HSC Mathematics 2nd Paper Model Question-4

Time : 3 hrs.                                                                                                                              Marks. 75

Group -A ( Mechanics)

1.         Prove that the algebraic sum of the resolved parts of two forces acting at a point in a given direction        is equal to the resolved parts of their resultant in the same direction.                                       4

Or,       The angle of inclination between two forces P and Q is q. If P and Q be interchanged in position,             show that the resultant will be turned through an angle f where tan

2.         A force P acts along AO where O is the circumcentere of the triangle ABC. Show that the parallel             components of P acting at B and C are in the ratio Sin2B: Sin2C.                                                    5

Or,       P and Q are two like parallel. If p is moved parallel to itself though a distances x, show that the    resultant of P and Q move through a distance .

3.         Prove that a force and a couple in the same plane are equivalent to a single force equal and parallel to       the given single force.                                                                                                                   5

Or,       Force P, Q and R act respectively along the sides BC, CA and AB of the triangle ABC. If the line of             action of their resultant passes through the centroid of ABC, show that .

4.         State triangle of velocities. Prove that the resultant of two equal velocities bisects the angle between             them.                                                                                                                                                   4

Or,       If a point moving with uniform acceleration describes successive equal distances in tines t₁, t₂, t₃,             show that, X .

5.         A particle is thrown in vacuo with a velocity a at an angle a, find its time of flight and greatest    horizontal range.                                                                                                                          4

Or,       From a baloon ascending with a velocity of 10 m/sec, a stone is let fall and reaches the ground in 10             secs. How high was the baloon when the stone was dropped?

6.         State Newton's laws of motion and under usual notations deduce the formula P = mf.                   4

Or,       The average velocities of a particle moving along a straight line are v₁, v₂, v₃ at tines t₁, t₂, t₃ .

            Show that. .

7.         A shot of mass 4 lbs fired into a wall stops after perotrating-10 inches. If the resistance is 42 tons, wt,

            find the velocity of the shot.                                                                                                              4

Or,       Two masses 14 kg and 18 kg are connceted by a light inextensible string passing over a smooth    pulley and hang freely. After 3 seconds, the string is cut. How for will the lighter body rise?

Group-B ( Calculus)

8.         If f(x) = tanx, prove that f(a -b ) = .                                                                                4

Or,       Find the limit ´ :   .

9.         Find the differential co-efficient of sinax with respect to x from the first principle.                         4

Or,       Find the differential co-efficient w. r to x, (any two).

            (i)            (ii)  tan ( sin^-1x        (ii)  ( x² + 1) tan^-1 x -x            (iv)  In .

10.       If  y = e^ax sinbx, show that  Y₂ - 2ay₁ + (a² +b² ) y = 0.                                                                    4

Or,       If   y  = show that 2yy₂ + 2y+ y² = 4.

11.       Expand tan^-1x  or Inx in an infinite series by Maclourin's theorem.                                                   4

12.       A curve y = ax² + bx + c passes through the origin and the point ( 2, 2). If the solpe of the curve at the

            origin is 2, find a, b and c.                                                                                                                  4

Or,       Find the minimum volume of y = 4e^x + 9e ^-x. 

13.       Integrate ( any two).                                                                                                               2.5x2=5

            (i)         ò dx                  (ii)     òSin^-1 x dx                    (iii)   ò dx

            (iv)       ò sin³x cos⁴x dx.

14.       Find the value of the following ( any two)                                                                             2.5x2=5

            (i)         ò   dx                   (ii)    ò  dx         

            (iii)   ò   dx       (iv)       ò x³  _edx

Or,       Find the area enclosed by the curves x - y + 2 = 0 and y = x².

Group -C : ( Discrete Mathematics)

15.       A man wishes to purchase 6 books and 4 pen within an amount of Tk. 500. The cost of each book and     pen are respectively Tk. 30 and Tk. 40 How many of each kind should be purchased so that under the            conditions, he may purchase the highest number of articles?                                                        5

Or,       Solve with the help of graph and find the maximum value of  z = 4x + 6y under the conditions

            x + y =5, x ³ 2, y £ 4, x, y ³ 0.

16.       A bag contains 3 black and 4 white balls. One ball is drawn at random and without replacing it     another is drawn. What is the probability of the second ball to be white?                                             5

Or,       Write down the sample space of two dice thrown simultameously and find the probability of        appearing two sixes. 

17.       What do you mean by flow chart and algorithm? Explain the different symbols of flow chart.       5

Or,       Convert (105)₁₀ into binary number and (1010101)₂ into decimal number.