# Maths 1a Imp Qstns

Topics: Linear algebra, Vector space, Euclidean vector Pages: 17 (3288 words) Published: January 14, 2012
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MATHS - FIRST YEAR 1 A VERY IMPORTANT QUESTIONS BY WWW.PAPERSHUNT.COM AND HUNT FOR SUCCESS PUBLICATIONS.

Functions :
01. Let f : A  B , g : B  C be bijections. Then gof : A  C is a bijection. 02. Let f : A  B , g : B  C be bijection. Then ( gof ) 1  f 1og 1 03. Let f : A  B , I A and I B be identify functions on A and B respectively. Then

foI A  f  I B of
04. Let f : A  B be a bijection. Then fof

05. Let f : A  B be a function. Then f is a bijection if and only if there exists a function

g : B  A such that fog  I B and gof  I A and, in this case, g  f 1

Mathematical Inductions :

06. Show that 49n  16n  1 is divisible by 64 for all positive integers n.

n(n 2  6n  11) 07. 2.3  3.4  4.5  ... upto n terms  3 08. 3.52 n 1  23 n 1 is divisible by 17

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1

 I B and f 1of  I A

09.

n 13 13  23 13  23  33  2n 2  9n  13    .... upto n terms   24  1 1 3 1 3  5 2 2 2 2 2 2

n(n  1)2 (n  2) 10. 1  (1  2 )  (1  2  3 )  ..... upto n terms  12

Multiplication of Vectos :
11. Let 1 and 2 be non-negative real numbers such that 1   2   . Then (i) cos(1   2 )  cos 1 cos  2  sin 1 sin  2 (ii) cos(1   2 )  cos 1 cos  2  sin 1 sin  2 12. If in a parallelogram, diagonals are equal, then it is a rectangle. 13. If 0  A, B   , then sin( A  B ) = sinA cosB - cosA sinB 14. Find the shortest distance between the shew lines

r  (6i  2 j  2k )  t (i  2 j  2k )

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and r  ( 4i  k )  s (3i  2 j  2k ) 15. If A  (1, 2, 1), B  (4, 0, 3), C  (1, 2, 1) and D  (2, 4, 5) , find the distance between AB and CD

Trigonometry upto Transformation :
16. If A is not an integral multiple of

 , prove that

cos A.cos 2 A.cos 4 A.cos8 A 

sin16 A and hence deduce that 16 sin A

cos

2 4 8 16 1 .cos .cos .cos  15 15 15 15 16

17. Suppose (   ) is not an odd multiple of

 , m is a non zero real number such that 2

sin(   ) 1  m     m  1 and cos(   )  1  m . Then prove that tan  4     m.tan  4  B      18. If A,B,C are angles in a tria ngle , then prove tha t

sin

A B C A  B  C  sin  sin  1  4sin .sin .sin 2 2 2 4 4 4

19. In triangle ABC, prove that (i) cos

A B C A  B  C  cos  cos  4 cos cos cos 2 2 2 4 4 4

(ii) cos (iii) sin

A B C  A  B  C  cos  cos  4 cos cos cos 2 2 2 4 4 4

A B C A  B  C  sin  sin  1  4 cos cos sin 2 2 2 4 4 4

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SA S B C cos sin 2 2 2 SA S B C cos cos 2 2 2

20. If A  B  C  2S , then prove that (i) sin( S  A)  sin( S  B )  sin C  4 cos

(ii) cos( S  A)  cos( S  B )  cos C  1  4 cos

Properties of Triangles :
2 21. Show that a cos

A B C   b cos 2  c cos 2  s  2 2 2 R

22. Prove that a3 cos( B  C )  b3 cos(C  A)  c 3 cos( A  B )  3abc 23. If a 2  b 2  c 2  8R 2 then prove that the triangle is right angled.

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24. Show that

r1 r2 r3 1 1     bc ca ab r 2 R A  C r  cos 2  cos 2  2  2 2 2 2R

2 25. Show that cos

Hights and Distances :
26. From the top of a tree on the bank of a lake, an aeroplane in the sky makes an angle of elevation  and its image in the river makes an angle of depression.  . If the height of the tree from the water surface is 'a' and the height of the aeroplane is h, show that

h

a sin(   ) sin(    )

27. One end of the ladder is in contact with a wall and the other end is in contact with the level ground making an angle  . When the foot of the...