MATHS - FIRST YEAR 1 A VERY IMPORTANT QUESTIONS BY WWW.PAPERSHUNT.COM AND HUNT FOR SUCCESS PUBLICATIONS.

LONG ANSWER QUESTIONS

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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SA S B C cos sin 2 2 2 SA 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...