AHSEC HS 2nd Year Question Paper 2023 Maths

ASSAM BOARD
QUESTION
PAPER
2023

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Total nuiber of pages - 16
33T MATH

2023

MATHEMATICS
Full Marks : 100
Pass Marks : 30

Time : Three hours
The figures in the nargin indicate full marks
for the questions.

mark each 1x10 = 10
Q. No. 1(i-x) carries 1
marks eacl1 4x12 = 48
0. Nos. 2-13 carry 4
narks each 6x7 = 42
Q. Nos. 14-20carny 6
Total = 100

Contd.

1x10=10
1. Answer the following questions :

State true or false :

On any finite set X, an f:X’X is
one-one function
necessarily onto.

Pa f:X -’ X N b1|

(i) If (Tf) Cos x=y, then the value of y is (CO8 y AA )

(a) 0 sys n
(b) 0<y< n

(c)sy<2
(d)y<2
(iüi) Fill in the blanks :

The number of all possible matrices of order 2x2 with cach
entry 0 or 1 is

(iv) What do you mean by critical point of a function ?

(v) Give an example of afunction which is continuous on but
not differentiable therein.

33T MATH [21

d 3
(vi) If dx f(x)= 4r. such that f()=0, then find f(x).

f(x)-4y
dx

(vii) Write the order and degree (if cxist) of the differential equation
d'y cos
dy
dx2 dx

d'y Cos dy
dx? dx

(vii) If is a non-zero vector of magnitude 'a' and 2 is a non
zero scalar, then 22ã is unit vector if

) a=1 (ii) 2 =-1
1
(ti) a =|2| (iv) a=
2|2|

(ix Find the Cartesian equation of the plane
F.(i+j-k )=2
where Y be the position vector of any arbitrary point.

(x) Define Bernoulli trials.

33T MATH [3] Contd.

is one-one. Find
2. Show that f:-1, 1J>R given by f(x) X+3
2+2=4
the inverse of the function
f:[-1,1 ’ Rangef.

X+3

<paAD| aTÀI f: (-1, 1]’ Range f

OR/qT
in
be the relation
Let L be the set of all lines
in xy -plane and R
that R is an
L defined as R=(4)/ is parallel to y . Show
to the line
equivalence relation. Find the set of all lines related
3+1=4
y=3x+1.

a
R={(4,,)/4, 4 | y= 3x+ l

Prove that 2 tanx= s i n x for xe-1, 1|. Also find the
value
3.
l+x2
of Sun
2+2=4
3

-1 2x
gaY 0 XE-1, 1] iG 2tan'x= sin 1+x?

G505 sin Sin
3

OR/ WAT
4
Show that (7Gq0 A)

Sin (12)
-1/1
+ COS + tan-l (63
13 16)

33T MATH [4]

of determinants, show that
Using propertics
4
4.

-a? ab ac

ba -b? bc 4a' b2
Ca cb - c?

OR/ AI

A with real entries, prove that A+ A' is
For any square matrix
is the
symmetric and A-A' is skew Symmetric matrix (where A'
2+2=4
transpose of A).

5. Find dy if 2+2=4
dx

dy
dx

log (log x) ,
1-2)
(i) y= sin 0<x<1
|1+2

OR/ WRT

If y* = x, find dy
4
dx

dy
dx
yea|
Contd.
33T MATH [5]

2+2=4
b. Evaluatc: (any two)

dx
X+xlog x

1- cos 2x
-dx
(iü) 1 +cos 2X
(üi)esinx dx
1x4=4
7. Integrate : (any one)

2x
(9 J2+3x+2
2/3 dx

4+9x?

8. For the diferential equation xy:dy = (x+2) (y +2),find the solution
dx
4
curve passing through the point (1, -1).
dy =(x+2)(y+2) A(1, -1 ) AaOP| 4| 1KA
Xy
dx

OR/ W

Find a particular solution of the differential equation
dy
dx
+ y cot x= 4xcosecx (x*0)

where u(3%)-0. 4

33T MATH [6]

dy +ucot x= 4x COsec x ( #0) ATA â e r gAa Giq0
dx
u5-0,
Answer (i) and (11) OR (a) and b).
9.

2+2=4
COS X - sin x 01
(i) If (A1) F(x) =|sinx COS X
1

show that ( 7 s R) F(x) F(y) =F(x+y)
(ü) Prove that ( )
a
2a

(f(oldx =Jf() dx+ r(2a -x) dx
OR/ WAt
2+2=4

X 6
(a) If |18 18 6 then find x.

2 6
18 x 18 6

(b) If x =a(cost + tsint ), y = a( sint-tcost)
find dy
dx

A7 dy taST||
r=a( cost + tsint ), y = a( sint - tcost) A, dx

33T MATH (7] Contd.

vectors
position
10. Show that the
points A
A, B and C with
respectively form
=3i -4j- 4X, b=2i -j+ -j+k and =i-3-5k 1x4=4
the vertices of a right angled triangle.

=i-3j- 5k I 8
3+1=4
11. vectors + b
the
Find a unit vector perpendicular to each of
and @-b, where =3i +2j+2k and

tAet T't å= 3i +2j +2k yiRs b=i+2j-2k!
fi) Evaluate the product

(3 -sõ).(2ã +75)
OR/ qaI

C(11, 3, 7 )
Show that the points A(1, -2, -8), B(5. 0, -2) and 4
are collinear and find the ratio in which B divides AC.

Ts A A(1,-2, -8), B(5, 0, -2) *C(u1,3, 7) foDi.AAAI

one of the digits fromn
12. A bag consists of 10balls each marked with
replacement from the
0to 9. If 4 balls are drawn successively with
marked with the
bag, what is the probability that one ball is 4
digit 1.

33T MATH [8]

13. Find all points
of discontinuity of f where fis defined by
||x|+4, if XS-4
f(x)= --6x2x,+2 if
if
-4< X<4 4
x4

|x|+4, af X<-4
f(x)= -2x, z1 4 <x< 4
6x +2 X>4

14 Using elementary operation, find the inverse of the matrix A
1 2
where A= 1 23
3 1

o 1 2
1 2 3
3 1 1

OR/ AI
Solve the following system of linear equations using matrix
6
method

2x+ 3y + 3z = 5
X-2y t z =-4

3x-y-22 = 3

15.
2+4=6

() The radius of a circle is increasing at the rate 05 cm/s. What
is the rate of increase of its circumference ?

33T MATH [9] Contd.

increasing
(i) Find the interval in which the yis strictly
and decreasing where y =
function
xe-*

OR/
3+3=6
the
(a) Find the points on the Curve x +y-2x-3 = 0 at which
tangents are parallel to the X-axis.

the
(b) Find all the points of local and local minima of
function fgiven by maxima
f(x)=2x -6x +6x+5 (if exist).

16. 3+3=6
(a) Evaluate :

5

-5
[|x+2| dx

(b) Prove that

2
sin xdx
3

33T MATH [10]

OR/
Find the area of the
y= X, X=0 and x-3
region bounded bby the curves y=x+2,
+2 4 y =X, 6
X=0 x=3

17.
(a) Form a differential 2+4=6
curves y=e e (a cos
x+ equation representing the given family of
constants a and b. bsinx) by eliminating arbitrary
anE fg Y=e(a cos x +
bsin x)a u
a 4 bya3-|

(b) Find the general
solution of the differential equation
xlog x dy +y=-log
2
x,
dx X

xlog x dy 2
dx X

OR/ a

Show that the differential equation 2ye dx + Y-2xe?y dy = 0
is homogeneous and find its particular solution when y(0) =1.
6

x/
(RGaI A 2yedx+y-2xedy =0 a a a i t g q

33T MATH [11] Contd.

through the
passing
18, Find
the vector cquation of the planc
planCs
interscctionof the
r.(2i +2)- 3Á )=7
F.(2i+5j+3k -9
and through the oint (2, 1,3).

Ai i.(2i +2j-3k )=7,
(2, 1, 3) fag
F.(2i +5j+3É)=9

OR/Qat
4+2=6

line that passes
Find the vector and Cartesian equations of the
through the points (3,-2, -5) and (3, -2, 6).
ARRU yK (U~4 FILbN
(3, -2, -5) *(3, -2,6) fr

lines X-5 y+2 Z
and are
(i) Show that the 7 -5 1 1 2 3

perpendicular to each other.

X-5_y +2 = =

7 -5 1 1 2 3

33T MATH [12]

19. Solve graphically the following lincar programming problem. 6

Maximize and minimize
Z=-x+ 2y
subject to the constraints
x2
X+y5
x+ 2y 6
y0

x+y>5
x+ 2y >6
y20

OR/1
A manufacturer makes two types of tovs A and B. Three machines
are needed for this purpose and the time (in minutes) required for
each toy of the machines is given below :
Machines
Types of Toys
I III

A 12 18 6

B 6

Each machine is available for amaximum of 6 hours per day. If the
profit on each toy of type A is Rs. 7-50and that on each toy of type
Bis Rs. 5, show that 15 toys of type A and 30 toys of type B should
be manufactured in a day to get maximum profit. 6

33T MATH [13] Contd.

cafbe
III
A 12 18 6
B 6

20. A doctor is to visit a patient. From the past experience, it 1s known
that the probabilities that he will come by train. bus, scooter or by

1 1 2
other means of transpOrt are respectively 3 and The
10' 5° 1O, 5
1 1 1
probabilities that he will be late are and if he comes by
4'3 12
train, bus and scooter respectively, but if he comes by other means
of transport, then he will not be late. When he arrives, he is late.
What is the probability that he comes by bus ? 6

1
10' 5' 10
2 1 1
5 4 3
1
12

33T MATH [14]

OR/ at
In a girls' hostel; 2+2=4
30% read English 70% of the
English newspapers,newspaper
A
students read Hindi newspaper,
and 20% read both Hindi and
(a) Find the student is selected at random.
probability
English newspapers. that she reads
neither Hindi
nor
(b) If she reads
she reads Hindi newspaper, find the probability that
English newspaper.
70% CA R. 30%ta a 20% f

(ü) Define independent events with an example. 2

33T MATH [15]