HSC Mathematics 2nd Paper Model Test-1

Class – XII

Subject: Mathematics 2nd Paper

Time : 3 hrs.                                                                                                                                         Marks.75

Group -A ( Machanics)

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 part of their resultant in the same direction.                                                             4

Or,       Forces P, Q, R acting along OA, OB, OC, where O is the incentre of the triangle ABC, are in   equilibrium. Show that  P: Q: R = Cos .                                                                    

2.         Find the magnitude and the point of action of the resultant of two unequal like parallel forces.       5

Or,       If two equal and opposite forces S in any two parallel lines at distance b apart in the plane of two like     parallel forces P, Q be combined with them, show that the resultant is displaced at a distance   

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

Or,       A uniform plank of length 2a and weight w is supported horizontally on two vertical props at a   distance b apart. The greatest weight that can be placed at the two ends in succession without hitting        the plank are W_­1 and W₂ respectively. Show that.    .

4.         State the parallelogram law of velocities. Find the magnitude and direction of the resultant of two            velocities u and v acting at a point at an angle .                                                                            4

Or,       If a point moving under uniform acceleration describes successive equal distances in times t₁, t₂, t₃,         then show that: .

5.         Find the time of flight and horizontal range of a particle projected in vacuo with velocity u at an

            angle with the horizon.                                                                                                                       4

Or,       If t₁ be the time in which a projectile reaches a point p of its path and t₂ be the time from p until it reaches the horizontal plane through the point of projection, then show that the height of p is h = gt₁ t₂.

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

Or,       A bullet leaves the muzzle of a rifle barrel 4 metres long with a velocity of 680 m/see. If the

            mass of the bullet is 5 gm. Find the uniform force acting on the bullet in the barrel. How much

            time will the bullet take to traverse the barrel?

7.         Two particles of masses P and Q are connected by a light inextensible string which passes over

            a light smooth fixed pulley and are allowed to hang freely. If  P> Q, find the acceleration of the   system and the tension of the string.                                                                                                       4

Or,       Two masses 14 kg and 18 kg are connected by a light inextensible string passing over a smooth

            pulley and hang freely. After 3 sees, the string is cut. How for will the lighter body rise?

Group -B ( Calculus)

8.         If   y = f (x) =  then show that f (y) = x                                                                             4

Or,         .

9.         Find from the first principle the differential coefficient of sin2x with respect to x .                         4

Or,       Find the differential coefficient with respect to x of any two.

            (i)         tan^-1               (ii)    ( x^x ) ^x                  (iii)   2x⁰ cos 3x⁰

            (iv)      

10.       If   y = e, show that ( 1 - x²) y₂ - xy₁ = a²y.                                                                                  4

Or,       If   y = sin ( m sin^-1x), then show that ( 1 - x²)

11.       Expand ln ( 1+x) or tan^-1x  in an infinite series with the help of Maclaurin's series.                                   4

12.       Find the abscissa of the points on the curve  y = x³ - 3x² - 2x +1  where its tangents make                     4

            equal angles with the axes of coordinates.

Or,       Show that the maximum value of  x +   is smaller than its minimum value.

13.       Find the integrals ( any two)                                                                                                      2.5x2 =5

            (i)         ò                              (ii)        ò  e^x Sin2x  dx

            (iii)       ò Sce² x cosce²x  dx                  (iv)       ò  Cos^-1x dx

14.       Find the value of any two:                                                                                                         2.5x2 =5

            (i)         òe^x( Sinx + Cosx) dx            (ii)        ò  x dx

            (iii)       ò                   (iv)       ò Cos²x dx.

Or,       Find the area of the surface surrounded by the curve 

Group - C (Discrete Mathematics)

15.       Solve the following linear programing with the help of graph and maximise  Z = 3x + 4y.             5

            Conditions : x + y £ 7,  2x + 5y = 20,  x, y ³ 0.

Or,       Solve the following linear programming with the help of graph and minimise  Z = 2x -y,

            Conditions:  x + y £ 5, x + 2y £ 8,  4x + 3y >12, x, y, ³ 0.

16.       State and prove Bayes theorem.                                                                                                            5

Or,       A box contains 10 blue and 15 red marbles. If a boy draws at random, then what is the

            probability of two marbles (i)  being of different colours and (ii) being of the same colour?

17.       Write the algorithm and construct the flow chart for finding the sum of 1 + 2+3 +4 + ........+ 1000             5

Or,       Convert (115)₁₀ into binary number and ( 11010011)₂ into decimal number.