Mathematics Past Paper 311302 - Winter 2023 PDF

23124 311302 3 Hours / 70 Marks Seat No. Instructions : (1) All Questions are compulsory. (2) Answer each next main Question on a new page. (3) Illustr...

23124 311302 3 Hours / 70 Marks Seat No. Instructions : (1) All Questions are compulsory. (2) Answer each next main Question on a new page. (3) Illustrate your answers with neat sketches wherever necessary. (4) Figures to the right indicate full marks. (5) Use of Non-programmable Electronic Pocket Calculator is permissible. (6) Mobile Phone, Pager and any other Electronic Communication devices are not permissible in Examination Hall. Marks 1. Attempt any FIVE of the following : 10 (a) Find the value of x if, log5 (x2 – 5x + 11) = 1 (b) Find the value of sin (15°) using compound angles. (c) Find the intercepts of the line 2x + 3y = 6 on both the axes. (d) State whether the function is even or odd if, f(x) = x3 + 4x + sin x. (e) At which point on the curve y = 3x – x2 the slope of the tangent is –5 ? (f) Divide 100 into two parts such that their product is maximum. (g) If mean is 34.5 and standard deviation is 5, find the co-efficient of variance. [1 of 4] P.T.O. 311302 [2 of 4] 2. Attempt any THREE of the following : 12 3  1  1 2 (a) If A =  ,B=  3 0 , then 2 4    Find the matrix ‘X’ such that 2X + 3A – 4B = I, where I is identity matrix of order 2. 2 1   2 0 2  3 5  , whether AB is singular or non-singular matrix ? (b) If A =  ,B=  3 4 5    0 2  3x  2 (c) Resolve into partial fraction. ( x  2) ( x 2  4) 5 –4 (d) If A and B are obtuse angle and sin A = and cos B = , then find 13 5 sin (A + B). 3. Attempt any THREE of the following : 12 sin 3A – sin A (a) Prove that, = tan A cos 3A + cos A 1 3 1 8 1  84  (b) Prove that sin    sin    cos  . 5  17   85  (c) Find the equation of straight line passing through the point of intersection of lines 4x + 3y = 8 and x + y = 1 ; and parallel to the line 5x – 7y = 3. dy (d) Find , if x3 + xy2 = y3 + yx2. dx 4. Attempt any THREE of the following : 12 dy  (a) If x = a ( + sin ) & y = a (1 – cos ), find at  =. dx 2 dy (b) If y = (x)sin x + (tan x)x, find. dx 311302 [3 of 4] (c) Find the range and co-efficient of range for the following data : Class Interval 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 Frequency 15 25 13 17 10 (d) Calculate the mean deviation about mean of the following data : 17, 15, 18, 23, 25, 22, 11, 5 (e) The following data pertains to two workers doing the same job in a factory : Details Worker A Worker B Mean time of completing job 40 42 Standard deviation 8 6 Who is more consistent worker ? 5. Attempt any TWO of the following : 12 (a) Solve the following system of equations by matrix inversion method : x + y + z = 3, 3x – 2y + 3z = 4, 5x + 5y + z = 11 A 1 (b) (i) If tan   , find the value of cos A. 2 3 (ii) Evaluate without using calculator tan 85   tan 40  1  tan 85   tan 40  (c) (i) Find the distance between the parallel lines 3x + 2y = 5 and 3x + 2y = 6. (ii) Find the acute angle between the line, 3x = y – 4 and 2x + y + 3 = 0. 6. Attempt any TWO of the following : 12 (a) A manufacturer can sell ‘x’ items at a price of ` (330 – x) each. The cost of producing x items in ` (x2 + 10x + 12). Determine the number of items to be sold so that the manufacturer can make the maximum profit. P.T.O. 311302 [4 of 4] (b) A beam is bent in the form of curve y = 2 sinx – sin 2x. Find radius of  curvature of the beam at x =. 2 (c) Find mean, standard deviation and co-efficient of variance of the following data : Class Interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 Frequency 14 23 27 21 15 _______________