AHSEC HS 2nd Year Question Paper 2019 Maths

Total number of pages – 16
29T MATH

2019

MATHEMATICS

Full Marks : 100

Pass Marks : 30

Time : Three hours

The figures in the margin indicate full marks
for the questions.

Q. No. 1 (a–j) carries 1 mark each 1×10 = 10

Q. Nos. 2 –13 carry 4 marks each 4×12 = 48

Q. Nos. 14– 20 carry 6 marks each 6×7 = 42
Total = 100

Contd.

1. Answer the following questions : 1×10=10

Ó¬˘1 õ∂ùüÀ¬ı±11 ά◊M√ 1 ø˘‡± :

(a) Let A   x : 1  x  10, x is an odd natural number  and
B   y : 90  y  100, y is a prime number  .
Write the number of relations from A to B. 1

Ò1± ˝√√í ˘ A   x : 1  x  10, x ¤È¬± ’˚≈¢¨ ¶§±ˆ¬±øªfl¬ ¸—‡…±  ’±1n∏
B   y : 90  y  100, y ¤È¬± Œ˜Ãø˘fl¬ ¸—‡…±  º
A 1 ¬Û1± B Δ˘ Œ˝±ª± ¸•§g1¬ ¸—‡…± ø˘‡±º

1
(b) Write down the range of f  x   cot x . 1

f  x   cot 1 x Ù¬˘Ú1 ¬Ûø1¸1 ø˘‡±º

(c) Find all the positive values of 2×2 determinants whose entries

are from the set   1, 0, 1 . 1

¸—˝øÓ¬   1, 0, 1 1 Œ˜Ã˘ ¸˜”À˝À1 ·øͬӬ 2×2 ¸±1øÌfl¬À¬ı±11 ¸fl¬À˘± ÒÚ±Rfl¬ ˜±Ú
øÚÌ«˚˛ fl¬1±º

(d) Let A be a skew-symmetric matrix of odd order. Write the

value of | A|. 1

Ò1± ˝í˘ A ¤È¬± ’˚≈¢¨ ˜±S±1 ø¬ı¯∏˜ ¸˜ø˜Ó¬ Œ˜Ã˘fl¬é¬º | A| 1 ˜±Ú ø˘‡±º

29T MATH [2]

(e) Let A be a matrix of order 3, such that | A |   9 . Find the

value of  3 A 1 . 1

Ò1± ˝í˘ A ¤È¬± 3 ˜±S±1 Œ˜Ã˘fl¬é¬ ˚íÓ¬ | A|   9 º  3 A 1 1 ˜±Ú øÚÌ«˚˛
fl¬í1±º

dy
(f) If 2 x  3 y , then find . 1
dx

˚ø 2 x  3 y , ŒÓ¬ÀôL dy øÚÌ«˚˛ fl¬1±º
dx

(g) Evaluate ∫ 2 x f   x 2  dx . 1

∫ 2 x f   x  dx øÚÌ«˚˛ fl¬1±º
2

(h) Find the order and degree of the differential equation

3
d 2y
 7 ⎛⎜
dy ⎞
⎟  6y  0 . 1
dx 2
⎝ dx ⎠

3
d 2y
 7 ⎛⎜
dy ⎞
⎟  6y  0 ’¬ıfl¬˘ ¸˜œfl¬1Ì1 Sê˜ ’±1n∏ ‚±Ó¬ øÚÌ«˚˛ fl¬1±º
dx 2
⎝ dx ⎠

(i) Write the interval in which the function f  x   cos x is strictly

decreasing. 1

f  x   cos x Ù¬˘ÚÀȬ± Œfl¬±ÚÀȬ± ’ôL 1±˘Ó¬ ¸Ó¬Ó¬ ˝ò±¸˜±Ú ˝˚˛ ø˘‡±º

29T MATH [3] Contd.

(j) Write the equation of the plane passing through  a , b , c  and

parallel to xy- plane. 1

 a , b, c  ø¬ıμ≈1 ˜±ÀÊÀ1 Œ˚±ª± ’±1n∏ xy -¸˜Ó¬˘1 ¸˜±ôL1±˘ Œ˝±ª± ¸˜Ó¬˘‡Ú1
¸˜œfl¬1Ì ø˘‡±º

2. Let the mapping f  x   ax  b , a  0 , maps   1, 1  onto  0, 2  ;

show that cot  cot 1 7  cot 1 8  cot 1 18   f  2 . 4

Ò1± ˝í˘ Ù¬˘Ú f  x   ax  b , a  0 ’±1n∏   1, 1  1 ’±2±Ú √õ∂øÓ¬ø‰¬S ˝í˘  0, 2  ;
Œ‡≈›ª± Œ˚, cot  cot 1 7  cot 1 8  cot 118   f 2 .

Find the value of
1 1 ⎧ 1 ⎛ 2 ⎞⎫ 1
cos x  cos ⎨ ⎜ x  3 1  x ⎟ ⎬,  x  1.
⎩ 2⎝ ⎠⎭ 2

⎧1 ⎫ 1
cos 1 x  cos 1 ⎨ ⎛⎜ x  3 1  x 2 ⎞⎟ ⎬ ,  x  11
⎩ 2⎝ ⎠⎭ 2
˜±Ú øÚÌ«˚˛ fl¬1±º

3. Let f : R  R is defined by f  x   3x  2
x 2
and g : R  R is defined by g  x   .
3
Show that fg  gf . 4

29T MATH [4]

Ò1± ˝í˘ f : R  R fl¬ f  x   3x  2 Œ1 ¸—:±¬ıX√√ fl¬1± Δ˝À ’±1n∏ g : R  R fl¬
x 2
g x   Œ1 ¸—:±¬ıX√√ fl¬1± Δ˝Àº
3

Œ‡≈›ª± Œ˚, f  g  g  f .

4. Show that
a b c 2a 2a
2b b c a 2b  a  b  c 3 4
2c 2c c a b

Œ‡≈›ª± Œ˚,
a b c 2a 2a
2b b c a 2b   a  b  c 3
2c 2c c a b

Without expanding show that

 a x  a x  2  a x  a x  2 2
 b x  b x  2  b x  b  x  2 2  0
 c x  c  x  2  c x  c x  2 2

ø¬ıô¶±1 Úfl¬1±Õfl¬ Œ‡≈›ª± Œ˚,

 a x  a x  2  a x  a x  2 2
 b x  b x  2  b x  b  x  2 2  0
 c x  c  x  2  c x  c x  2 2
29T MATH [5] Contd.

5. Show that the function f defined by
f  x   1  x  x , x  R is a continuous function. 4

Œ‡≈›ª± Œ˚, f  x   1  x  x , x  R Œ1 ¸—:±¬ıX√√ Ù¬˘Ú f , ¤È¬± ’ø¬ıø2Âiß
Ù¬˘Úº

Discuss the applicability of Rolle’s theorem to the function
f  x   x 2  1 on  2, 2  .

Ù¬˘Ú f  x   x 2  1 , ’ôL 1±˘  2, 2  Ó¬ 1˘‰ƒ¬ Î◊¬¬Û¬Û±…1 √õ∂À˚˛±· ’±À˘±‰¬Ú± fl¬1±º

dy
6. If y  e x , find . 4
dx

dy
˚ø y  e x , ŒÓ¬ÀôL øÚÌ«˚˛ fl¬1±º
dx

1 ⎛ 1  4x 3 ⎞
7. If y  cos 1 ⎜ ⎟, x  0,
⎝ 1  4x ⎠
3
2

dy
find . 4
dx

1 ⎛ 1  4 x ⎞
3
1
˚ø y  cos ⎜ ⎟, x  0,
⎝ 1  4x ⎠
3
2
dy
ŒÓ¬ÀôL dx
1 ˜±Ú øÚÌ«˚˛ fl¬1±º

29T MATH [6]

Determine the set of all points where the function f  x   x | x | is
differentiable.

Ù¬˘Ú f  x   x | x | ’ªfl¬˘Úœ˚˛ Œ˝±ª±1 ø¬ıμ≈¸˜”˝1 ¸—˝øÓ¬ øÚÒ«±1Ì fl¬1±º

1 ⎛ 1 x ⎞
8. Evaluate ∫ x ⎜⎝ 1  x 2 ⎟⎠ dx 4

1 ⎛ 1 x ⎞
∫ x ⎜⎝ 1  x 2 ⎟⎠ dx 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

cos 8x  1
Evaluate ∫ tan 2x  cot 2x dx .

cos 8x  1
∫ tan 2x  cot 2x dx 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

1 3x2
9. Evaluate ∫0 3  x 2  2 dx . 4

1 3x2
∫0 3  x 2  2 dx 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

29T MATH [7] Contd.

 1
Evaluate ∫ 3 dx
6 1  tan x

 1
√ ∫ 3 dx 1 ˜±Ú øÚÌ«˚˛ fl¬1±º
6 1  tan x

10. Solve the differential equation 4
dy
x  2y  x 2 log x
dx

dy
’ªfl¬˘ ¸˜œfl¬1Ì x dx  2y  x 2 log x ¸˜±Ò±Ú fl¬1±º

11. If y  3cos  log x   4 sin  log x  ,

d 2y dy
show that x2 2
x  y  0. 4
dx dx

˚ø y  3cos  log x   4 sin  log x  ,

d 2y dy
Œ‡≈›ª± Œ˚, x 2 2
x y  0º
dx dx

 
12. If a  6 î  8 ĵ and b  3 ĵ  4k̂ then determine the vector component
 
of a along b . 4
   
˚ø a  6 î  8 ĵ ’±1n∏ b  3 ĵ  4k̂ ŒÓ¬ÀôL b 1 ø˙Ó¬ a 1 Œˆ¬"√1 Î◊¬¬Û±—˙ øÚÌ«˚˛ fl¬1±º

29T MATH [8]

 
Find a unit vector perpendicular to each of the vectors a  b and
   
a  b , where a  3 î  2 ĵ  2k̂ and b  î  2 ĵ  2k̂ .

     
a  3 î  2 ĵ  2k̂ , b  î  2 ĵ  2k̂ √ŒÓ¬ÀôL a  b ’±1n∏ a  b ≈À˚˛±È¬± Œˆ¬"√ 11 ›¬Û1Ó¬

˘•§ Œ˝±ª± ¤fl¬fl¬ Œˆ¬"√1 øÚÌ«˚˛ fl¬1±º

13. A natural number is selected at random from the set

A   x : 1  x  50  . Find the probability such that the number

satisfies the inequation x 2  13 x  30 . 4

¸—˝øÓ¬ A   x : 1  x  50  1 ¬Û1± ˚±‘ø˙fl¬ ˆ¬±À¬ı ¤È¬± ¶§±ˆ¬±øªfl¬ ¸—‡…± øÚ¬ı«±‰¬Ú
fl¬1± ˝í˘º ¸—‡…±ÀȬ±Àª ’¸˜Ó¬± ¸˜œfl¬1Ì x 2  13x  30 ¸˜±Ò±Ú fl¬1±1 ¸y±øªÓ¬± øÚÌ«˚˛
fl¬1±º

⎡ 0 
 tan ⎤
⎢ 2 ⎥ , then
14. If A⎢ ⎥

⎢ tan 0 ⎥
⎣ 2 ⎦

⎡ cos   sin  ⎤
show that I  A   I  A  ⎢ ⎥,
⎢⎣ sin  cos  ⎥⎦

where I is the identity matrix of order 2. 6

29T MATH [9] Contd.

⎡ 0 
 tan ⎤
⎢ 2 ⎥ , ŒÓ¬ÀôL
˚ø A  ⎢  ⎥
⎢ tan 0 ⎥
⎣ 2 ⎦

⎡ cos   sin  ⎤
Œ‡≈›ª± Œ˚, I  A   I  A  ⎢ ⎥ ,
⎣⎢ sin  cos  ⎦⎥

˚íÓ¬ I ¤È¬± 2 ˜±S±1 ¤fl¬fl¬ Œ˜Ã˘fl¬é¬º

⎡ 1 2 3 ⎤
⎢ ⎥
If A⎢2 3 2 ⎥ , then find A 1 ; and hence solve the
⎢ ⎥
⎢⎣ 3  3  4 ⎥⎦

system of equations
x  2y  3z   4
2x  3y  2z  2
3x  3y  4z  11

⎡ 1 2 3 ⎤
⎢ ⎥
˚ø A  ⎢ 2 3 2 ⎥ , ŒÓ¬ÀôL A 1 øÚÌ«˚˛ fl¬1± ; ’±1n∏
⎢ ⎥
⎢⎣ 3  3  4 ⎥⎦

¸˜œfl¬1Ì ¬ÛX√√øÓ¬
x  2y  3z   4
2x  3y  2z  2
3x  3y  4z  11
¸˜±Ò±Ú fl¬1±º

29T MATH [ 10 ]

15. Form the differential equation satisfied by  x  a  2  y  b  2  r 2 ,
where a and b are arbitrary constants. 6

 x  a  2  y  b  2  r 2 ¸˜œfl¬1ÀÌ ø¸X√√ fl¬1± ’¬ıfl¬˘Ê ¸˜œfl¬1Ì ·Í¬Ú fl¬1±, ˚íÓ¬ a ’±1n∏
b ‹ø2Âfl¬ ÒËn¬ıfl¬º

Find the maximum and minimum values of the function

f  x   x  sin 2x on  0, 2  .

 0, 2  ’ôL 1±˘Ó¬ f  x   x  sin 2x Ù¬˘Ú1 ¸À¬ı«±2‰¬ ’±1n∏ ¸¬ı«øÚ•ß ˜±Ú øÚÌ«˚˛
fl¬1±º

16. Prove that the area of a right angled triangle of a given hypotenuse
is maximum when the triangle is isosceles. 6

√õ∂˜±Ì fl¬1± Œ˚ √õ∂M ’øÓ¬ˆ≈¬Ê√ ø¬ıø˙©Ü ¤È¬± ¸˜Àfl¬±Ìœ øSˆ≈¬Ê1 fl¬±ø˘ ¸À¬ı«±2‰¬ ˝¬ı Œ˚øÓ¬˚˛±
øSˆ≈¬ÊÀȬ± ¸˜øX√¬ı±U ˝¬ıº

Find the area of the smaller portion enclosed by the curves

x 2  y 2  9 and y
2
 8x .

x 2  y 2  9 ’±1n∏ y 2  8x ¬ıSê˝◊ ’±&1± ¸1n∏ ’—˙1 fl¬±ø˘ øÚÌ«˚˛ fl¬1±º

29T MATH [ 11 ] Contd.

17. Find the shortest distance between the lines

r  6 î  2 ĵ  2k̂    î  2 ĵ  2k̂ 


and r  4 î  k̂   3 î  2 ĵ  2k̂  .

6

r  6 î  2 ĵ  2k̂    î  2 ĵ  2k̂  ’±1n∏ r  4 î  k̂   3 î  2 ĵ  2k̂  Œ1‡±
 

≈ά±˘1 ˜±Ê1 ¸¬ı«øÚ•ß ”1Q øÚÌ«˚˛ fl¬1±º

Find the equations of two lines through the origin which intersect

x 3 y 3 z 
the line   at .
2 1 1 3

x 3 y 3 z 
˜”˘ø¬ıμ≈1 ˜±ÀÊÀ1 ¬Û±1Õ˝ Œ˚±ª±   Œ1‡±Î¬±˘fl¬ Œfl¬±ÌÓ¬ ŒÂ fl¬1± Œ1‡±
2 1 1 3

≈ά±˘1 ¸˜œfl¬1Ì øÚÌ«˚˛ fl¬1±º

  
18. Prove that  a  b    a  b   2  a  b . Hence find the area of the
parallelogram whose diagonals are the vectors

3 î  ĵ  2k̂ and î  3 ĵ  4k̂ . 6

29T MATH [ 12 ]

  
√õ∂˜±Ì fl¬1± Œ˚  a  b    a  b   2  a  b º ¤ÀÓ¬Àfl¬ Œˆ¬"√1 3 î  ĵ  2k̂ ’±1n∏

î  3 ĵ  4k̂ fl¬Ì« ø¬ıø˙©Ü ¸˜±ôLø1fl¬1 fl¬±ø˘ øÚÌ«˚˛ fl¬1±º

Find the vector equation of the line passing through  1, 2, 3  and

parallel to the planes r .  î  ĵ  2k̂   5 and r .  3 î  ĵ  k̂   6 .
 

 1, 2, 3  ø¬ıμ≈1 ˜±ÀÊÀ1 ¬Û±1Õ˝ Œ˚±ª± ’±1n∏ r .  î  ĵ  2k̂   5 ’±1n∏


r .  3 î  ĵ  k̂   6 ¸˜Ó¬˘1 ¸˜±ôL 1±˘ Œ1‡±1 Œˆ¬"√ 1 ¸˜œfl¬1Ì øÚÌ«˚˛ fl¬1±º


19. Solve the linear programming problem graphically. 6

Δ˘ø‡fl¬ øÚ˚˛À˜À1 Ó¬˘1 Δ1ø‡fl¬ √õ∂À·Ëø˜— ¸˜¸…±ÀȬ±1 ¸˜±Ò±Ú Î◊¬ø˘›ª±º

Maximize z  20 x  15y , subject to the conditions
2x  y  200 ,
x  y  150 and x  0, y  0 .

z  20x  15y 1 ¸À¬ıı«±‰¬ ˜±Ú Î◊¬ø˘›ª±

˚íÓ¬,
2x  y  200 ,
x  y  150 ’±1n∏ x  0, y  0 º

29T MATH [ 13 ] Contd.

Maximize and minimize

z  5x  2y , subject to the conditions,

x  2y  2 ,
3x  2y  12 ,
 3x  2y  3 and x  0, y  0 .

z  5 x  2y 1 ¸À¬ı«±2‰¬ ’±1n∏ ¸¬ı«øÚ•ß ˜±Ú Î◊¬ø˘›ª±
˚íÓ¬
x  2y  2 ,
3x  2y  12 ,
 3x  2y  3 ’±1n∏ x  0, y  0 º

20. Two numbers are selected at random from a set of first 90 natural
numbers. Find the probability that the product of randomly selected
numbers is divisible by 3. 6

√õ∂Ô˜ 90 Ȭ± ¶§±ˆ¬±øªfl¬ ¸—‡…±1 ¬Û1± 2 Ȭ± ¸—‡…± ˚±‘ø˙fl¬ˆ¬±À¬ı øÚ¬ıı«±‰¬Ú fl¬1± ˝í˘º ˚±‘ø˙fl¬ˆ¬±Àª
øÚ¬ıı«±‰¬Ú fl¬1± ¸—‡…± ≈Ȭ±1 &ÌÙ¬˘ 3Œ1 ø¬ıˆ¬±Ê… Œ˝±¬ı±1 ¸y±øªÓ¬± øÚÌ«˚˛ fl¬1±º

In a 3×3 matrix, entries aij are selected randomly from the digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with replacement where each element aij
is a three digit number. Find the probability that each element in
each row is divisible by 15.

29T MATH [ 14 ]

¤È¬± 3×3 ˜±S±1 Œ˜Ã˘fl¡é¬1 Œ˜Ã˘¸˜”˝√ aij fl¬ ’—fl¬ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 1
¬Û1± ¬Û≈Ú1¶ö±ø¬ÛÓ¬ ø˝‰¬±¬ıÓ¬ ˚±‘ø˙fl¬ˆ¬±À¬ı øÚ¬ıı«±‰¬Ú fl¬1± Δ˝À , ˚íÓ¬ √õ∂ÀÓ¬…fl¬ aij ¤È¬± øÓ¬øÚȬ± ’—fl¬
ø¬ıø˙©Ü ¸—‡…±º √õ∂ÀÓ¬…fl¬ ˙±1œ1 √õ∂ÀÓ¬…fl¬ Œ˜Ã˘ 15Œ1 ø¬ıˆ¬±Ê… Œ˝±ª±1 ¸y±øªÓ¬± øÚÌ«˚˛ fl¬1±º

———— ×————

29T MATH [ 15 ]

29T MATH [ 16 ] 45+