HSC Board Mathematics Model Question Paper 1

Following are HSC Board  Mathematics and Statistics Model Question Paper 1 for Arts and Science.

Model Question Paper 1

Time 3 hours                                                                                               Max Marks 80

Note: (1) All questions are compulsory.

            (2) The question paper consists of 30 questions divided in to four  

                 sections A, B, C, D.

            (3) Section A content questions of 1 mark each

                Section B content questions of 2 mark each

                Section C content questions of 3 mark each

                Section D content questions of 4 mark each

           (4) Use of logarithmic table is allowed.

           (5) Use of calculator is not allowed.

          (6) In LPP only rough sketch of graph is expected. Graph paper is not   necessary

                                           SECTION A                                           

Select and write the most appropriate answer from alternatives in each of  the following questions
1. If p→q is true and p⋀q is false , then the truth values of p and q are
    a) T,F        b) T,T        c) F,T      d ) F,F
2. If the vectors $2\hat i - q\hat j + 3\hat k$      and  $4\hat i - 5\hat j + 6\hat k$     are      collinear then the value of q is
  a) 5         b) 10          c) 5/2       d) 5/4
3.   The two value of k for which the lines with direction ratios k,-6,-2 and  k-1 ,k,4 are        perpendicular to each other are     

  a )8,-1          b)2,3          c) 8,1         d)-8,-1

4. If the function 

       f(x) = $(cosx)^\frac{1}{x}$                      x≠0

              = k,                     x=0

       is  continuous at x = 0, then the value of k is 

  a) 1        b) -2         c) 3          d) 4
5. $\int\frac{dx}{x+x^n}$ is
  a) $\frac{1}{n}log|x^n|$   + c    b)$\frac{1}{1-n}log|x^{1-n}|$   +c 
  c)  $\log|x^n+1|$   + c   d)$\frac{1}{n}log|x^n+1|$   + c 
6. The differential equation $y\frac{dy}{dx}$  + x = 0 represent family of 
  a) circle     b) parabola     c) ellipse    d) hyperbola
                                                       
                             SECTION B                                       

7. Prepare the truth table for ∼p⋀q
8. Find the general solution of sin(x +⫪/5 ) = 0
9. In ∆ABC, show that   $tan\frac{A}{2}tan\frac{B}{2} = \frac{a + b - c}{a + b + c}$
10. Find the value of ⋋ for which the points (6,-1, 2) (8,-7, ⋋) and (5, 2, 4) are collinear 
10. Differentiate $sin^{-1}(2x\sqrt{1-x^2})$             w.r.t x
11. The displacement S of a particle at time is given by S = $t^3 - t^2 -5t$         find the velocity and acceleration at time t = 2

13. Evaluate  $$\int_{0}^{\frac{∏}{2}}{\frac{sinx}{1 + (cosx)^2}}dx$$
14. Solve the differential question      y-x$\frac{dy}{dx}$ = 0 

                                   OR

14. Find the area of the region bounded by the curve y  = $x^2$ , the x- axis and given lines x=1, and x=5

                                                          SECTION C                               

15. Using truth table, prove the equivalence ∼p∧ q  = (p∨ q)∧∼q 

16.. Find the vector equation of the line passing through the point (2,3,-4) and perpendicular to the XZ-plane .Hence find its equation in Cartesian plane

17. Find the vector equation of the plane  $\bar r = ( 2\hat i + \hat k) +\lambda\hat i + \mu(\hat i +2\hat j - 3\hat k)$                     in scalar product form

                                           OR

17. Find the equation of the plane passing through the intersection of the  planes 3x=2y- z+1 = 0 and x+y+z-2 = 0 and the point (2, 2, 1).

18.  Find the value of k, if the function

        f(x) = log (1+2x),              for x ≠ 0

               = k,                               for x = 0

              Is continuous at x = 0

19.  Obtain the probability distribution of the number of sixes in two tosses of fair die

                                      OR

19.Let X have p.m.f. 

        P(x)  = $kx^2$                     x = 1,2,3.4

                = 0,                              otherwise 

       .Find mean and variance of X

20. The probability that a certain kind of component will survive a check

       test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive.

                                     SECTION D

21. If three numbers are added the sum is 15. If the second is subtracted from the sum of first and third number the we get '5' and if twice the first number is added to the second and the third number is subtracted from the sum we get '4'. Use  matrices to  find the number. 

.22.In △ABC prove that (b + c -a) tan A/2 = (c + a -b) tan B/2 = tan ( a +b - c) tan C/2  

                                 OR

22.Find the general solution of $\sqrt {3}$cosx - sinx = 0
23.Find p and q, if the equation p$x^2$ -8xy = 3$y^2$ = 14x + 2y + q =0 represent a pair of perpendicular lines.   

24.Find the volume of tetrahedron whose vertices are 

            A (-1, 2, 3), B (3, -2, 1), C (2, 1, 3), and D (-1, -2, 4)

25.Solve by graphically

           Maximize z = 15x + 30y subject to 3x + y ≤ 12, x + 2y  ≤ 10, x ≥ 0, y ≥ 0.

26.If $x^{m}y^{n} = ( x + y)^{m + n}$      then show that  $\frac{d^2 y}{dx^2}$ = 0.

27. A manufacture can sell x items at the rate of   ₹(330 -x ) each. The cost of

      producing x items is  ₹($x^2$+ 10x + 12 ) How many items must be sold so that his profit is maximum?

28.Find the area of the region lying between the parabolas $y^2$ = x and $x^2$ = y.

29.Prove that
∫$\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{2}$     +c
30.Solve the equation $x^{-1}cos^2ydy + y^{-1}co^2xdx$       = 0.

                                              OR

30. In a culture of yeast,the active ferment double itself in 3 hours. Assuming that the quantity increases at a rate proportional  to itself, determine the number of times it multiplies itself in 15 hours. 

                                                            END

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