Maharashtra HSC Board Mathematics Question Paper 2023 PDF

 Board Question Paper: March 2023 Maharashtra HSC BOARD QUESTION PAPER 2023 Mathematics Time: 3 Hrs....

 Board Question Paper: March 2023 Maharashtra HSC BOARD QUESTION PAPER 2023 Mathematics Time: 3 Hrs. Max. Marks: 80 General instructions: The question paper is divided into FOUR sections. (1) Section A: Q. 1 contains Eight multiple choice type of questions, each carrying Two marks. Q. 2 contains Four very short answer type questions, each carrying one mark. (2) Section B: Q. 3 to Q. 14 contain Twelve short answer type questions, each carrying Two marks. (Attempt any Eight) (3) Section C: Q. 15 to Q. 26 contain Twelve short answer type questions, each carrying Three marks. (Attempt any Eight) (4) Section D: Q. 27 to Q. 34 contain Eight long answer type questions, each carrying Four marks. (Attempt any Five) (5) Use of log table is allowed. Use of calculator is not allowed. (6) Figures to the right indicate full marks. (7) Use of graph paper is not necessary. Only rough sketch of graph is expected. (8) For each multiple choice type of question, it is mandatory to write the correct answer along with its alphabet, e.g. (a)……. / (b)…….. /(c)…….. /(d)…….., etc. No marks shall be given, if ONLY the correct answer or the alphabet of correct answer is written. Only the first attempt will be considered for evaluation. (9) Start answer to each section on a new page. SECTION  A Q.1. Select and write the correct answer for the following multiple choice type of questions: i. If p  q is F, p q is F then the truth values of p and q are _______ respectively. (a) T, T (b) T, F (c) F, T (d) F, F (2) ii. In ABC, if c2 + a2 – b2 = ac, then B = _______.     (a) (b) (c) (d) (2) 4 3 2 6 iii. The area of the triangle with vertices (1, 2, 0), (1, 0, 2) and (0, 3, 1) in sq. unit is _______. (a) 5 (b) 7 (c) 6 (d) 3 (2) iv. If the corner points of the feasible solution are (0, 10), (2, 2) and (4, 0) then the point of minimum z = 3x + 2y is _______. (a) (2, 2) (b) (0, 10) (c) (4, 0) (d) (3, 4) (2) v. If y is a function of x and log (x + y) = 2xy, then the value of y(0) = _______. (a) 2 (b) 0 (c) –1 (d) 1 (2)  cos 3 vi. x dx = _______. 1 3 1 1 (a) sin 3 x  sin x  c (b) sin 3 x  sin x  c 12 4 12 4 1 3 1 1 (c) sin 3 x  sin x  c (d) sin 3 x  sin x + c (2) 12 4 12 4 dx x log x vii. The solution of the differential equation  is _______ dt t (a) x = ect (b) x + ect = 0 (c) x = et + t (d) xect = 0 (2) 1   Mathematics and Statistics x 4 x 5  4 viii. Let the probability mass function (p.m.f.) of a random variable X be P(X = x) = 4Cx      , 9 9 for x = 0, 1, 2, 3, 4 then E(X) is equal to _______ 20 9 12 9 (a) (b) (c) (d) (2) 9 20 9 25 Q.2. Answer the following questions: i. Write the joint equation of co-ordinate axes. (1) ii. Find the values of c which satisfy c u = 3 where u  ∃i  2∃j  3k∃. (1) iii. Write  cot x dx. (1) dy dy iv. Write the degree of the differential equation e dx  =x (1) dx SECTION  B Attempt any EIGHT of the following questions: Q.3. Write inverse and contrapositive of the following statement: If x < y then x2 < y2 (2)  x 0 0 Q.4. If A =  0 y 0  is a non singular matrix, then find A–1 by elementary row transformations.  0 0 z  2 0 0 Hence write the inverse of  0 1 0  (2)  0 0 1  Q.5. Find the cartesian co-ordinates of the point whose polar co-ordinates are  2, . (2) 4  Q.6. If ax2 +2hxy + by2 = 0 represents a pair of lines and h2 = ab ≠ 0 then find the ratio of their slopes. (2) Q.7. If a , b , c are the position vectors of the points A, B, C respectively and 5a  3b  8c  0 then find the ratio in which the point C divides the line segment AB. (2) Q.8. Solve the following inequations graphically and write the corner points of the feasible region: 2x + 3y ≤ 6, x + y ≥ 2, x ≥ 0, y ≥ 0 (2) Q.9. Show that the function f(x) = x3 + 10x + 7, x ∊ R is strictly increasing. (2)  2 Q.10. Evaluate:  0 1  cos 4 x dx (2) Q.11. Find the area of the region bounded by the curve y2 = 4x, the X-axis and the lines x = 1, x = 4 for y ≥ 0. (2) Q.12. Solve the differential equation cos x cos y dy – sin x sin y dx = 0 (2) Q.13. Find the mean of number randomly selected from 1 to 15. (2) Q.14. Find the area of the region bounded by the curve y = x2 and the line y = 4. (2) 2   Board Question Paper: March 2023 SECTION  C Attempt any EIGHT of the following questions: Q.15. Find the general solution of sin  + sin 3 + sin 5 = 0 (3)  Q.16. If 1  x  1, the prove that sin1 x + cos1 x = (3) 2 Q.17. If  is the acute angle between the lines represented by ax2 + 2hxy + by2 = 0 then prove that 2 h 2  ab tan  = (3) ab Q.18. Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are –2, 1, –1 and –3, –4, 1. (3) x 1 y  2 z  3 x 2 y 4 z 5 Q.19. Find the shortest distance between lines   and  . (3) 2 3 4 3 4 5         Q.20. Lines r  ˆi  ˆj  kˆ   2iˆ  2 ˆj  kˆ and r  4iˆ  3 ˆj  2kˆ   ˆi  2 ˆj  2kˆ are coplanar. Find the equation of the plane determined by them. (3) Q.21. If y  tan x  tan x  tan x .....   , then dy sec 2 x show that . dx 2 y  1 dy Find at x = 0. (3) dx Q.22. Find the approximate value of sin (3030′). Give that 1 = 0.0175c and cos 30 = 0.866 (3) Q.23. Evaluate  x tan 1 x dx (3) dy  Q.24. Find the particular solution of the differential equation = e2y cos x, when x = , y = 0 (3) dx 6 Q.25. For the following probability density function of a random variable X, find (a) P(X < 1) and (b) P(| X| < 1). x2 f(x) = ; for –2 < x < 4 18 =0 , otherwise (3) Q.26. A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes. (3) SECTION  D Attempt any FIVE of the following questions: Q.27. Simplify the given circuit by writing its logical expression. Also write your conclusion. S1' S1 S2 S'2 (4) 1 2  Q.28. If A =   verify that A(adjA) = (adjA)A = |A|I (4) 3 4  3   Mathematics and Statistics 1 Q.29. Prove that the volume of a tetrahedron with coterminus edges a , b and c is a b c . 6  Hence, find the volume of tetrahedron whose coterminus edges are a  i  2 j  3k , b  ˆi  ˆj  2kˆ and ˆ ˆ ˆ c  2iˆ  ˆj  4kˆ. (4) Q.30. Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line    r  7iˆ  7ˆj  6kˆ   2iˆ  2ˆj  3kˆ  (4) d2 y dy Q.31. If y = cos(m cos–1 x) then show that (1 – x2) –x + m2y = 0 (4) dx 2 dx Q.32. Verify Lagrange’s mean value theorem for the function f(x) = x  4 on the interval [0, 5]. (4) 2x2  3 Q.33. Evaluate:   x 2  5 x 2  4  dx (4) 2a a a Q.34. Prove that:  f  x  dx   f  x  dx   f  2a  x  dx 0 0 0 (4) 4