 4 45. the plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through

which one of the following points?

[2019] (1) (2, 1, 3) (2) (4, –1, 2) (3) (4, 1, –2) (4) (–2, 3, 5)

46. Let a = i + 2 j + 4 k , b = i + j + 4 k and c = 2 i + 4 j + (– 1) k be coplanar vectors. Then the non-zero

vector a × c is :

[2019]

(1) –10 i – 5 j (2) –10 i + 5 j (3) –14 i + 5 j (4) –14 i – 5 j

47. The plane containing the line x – 3

2 =

y 2

–1

=

z –1

3 and also containing its projection on the plane

2x + 3y – z = 5, contains which one of the following points?

[2019] (1) (–2, 2, 2) (2) (2, 2, 0) (3) (2, 0, –2) (4) (0, –2, 2)

48. The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle 4

with

the plane y – z + 5 = 0 are : [2019]

(1) 2, –1, 1 (2) 2 3 , 1, –1 (3) 2 , 1, –1 (4) 2, 2 , – 2

49. Two lines x – 3

1 =

y 1

3

=

z – 6

–1 and

x 5

7

=

y 2

–6

=

z – 3

4 intersect at the point R. The reflection of R in the

xy-plane has coordinates:

[2019] (1) (2, 4, 7) (2) (2, –4, –7) (3) (2, –4, 7) (4) (–2, 4, 7)

50. If the point (2, , ) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to

the plane 2x – 5y = 15, then 2 – 3 is equal to [2019]

(1) 12 (2) 7 (3) 17 (4) 5

51. Let ˆ3 i + j , i + ˆ3 j and i + (1 –) j respectively be the position vectors of the points A, B and C with

respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is3

2, then

the sum of all possible values of is. [2019]

(1) 4 (2) 1 (3) 2 (4) 3

52. The sum of the distinct real values of , for which the vectors, i + j + k , i + j + k , i + j + k are co-

planar, is [2019]

(1) 2 (2) –1 (3) 0 (4) 1



https://t.me/omsir





53. The perpendicular distance from the origin to the plane containing the two lines, x 2

3

=

y – 2

5 =

z 5

7

and

x –1

1 =

y – 4

4 =

z 4

7

is :

[2019]

(1) 6 11 (2) 11

6 (3) 11 (4) 11 6

54. A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(–1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and

PQR is :

[2019]

(1) cos–117

31

(2) cos–19

35

(3) cos–119

35

(4) cos–17

31

55. Let S be the set of all real values of such that a plane passing through the points (–2, 1, 1), (1, –2, 1) and

(1, 1, –2) also passes through the point (–1, –1, 1). Then S is equal to :

[2019]

(1) {1, –1} (2) {3, –3} (3) { 3 } (4) { 3 , – 3 }

56. If an angle between the line, x 1

2

=

y 2

1

=

z 3

–2

and the plane, x – 2y – kz = 3 is cos–1

2 2

3

, then a value of

k is :

[2019]

(1) 3

5 (2) –

5

3 (3) –

3

5 (4)

5

3

57. Let a , b and c be three unit vectors, out of which vectors b and c are non-parallel. If and are the angles

which vector a makes with vectors b and c respectively and a × b c = 1

2b , then | – | is equal to :

[2019]

(1) 90º (2) 30º (3) 45º (4) 60º

58. The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and

passing through the point (1, 1, 0) is –

[2019]

(1) x – 3y – 2z = – 2 (2) x – y – z = 0 (3) x + 3y – z = 4 (4) 2x –z = 2

59. The length of the perpendicular from the point (2, –1, 4) on the straight line, x 3

10

=

y – 2

–7 =

z

1 is –

[2019]

(1) greater than 3 but less then 4 (2) greater than 2 but less than 3

(3) greater than 4 (4) less than 2

60. The magnitude of the projection of the vector ˆ2i + ˆ3j + k on the vector perpendicular to the plane containing the

vectors i + j + k and i + ˆ2 j + ˆ3k , is -

[2019]

(1) 3

2 (2)

3

2 (3) 3 6 (4) 6



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61. If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R

from the origin is –

[2019]

(1) 6 (2) 53 (3) 2 14 (4) 2 21

62. The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5

which is perpendicular to the plane x – y + z = 0 is –

[2019]

(1) r .( i – k ) – 2 = 0 (2) r × ( i + k ) + 2 = 0 (3) r ( i – k ) + 2 = 0 (4) r × ( i + k ) + 2 = 0

63. Let a = ˆ3i + ˆ2 j + ˆxk and b = i – j – k , for some real x. Then | a × b | = r is possible if –

[2019]

(1) 3

2 < r

3

2 (2) r

35

2 (3)

33

2 < r <

35

2 (4) 0 < r

3

2

64. Let = ˆ3i + j and = ˆ2i – j + ˆ3k . If = 1 – 2 , where 1 is parallel to and 2 is perpendicular to

then 1 × 2 is equal to :

[2019]

(1) – ˆ3i + ˆ9 j + ˆ5k (2) 1

2(– ˆ3i + ˆ9 j + ˆ5k ) (3) ˆ3i – ˆ9 j – ˆ5k (4)

1

2( ˆ3i – ˆ9 j + ˆ5k )

65. If the line, x –1

2 =

y 1

3

=

z – 2

4 meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the

origin is :

[2019]

(1) 5 /2 (2) 7/2 (3) 2 5 (4) 9/2

66. A plane passing through the points (0, –1, 0) and (0, 0, 1) and making an angle 4

with the plane y – z + 5 = 0,

also passes through the point :

[2019]

(1) ( 2 , –1, 4) (2) (– 2 , 1, –4) (3) (– 2 , –1, –4) (4) ( 2 , 1, 4)

67. If a unit vector a makes angles /3 with i , /4 with j and (0, ) with k , then a value of is :

[2019]

(1) 5

12

(2)

5

6

(3)

4

(4)

2

3

68. Let P be the plane, which contains the line of intersection of the planes, x + y + z – 6 = 0 and 2x + 3y + z + 5 = 0

and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to :

[2019]

(1) 205 5 (2) 11/ 5 (3) 63 5 (4) 17/ 5

69. The vertices B and C of ABC lie on the line, x 2

3

=

y –1

0 =

z

4 such that BC = 5 units. Then the area(in sq.

units) of this triangle, given that the point A(1, –1, 2), is :

[2019]

(1) 5 17 (2) 34 (3) 2 34 (4) 6



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70. If the length of the perpendicular from the point (, 0, ) ( 0) to the line, x

1 =

y –1

0 =

z 1

–1

is

3

2, then is

equal to :

[2019] (1) 2 (2) 1 (3) – 2 (4) – 1

71. If Q(0, –1, –3) is the image of the point P in the plane 3x – y + 4z = 2 and R is the point (3, –1, –2), then the area

(in sq. units) of PQR is :

[2019]

(1) 65

2 (2) 2 13 (3)

91

2 (4)

91

4

72. Let A (3,0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC, If G divides

BM in the ratio, 2 : 1, then cos ( GOA) (O being the origin) is equal to :

[2019]

(1) 1

15 (2)

1

6 10 (3)

1

30 (4)

1

2 15

73. If the system of linear equations x + y + z = 5, x + 2y + 2z = 6, x + 3y + z = , (, R), has infinitely many

solutions, then the value of + is :

[2019] (1) 10 (2) 9 (3) 12 (4) 7

74. If the plane 2x – y + 2z + 3 = 0 has the distances 1

3 and

2

3 units from the planes 4x – 2y + 4z + = 0 and

2x – y + 2z + = 0, respectively, then the maximum value of + is equal to :

[2019] (1) 13 (2) 9 (3) 5 (4) 15

75. The distance of the point having position vector – i + ˆ2 j + ˆ6k from the straight line passing through the point (2,

3, – 4) and parallel to the vector ˆ6i + ˆ3j – ˆ4k is :

[2019] (1) 6 (2) 7

(3) 2 13 (4) 4 3

76. A perpendicular is drawn from a point on the line x –1

2 =

y 1

–1

=

z

1 to the plane x + y + z = 3 such that the foot

of the perpendicular Q also lies on the plane x – y + z = 3. Then the co-ordinates of Q are :

[2019] (1) (4, 0, –1) (2) (2, 0, 1) (3) (1, 0, 2) (4) (–1, 0, 4)

77. Let a = ˆ3i + ˆ2 j + ˆ2k and b = i + ˆ2 j – ˆ2k be two vectors. If a vector perpendicular to both the vectors a + b

and a – b has the magnitude 12 then one such vector is :

[2019]

(1) 4( ˆ2i – ˆ2 j – k ) (2) 4(– ˆ2i – ˆ2 j + k ) (3) 4( ˆ2i + ˆ2 j – k ) (4) 4( ˆ2i + ˆ2 j – k )

78. If the volume of parallelepiped formed by the vectors i + j + k , j + k and i + k is minimum, then is

equal to :

[2019]

(1) –1

3 (2) 3 (3)

1

3 (4) – 3



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79. If the line x – 2

3 =

y 1

2

=

z –1

–1 intersects the plane 2x + 3y – z + 13 = 0 at a point P and the plane 3x + y + 4z =

16 at a point Q, then PQ is equal to :

[2019]

(1) 2 7 (2) 14 (3) 2 14 (4) 14

80. A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0. Passes

through the point :

[2019]

(1) (1, –4, 1) (2) (1, 4, –1) (3) (2, 4, 1) (4) (2, –4, 1)

81. The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines r = ( i + j )

+ ( i + 2 j – k ) and r = ( i + j ) + (– i + j – 2 k ) is :

[2019]

(1) 1

3 (2)

1

3 (3) 3 (4) 3

82. Let R and the three vectors a = i + j + ˆ3k , b = ˆ2i + j – k and c = i – ˆ2 j + ˆ3k . Then the set

S = { : a , b and c are coplanar}

[2019]

(1) contains exactly two numbers only one of which is positive (2) is singleton

(3) contains exactly two positive numbers (4) is empty

83. Let u be a vector coplanar with the vectors ˆ ˆ ˆa 2i 3j – k and ˆ ˆb j k . If u is perpendicular to a and

.u b – 24, then 2

u is equal to:

[JEE MAIN 2018] (1) 315 (2) 256 (3) 84 (4) 336

84. The length of the projection of the line segment joining the points (5, –1, 4) and (4, –1, 3) on the plane, x + y + z = 7 is

[JEE MAIN 2018]

(1) 2

3 (2)

1

3 (3)

2

4 (4)

2

3

85. If L1 is the line of intersection of the planes 2x – 2y + 3z – 2 = 0, x – y + z + 1 = 0 and L2 is the line of intersection

of the planes x + 2y – z – 3 = 0, 3x – y + 2z – 1 = 0, then the distance of the origin from the plane, containing the

lines L1 and L2, is :

[JEE MAIN 2018]

(1) 1

3 2 (2)

1

2 2 (3)

1

2 (4)

1

4 2

86. Let ˆ ˆ ˆa 2i j – 2k and ˆ ˆb i j . Let c be a vector such that c – a 32, a b c 3 and the angle

between c and a b be 30o. Then a.c is equal to

[JEE MAIN 2017]

(1) 1

8 (2)

25

8 (3) 2 (4) 5



https://t.me/omsir





87. If the image of the point P(1, – 2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to line,

x y z

1 4 5 is Q, then PQ is equal to

[JEE MAIN 2017]

(1) 6 5 (2) 3 5 (3) 2 42 (4) 42

88. The distance of the point (1, 3, –7) from the plane passing through the point (1, –1, –1), having normal

perpendicular to both the lines x –1 y 2 z – 4 x – 2 y 1 z 7

and1 –2 3 2 –1 –1

, is

[JEE MAIN 2017]

(1) 10

74 (2)

20

74 (3)

10

83 (4)

5

83

89. Let a,b and c be three unit vectors such that 3

a b c b c .2

If b is not parallel to c , then the

angle between a and b is :

[JEE Main- 2016]

(1) 5

6

(2)

3

4

(3)

2

(4)

2

3

90. Let a,b and c be three non-zero vectors such that no two of them are collinear

and 1

a b c | b || c | a3

. If is the angle between vectors b and c , then a value of sin is :

[JEE Main- 2015]

(1) 2

3 (2)

–2 3

3 (3)

2 2

3 (4)

– 2

3

91. The equation of the plane containing the line 2x – 5y + z = 3 ; x + y + 4z = 5, and parallel to the plane, x

+ 3y + 6z = 1, is [JEE- MAIN - 2015]

(1) x + 3y + 6z = 7 (2) 2x + 6y + 12z = –13 (3) 2x + 6y + 12z = 13 (4) x + 3y + 6z = –7

92. The distance of the point (1, 0, 2) from the point of intersection of the line x – 2 y 1 z – 2

3 4 12

and the

plane x – y + z = 16, is : [JEE- MAIN - 2015]

(1) 3 21 (2) 13 (3) 2 14 (4) 8

93. If 2

a a ab b c c bc then is equal to :

[JEE-Main 2014] (1) 2 (2) 3 (3) 0 (4) 1

94. The angle between the lines whose direction cosines satisfy the equation + m + n = 0 and 2 = m2 + n2 is :

[JEE- MAIN - 2014]

(1) 3

(2)

4

(3)

6

(4)

2



https://t.me/omsir





95. The image of line x – 1 y – 3 z – 4

3 1 –5 in the plane 2x – y + z + 3 = 0 is the line :

[JEE- MAIN - 2014]

(1) x 3 y – 5 z – 2

3 1 –5

(2)

x 3 y – 5 z 2

–3 –1 5

(3) x – 3 y 5 z – 2

3 1 –5

(4)

x – 3 y 5 z – 2

–3 –1 5

96. If the vectors ˆ ÂB 3i 4k and ˆ ˆ ÂC 5i – 2j 4k are the sides of a triangle ABC, then the length of the

median through A is : [AIEEE- 2013]

(1) 18 (2) 72 (3) 33 (4) 45

97. Let ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ â 2i – j k,b i 2j – k and c i j – 2k be the three vectors. A vectors of the type b c for

some scalar , whose projection on a is of magnitude 2

3, is : [JEE-Mains 2013]

(1) ˆ ˆ ˆ2i 3j 3k (2) ˆ ˆ ˆ2i j 5k (3) ˆ ˆ ˆ2i – j 5k (4) ˆ ˆ ˆ2i 3j 3k

98. Let ˆ ˆ ˆ ˆ â 2i j – 2k, b i j. If c is a vector such that a a.c | c |,| c – | 2 2 and the angle between

a b and c is 30°, then a b c equals :

[JEE-Mains 2013]

(1) 3

2 (2) 3 (3)

1

2 (4)

3 3

2

99. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is :

[JEE- MAIN - 2013]

(1) 3

2 (2)

5

2 (3)

7

2 (4)

9

2

100. If the lines x – 2 y – 3 z – 4 x –1 y – 4 z – 5

and1 1 –k k 2 1

are coplanar, then k can have :

[JEE- MAIN - 2013] (1) any value (2) exactly one value (3) exactly two values (4) exactly three values. 101. A vector n is inclined to x-axis at 45°, to y-axis at 60° and at an acute angle to z-axis. If n is a normal to

a plane passing through the point ( 2,–1, 1) , then the equation of the plane is :

[JEE- MAIN - 2013]

(1) 2 x – y –z = 2 (2) 2 x + y + z = 2 (3) 3 2 x – 4y – 3z = 7 (4) 4 2 x + 7y + z = 2 102. The acute angle between two lines such that the direction cosines , m, n of each of them satisfy the

equation + m + n = 0 and 2 + m2 –n2 = 0 is : [JEE- MAIN - 2013]

(1) 30° (2) 45° (3) 60° (4) 15° 103. Let Q be the foot of perpendicular from the origin to the plane 4x –3y + z + 13 = 0 and R be a point

(–1, 1, –6) on the plane. Then length QR is : [JEE- MAIN - 2013]

(1) 7

32

(2) 14 (3) 19

2 (4)

3

2



https://t.me/omsir





104. If the projections of a line segment of the x, y and z-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :

[JEE- MAIN - 2013] (1) 7 (2) 9 (3) 12 (4) 6

105. If two lines L1 and L2 in space, are defined by [JEE- MAIN - 2013]

L1 = x y ( –1) ; z = –1 y and L2 = x y (1– )

z = 1– y , then L1 is perpendicular to L2, for al non-negative reals and , such that :

(1) = (2) (3) 1 (4) + = 0

106. The equation of a plane through the line of intersection of the planes x + 2y = 3, y –2z + 1 = 0 and

perpendicular to the first plane is : [JEE- MAIN - 2013]

(1) 2x – y + 7z = 11 (2) 2x – y + 10 z = 11 (3) 2x – y –9z = 10 (4) 2x – y – 10z = 9



https://t.me/omsir





EXERCISE # 6 [JEE ADVANCE PYQ]

1. Let L1 and L2 be the following straight lines. L1 :

x –1

1 =

y

–1 =

z –1

3 and L2 :

x –1

–3 =

y

–1 =

z –1

1 Suppose the straight line L :

x – =

y –1

m =

z –

–2

lies in

the plane containing L1 and L2, and passes through the point of intersection of L1 and L2. If the line L bisects the

acute angle between the lines L1 and L2, then which of the following statements is/are TRUE? [JEE Adv 2020]

(A) – = 3 (B) + m = 2 (C) – = 1 (D) + m = 0

2. In a triangle PQR, let a = —

QR

, b = —

RP

and c = —

PQ

. If | a | = 3, | b | = 4 and a (c – b)

c (a – b)

=

| a |

| a | | b |, then the

value of 2

a b is …

[JEE Adv 2020]

3. Let , , , be real numbers such that 2 + 2 + 2 0 and + = 1. Suppose the point (3, 2, –1) is the mirror

image of the point (1, 0, –1) with respect to the plane x + y + z = . Then which of the following statements

is/are TRUE? [JEE Adv 2020]

(A) + = 2 (B) – = 3 (C) + = 4 (D) + + =

4. Let a and b be positive real numbers. Suppose —

PQ

= ˆai + bj and —

PS

= ˆai – bj are adjacent sides of a

parallelogram PQRS. Let u and v be the projection vectors of w = i + j along —

PQ

and —

PS

respectively.

[JEE Adv 2020] If | u | + | v | = | w | and if the area of the parallelogram PQRS is 8, then which of the following is/are TRUE?

(A) a + b = 4

(B) a – b = 2

(C) The length of the diagonal PR of the parallelogram PQRS is 4

(D) w is an angle bisector of the vectors —

PQ

and —

PS

5. Let L1 and L2 denote the lines ˆ ˆ ˆ ˆr i (–i 2j 2k), R and

ˆ ˆ ˆr (2i – j 2k), R respectively. If L3 is a line

which is perpendicular to both L1 and L2 and cuts both of them, then which of the following option describe(s) L3?

[JEE Adv 2019]

(A) 2 ˆ ˆ ˆ ˆ ˆ ˆr (4i j k) t(2i 2j – k), t R9

(B) 2 ˆ ˆ ˆ ˆ ˆ ˆr (2i – j 2k) t(2i 2j – k), t R9

(C) 1 ˆ ˆ ˆ ˆ ˆr (2i k) t(2i 2j – k), t R3

(D) ˆ ˆ ˆr t(2i 2j – k), t R

6. Three lines are given by ˆr i, R , ˆ ˆr (i j), R and ˆ ˆ ˆr v(i j k),v R

Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the are of the triangle ABC is then

the value of (6)2 equals _______

[JEE Adv 2019]



https://t.me/omsir





7. Three lines L1 : ˆr i, R ,L2 : ˆ ˆr k j , R and L3 :

ˆ ˆ ˆr i j vk,v R are given. For which point(s) Q on

L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?

[JEE Adv 2019]

(A) 1ˆ ˆk – j2

(B) k (C) 1ˆ ˆk j2

(D) ˆ ˆk j

8. Let ˆ ˆ ˆa 2i j – k and

ˆ ˆ ˆb i 2j k be two vectors. Consider a vector c a b, , R . If the projections

of c on the vector (a b) is 3 2 , then the minimum value of (c (a b)).c equals ________

[JEE Adv 2019]

9. Let P1 : 2x + y – z = 3 and P2 : x + 2y + z = 2 be two planes Then, which of the following statement(s) is (are) TRUE ?

[JEE Adv 2018] (A) The line of intersection of P1 and P2 has direction ratios 1, 2, –1

(B) The line 3x – 4 1– 3y z

9 9 3 is perpendicular to the line of intersection of P1 and P2

(C) The acute angle between P1 and P2 is 60o

(D) If P3 is the plane passing through the point (4, 2, –2) and perpendicular to the line of intersection of P1 and P2,

then the distance of the point (2, 1, 1) from the plane P3 is 2

3

10. Let a and b be two unit vectors such that .a b 0 . For some x, y R, let c xa yb +(a b). If | c | 2 and the

vector c is inclined at the same angle to both a and b , then the value of 8 cos2 is ________.

[JEE Adv 2018]

11. Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is

perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the z-axis. Let the

distance of P from the x-axis be 5 If R is the image of P is the xy-plane, then the length of PR is ____________.

[JEE Adv 2018]

12. Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the x-axis, y-axis and z-axis,

respectively, where O(0, 0, 0) is the origin. Let S 1 1 1

, ,2 2 2

be the centre of the cube and T be the vertex of the

cube opposite to the origin O such that S lies on the diagonal OT. If P SP,q SQ, r SR and t ST, then

the value of | (p q) (r t) | is __________

[JEE Adv 2018]

13. The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes

2x + y – 2z = 5 and 3x – 6y – 2z = 7, is

[JEE Adv 2017] (A) 14 x + 2y – 15z = 1 (B) 14 x – 2y + 15z = 27

(C) 14 x + 2y + 15z = 31 (D) –14x + 2y + 15z = 3

14. Let O be the origin and let PQR be an arbitrary triangle. The point S is such that

OP.OQ OR.OS OR.OP OQ.OS OQ.OR OP.OS Then the triangle PQR has S as its

[JEE Adv 2017] (A) Centroid (B) Circumcentre (C) Incentre (D) Orthocentre



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PARAGRAPH # 1

Let 𝑂 be the origin, and OX,OY,OZ be three unit vectors in the directions of the sides QR,RP,PQ ,

respectively, of a triangle 𝑃𝑄𝑅.

15. OX OY

(A) sin(P + Q) (B) sin 2R (C) sin(P + R) (D) sin(Q + R) 16. If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is

[JEE Adv 2017]

(A) 5

3 (B)

3

2 (C)

3

2 (D)

5

3

17. Consider a pyramid OPQRS located in the first octant (x0, y 0, z 0) with O as origin, and OP and OR along the

x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3 Then

[JEE Adv 2016]

(A) the acute angle between OQ and OS is 3

(B) the equation of the plane containing the triangle OQS is x – y = 0

(C) the length of the perpendicular from P to the plane containing the triangle OQS is 3

2

(D) the perpendicular distance from O to the straight line containing RS is 15

2

18. Let P be the image of the point (3,1,7) with respect to the plane x – y + z = 3. Then the equation of the plane passing

through P and containing the straight line x y z

1 2 1 is

[JEE Adv 2016] (A)x + y – 3z = 0 (B) 3x + z = 0 (C) x– 4y + 7z = 0 (D) 2x – y= 0

19. Let1 2 3ˆ ˆ ˆu u i u j u k be a unit vector in R3and

1 ˆ ˆ ˆw (i j 2k)6

. Given that there exists a vector in R3such

that u = 1 and ˆ ˆw.(u ) 1 . Which of the following statement(s) is(are) correct?

[JEE Adv 2016] (A)There is exactly one choice for such

(B) There are infinitely many choices for such

(C)If u lies in the xy-plane then |u1| = |u2|

(D) If u lies in the xz-plane then 2|u1| = |u3|

20. In R3, consider the planes P1 : y = 0 and P2 : x + z = 1. Let P3 be a plane, different from P1 and P2, which passes

through the intersection of P1 and P2. If the distance of the point (0,1,0) from P3 is 1 and the distance of a point

(,,) from P3 is 2, then which of the following relations is (are) true?

[JEE Adv 2015] (A) 2+ + 2+ 2 = 0 (B) 2– + 2+ 4 = 0

(C) 2+ – 2–10 = 0 (D) 2– + 2– 8 = 0

21. In R3, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance

from the two planes P1: x + 2y – z + 1 = 0 and P2 : 2x – y + z – 1 = 0 . Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1 .which of the following points lie(s) on M?

[JEE Adv 2015]

(A)5 2

0, ,6 3

(B)

1 1 1, ,

6 3 6

(C)5 1

,0,6 6

(D)1 2

,0,3 3



https://t.me/omsir





22. Let PQR be a triangle. Let a QR, b RP and c PQ . If | a | 12, |b | 4 3 and b.c 24 , then which of the

following is (are) true?

[JEE Adv 2015]

(A)2| c |

| a | 122

(B)2| c |

| a | 302

(C) | a b c a | 48 3 (D) a.b 72

23. Suppose that p,q and r are three non-coplanar vectors in R3. Let the components of a vector s along p,q and r be

4,3 and 5, respectively. If the component of this vector s along ( p q r) , (p q r) and ( p q r) are x, y and

z respectively, then the value of 2x + y + z is

[JEE Adv 2015] 24. From a point (𝜆, 𝜆, 𝜆), perpendiculars 𝑃Q and 𝑃R are drawn respectively on the lines 𝑦 = 𝑥, 𝑧 = 1 and 𝑦 = −𝑥, 𝑧 = −1.

If 𝑃 is such that ∠QPR is a right angle, then the possible value(s) of 𝜆 is (are) [JEE Adv 2014]

(A) 2 (B)1 (C) –1 (D) 2

25. Let x, y and z be three vectors each of magnitude 2 and the angle between each pair of them is3

. If a is a

nonzero vector perpendicular to x and y z and b is a nonzero vector perpendicular to y and z x , then

[JEE Adv 2014] (A) b (b.z)(z – x) (B) a (a. y)(y z)

(C) a.b (a. y)(b. z) (D) a (a. y)(z y)

26. Let a , b and c be three non-coplanar unit vectors such that the angle between every pair of them is 3

. If

a b b c pa qb rc , where p,q and r are scalars, then the value of 2 2 2

2

2p q r

q

is

[JEE Adv 2014]

27. Perpendiculars are drawn from points on the line x 2 y 1 z

2 1 3

to the plane x + y + z = 3.The feet of

perpendiculars lie on the line [JEE Adv 2013]

(A)x y 1 z 2

5 8 13

(B)

x y 1 z 2

2 3 5

(C)x y 1 z 2

4 3 7

(D)

x y 1 z 2

2 7 5

28. Let ˆ ˆ ˆPR 3i j 2k and ˆ ˆ ˆSQ i 3j 4k determine diagonals of a parallelogram PQRS and ˆ ˆ ˆPT i 2j 3k be

another vector. Then the volume of the parallelepiped determined by the vectors PT,PQ and PS is

[JEE Adv 2013] (A) 5 (B) 20 (C) 10 (D) 30

29. A line l passing through the origin is perpendicular to the lines

1

ˆ ˆ ˆ: (3 t)i ( 1 2t)j (4 2t)k, t l

2

ˆ ˆ ˆ: (3 2s)i (3 2s)j (2 s)k, t l

Then, the coordinates(s) of the point(s) on l2 at a distance of 17 from the point of intersection of l and l2is(are)

[JEE Adv 2013]

(A)7 7 5

, ,3 3 3

(B) (–1, –1, 0) (C) (1, 1, 1) (D)7 7 8

, ,9 9 9



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30. Consider the set of eight vector V = { ˆ ˆ ˆai bj ck : a,b,c, { 1,1} }. Three non-coplanar vectors can be chosen from

V in 2p ways. Then p is

[JEE Adv 2013]

31. Two lines L1 :y z

x 5,3 2

and L2 : x = , y z

1 2

are coplanar. Then can take value(s)

[JEE Adv 2013] (A) 1 (B) 2 (C) 3 (D) 4

32. Consider the lines L1 : 2

x 1 y z 3 x 4 y 3 z 3,L :

2 1 1 1 1 2

and the planes

P1: 7x + y + 2z = 3, P2 : 3x + 5y – 6z = 4. Let ax + by + cz = d be the equation of the plane passing through the point

of intersection of lines L1 and L2, and perpendicular to planes P1 and P2

Match List-I with List-II and select the correct answer using the code given below the lists : [JEE Adv 2013]

List-I List-II

(P) a = 1. 13

(Q) b = 2. –3

(R) c = 3.1

(S) d = 4.–2

Codes :

P Q R S

(A) 3 2 4 1

(B) 1 3 4 2

(C) 3 2 1 4

(D) 2 4 1 3

33. Match List-I with List-II and select the correct answer using the code given below the lists :

[JEE Adv 2013] List-I List-II

(P) Volume of parallelepiped determined by vectors a,b and c is 2. 1. 100

Then the volume of the parallelepiped determined by vectors

2 a b ,3 b c and c a is

(Q) Volume of parallelepiped determined by vectors a , b and c is 5. 2. 30

Then the volume of the parallelepiped determined by vectors

3 a b , b c and 2 c a is

(R) Area of a triangle with adjacent sides determined by vectors 3. 24

a and b is 20. Then the area of the triangle with adjacent sides

determined by vectors 2a 3b and a b is

(S) Area of a parallelogram with adjacent sides determined by 4. 60

vectors a and b is 30. Then the area of the parallelogram with adjacent

sides determined by vectors a b and a is

Codes :

P Q R S

(A) 4 2 3 1

(B) 2 3 1 4

(C) 3 4 1 2

(D) 1 4 3 2



https://t.me/omsir





34.(a) The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, –1, 4) with the plane 5x –4y –z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is -

[JEE- Adv - 2012]

(A) 1

2 (B) 2 (C) 2 (D) 2 2

(b) The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x –y + z

= 3 and at a distance 2

3from the point (3, 1, –1) is

[JEE- Adv - 2012]

(A) 5x –11y + z = 17 (B) 2 x + y = 3 2 –1

(C) x + y + z = 3 (D) x – 2y 1– 2

(c) If the straight lines x – 1 y 1 z

2 k 2

and

x 1 y 1 z

5 2 k

are coplanar, then the plane(s) containing these

two lines is(are) [JEE- Adv – 2012]

(A) y + 2z = –1 (B) y + z = –1 (C) y –z = –1 (D) y –2z = –1

35.(a) If a,b and c are unit vectors satisfying 2 2 2a a| – b | | b – c | | c – | 9 , then a| 2 5b 5c | is

(b) If a and b are vectors such that a| b | 29 and ˆ ˆ ˆ ˆ ˆ â (2i 3j 4k) (2i 3j 4k) b , then a possible

value of ˆ ˆ â b . –7i 2j 3k is :

[JEE 2012] (A) 0 (B) 3 (C) 4 (D) 8

36.(a) Let ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ â i j k i j k i j k, b – and c – – be three vectors. A vector v in the plane of a and b , whose

projection on c is 1

3, is given by -

[JEE 2011]

(A) ˆ ˆ î – 3j 3k (B) ˆ ˆ ˆ–3i – 3j k (C) ˆ ˆ ˆ3i – j 3k (D) ˆ ˆ î 3 j – 3k

(b) The vector(s) which is/are coplanar with vectors ˆ ˆ î j 2k and ˆ ˆ î 2j k , and perpendicular to the vector

ˆ ˆ î j k is/are :

[JEE 2011]

(A) ˆ ˆj – k (B) ˆ ˆ–i j (C) ˆ î – j (D) ˆ ˆ– j k

(c) Let ˆ â i k– – , ˆ ˆ ˆ ˆ î j i j kb – and c 2 3 be three given vectors. If r is a vector such that

r b c b and ar. 0 , then the value of r.b is

[JEE 2011]

37.(a) Two adjacent sides of a parallelogram ABCD are given by ˆ ˆ ÂB 2i 10j 11k and ˆ ˆ ÂD –i 2j 2k . The

sides AD is rotated by an acute angle in the plane of the parallelogram so that AD becomes AD’ If AD’

makes a right angle with the side AB, then the cosine of the angle is given by -

(A) 8

9 (B)

17

9 (C)

1

9 (D)

4 5

9



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(b) If a and b are vectors in space given by ˆ ˆi j– 2

a5

and ˆ ˆ ˆi j2 3k

b14

, then the value of

a a a2 b . b – 2b

is

[JEE 2010] (A) (B) (C) (D)

38.(a) Equation of the plane containing the straight line x y z

2 3 4 and perpendicular to the plane containing the

straight lines x y z

3 4 2 and

x y z

4 2 3 is –

[JEE- Adv - 2010] (A) x + 2y – 2z = 0 (B) 3x + 2y – 2z = 0 (C) x – 2y + z = 0 (D) 5x + 2y – 4z = 0

(b) If the distance of the point P(1, –2,1) from the plane x + 2y – 2z = , where > 0, is 5, then the foot of the perpendicular from P to the plane is -

[JEE- Adv - 2010]

(A) 8 4 7

, ,–3 3 3

(B) 4 4 1

,– ,3 3 3

(C) 1 2 10

, ,3 3 3

(D) 2 1 5

,– ,3 3 2

(c) If the distance between the plane Ax – 2y + z = d and the plane containing the lines

x –1 y – 2 z – 3 x – 2 y – 3 z – 4and

2 3 4 3 4 5 is 6 , then |d| is –

[JEE- Adv - 2010] (d) Match the statement in Column-I with the values in Column-II.

[JEE- Adv - 2010] Column-I Column-II (A) A line from the origin meets the lines (p) –4

8x –

x – 2 y – 1 z 1 y 3 z – 13and1 –2 1 2 –1 1

at P and Q

Respectively. If length PQ = d, then d2 is (B) The values of x satisfying (q) 0

tan–1(x + 3) –tan–1(x –3) = sin–1 3

5

are

(C) Non-zero vectors a,b and c satisfy a.b 0 , (r)

b – a . b c 0 and 2 b c b – a .

If a b 4c, then the possible values of are

(D) Let f be the function on [–] given by (s) 5

f(0) = 9 and f(x) = sin9x x

/ sin2 2

for x 0.

The value of –

2f(x)dx

is (t) 6

39.(a) If a,b,c and d are unit vectors such that a b . c d 1 and 1

a.c2

, then –

[JEE 2009]

(A) a,b,c are non-coplanar (B) b,c,d

(C) b,d are non-parallel (D) a,d are parallel and b,c are parallel



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(b) Match the statements/ expressions given in Column-I with the values given in Column-II.

[JEE 2009] Column-I Column-II

(A) Root(s) of the equation 2sin2+ sin22= 2 (P) 6

(B) Points of discontinuity of the function f(x) = 6x 3x

cos

(Q) 4

Where [y] denotes the largest integer less than or equal to y

(C) Volume of the parallelepiped with its edges represented (R) 3

by the vectors ˆ ˆ ˆ ˆ ˆ ˆ î j, i 2j and i j, k

(D) Angle between vectors a and b where a,b and c are unit (S) 2

vectors satisfying a b 3c 0 (T)

40.(a) Let P(3, 2, 6) be a point in space and Q be a point on the line ˆ ˆ ˆ ˆ ˆ ˆr ( i – j 2k) (–3i j 5k). Then the

value of for which the vector PQ is parallel to the plane x – 4y + 3z = 1 is –

[JEE- Adv - 2009]

(A) 1

4 (B)

1–

4 (C)

1

8 (D)

1–

8 (b) A line with positive direction cosines passes through the point P(2, –1, 2) and makes equal angles with

the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals –

[JEE- Adv - 2009]

(A) 1 (B) 2 (C) 3 (D) 2

(c) Let (a, y, z) be points with integer coordinates satisfying the system of homogeneous equations : 3x – y – z = 0 ; –3x + z = 0 ; –3x + 2y + z = 0. Then the number of such points for which x2 + y2 + z2

100 is - [JEE- Adv - 2009]

41.(a) The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors

ˆˆ â,b,c such that 1ˆ ˆˆ ˆ ˆ â.b b.c c.a2

. Then, the volume of the parallelepiped is :

[JEE 2008]

(A) 1

2 (B)

1

2 2 (C)

3

2 (D)

1

3

(b) Let two non-collinear unit vectors a and b form an acute angle. A point P moves so that at any time t the

position vector OP (where O is the origin) is given by a cost + b sint. When P is farthest from origin O,

let M be the length of OP and u be the unit vector along OP . Then -

[JEE 2008]

(A) 1/2â b ˆˆ â and M (1 a.b)ˆˆ| a b |

(B) 1/2

â – b ˆû and M (1 a.b)ˆˆ| a – b |

(C) 1/2â b ˆû and M (1 2a.b)ˆˆ| a b |

(D) 1/2

â – b ˆû and M (1 2a.b)ˆˆ| a – b |



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ANSWER KEY

EXERCISE # 1

Qus. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. B A D B C A C B A B B A D B B

Qus. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Ans. C A,C A D A C C A C D D B B C B,C,D

Qus. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Ans. B,D A A A C C A A A,C,D C B B,C A,B,C B C

46. False 47. (i) True (ii) True 48. False 49. False

50. cos–1 10

3 51. 5 2 52. 0 53. Orthocentre 54. 2 i – j

55. 4

or

4

3 56. a

EXERCISE # 2

Qus. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ans. B D A B A C C A A B B A B A A

Qus. 16 17 18 19 20 21 22 23 24 25 26 27

Ans. B A,B,C,D C C A B A C D B A,B,C,D B

28. True 29. True 30. a + bm + cn = 0 31. 2

1

222222 baaccb

EXERCISE # 3

PART- A

Qus. 1 2 3 4 5 6 7 8 9 10

Ans. B D C C C D B B D A

Qus. 11 12 13 14 15 16 17 18 19

Ans. C C C C C A B D A

PART- B

Qus. 20 21 22 23 24 25 26 27 28 29

Ans. A,C A A,B C,D A,B A A,C,D B,C,D A,C A,C,D

PART- C

Qus. 30 31 32 33

Ans. B C D A PART- D

34. A R; B S; C P; D Q 35. A P; B P,R; C R; D T



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EXERCISE # 4

(PART- A)

Qus. 1 2 3 4 5 6 7 8 9

Ans. B A A A C D D A A

(PART- B)

Qus. 10 11 12 13 14 15 16 17

Ans. A,B,C,D B,C A,B,C A,C B,C A,B,C,D A,B,C,D A,C

(PART- C)

18. A R; B P, S; C S; D R 19. A S; B R, S; C Q; D R, S

20. A R,; B S; C P; D Q

EXERCISE # 5

1 2 3 4 5 6 7 8 9 10

4.00 1.00 6.00 3 18 5 3 2.00 8 1.00

11 12 13 14 15 16 17 18 19 20

30 3 4.00 3 3 1 2 1 1 1

21 22 23 24 25 26 27 28 29 30

B B A A C D B D D B

31 32 33 34 35 36 37 38 39 40

B B D 1 1 4 3 2 3 3

41 42 43 44 45 46 47 48 49 50

2 3 3 4 3 2 3 3, 4 2 2

51 52 53 54 55 56 57 58 59 60

2 2 2 3 4 4 2 2 1 2

61 62 63 64 65 66 67 68 69 70

3 3 2 2 4 4 4 2 2 4

71 72 73 74 75 76 77 78 79 80

3 3 1 1 3 2 1 3 3 4

81 82 83 84 85 86 87 88 89 90

4 4 1 1 3

91 92 93 94 95 96 97 98 99 100

1 2 4 1 1 3 1 1 3 3

101 102 103 104 105 106

2 3 1 1 1,4 2



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EXERCISE # 6

1 2 3 4 5 6 7 8 9 10

A,B 108 A,B,C A,C A,B,C 0.75 A,C 18.00 C,D 3

11 12 13 14 15 16 17 18 19 20

8 0.5 C D A B B,C,D C B,C B,D

21 22 23 24 25 26 27 28 29 30

A,B A,C,D 9 C A,B,C 4 D C B,D 5

31 32 33

A,D A C

34. (a) A; (b) A ; (c) B,C

35. (a) 3 ; (b) C

36. (a) C ; (b) A,D ; (c) 9

37. (a) B; (b) 5

38. (a) C; (b) A ; (c) 6 ; (d) (A) t (B) p,r (C) q (D) r

39. (a) C; (b) (A) Q, S; (B) P,R,S,T (C) T, (D) R

40. (a) A; (b) C ; (c) 7

41. (a, b) (A,A)



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mastersheet: binomial theorem exercise # 1 [vector]

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