Maharashtra 12th Std Model Question Paper 2027 Mathematics and Statistics

MODEL QUESTION PAPER 2027 MATHEMATICS AND STATISTICS Std. XII (H.S.C.) · Maharashtra State Board Time: 3 Hours Max. Marks: 80 General instructions: (1) All questions are compulsory. (2) The numbers to the right indicate full marks. (3) Use of a logarithmic table is allowed; use of a calculator is not allowed. (4) For each MCQ, only the first attempt will be evaluated. (5) Start each section on a new page. SECTION A Q. 1. Select and write the correct answer of the following (each 2 marks): 16 3 (i) The approximate value of the function f(x) = x - 3x + 5 at x = 1.99 is ____. (a) 6.09 (b) 6.91 (c) 7.09 (d) 7.91 (ii) If f(x) = sin-1 x/sqrt(1 - x2) and g(x) = e^(sin-1 x) then integral f(x).g(x) dx = ____. (a) e^(sin-1 x)(1 - sin-1 x) + c (b) e^(sin-1 x)(sin-1 x - 1) + c (c) e^(sin-1 x)(sin-1 x + 1) + c (d) e^(cos-1 x)(cos-1 x - 1) + c (iii) The value of sin[sin-1(-1/sqrt(2)) - 3 sin-1(sqrt(3)/2)] = ____. (a) -1/sqrt(2) (b) 1/sqrt(2) (c) sqrt(3)/2 (d) -sqrt(3)/2 (iv) Write the dual of p ^ t. (v) Find the angle between the lines r = (i + 2j + 3k) + lambda(2i - j + k) and r = (i + 2j + 3k) + mu(i + 2j + k). (vi) In a Binomial Distribution, E(X) = 6, Var(X) = 4.2 then the value of n is ____. (a) 22 (b) 24 (c) 20 (d) 26 (vii) If tan-1(2x) + tan-1(3x) = pi/4, then x = ____. (a) -1 (b) 1/6 (c) 1/3 (d) 3/2 (viii) If the p.d.f. of a continuous r.v. X is f(x) = (x + 2)/18, for -2 < x < 4, = 0 otherwise then P(|X| < 1) = ____. (a) 1/9 (b) 2/9 (c) 1/27 (d) 2/27 Q. 2. Answer the following (each 1 mark): 4 x 2 x (i) Evaluate: integral e (1 + x)/cos (x e ) dx. (ii) Find the general solution of equation cos 5(theta) = 1/2. (iii) Write the order of differential equation (d2y/dx2)^(5/2) = cube root of (dy/dx). (iv) Evaluate: integral 1/(25 - 9x2) dx. SECTION B Q. 3 to Q. 14. Attempt any EIGHT of the following (each 2 marks): 16 (i) Evaluate: integral from 0 to -1 of e^(-x) dx. (ii) Using truth table, determine whether statement pattern [(p v q) ^ ~p] ^ ~q is a tautology or a contradiction or a contingency. (iii) If a, b, c are the position vectors of points A, B, C respectively and 10a = 7b + 3c then find the ratio in which the point C divides the line segment AB. (iv) Find the area bounded by curve x2 + y = 0, X-axis and lines x = 1 and x = 4. (v) Using truth table, prove that ~p ^ q = (p v q) ^ ~p. (vi) Solve the D.E. 3ex tan y dx + (1 + ex) sec2 y dy = 0. (vii) Construct the switching circuit for the statement (~p ^ q) v (p ^ ~r). (viii) Test whether the function f(x) = x - 1/x, x in R, x != 0 is increasing or decreasing. (ix) Find dy/dx, if sqrt(x) + sqrt(y) = sqrt(a). (x) Verify Rolle's theorem for the function f(x) = x2 - 5x + 9, x in [1, 4].

(xi) A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3 cm/sec. How fast the area enclosed is increasing when the radius is 8 cm? (xii) If A = [[2, -4], [3, 1]], then the adjoint of matrix A is ____. (a) [[1, 3], [4, -2]] (b) [[-1, 3], [-4, 1]] (c) [[-1, -3], [-4, 2]] (d) [[1, 4], [-3, 2]] SECTION C Q. 15 to Q. 26. Attempt any EIGHT of the following (each 3 marks): 24 (i) A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes. (ii) Solve the differential equation x dy/dx - sqrt(x2 + y2) = y. (iii) Prove that: sin-1(3/5) + cos-1(12/13) = sin-1(56/65). (iv) Find the shortest distance between the lines, (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-2)/3 = (y-4)/4 = (z-5)/5. (v) If y = sin-1 x, then show that: (1 - x2) d2y/dx2 - x dy/dx = 0. (vi) Find the vector equation of the plane passing through the points A(2, 1, 1), B(0, 2, 3) and C(4, 5, 6). (vii) Find E(X) and V(X), where X is the number obtained on uppermost face, when a fair die is thrown. (viii) Are the four points A(1, 2, 1), B(2, -3, 4), C(3, 4, -5), D(2, 3, -2) coplanar? Justify your answer. (ix) In triangle ABC, prove that a(b cos C - c cos B) = b2 - c2. (x) The angle between the line r = (i + 2j + k) + lambda(i + j + k) and the plane r.(2i - j + k) = 8 is ____. (a) sin-1(sqrt(2)/3) (b) sin-1(sqrt(3)/2) (c) sin-1(1/2) (d) sin-1(1/sqrt(2)) (xi) Verify LMVT for the function f(x) = log x, on [1, e]. (xii) The probability distribution of X is as follows: x: 0, 1, 2, 3, 4; P[X=x]: 0.1, k, 2k, 2k, k. Find (i) k, (ii) P(X < 2), (iii) P[1 <= X < 4]. SECTION D Q. 27 to Q. 34. Attempt any FIVE of the following (each 4 marks): 20 (i) Solve the linear programming problem (L.P.P.) by graphical method. Minimize z = 8x + 10y Subject to 2x + y >= 7, 2x + 3y >= 15, x >= 0, y >= 2. (ii) If x = f(t) and y = g(t) are differentiable functions of t, so that y is differentiable function of x and dx/dt != 0 then prove that dy/dx = (dy/dt)/(dx/dt). Hence find dy/dx, if x = sin t, y = cos t. (iii) Prove that the homogeneous equation of degree two in x and y, ax2 + 2hxy + by2 = 0 represents a pair of lines passing through the origin if h2 - ab >= 0. (iv) integral from 1 to 2 of (1/x2) . e^(1/x) dx = ____. (a) sqrt(e) + 1 (b) sqrt(e) - 1 (c) sqrt(e)(sqrt(e) - 1) (d) (sqrt(e) - 1)/e (v) Prove that: integral from a to b of f(x) dx = integral from a to b of f(a + b - x) . dx. Hence, find integral from pi/6 to pi/3 of sin2 x . dx. (vi) A body cools according to Newton's law from 100 deg C to 60 deg C in 20 minutes. The temperature of the surrounding being 20 deg C. How long will it take to cool down to 40 deg C? (vii) If A = [[1, 2], [3, 4]], prove that A.(adj A) = (adj A).A = |A|.I. (viii) Evaluate: integral (3x2 + 4x - 5)/((x2 - 1)(x + 2)) . dx. This is a model / practice paper built by AglaSem from an analysis of past MSBSHSE board papers. It follows the current board pattern for practice purpose only.