MATHEMATICS IA PART-1 LONG ANSWER QUESTIONS( 7 MARKS)

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1 MATHEMATICS IA PART-1 LONG ANSWER QUESTIONS( 7 MARKS) 1. A line makes angles 1 2 3 4 , , and θθ θ θ with the diagonals of a cube. Show that 2 2 2 2 1 2 3 4 4 cos cos cos cos 3 θ θ θ θ + + + = 2. If b c d + c a d + a b d = a b c , then show that the points with position vectors a,b,c and d are coplanar. 3.If ( ) ( ) ( ) ( ) 1, 2, 1 , 4, 0 3, 1, 2, 1 2, 4, 5 A B C and D = = , find the distance between AB and CD. 4. If 2 , 2 , 2 a i j k b i j k c i j k = + = + + = + find ( ) ( ) a b c and a b c × × × × 5. If 2 3, 2 3 2 a i j k v i j k and c i j k = = + = + , verify that ( ) ( ) a b c a b c × × × × 6. If 2 3, 2 , 4 a i j kb i j kc i j k = + = + = + and d i j k =+ + , then compute ( ) ( ) a b c d × × × 7. Find the shortest distance between the skew lines ( ) ( ) 6 2 2 2 2 r i j k ti j k = + + + + and ( ) ( ) 4 3 2 2 r i k s i j k = + 8. Find the equation of the plane passing through the point ( ) 3, 2, 1 A = and parallel to the vectors 2 4 3 2 5 b i j k and c i j k = + = + 9. Prove that the smaller angle θ between any two diagonals of a cube is given by 1 cos 3 θ = . 10.Find the vector equation of the plane passing through the point and show that point lies in the plane 11. If 2 3, 2 , 2 a i j k b i j k c i j k = + = + + = + + then find ( ) ( ) a b c and a b c × × × × 12. If 7 2 3, 2 8 a i j k b i k and c i j k = + = + = + + , then compute ( ) , a ba c and a b c × × × + . Verify whether the cross product is distributive over vector addition.

Transcript of MATHEMATICS IA PART-1 LONG ANSWER QUESTIONS( 7 MARKS)

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MATHEMATICS IA PART-1 LONG ANSWER QUESTIONS( 7 MARKS) 1. A line makes angles 1 2 3 4, , andθ θ θ θ with the diagonals of a cube. Show

that 2 2 2 2

1 2 3 44cos cos cos cos3

θ θ θ θ+ + + =

2. If b c d⎡⎣ ⎤⎦ + c a d⎡⎣ ⎤⎦ + a b d⎡⎣ ⎤⎦ = a b c⎡⎣ ⎤⎦ , then show that the points with position vectors a,b,c and d are coplanar.

3.If ( ) ( ) ( ) ( )1, 2, 1 , 4,0 3 , 1,2, 1 2, 4, 5A B C and D= − − − − = − − , find the distance between AB and CD.

4. If 2 , 2 , 2a i j k b i j k c i j k= − + = + + = + − find ( ) ( )a b c and a b c× × × × 5. If 2 3 , 2 3 2a i j k v i j k and c i j k= − − = + − = + − , verify that

( ) ( )a b c a b c× × ≠ × × 6. If 2 3 , 2 , 4a i j k b i j k c i j k= + − = − + = − + − and d i j k= + + , then compute

( ) ( )a b c d× × × 7. Find the shortest distance between the skew lines

( ) ( )6 2 2 2 2r i j k t i j k= + + + − + and ( ) ( )4 3 2 2r i k s i j k= − − + − − 8. Find the equation of the plane passing through the point ( )3, 2, 1A = − − and

parallel to the vectors 2 4 3 2 5b i j k and c i j k= − + = + − 9. Prove that the smaller angle θ between any two diagonals of a

cube is given by 1cos3

θ =.

10.Find the vector equation of the plane passing through the point and show that point lies in the

plane 11. If 2 3 , 2 , 2a i j k b i j k c i j k= − + = + + = + + then find ( ) ( )a b c and a b c× × × × 12. If 7 2 3 , 2 8a i j k b i k and c i j k= − + = + = + + , then compute

( ),a b a c and a b c× × × + . Verify whether the cross product is distributive over vector addition.

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13. Find the vector equation of the plane passing through points

4 3 , 3 7 10i j k i j k− − + − and 2 5 7i j k+ − and show that the point 2 3i j k+ − lies in the plane

14. Solve the following simultaneous linear equations by using ‘Cramer’s

rule. (i)x +y +z=1 2x +2y +3z=6 x +4y +9z=4 (ii) x + 2y + 3x = 6, 2x + 4y + z = 7 3x + 2y + 9z = 14 (iii) 3 4 5 18x y z+ + = 2 8 13x y z− + = 5 2 7 20x y z− + = 15. Solve the following equations by matrices inverse method 2 x- y +3z = 9

x + y+ z = 6 x –y + z = 2

16. If

1 1 1

2 2 2

3 3 3

a b cA a b c

a b c

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦ is a non-singular matrix then

A is invertible and 1

detAdjAAA

− =

17.

(i) ⎥⎥⎥

⎢⎢⎢

−−=113110121

A

, then P.T OIAAA =+−− 323 93

(ii)

IfA =1 −1 1−1 2 −11 −1 2

⎢⎢⎢

⎥⎥⎥then

prove that A3 − 6A2 + 9A− 4I = 0

18. If ( ) ( ) ( )2 2 21, , , 1, , 1, ,A a a B b b and C c c= = = are non-coplanar

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vectors and

2 3

2 3

2 3

11 01

a a ab b bc c c

++ =+ , then show that 1 0a b c + =

19 a = 13, b = 14, c = 15, show that 1 2 365 21, 4, , 12, 148 2

R r r r r= = = = =

20. If 1 2 38, 12, 24r r r= = = show that a = 12, b = 16, c= 20

SHORT ANSWER TYPE QUESTIONS( 4 MARKS)

01.If

1 0 0 1and E

0 1 0 0I ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ then show that ( )3 3 23 .aI bE a I a bE+ = +

02. If

0 2 12 0 21 0

Ax

⎡ ⎤⎢ ⎥= − −⎢ ⎥⎢ ⎥−⎣ ⎦ is a skew symmetric matrix, then find x.

03.

1 5 32 4 03 1 5

A⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥− −⎣ ⎦ and

2 1 00 2 51 2 0

B−⎡ ⎤

⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦then find '3 4A B−

04. If

1 2 22 1 22 2 1

A− − −⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦then show that the

ad joint of A is 3AT. Find 1A−

05.If

3 3 42 3 40 1 1

A−⎡ ⎤

⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦ then show that 1 3A A− =

(ii) If 3A= then show that A-1=AT

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06. If

cos sinsin cos

Aθ θθ θ

⎡ ⎤= ⎢ ⎥−⎣ ⎦ then show that for all the positive integers n,

cos sinsin cos

n n nA

n nθ θθ θ

⎡ ⎤= ⎢ ⎥−⎣ ⎦

7. If

1 22 1 2

3 01 3 4

5 4A and B

−⎡ ⎤−⎡ ⎤ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎢ ⎥⎣ ⎦ then verify that (AB)T =BT AT

8. If , ,a b c are non-coplanar vectors, then test for the collinearity of the

following points whose position vectors are given 3 4 3 , 4 5 6 ,4 7 6a b c a b c a b c− + − + − − + 9. If the points whose position vectors are

3 2 , 2 3 4 , 2i j k i j k i j k− − + − − + + and 4 5i j kλ+ + are coplanar, then

show that 1467

λ −=

10. Let A B C D E F be a regular hexagon with centre ‘O’. Show that

AB + AC + AD + AE + AF=3 AD=6 AO 11. In the two dimensional plane, prove by using vector methods, the

equation of the line whose intercepts on the axes are ‘a’ and ‘b’ is 1x y

a b+ = .

12. If the vectors a= 2i-j+k , b = i+2j-3k, c=3i+pj+5k are coplanar find p 13. Find the vector equation of the line passing through the point 2i +3j -k and perpendicular to the vector 3i -2j -2k.Also find distance from origin to plane. 14. Find the volume of the tetrahedron whose vertices are

( ) ( ) ( )1,2,1 , 3,2,5 , 2, 1,0− and ( )1,0,1− .

15. Find unit vector perpendicular to the plane passing through the points ( ) ( ) ( )1,2,3 , 2, 1,1 1,2, 4and− − .

16. If 2 3 3 ,a i j k b i j k and c i j k= + + = + − = − + , then compute ( )a b c× × and verify that it is perpendicular to a.

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17. , ,a b c are non-zero vectors and a is perpendicular to both b and c. If

( ) 22, 3, 4 ,3

a b c and b c π= = = =, then find a b c⎡ ⎤⎣ ⎦

18.If 2πθ φ− =

, show that

2

2

cos cos sincos sin sin

θ θ θθ θ θ

⎡ ⎤⎢ ⎥⎣ ⎦

2

2

cos cos sin0

cos sin sinφ φ φ

φ φ φ⎡ ⎤

=⎢ ⎥⎣ ⎦

VERY SHORT ANSWER TYPE QUESTIONS( 2 MARKS)

1. If 3sinh4

x =, find ( )cosh 2x and ( )sinh 2x

2. Prove that (i) ( ) ( ) ( )cosh sinh cosh sinhnx x nx nx− = − , for any n R∈

3. Prove that tanh tanh 2cossec 1 sec 1

x x echxhx hx

+ = −− + for 0x ≠

4. For any x R∈ , Prove that ( )4 4cosh sinh cosh 2x x x− = 5.prove that Coshx + Sinhx 1 - Tanhx 1 - cothx = sinhx + coshx

6. If 5cosh2

x =, find the values of (i) ( )cosh 2x and (ii) ( )sinh 2x

7. If sinh 5x = , show that ( )log 5 26ex = +

8. Show that 1 1 1tanh log 32 2 e

− ⎛ ⎞ =⎜ ⎟⎝ ⎠ 9. Find the following products wherever possible

2 2 1 2 3 41 0 2 2 2 32 1 2 1 2 2

− −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

10 . Find product

3 4 913 2 0

0 1 50 4 1

2 6 12

⎡ ⎤−⎡ ⎤⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

11. If A =

1 2 32 5 63 7x

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ is a symmetric matrix, then find x.

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12. If A =

26

3−1

15

⎣⎢⎢

⎦⎥⎥

and B =10

2−1

−13

⎣⎢⎢

⎦⎥⎥ the find the matrix X

such that A + B – X = 0 What is the order of the matrix X ?

13. Construct a 3 x 2 matrix whose elements are defined by 1 32ija i j= −

14.If ⎥⎦

⎤⎢⎣

⎡=

ii

A0

0

find A2

15. A certain bookshop has 10 dozen chemistry books, 8 dozen economics books. Their selling prices are Rs.80, Rs. 60 and Rs.40 each respectively. Using matrix algebra, find the total value of the books in the shop.

16. If

3 2 8 5 22 6 2 4

x yz a− −⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥+ − −⎣ ⎦ ⎣ ⎦ then find the values of x, y, z and a

17. Find the trace of

1 3 52 1 52 0 1

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

18. Let 2 3a i j k= + + and 3b i j= + . Find the unit vector in the direction of a b+ . 19. If , 3 2 , 2 2OA i j k AB i j k BC i j k= + + = − + = + − and

2 3CD i j k= + + , then find the vector OD . 20. 2 5a i j k= + + and 4b i mj nk= + + are collinear vectors,

then find m and n . 21. Find the vector equation of the line passing through the point 2 3i j k+ + and parallel to the vector 4 2 3i j k− + . 22. Find the vector equation of the line joining the points 2 3i j k+ + and 4 3i j k− + − . 23. Find the vector equation of the plane passing through the points 2 5 , 5 ,i j k j k− + − − and 3 5i j− +

. 24,Find the angle between the vectors 2 3i j k+ + and 3 2i j k− +

25. Find the angle between the planes ( )2 2 3r i j k− + = and ( )3 6 4r i j k+ + =

26. If 2 3a i j k= + − and 3 2b i j k= − + , then show that a b+ and a b− are perpendicular to each other

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27.Find vector in the direction of a = i -2j that has magnitude 7 28. Find the vector equation of the plane passing through the points

(0,0,0) ,(0,5 ,0) and (2,0,1) 29. Find the area of the parallelogram having 2a j k and b i k= − = − + and adjacent sides

30. If the vectors 2i +p j –k and 4i -2j +2k are perpendicular find p

30.If A= and det A=45 then Find x

31. If

3 2 8 5 22 6 2 4

x yz a− −⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥+ − −⎣ ⎦ ⎣ ⎦ find the values of x,y,z and a.

32. If A =

1 2 32 5 63 7x

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ is a symmetric matrix, then find x.

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