\n| <\/td>\n | Maximum available time 10 hrs\/day<\/td>\n | Maximum available time 15 hrs\/day<\/td>\n | <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n i) Write the objective function.
\nii) Whether it is a maximization case or a minimization case. Justify.
\niii) Write the constraints.
\nAnswer:
\nLet x units = Machine G and y units = Machine H
\ni) Objective function: Z = 20x + 30y
\nii) It is a maximization problem.
\niii) Constraints are:
\n3x + 4y \u2264 10; 5x + 6y \u2264 15; x, y \u2265 0<\/p>\n Questions 8 to 17 carry 4 scores each. Answer any 8 questions. (8 \u00d7 4 = 32)<\/span><\/p>\nQuestion 8.
\ni) A function f : A \u2192 B, where A = {1, 2, 3} and B = {4, 5, 6} defined by f(1) = 5, f(2) = 6, f(3) = 4, Check whether f is a bijection. If it is a bijection, write f-1<\/sup> as set of ordered pair.
\nii) The operation table for an operation * is given below. Given that 1 is the identity element. Then which among the following is true regarding the element in the first column?<\/p>\n\n\n\n| *<\/td>\n | 1<\/td>\n | 2<\/td>\n | 3<\/td>\n<\/tr>\n | \n| 1<\/td>\n | <\/td>\n | 2<\/td>\n | 3<\/td>\n<\/tr>\n | \n| 2<\/td>\n | <\/td>\n | 2<\/td>\n | 3<\/td>\n<\/tr>\n | \n| 3<\/td>\n | <\/td>\n | 3<\/td>\n | 3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n a) 3, 2, 2
\nb) 1, 2, 3
\nc) 1, 1, 2
\nd) 2, 2, 2
\nCheck whether * is commutative.
\nAnswer:
\ni) A = {1, 2, 3}; B = {4, 5, 6}
\nf(1) = 5, f(2) = 6, f(3) = 4
\nHence f= {(1, 5),(2, 6),(3, 4)}
\nf-1<\/sup> = {(5, 1), (6, 2), (4, 3)}
\nii) b) 1, 2, 3
\nSince; 1*2 = 2 and 2*1 =2
\n3 * 2 = 3 and 2*3 = 3
\nTherefore * is commutative<\/p>\nQuestion 9.
\ni) If Sin-1<\/sup> x = y, then
\n(a) 0 \u2264 y \u2264 \u03c0
\nb) \\(-\\frac{\\pi}{2}\\) \u2264 y \u2264 \\(\\frac{\\pi}{2}\\)
\nc) 0 < y < \u03c0
\nd) \\(-\\frac{\\pi}{2}\\) \u2264 y \u2264 \\(\\frac{\\pi}{2}\\)
\nii) Find the principal value of sin-1<\/sup>\u00a0(\\(\\frac{1}{2}\\))
\niii) sin-1<\/sup>x = \\(\\frac{3}{4}\\) find the value of cos-1<\/sup> x
\nAnswer:
\ni) \\(-\\frac{\\pi}{2}\\) \u2264 y \u2264 \\(\\frac{\\pi}{2}\\)
\nii) sin-1<\/sup> \\(\\frac{1}{2}\\) = \\(\\frac{\\pi}{6}\\)
\niii) sin-1<\/sup>x + cos-1<\/sup>x = \\(\\frac{\\pi}{2}\\)
\n\\(\\frac{3}{4}\\) + cos-1<\/sup>x = \\(\\frac{\\pi}{2}\\) \u21d2 cos-1<\/sup>x = \\(\\frac{\\pi}{2}\\) – \\(\\frac{3}{4}\\)<\/p>\nQuestion 10.
\ni) Find the relation between \u2018a\u2019 and \u2018b\u2019 so that the function defined by
\nf(x) = \\(\\left\\{\\begin{array}{ll}
\na x+1, & x \\leq 3 \\\\
\nb x+3, & x>3
\n\\end{array}\\right.\\) is continuous at x = 3
\nii) \u201cAll continuous function are not differentiable.\u201d Justify your answer with an example.
\nAnswer:
\ni) Since f(x) is continuous at x = 3
\n\\(\\lim _{x \\rightarrow 3^{-}}\\) f(x) = \\(\\lim _{x \\rightarrow 3^{+}}\\) f(x) = f(3)
\n\\(\\lim _{x \\rightarrow 3}\\) (ax + 1) = \\(\\lim _{x \\rightarrow 3}\\) (bx + 3) = 3a +1
\n\u21d2 3a + 1 = 3b + 3 = 3a +1
\n\u21d2 3a + l = 3b + 3 \u21d2 3a – 3b – 2 = 0<\/p>\n ii) Consider the function f(x) = |x|.
\nLet us check the continuity and differentiability at x = 0.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-7.png\")
\nLeft derivative * Right derivative
\nTherefore not differentiable at x = 0.<\/p>\n Question 11.
\ni) Find the equation to the tangent to the curve y = x2<\/sup> – 2x + 7 at (2, 7)
\nii) Find the maximum value of the function?
\nAnswer:
\nf(x) = sin x + cos x, 0 < x < \\(\\frac{\\pi}{2}\\)
\ni) y = x2<\/sup> – 2x + 7 \u21d2 \\(\\frac{d y}{d x}\\) = 2x – 2
\nSlope at x = 2 is 2 \u00d7 2 – 2 – 2
\nEquation of the tangent is
\n(y – y1<\/sub>) = m(x – x1<\/sub>)
\n\u21d2 (y – 7) = 2(x- 2)
\n\u21d2 y – 7 = 2x – 4
\n\u21d2 2x – y + 3 = 0<\/p>\nii) c) f(x) = sin x + cos x
\nf'(x) = cos x – sin x
\nf”(x) = -sinx-cosx
\nFor turning points;
\nf'(x) = cos x – sin x = 0 \u21d2 tan x = 1
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-8.png\") <\/p>\n Question 12.
\nIntegrate \\(\\int \\frac{x+2}{2 x^{2}+6 x+5} d x\\)
\nAnswer:
\nPut x + 2 = A(4x + 6) + B
\n4A = 1 \u21d2 A = \\(\\frac{1}{4}\\)
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-9.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-10.png\") <\/p>\n Question 13.
\nConsider the differential equation
\nx \\(\\frac{d y}{d x}\\) + y = \\(\\frac{1}{x^{2}}\\)
\ni) Find the integrating factor.
\nii) Solve the above differential equation.
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-11.png\") <\/p>\n Question 14.
\nlf the vectors \\(\\overline{P Q}\\) = -3i + 4j + 4k and \\(\\overline{P R}\\) = -5i + 2j + 4k are the sides of a \u0394PQR
\ni) Find the angle between \\(\\overline{P Q}\\) and \\(\\overline{P R}\\)
\nii) Find the length of the median through the vertex P.
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-12.png\")
\nii)
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-13.png\")
\nWith respect to the initial P the position vector of Q and R will be \\(\\overline{P Q}\\) = -3i + 4j + 4k
\nand \\(\\overline{P R}\\) = -5i + 2j + 4k respectively.
\nSince M is the midpoint of QR. The position vector of M will be
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-14.png\") <\/p>\n Question 15.
\ni) If \\(\\bar{a}\\) = 5i – j – 3k and \\(\\bar{b}\\) = i + 3j + 5k, then show that the vectors \\(\\bar{a}+\\bar{b}, \\bar{a}-\\bar{b}\\) are perpendicular.
\nii) If \\(\\bar{a}\\) = i – 2j + 3k, \\(\\bar{b}\\) = 2i + 3j – 4k and
\n\\(\\bar{c}\\) = i – 3j + 5k, then check whether \\(\\bar{a}\\), \\(\\bar{b}\\), \\(\\bar{c}\\) are coplanar.
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-15.png\") <\/p>\n Question 16.
\ni) Find the Cartesian equation of the line passing through origin and (5, -2, 3) ii) The point P(x, y, z) lies in the first octant and its distance from the origin is 12 units. If the position vector of P makes angles 45\u00b0, 60\u00b0with x and y axes respectively, find coordinates of P.
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-16.png\")
\nSince P lies in the first octant, we take n = \\(\\frac{1}{2}\\)
\nTherefore the coordinate of P is
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-17.png\") <\/p>\n Question 17.
\nSolve graphically Maximise Z = 5x + 3y Subject to the constraints 3x + 5y \u2264 15; 5x + 2y \u2264 10; x \u2265 0, y \u2265 0.
\nAnswer:
\nIn the figure the shaded region OABC is the fesible region. Here the region is bounded.
\nThe corner points are
\nO(0, 0), A(2, 0), B(\\(\\frac{20}{19}, \\frac{45}{19})\\), C(0, 3)
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-18.png\")
\nGiven Z = 5x + 3y<\/p>\n \n\n\n| Corner points<\/td>\n | Value of Z<\/td>\n<\/tr>\n | \n| O<\/td>\n | Z = 0<\/td>\n<\/tr>\n | \n| A<\/td>\n | Z = 5(2) + 3(0) = 10<\/td>\n<\/tr>\n | \n| B<\/td>\n | Z = \\(5\\left(\\frac{20}{19}\\right)+3\\left(\\frac{45}{19}\\right)=\\frac{235}{19}\\)<\/td>\n<\/tr>\n | \n| C<\/td>\n | Z = 5(0) + 3(3) = 9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n Since maximum value of Z occurs at B, the soluion is z = \\(\\frac{235}{19}\\), (\\(\\frac{20}{19}\\), \\(\\frac{45}{19}\\))<\/p>\n Questions 18 to 24 carry 6 scores each. Answer any 5 questions. (5 \u00d7 6 = 30)<\/span><\/p>\nQuestion 18.
\nA = \\(\\left[\\begin{array}{ccc}
\n3 & 3 & -1 \\\\
\n-2 & -2 & 1 \\\\
\n-4 & -5 & 2
\n\\end{array}\\right]\\)
\na) Find AT<\/sup>
\nb) Express A as the sum of a symmetric and skew symmetric matrix.
\nii) If AT<\/sup> = \\(\\left[\\begin{array}{cc}
\n\\cos x & \\sin x \\\\
\n-\\sin x & \\cos x
\n\\end{array}\\right]\\), verift that AT<\/sup> A = I
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-19.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-20.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-21.png\") <\/p>\nQuestion 19.
\ni) Without expanding prove that
\n\\(\\left|\\begin{array}{ccc}
\nx+y & y+z & z+x \\\\
\nz & x & y \\\\
\n1 & 1 & 1
\n\\end{array}\\right|\\) = 0
\nii) Consider the following system of equations
\n2x – 3y + 5z = 11; 3x + 2y – 4z = -5; x + y – 2z = -3
\na) Express the system in Ax = B form.
\nb) Solve the system by matrix method.
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-22.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-23.png\") <\/p>\n Question 20.
\nFind \\(\\frac{d y}{d x}\\) of the following
\ni) x2<\/sup> + 2xy + 2y2<\/sup> = 1
\nii) yx<\/sup> = 2x<\/sup>
\niii) x = cos \u03b8; y = sin \u03b8 at \u03b8 = \\(\\frac{\\pi}{4}\\)
\nAnswer:
\ni) x2<\/sup> + 2xy + 2y2<\/sup> = 1
\nDifferentiating w.r.to x;
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-24.png\") <\/p>\nii) yx<\/sup> = 2x<\/sup>
\nTake log on both sides;
\nx log y = x log 2
\nDifferentiating w.r.to x;
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-25.png\") <\/p>\nQuestion 21.
\nEvaluate the following
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-26.png\")
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-27.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-28.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-29.png\") <\/p>\n Question 22.
\nConsider the parabolas y2<\/sup> = 4x, x2<\/sup> = 4y
\ni) Draw a rough figure for the above parabolas.
\nii) Find the point of intersection of the two parabolas.
\niii) Find the area bounded by these two parabolas.
\nAnswer:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-30.png\")
\nii) solving the two conics we get the point of intersection as (0, 0) and (4, 4).
\niii) Area of the enclosed region
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-31.png\") <\/p>\nQuestion 23.
\ni) Find the shortest distance between the lines whose vector equations are
\n\\(\\bar{r}\\) = (i + 2j + 3k) + \u03bb(i – 3j + 2k) and \\(\\bar{r}\\) =(4i + 5j + 6k) + \u03bc(i – 3j + 2k)
\nii) If a plane meets positive x axis at a distance of 2 units from the origin, positive y axis at a distance of 3 units from the origin and positive z axis at a distance of 4 units from the origin. Find the equation of the plane.
\niii) Find the perpendicular distance of (0, 0, 0) from the plane obtained in part (ii)
\nAnswer:
\ni) \\(\\overline{a_{1}}\\) = i + 2j + 3k; \\(\\bar{b}\\) = i – 3j + 2k
\n\\(\\overline{a_{2}}\\) = 4i + 5j + 6k Both lines are parallel.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-32.png\")
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-33.png\") <\/p>\n Question 24.
\ni) A die is thrown twice let the event A be ‘odd number on first throw’ and B be ‘odd number on the second throw’ check whether A and B are independent.
\nii) Coloured balls are distributed in three boxes as shown in the following table:<\/p>\n \n\n\n| Box<\/td>\n | Red<\/td>\n | Black<\/td>\n<\/tr>\n | \n| 1<\/td>\n | 2<\/td>\n | 0<\/td>\n<\/tr>\n | \n| II<\/td>\n | 0<\/td>\n | 2<\/td>\n<\/tr>\n | \n| III<\/td>\n | 1<\/td>\n | 1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n A box is selected at random and a ball is taken out. If the ball taken is of red colour, What is the probability that the other ball in the box is also of red colour?
\nAnswer:
\ni) n(S) = 36
\nA = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
\nB = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (1, 5), (2, 5), (3,5), (4, 5), (5, 5), (6, 5)}
\n(A \u2229 B) = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-34.png\")
\nTherefore A and B are independent.<\/p>\n ii) Let B1<\/sub>, B2<\/sub>, B3<\/sub> are the event of getting the boxes.
\nP(B1<\/sub>) = P(B2<\/sub>) = P(B3<\/sub>) = \\(\\frac{1}{3}\\)
\nLet E be the event of getting a red ball.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-35.png\") <\/p>\n | | | |
![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-1.png\")
<\/p>\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-2.png\")
<\/p>\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-3.png\")
![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-4.png\")
<\/p>\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-5.png\")
<\/p>\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/01\/Plus-Two-Maths-Model-Question-Papers-Paper-1-6.png\")
<\/p>\n