{"id":19155,"date":"2020-01-09T14:32:40","date_gmt":"2020-01-09T09:02:40","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=19155"},"modified":"2021-07-02T15:30:32","modified_gmt":"2021-07-02T10:00:32","slug":"2nd-puc-maths-model-question-paper-4","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-model-question-paper-4\/","title":{"rendered":"2nd PUC Maths Model Question Paper 4 with Answers"},"content":{"rendered":"

Students can Download 2nd PUC Maths Model Question Paper 4 with Answers, Karnataka 2nd PUC Maths Model Question Papers with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n

Karnataka 2nd PUC Maths Model Question Paper 4 with Answers<\/h2>\n

Time: 3 Hrs 15 Min
\nMax. Marks: 100<\/p>\n

Part-A<\/span><\/p>\n

Answer all the questions: (10 \u00d7 1 = 10)<\/span><\/p>\n

Question 1.
\nLet * be a operation defined on the set of rational numbers by a* b = \\(\\frac{\\mathrm{ab}}{4}\\) find the identify element.
\nAnswer:
\na*e = e*a=a
\n\"2nd
\ne = 4<\/p>\n

Question 2.
\nWrite the values of x for which 2 tan-1<\/sup> x = cos-1<\/sup>\\(\\frac{1-x^{2}}{1+x^{2}}\\) holds
\nAnswer:
\nx \u2265 1<\/p>\n

Question 3.
\nConstruct a 2 \u00d7 2 matrix A = [aij<\/sub>] whose elements are given by \\(\\frac{1}{2}|-3 i+j|\\)
\nAnswer:
\nA = \\(\\left[ \\begin{matrix} 1 & 1\/2 \\\\ 5\/2 & 2 \\end{matrix} \\right] \\)<\/p>\n

Question 4.
\nFind the values of x for which \\(\\left| \\begin{matrix} 3 & x \\\\ x & 1 \\end{matrix} \\right| \\) = \\(\\left| \\begin{matrix} 3 & 2 \\\\ 4 & 1 \\end{matrix} \\right|\\)
\nAnswer:
\n\"2nd
\nx2<\/sup> = 8
\nx = \u00b1 \\(\\sqrt{8}\\) = \u00b1 2\\(\\sqrt{2}\\)<\/p>\n

\"KSEEB<\/p>\n

Question 5.
\nFind \\(\\frac{d y}{d x}\\), if y = sin (x2<\/sup> + 5)
\nAnswer:
\n\\(\\frac{d y}{d x}\\) = cos(x2<\/sup> + 5).2x<\/p>\n

Question 6.
\nEvaluate : \\(\\int e^{x}\\left(\\frac{x-1}{x^{2}}\\right) d x\\)
\nAnswer:
\n\\(=\\int e^{x}\\left(\\frac{1}{x}-\\frac{1}{x^{2}}\\right) \\cdot d x=e^{x}\\left(\\frac{1}{x}\\right)+c\\)<\/p>\n

Question 7.
\nDefine negative of a vector.
\nAnswer:
\nNegative vector is a vector whose magnitude is same with opposite direction<\/p>\n

Question 8.
\nWrite the direction cosines of x – axis.
\nAnswer:
\n1, 0, 0<\/p>\n

Question 9.
\nDefine feasible region in LPP.
\nAnswer:
\nThe common region determined by all the constraints including non-negative constraints.<\/p>\n

Question 10.
\nIf P(A) = \\(\\frac { 3 }{ 5 }\\), P(B) = \\(\\frac { 1 }{ 5 }\\)
\nfind p(A \u2229 B) if A and B are independent events
\nAnswer:
\np(A \u2229 B) = p(A), p(B)
\np(A \u2229 B) = \\(3 \/ 5 \\cdot 1 \/ 5\\) = \\(\\frac { 3 }{ 25 }\\)<\/p>\n

Part-B<\/span><\/p>\n

Answer any ten questions: (10 \u00d7 2 = 20)<\/span><\/p>\n

Question 11.
\nShow that if f: A \u2192 B and g: B \u2192 C are one – one, then gof : A \u2192 C is also one – one
\nAnswer:
\ngof (x1<\/sub>) = gof(x2<\/sub>)
\ng[f(x1<\/sub>)]=g[(x2<\/sub>)]
\nf(x1<\/sub>) = f(x2<\/sub>)
\nx1<\/sub> = x2<\/sub>
\nsince g and f is one – one, gof is one – one<\/p>\n

Question 12.
\nShow that
\nsin-1<\/sup> \\((2 x \\sqrt{1-x^{2}})\\) = 2sin-1<\/sup> x for \\(\\frac{-1}{\\sqrt{2}} \\leq x \\leq \\frac{1}{\\sqrt{2}}\\)
\nAnswer:
\nLet x = sin \u03b8 \u21d2 \u03b8 = sin-1<\/sup> x
\n2x \\(\\sqrt{1-x^{2}}\\) = 2 sin \u03b8 \\(\\sqrt{1-\\sin ^{2}} \\theta\\)
\n2 sin \u03b8. cos \u03b8 = sin 2\u03b8
\n\u21d2 sin-1<\/sup> (2x\\(\\sqrt{1-x^{2}}\\))
\n= sin-1<\/sup>(sin 2\u03b8) = 2 sin-1<\/sup> x<\/p>\n

Question 13.
\nShow that
\n2 tan-1<\/sup> \\(\\frac { 1 }{ 2 }\\) + tan-1<\/sup> \\(\\frac { 1 }{ 7 }\\) = tan-1<\/sup> \\(\\frac { 31 }{ 17 }\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 14.
\nIf the area of the triangle with vertices (-2,0),(0,4) and (0,k) is 4 square units, find the values of k using determinants.
\nAnswer:
\n\"2nd
\n\u00b14 = \\(\\frac { 1 }{ 2 }\\) {-2 (4 – k) + (0) }
\n\u00b14 = K – 4
\n+4 = k – 4 -4 = k – 4
\nk = 4 + 4 k = 0
\nk = 8<\/p>\n

Question 15.
\nDifferentiate \\(\\left(x+\\frac{1}{x}\\right)^{x}\\) with respect to x,
\nAnswer:
\n\"2nd<\/p>\n

Question 16.
\nFind the slope of the tangent to the curve y = \\(\\frac{\\mathbf{x}-1}{\\mathbf{x}-2}\\) x \u2260 2 at x = 0
\nAnswer:
\n\"2nd<\/p>\n

Question 17.
\nFind \\(\\frac{d y}{d x}\\), if x2<\/sup> + xy + y2<\/sup> = 100
\nAnswer:
\n2x + x\\(\\frac{d y}{d x}\\) + y + 2y \\(\\frac{d y}{d x}\\) = 0
\n(x + 2y)\\(\\frac{d y}{d x}\\) = -(2x + y)
\n\\(\\frac{d y}{d x}=\\frac{-(2 x+y)}{x+2 y}\\)<\/p>\n

Question 18.
\nEvaluate :\\(\\int \\frac{\\cos 2 x-\\cos 2 \\alpha}{\\cos x-\\cos \\alpha} d x\\)
\nAnswer:
\n\\(=\\int \\frac{\\left(2 \\cos ^{2} x-1\\right)-\\left(2 \\cos ^{2} \\alpha-1\\right)}{\\cos x-\\cos \\alpha}, d x\\)
\n\\(=\\int 2 \\frac{(\\cos x-\\cos \\alpha)(\\cos x+\\cos \\alpha)}{\\cos x-\\cos \\alpha}, d x\\)
\n\\(=2 \\int \\cos x+\\cos \\alpha, d x\\)
\n= 2 sin x + 2x cos \u03b1 + c<\/p>\n

Question 19.
\nEvaluate \\(\\int \\frac{d x}{x-\\sqrt{x}}\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 20.
\nFind the order and degree, if defined of the differential equation
\n\\(\\left(\\frac{\\mathbf{d}^{2} \\mathbf{y}}{\\mathbf{d x}^{2}}\\right)^{3}+\\left(\\frac{\\mathbf{d y}}{\\mathbf{d x}}\\right)^{2}+\\sin \\frac{\\mathbf{d y}}{\\mathrm{dx}}+1=\\mathbf{0}\\)
\nAnswer:
\norder is 2
\ndegree is not defined<\/p>\n

\"KSEEB<\/p>\n

Question 21.
\nFind \\(|\\overrightarrow{\\mathrm{b}}|\\), if \\((\\vec{a}+\\vec{b}) \\cdot(\\vec{a}-\\vec{b})=8\\) and \\(|\\overrightarrow{\\mathrm{a}}|\\) = 8\\(|\\overrightarrow{\\mathrm{b}}|\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 22.
\nFind the area of the parallelogram whose adjacent sides are determined by the vectors \\(|\\overrightarrow{\\mathrm{a}}|\\) = \u00ee – \u0135 + 3k\u0302 and \\(|\\overrightarrow{\\mathrm{b}}|\\) = 2\u00ee – 7\u0135 + k\u0302
\nAnswer:
\n\"2nd<\/p>\n

Question 23.
\nFind the angle between the pair of lines given by
\n\\(|\\overrightarrow{\\mathrm{r}}|\\) = 3\u00ee + 2\u0135 – 4k\u0302 + \u03bb(\u00ee + 2\u0135 + 2k\u0302) and \\(|\\overrightarrow{\\mathrm{r}}|\\) = 5\u00ee – 2\u0135 + \u00b5(3\u00ee + 2\u0135 + 6k\u0302)
\nAnswer:
\n\"2nd<\/p>\n

Question 24.
\nLet X denote the number of hours you study during a randomly selected school day. The probability that X can take the values of x, has the following form, where k is some constant
\n\"2nd
\nfind the value of k.
\nAnswer:
\n\u03a3P(x) = 1
\n\"2nd
\nk = \\(\\frac { 1 }{ 5 }\\)<\/p>\n

Part-C<\/span><\/p>\n

Answer any ten questions: (10 \u00d7 3 = 30)<\/span><\/p>\n

Question 25.
\nDetermine whether the relation R in the set A = {l,2,3, ……… 13,14} defined as R = {(x,y) : 3x = y = o}, is reflexive, symmetric arid transitive.
\nAnswer:
\ny = 3x
\nR – {(1,3) (2,6) (3, 9) (4,12)}
\nR is not reflexive as (l,l) \u2209 R
\nR is not symmetric as
\n(1,3) \u2208 R but (3,1) \u2209 R
\nR is not transitive as
\n(1,3) \u2208 R, (3,9) \u2208R but (l,9)\u2209 R<\/p>\n

\"KSEEB<\/p>\n

Question 26.
\nIf tan-1<\/sup> \\(\\frac{x-1}{x-2}\\) + tan-1<\/sup> \\(\\frac{x+1}{x+2}\\) = \\(\\frac{\\pi}{4}\\) , then find the values of x.
\nAnswer:
\n\"2nd<\/p>\n

Question 27.
\nIf A and B are invertible matrices of the same order, then prove that (AB)-1<\/sup> = B-1<\/sup> A-1<\/sup>.
\nAnswer:
\nFrom definition of inverse of matrix
\n(AB) (AB)-1<\/sup> =1
\nA-1<\/sup> (AB)(AB)-1<\/sup> = A-1<\/sup>l
\n(A-1<\/sup>A) B(AB)-1<\/sup>=A-1<\/sup>.
\nIB (AB)-1<\/sup> = A-1<\/sup>
\nB(AB)-1<\/sup> = A-1<\/sup>
\nB-1<\/sup>B(AB)-1<\/sup> = B-1<\/sup> A-1<\/sup>
\nl(AB)-1<\/sup> = B-1<\/sup> A-1<\/sup>
\n(AB)-1<\/sup> = B-1<\/sup>A-1<\/sup><\/p>\n

Question 28.
\nverify Rolles theorem for the funciton f(x)= x2<\/sup> + 2X – 8, x \u2208[-4,2]
\nAnswer:
\nf (x) =x2<\/sup> +2x – 8
\n1. f (x) being a polynominal function it is continuous on (-4,2)
\n2. Also f1<\/sup> (x) = 2x + 2 hence derivable on (-4,2)
\n3. f(-4) = (-4)2<\/sup> + 2(-4) -8 = 0
\nf(2) = 22<\/sup> + 2(2) – 8 =0
\n\u2234 f(-4) = f(2)
\n\u2234 All three conditions of Rolle\u2019s theorem are verified.
\nf1<\/sup> (c) = 0
\nf1<\/sup> (c) = 2c + 2 \u21d2 0 = 2c + 2
\n\u21d2 c = -1 \u2208(-4,2)<\/p>\n

Question 29.
\nIf x = \\(\\sqrt{\\mathbf{a}^{\\sin ^{-1} \\mathbf{t}}}\\) and \\(\\sqrt{\\mathbf{a}^{\\cos ^{-1} \\mathbf{t}}}\\) then prove that \\(\\frac{d y}{d x}=\\frac{-y}{x}\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 30.
\nFind the two positive numbers whose sum is 15 and sum of whose squares is minimum.
\nAnswer:
\nx + y = 15 \u21d2 y = 15 – x
\ns = x2<\/sup> + y2<\/sup> \u21d2 s = x2<\/sup> + (15 – x)2<\/sup>
\n\\(\\frac{d s}{d x}\\) = 2x + 2(15 – x) (-1)
\n\\(\\frac{d s}{d x}\\) = 0 2x – 30 + 2x
\n0 = 4x – 30
\nx = \\(\\frac { 30 }{ 4 }\\) = 7.5
\n\\(\\frac{d^{2} s}{d x^{2}}=2+2=4>0\\)
\n\u2234 x = 7.5 require no is y = 7.5 & x = 7.5<\/p>\n

\"KSEEB<\/p>\n

Question 31.
\nEvaluate : \\(\\int x \\tan ^{-1} x d x\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 32.
\nEvaluate \\(\\int_{0}^{2} e^{x}\\) dx as a limit of a sum
\nAnswer:
\n\\(\\int_{0}^{2} e^{x}\\) .dx
\n= \\(\\left.e^{x}\\right]_{0}^{2}\\) = e2<\/sup> – e0<\/sup> = e2<\/sup> – 1<\/p>\n

Question 33.
\nFind the area of the region bounded by the curve y2<\/sup> = 4x and the line x = 3
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 34.
\nShow that the position vector of the point P, which divides the line joining the points A and B having position vectors \\(\\overrightarrow{\\mathrm{a}}\\) and \\(\\overrightarrow{\\mathrm{b}}\\) internally in the ratio m : n is \\(\\frac{m \\vec{b}+n \\vec{a}}{m+n}\\)
\n\"2nd
\nAnswer:
\n\u2018O\u2019 is the fixed point, \\(\\overrightarrow{\\mathrm{OA}}=\\overrightarrow{\\mathrm{a}}\\)
\n\\(\\overrightarrow{\\mathrm{OA}}=\\overrightarrow{\\mathrm{b}}\\) P divides the line AB internally in the ratio m : n
\n\"2nd
\n\"2nd<\/p>\n

Question 35.
\nShow that the four points with position vectors 4\u00ee + 8\u0135 + 12k\u0302, 2\u00ee + 4\u0135 + 6k\u0302, 3\u00ee + 5\u0135 + 4k\u0302 and 5\u00ee + 8\u0135 + 5k\u0302 are coplanar.
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 36.
\nFind the equation of the plane passing through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z + 2=0 and the point (2,2,1)
\nAnswer:
\nEquation of plane passing through intersection of the plane is
\n(3x – y + 2z – 4) + \u03bb (x + y + z + 2) = 0 This plane pass through (2,2,1)
\n3(2) – 2 + 2(1) -4 + \u03bb(2 + 2 + 1 + 2)=0
\n2 + 7\u03bb = 0
\n\u03bb = \\(\\frac { -2 }{ 7 }\\)
\nRequired equqtion of plane is
\n(3x – y + 2z – 4) + \\(\\left(\\frac{-2}{7}\\right)\\) (x + yz + 2) + 0 [xly<\/sup> by 7]
\n21x – 7y + 14z – 28 – 2x – 2y – 2z – 4 = 0
\n19x – 9y + 12z = 32<\/p>\n

Question 37.
\nForm the differential equation of the circles touching the x – axis at origin.
\nAnswer:
\n(x – h)2<\/sup> +(y – k)2<\/sup> = r2<\/sup>
\n(x – 0)2<\/sup> +(y – k)2<\/sup> =a2<\/sup>
\nx2<\/sup> + y2<\/sup> = 2ay \u21d2 2a = \\(\\frac{x^{2}+y^{2}}{y}\\)
\ndiff Equ w.r. to x
\n\"2nd<\/p>\n

Question 38.
\nAn insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01,0.03 and 0.15 respectively. One of the insured person meets with an accidents. What is the probability that he is a scooter driver?
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Part-D<\/span><\/p>\n

Answer any six questions: (6 \u00d7 5 = 30)<\/span><\/p>\n

Question 39.
\nLet R+<\/sub> be the set of all non- negative real numbers. Show that the function f: R+<\/sub> \u2192 [4,\u221e)given by f(x) = x2<\/sup> +4 is invertible and write the inverse of f.
\nAnswer:
\nf(x1<\/sub>) = f(x2<\/sub>)
\nX1<\/sub>2<\/sup> + 4 = X2<\/sub>2<\/sup> + 4
\nx1<\/sub> = x2<\/sub> \u2234 f (x) is one-one
\nLet y = f(x)
\ny = 4x +3
\nx = \\(\\frac{y-3}{4}\\) \u2208R+<\/sub> \u2234 f (x) is onto
\nLet
\nf-1<\/sup> (x) = y
\nf(y) = x
\n4y +3 =x
\n\\(y=\\frac{x-3}{4}\\)
\n\u2234 f-1<\/sup> : [4,8) \u2192 R+<\/sub> is defined by
\nf-1<\/sup> (x) = \\(\\frac{x-3}{4}\\)<\/p>\n

Question 40.
\nIf A = \\(\\left[ \\begin{matrix} 1 & 2 & 3 \\\\ 3 & -2 & 1 \\\\ 4 & 2 & 1 \\end{matrix} \\right]\\) , then show that A3<\/sup> -23A -40I = 0
\nAnswer:
\n\"2nd<\/p>\n

Question 41.
\nSolve by Matrix method :
\n2x + 3y + 3z = 5
\nx – 2y + z = 5
\n3x – y – 2z = 3
\nAnswer:
\n\"2nd<\/p>\n

Question 42.
\nIf y = Aemx<\/sup> + Benx<\/sup> show that \\(\\frac{d^{2} y}{d x^{2}}\\) -(m + n) \\(\\frac{d y}{d x}\\) + mny = 0
\nAnswer:
\n\\(\\frac{d y}{d x}\\) = Ameex<\/sup> + Bnexn<\/sup>
\n\\(\\frac{d^{2} y}{d x^{2}}\\) = Am2<\/sup>emx<\/sup> + Bn2<\/sup>enx<\/sup>
\n\\(\\frac{d^{2} y}{d x^{2}}\\) – (m + n) \\(\\frac{d y}{d x}\\) + my
\n= Am2<\/sup>emx<\/sup>+ Bn2<\/sup>enx<\/sup> – m2<\/sup>Aemx<\/sup> – Amnemx<\/sup> – Bn2<\/sup>enx<\/sup> – mnBenn<\/sup> + Amnemx<\/sup> + mnBenx<\/sup> = 0<\/p>\n

\"KSEEB<\/p>\n

Question 43.
\nA particle moves along the curve 6y = x3<\/sup> + 2. Find the points on the curve at which the y – coordinate is changing 8 times as fast as the x – coordinate.
\nAnswer:
\n6y = x3<\/sup> + 2 \u2192 (l)
\n\\(\\frac{d y}{d x}=\\frac{x^{2}}{2}\\)
\nGiven that \\(\\frac{d y}{d t}\\) = 8 \\(\\frac{d x}{d t}\\) \u21d2 \\(\\frac{d x}{d t}\\) = 8
\n\u2234 8 = \\(\\frac { 1 }{ 2 }\\) x2<\/sup>
\nx = \u00b14
\nput x in Equation (01) 6y = 43<\/sup> + 2
\n\u2234 x = 4 y = 11
\n\u2234 x = -4 , 6y = (-4)3<\/sup> + 2 \u21d2 y = -10,3 points are (4,11) (-4, -10,3)<\/p>\n

Question 44.
\nFind the integral of \\(\\frac{1}{\\sqrt{x^{2}-a^{2}}}\\) with respect to x and hence evaluate \\(\\frac{d x}{\\sqrt{x^{2}+6 x-7}}\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 45.
\nUsing integration find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4
\n\"2nd
\nAnswer:
\ny = 2x – 1
\npoints are (0,1) (1,3) (4,9) etc
\nx = 4, (4,0) (4,1) (4,2) etc
\nArea = \\(\\int_{0}^{4}(3 x+1), d x-\\int_{0}^{4}(2 x+1), d x\\)
\n\\(A=\\left(\\frac{3 x^{2}}{2}+x\\right)^{4}-\\left(\\frac{2 x^{2}}{2}+x\\right)^{4}\\)
\nA = 24 + 4 – 16 – 4 = 8 sq units<\/p>\n

Question 46.
\nSolve the differential equation \\(\\frac{d y}{d x}+y\\) sec x = tan x,0 \u2264 x < \\(\\frac{\\pi}{2}\\)
\nAnswer:
\nP = secx Q = tan x
\nIF = \\(e^{\\int \\sec x d x}=e^{\\log (\\sec x+\\tan x)}\\) = secx + tanx
\ny x IF = \u222b Q x IF
\ny (sec + tan x) = \u222b(sec x + tan x) tan x.dx = \u222bsec x.tan x + \u222b tan2<\/sup> x.dx
\n=secx + \u222b(sec2<\/sup> x – 1) dx
\ny(sec x+tan x) = sec x + tan x – x + c<\/p>\n

\"KSEEB<\/p>\n

Question 47.
\nDerive the equation of the line in space passing through a point and parallel to a vecor both in vector and Cartesian form.
\nAnswer:
\n\"2nd
\nLet \u2018A\u2019 be the point, \\(\\overrightarrow{\\mathrm{a}}\\) is P V of \\(\\overrightarrow{\\mathrm{A}}\\)
\n\\(\\overrightarrow{\\mathrm{AP}}=\\overrightarrow{\\mathrm{OP}}-\\overrightarrow{\\mathrm{OA}}=\\overrightarrow{\\mathrm{r}}-\\overrightarrow{\\mathrm{a}}\\)
\nAP is parallel to vector \\(\\overrightarrow{\\mathrm{b}}\\)
\n\\(\\overrightarrow{\\mathrm{AP}}=\\lambda \\overrightarrow{\\mathrm{b}}\\)
\n\\(\\overrightarrow{\\mathrm{r}}-\\overrightarrow{\\mathrm{a}}=\\lambda \\overrightarrow{\\mathrm{b}}\\)
\n\\(\\overrightarrow{\\mathrm{r}}=\\overrightarrow{\\mathrm{a}}+\\lambda \\overrightarrow{\\mathrm{b}}\\) is vector form
\nA=(x1<\/sub> y1<\/sub>, z1<\/sub>) p (x,y,z) \\(\\overrightarrow{\\mathrm{b}}\\) =b1<\/sub>i+b2<\/sub>j+b3<\/sub>k
\nNow \\(\\overrightarrow{\\mathrm{r}}\\) = xi + yj + zk, \\(\\overrightarrow{\\mathrm{a}}\\) =x1<\/sub>i + y1<\/sub>j + z1<\/sub>k
\n\\(\\overrightarrow{\\mathrm{r}}=\\overrightarrow{\\mathrm{a}}+\\lambda \\overrightarrow{\\mathrm{b}}\\) takes the form
\nxi + yj + zk=(x1<\/sub>i + y1<\/sub>j + z1<\/sub>k) + \u03bb(b1<\/sub>i + b2<\/sub>j + b3<\/sub>k)
\nxi + yj + zk=(x1<\/sub> +\u03bbb1<\/sub>)i+(y1<\/sub> +\u03bbb2<\/sub>) j + (z1<\/sub>+ \u03bbb3<\/sub>)k
\nx =X1<\/sub> +\u03bbb1<\/sub> y= y1<\/sub> + \u03bbb2<\/sub> z = z1<\/sub> + \u03bbb3<\/sub>
\n\"2nd<\/p>\n

Question 48.
\nA die is thrown 6 times. If \u2018getting an odd number\u2019is a success, what is probability of:
\na) 5 successes?
\nAnswer:
\n\\(P(5)=6_{c_{3}}\\left(\\frac{1}{2}\\right)^{6-5}\\left(\\frac{1}{2}\\right)^{5}=6\\left(\\frac{1}{2}\\right)\\left(\\frac{1}{32}\\right)=\\frac{3}{32}\\)<\/p>\n

b) at least 5 successes?
\nAnswer:
\n\\(P(x \\geq 5)=P(5)+P(6)=\\left(\\frac{1}{2}\\right)^{6}\\left[6_{C_{5}}+6_{C_{6}}\\right]=\\frac{6}{64}+\\frac{1}{64}=\\frac{7}{64}\\)<\/p>\n

c) at most successes?
\nAnswer:
\nP(X\u2264 5) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) = \\(\\left(\\frac{1}{2}\\right)^{6}\\) [1 + 6 + 15 + 20 + 15 + 6] = \\(\\frac { 63 }{ 64 }\\)<\/p>\n

\"KSEEB<\/p>\n

Part-E<\/span><\/p>\n

Answer any one question : (1 \u00d7 10 = 10)<\/span><\/p>\n

Question 49.
\na) Prove that
\n\"2nd
\nand hence evaluate \\(\\int_{-1}^{1} \\sin ^{5} x \\cos ^{4} x d x\\)
\nAnswer:
\n\"2nd
\nf(x) = sin5<\/sup>x.cos4<\/sup>x
\nf(-x) = sin5<\/sup>(-x) cos4<\/sup>(-x) = -sin5<\/sup>x.cos4<\/sup>x = -f(x)
\n\u2234 f is odd function, I = 0<\/p>\n

b) prove that \\(\\left| \\begin{matrix} { a }^{ 2 }+1 & ab & ac \\\\ ab & { b }^{ 2 }+1 & bc \\\\ ca & cb & { c }^{ 2 }+1 \\end{matrix} \\right| \\) = 1 + a2<\/sup> + b2<\/sup> + c2<\/sup>
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 50.
\na) A manufacturing company makes two models A and B of a product. Each piece Of model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 labour hours for finishing and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8,000 on each piece of model A and Rs. 12,000 on each piece of model B. How many pieces of model A and model B should be manufactured per week to realize a maximum profit? What is the maximum profit per week?
\nAnswer:
\n\"2nd
\nZ = 8000x + 12000y
\nax + 12y \u2264 180 \u21d2 180 \u21d2 3x + 4y \u2264 160
\nx + 3y \u2264 30
\n\"2nd<\/p>\n

b)
\nFind the value of K so that the function
\n\"2nd
\ncontinuous function.
\nAnswer:
\n\"2nd<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can Download 2nd PUC Maths Model Question Paper 4 with Answers, Karnataka 2nd PUC Maths Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 2nd PUC Maths Model Question Paper 4 with Answers Time: 3 Hrs 15 Min Max. Marks: …<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[82],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/19155"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=19155"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/19155\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=19155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=19155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=19155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}