HSC Mathematics -2nd Paper (Theoretical) Model Question-2
Group A-Mechanics
1. State and prove the theorem on Resolved parts. 4
Or,
Two
forces P and Q acting parallel to the length and base of a plane inclined at an
angle
to the horizon.
Support together a body placed on the plane. If
, P, Q are halved, the body is still equilibrium. Show
that, P: Q =



2. Find
the magnitude, direction and the line of action of the resultant of two like
parallel forces. 5 Or,
Three like parallel forces P, Q, R act at the
vertices A, B, C of a triangle ABC. If their resultant passes through the
circum-centre in all cases, whatever be the common directions of the forces,
show that,
.

3. A
uniform plank of length 2a and weight W is supported horizontally on two
vertical props at a distance b apart. The greatest weights that can be placed
at the two ends in succession without hitting the plank are W1 and W2
respectively. Show that
5
Or,

Forces
of magnitudes 3,2,4,3, Ö2 units act along AB, CB, CD,
AD, DB respectively, where ABCD is a square of side 5 units. Prove that, they
are equivalent to a couple and find its moment.
4. Under usual notations, deduce v2 = u2 + 2fs. 4
Or,
If a point moving under uniform
acceleration describes successive equal distances in times t1,t2
, t3, then show
that, 

5. If a
body be projected with velocity u making angle a with the
horizon. Find the greatest height attained, horizontal range and time of flight. 4
Or,
A
particle projected vertically upwards with a given velocity rises to a height h in time t
secs. and reaches the ground after anothert1secs. Prove that h = 1/2
gtt1
6. Under
usual notations, deduce P = mf . 4
Or,
A body of mass 4 gms. Falls from a height
of 6 meters and penetrates 5 cms. in mud.
Find the average thrust of
mud on the body.
7. Two
particles of mass m1 and m2 are connected by a light
inextensible string which passes over a light
smooth fixed pulley and are allowed to hang freely. If m1>m2
, find the resulting motion and the tension of the string. 4
Or,
If
common acceleration of two masses attached to the ends of an inextensible light
string passing over a fixed smooth pulley is 109 cm/sec2, find the
ratio of the masses .(where g = 981 cm/sec2 ).
Group
B –Calculus
8. If f(x)=ln(sinx) and g(x)=ln(cosx), show
that
4

Or,
If
then find the value of
.


9. Evaluate :-
4

Or,


10. Find from first principle the differential
co-efficient of secx or ax with respect to x 4
11. Find the differential
co-efficient of any two of the followings
with respect to x :- 3´2= 6
(a)
(b)
(c) logx a


12. If y = √(4+3sinx),
show that, 2y2+2y12+ y2 = 4. 4
Or,
If
, show that, (1 - x2
) y2 – xy1= a2y.

13. Using
Maclaurin’s formula, expand cosx Or,
ax in an infinite series. 4
14. Find
the equation of the normal at the point (1 , -1) of the curve 4
x3 + xy2 - 3x2 +
4x + 5y + 2 = 0.
Or,
Find the least value of 

Group C – Discrete Mathematics.
15. Solve the following linear programming with
the help of graph and minimize Z = 2y
–x,
conditions:
5

Or,
The protein and carbohydrate constituents
and the price per Kg. of two kinds of diets A and
B
are as follows :
Type of diet
|
Protein units per Kg.
|
Carbohydrate units per Kg.
|
Price per Kg.
|
A
|
8
|
10
|
TK.40.00
|
B
|
12
|
6
|
TK.50.00
|
Minimum daily requirement
|
32
|
22
|
|
How can the minimum requirements be
satisfied at the least cost ?
16. State
and prove the addition law of probabilities for exclusive events. 3
Or,
A bag
contains 4 white and 5 black balls. A man drew 3 balls at randam. Find the
probability of 3balls to be black.
17. What is
computer? Discuss in short the use of computer. 5
Or,
Convert
(115)10 into binary number and (11010011)2 into decimal
number.
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