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HSC Mathematics -2nd Paper (Theoretical) Model Question-2


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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