**Answer is not given due to the change in present syllabus<\/span><\/span><\/p>\n Section -A<\/strong><\/p>\n Question 1. Question 2. Question 3. Question 4. Question 5. Question 6. Question 7. Question 8. Question 9. Question 10. Section – B<\/strong><\/p>\n Question 11. Question 12. Question 13. Question 14. Question 15. Question 16. Question 17. Question 18. Question 19. Question 20. Question 21. Question 22. Section – C<\/strong><\/p>\n Question 23. Question 24. Question 25. Question 26. Question 27. Question 28. Question 29. Note: Except for the following questions, all the remaining questions have been asked in previous set.<\/p>\n Section – A<\/strong><\/p>\n Question 10. Section – B<\/strong><\/p>\n Question 19. Question 20. Question 21. Question 22. Section – C<\/strong><\/p>\n Question 28. Question 29. Note: Except for the following questions, all the remaining questions have been asked in previous sets.<\/p>\n Section – A<\/strong><\/p>\n Question 9. Question 10.CBSE Previous Year Question Papers Class 12 Maths 2012 Outside Delhi Set I<\/h3>\n
\nThe binary operation *: R \u00d7 R \u2192 R is defined as a * b = 2a + b. Find (2 * 3) * 4.** [1]<\/p>\n
\nFind the principal value of tan-1<\/sup>\\( \\sqrt{{3}} \\) – sec-1<\/sup> (-2) [1]
\nSolution:
\ntan-1<\/sup>\\( \\sqrt{{3}} \\) – sec-1<\/sup> (-2) …(i)
\nWe know that the range of principal value of
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-1.png?resize=373%2C311&ssl=1\")
<\/p>\n
\nFind the value of x+y from the following equation:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-2.png?resize=253%2C59&ssl=1\")
[1]
\nSolution:
\nGiven that,
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-3.png?resize=305%2C382&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-4.png?resize=306%2C150&ssl=1\")
<\/p>\n
\nIf AT<\/sup> = \\(\\left[\\begin{array}{rr}{3} & {4} \\\\ {-1} & {2} \\\\ {0} & {1}\\end{array}\\right]\\) and B = \\(\\left[\\begin{array}{rrr}{-1} & {2} & {1} \\\\ {1} & {2} & {3}\\end{array}\\right]\\), then find AT<\/sup> = BT<\/sup>. [1]
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-5.png?resize=312%2C445&ssl=1\")
<\/p>\n
\nLet A ba a square matrix of order 3 \u00d7 3. Write the value of |2A|, where |A| = 4. [1]
\nSolution:
\nIn a square matrix of order 3 \u00d7 3,
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-6.png?resize=234%2C94&ssl=1\")
<\/p>\n
\nEvaluate:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-7.png?resize=96%2C57&ssl=1\")
[1]
\nSolution:
\nWe know that,
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-8.png?resize=297%2C55&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-9.png?resize=340%2C250&ssl=1\")
<\/p>\n
\nGiven \\(\\int e^{x}(\\tan x+1) \\sec x d x=e^{x} f(x)+c\\). Write f(x) satisfying the above. [1]
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-10.png?resize=369%2C253&ssl=1\")
<\/p>\n
\nWrite the value of \\((\\hat{i} \\times \\hat{j}) \\cdot \\hat{k}+\\hat{i} \\cdot \\hat{j}\\) [1]
\nSolution:
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-11.png?resize=292%2C113&ssl=1\")
<\/p>\n
\nFind the scalar components of the vector \\(\\overrightarrow{\\mathbf{A B}}\\) with initial point A (2, 1) and terminal point B (- 5, 7). [1]
\nSolution:
\nPosition vector of A
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-12.png?resize=342%2C211&ssl=1\")
<\/p>\n
\nFind the distance of the plane 3x – 4y + 12z = 3 from the origin. [1]
\nSolution:
\nWe know that the distance of the point (x1<\/sub>, y1<\/sub>, z1<\/sub>) from the plane ax + by + cz + d = 0 is
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-13.png?resize=173%2C229&ssl=1\")
<\/p>\n
\nProve the following:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-14.png?resize=225%2C51&ssl=1\")
[4]
\nSolution:
\nTaking L. H. S
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-15.png?resize=365%2C537&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-16.png?resize=320%2C199&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-17.png?resize=320%2C51&ssl=1\")
<\/p>\n
\nUsing properties of determinants, show that
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-18.png?resize=194%2C80&ssl=1\")
[4]
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-19.png?resize=372%2C588&ssl=1\")
<\/p>\n
\nShow that f : N \u2192 N, given by
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-20.png?resize=190%2C55&ssl=1\")
\nis both one-one and onto. [4]
\nSolution:
\nLet x, y \u03f5 R such that f(x) = f(y)
\nm=m
\nIf x and y are odd, then
\n\u2234 f(x) = f(y)
\n\u21d2 x + 1 = y + 1
\n\u21d2 x = y
\nIf x and y are even, then
\nf(x) = f(y)
\n\u21d2 x – 1 = y – 1
\n\u21d2 x = y
\nf(x) = x + 1 is even and f(y) = y + 1 is odd.
\n\u2234 x \u2260 y \u21d2 f(x) \u2260 f(y)
\nSimilarly if x is even and y is odd, then
\nx \u2260 y \u21d2 f(x) \u2260 f(y)
\nHence, f : N \u2192 N is one-one .
\nAlso, f(1) = 1 + 1 = 2
\nf(1) = 2 (\u2235 is odd)
\nIf x is odd number, then 3 an even natural number, x + 1 \u03f5 N such that,
\nf(x + 1) = x + 1 – 1
\n= x
\nIf x is even number, then there exist a odd natural number x – 1 \u03f5 N such that,
\nf(x – 1) = x – 1 + 1
\n= x
\nHence for every y \u03f5 N \u018e x \u03f5 N such that f(x) = y,
\nso f is onto.
\nHence f is both one-one and onto.
\nHence Proved.
\nOR
\nConsider the binary operations * : R \u00d7 R \u2192 R and o : R \u00d7 R\u2192R defined as a*b= | a-b | and aob=a for all a, b \u03f5 R. Show that is commutative but not associative, ‘o’ is associative but not commutative.**<\/p>\n
\nIf x= \\(x=\\sqrt{a^{\\sin ^{-1} t}}, y=\\sqrt{a^{\\cos ^{-1} t}}, \\text { show that } \\frac{d y}{d x}=-\\frac{y}{x}\\). [4]
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-21.png?resize=371%2C517&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-22.png?resize=244%2C172&ssl=1\")
\nOR
\nDifferentiate \\(\\tan ^{-1}\\left[\\frac{\\sqrt{1+x^{2}}-1}{x}\\right]\\) with respect to x.
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-23.png?resize=368%2C520&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-24.png?resize=279%2C160&ssl=1\")
<\/p>\n
\nIf x = a(cos t + t sin t) and y = a(sin t – t cos t), 0 < t < \\(\\frac{\\pi}{2}\\), find \\(\\frac{d^{2} x}{d t^{2}}, \\frac{d^{2} y}{d t^{2}} \\text { and } \\frac{d^{2} y}{d x^{2}}\\). [4]
\nSolution:
\nGiven, x = a(cos t + t sin t)
\nOn differentiating w.r.t. t, we get
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-25.png?resize=230%2C46&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-26.png?resize=373%2C420&ssl=1\")
<\/p>\n
\nA ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm\/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ? [4]
\nSolution:
\nLet AC = 5 m be the ladder and y be the height of the wall at which the ladder touches. Also, let the foot of the ladder be at C whose distance from the wall is x m
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-27.png?resize=376%2C500&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-28.png?resize=371%2C269&ssl=1\")
<\/p>\n
\nEvaluate: [4]
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-29.png?resize=104%2C57&ssl=1\")
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-30.png?resize=399%2C461&ssl=1\")
\nOR
\nEvaluate:
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-31.png?resize=108%2C53&ssl=1\")
\nSolution:
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-32.png?resize=371%2C258&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-33.png?resize=372%2C422&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-34.png?resize=361%2C303&ssl=1\")
<\/p>\n
\nForm the differential equation of the family of circles in the second quadrant and touching the coordinate axes. [4]
\nSolution:
\nThe equation of circle in second quadrant which touches the coordinate axis is
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-35.png?resize=367%2C310&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-36.png?resize=368%2C495&ssl=1\")
\nOR
\nFind the particular solution of the differential equation x(x2<\/sup> – 1)\\(\\frac{d y}{d x}\\) = 1; y = 0 when x = 2.
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-37.png?resize=371%2C550&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-38.png?resize=373%2C529&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-39.png?resize=302%2C328&ssl=1\")
<\/p>\n
\nSolve the following differential equation :
\n(1 + x2<\/sup>) dy + 2xy dx = cot x dx; x \u2260 0. [4]
\nSolution:
\n(1 + x2<\/sup>) dy + 2xy dx = cot x dx
\nDividing with (1 + x2<\/sup>) dx on both sides, we get
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-40.png?resize=252%2C176&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-41.png?resize=335%2C252&ssl=1\")
<\/p>\n
\nLet \\(\\vec{a}=\\hat{i}+4 \\hat{j}+2 \\hat{k}, \\vec{b}=3 \\hat{i}-2 \\hat{j}+7 \\hat{k} \\text { and } \\vec{c}=2 \\hat{i}-\\hat{j}+4 \\hat{k}\\) Find a vector \\(\\vec{p}\\) which is perpendicular to both \\(\\vec{a} \\text { and } \\vec{b}\\) and \\(\\vec{p} \\cdot \\vec{c}\\) = 18. [4]
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-42.png?resize=371%2C526&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-43.png?resize=208%2C78&ssl=1\")
<\/p>\n
\nFind the coordinates of the point where the line through the points A = (3, 4, 1) and B = (5, 1, 6) crosses the XY-plane. [4]
\nSolution:
\nGiven, A = (3, 4, 1), B = (5, 1, 6)
\nThe equation of the line passing through above
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-44.png?resize=286%2C561&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-45.png?resize=369%2C92&ssl=1\")
<\/p>\n
\nTwo cards are drawn simultaneously (without replacement) from a well-shuffled pack of 52 cards. Find the mean and variance of the number of red cards. [4]
\nSolution:
\nLet P (A) = Probability of getting one red card.
\nNumber of red cards = 26
\nLet X be the random variable which can take values 0, 1, 2 where X is the number of red cards selected
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-46.png?resize=361%2C281&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-47.png?resize=343%2C258&ssl=1\")
<\/p>\n
\nUsing matrices, solve the following system of equations:
\n2x + 3y + 3z = 5, x – 2y + z = – 4, 3x – y – 2z = 3. [6]
\nSolution:
\nGiven, 2x + 3y + 3z = 5,
\nx – 2y + z = – 4,
\n3x – y – 2z = 3
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-48.png?resize=339%2C513&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-49.png?resize=231%2C167&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-50.png?resize=233%2C110&ssl=1\")
<\/p>\n
\nProve that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the
\ncone. [6]
\nSolution:
\nLet r = radius of cylinder
\nR = radius of cone
\nh = height of cylinder
\nH = height of cone
\nCurved surface area of cylinder = 2\u03c0rh
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-51.png?resize=279%2C541&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-52.png?resize=369%2C303&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-53.png?resize=367%2C170&ssl=1\")
\nOR
\nAn open box with a square base is to be made out of a given quantity of cardboard of area c2<\/sup> square units. Show that the maximum volume of the box is \\(\\frac{c^{3}}{6 \\sqrt{3}}\\) cubic units.
\nSolution:
\nLet l = x, b = x, h = y
\n\u2234 Area of open box = Area of cardboard
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-54.png?resize=372%2C523&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-55.png?resize=310%2C241&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-104.png?resize=272%2C87&ssl=1\")
<\/p>\n
\nEvaluate: [6]
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-57.png?resize=107%2C56&ssl=1\")
\nSolution:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-58.png?resize=364%2C398&ssl=1\")
\nOR
\nEvaluate:
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-59.png?resize=137%2C53&ssl=1\")
\nSolution:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-60.png?resize=368%2C531&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-61.png?resize=305%2C52&ssl=1\")
<\/p>\n
\nFind the area of the region {(x, y) : x2<\/sup> + y2<\/sup> \u2264 4, x + y \u2265 2}. [6]
\nSolution:
\nWe have two equations
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-62.png?resize=298%2C270&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-63.png?resize=364%2C352&ssl=1\")
<\/p>\n
\nIf the lines \\(\\frac{x-1}{-3}=\\frac{y-2}{-2 k}=\\frac{z-3}{2} \\text { and } \\frac{x-1}{k}=\\frac{y-2}{1}\\) \\(=\\frac{z-3}{5}\\) are perpendicular, find the value of k and hence find the equation of plane containing these lines. [6]
\nSolution:
\nThe equation of the given lines are
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-64.png?resize=317%2C204&ssl=1\")
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-65.png?resize=372%2C417&ssl=1\")
<\/p>\n
\nSuppose a girl throws a die. If she gets a 5 or 6, she tosses a coin 3 times and notes the number of heads. If she gets 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die? [6]
\nSolution:
\nThe outcome of an experiment can be represented as
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-66.png?resize=386%2C53&ssl=1\")
\nIf she gets 1, 2, 3 or 4, sample space will be (1H), (2H), (3H), (4H), (1T), (2T), (3T), (4T)
\nIf she gets 5 or 6, sample space will be
\n(5 HHH), (5 HTT), (5 HHT), (5 HTH), (5 THT), (5 TTH), (5 THH), (5 TTT), (6 HHH), (6 HTT), (6 HHT), (6 HTH), (6 THT), (6 TTH), (6 THH), (6 TTT)
\nLet
\nA = Getting 1, 2, 3 or 4 on die
\nB = Getting exactly 1 Head
\nA = (1H), (2H), (3H), (4H), (1T), (2T), (3T), (4T)
\nB = (1H), (2H), (3H), (4H), (5HTT), (5THT), (5TTH), (6HTT), (6TTH), (6THT)
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-67.png?resize=354%2C136&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-68.png?resize=377%2C561&ssl=1\")
<\/p>\n
\nA dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units\/kg of vitamin A and 1 unit\/kg of vitamin C while food II contains 1 unit\/kg of vitamin A and 2 units\/kg of vitamin 1 unit\/kg of vitamin C. It costs \u20b9 5 per kg to purchase food I and \u20b9 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically. [6]
\nSolution:
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-69.png?resize=362%2C109&ssl=1\")
\nLet the amount of food I = x kg
\nLet the amount of food II = y kg
\nIf Z denotes the total cost
\nTo minimize the cost we have to minimize Z.
\nSubject to the constraints,
\n2x + y \u2265 8
\nx + 2y \u2265 10
\nx \u2265 0
\ny \u2265 0
\nMinimize Z = 5x + 7y
\nFirst we draw the lines AB and CD whose equations are
\n2x + y = 8 ..(ii)
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-70.png?resize=390%2C543&ssl=1\")
\nThe feasible region is shaded in the figure. The lines are intersecting at the point E (2, 4).
\n\u2234 The vertices of the feasible region are A (0, 8), E (2, 4) and D (10, 0).
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-71.png?resize=362%2C101&ssl=1\")
\nSince the feasible region is unbounded 38 may or may not be minimum value of total cost for this we draw graph of inequality.
\n5x + 7y < 38
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-72.png?resize=249%2C65&ssl=1\")
\nClearly graph of L has no common point with the feasible region.
\n\u2234\u00a0 The minimum value of Z is 38 at the point E (2, 4). Hence, the amount of food I is 2 kg and amount of food II is 4 kg should be included in the mixture for minimum cost of \u20b9 38.<\/p>\nCBSE Previous Year Question Papers Class 12 Maths 2012 Outside Delhi Set II<\/h3>\n
\nWrite the value of \\(\\hat{(k \\times \\hat{j})} \\cdot \\hat{i}+\\hat{j} \\cdot \\hat{k}\\) [1]
\nSolution:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-73.png?resize=284%2C146&ssl=1\")
<\/p>\n
\nProve the following:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-74.png?resize=275%2C51&ssl=1\")
[4]
\nSolution:
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-75.png?resize=369%2C249&ssl=1\")
<\/p>\n
\nIf y = (tan-1<\/sup>x)2<\/sup>, show that
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-76.png?resize=244%2C50&ssl=1\")
[4]
\nSolution:
\nGiven. y = (tan-1<\/sup> x)2<\/sup>
\nDifferentiating w.r.t. x, we get
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-77.png?resize=369%2C290&ssl=1\")
<\/p>\n
\nFind the particular solution of the differential equation \\(\\frac{d y}{d x}\\) + y cot x = 4x cosec x, (x \u2260 0), given that y = 0 when x = \\(\\frac{\\pi}{2}\\) [4]
\nSolution:
\nGiven;
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-78.png?resize=373%2C565&ssl=1\")
\nThis is the required particular solution of the given differential equation.<\/p>\n
\nFind the coordinates of the point where the line through the point (3, -4, -5) and (2, – 3, 1) crosses the plane 2x + y + z = 7. [4]
\nSolution:
\nEquation of the line passes through the points (3, -4, -5) and (2, – 3, 1) is
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-79.png?resize=370%2C186&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-80.png?resize=277%2C121&ssl=1\")
<\/p>\n
\nUsing matrices, solve the following system of equations:
\nx + y – z = 3; 2x + 3y + z = 10; 3x – y -7z = 1 [6]
\nSolution:
\nThe given system of equations are
\nx + y – z = 3;
\n2x + 3y + z = 10;
\n3x – y – 7z = 1.
\nThese equations can be written in matrix form
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-81.png?resize=372%2C537&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-82.png?resize=221%2C170&ssl=1\")
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-83.png?resize=338%2C309&ssl=1\")
<\/p>\n
\nFind the length and the foot of the perpendicular from the point P (7, 14, 5) to the plane 2x + 4y – z = 2. Also find the image of point P in the plane. [6]
\nSolution:
\nThe given plane is
\n2x + 4y – z = 2 …(i)
\nThe d.r.s. of the normal to (i) are 2, 4, -1.
\n\u2234 Equation of a line perpendicular to (i) passing through P(7, 14, 5) is
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-84.png?resize=375%2C518&ssl=1\")
\nwhich is the length of the perpendicular from P on(i)
\nAgain let R(x, y, z) be the image of P in the plane
\n(i). Then Q is the mid-point of PR.
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-85.png?resize=293%2C135&ssl=1\")
\n![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-86.png?resize=288%2C73&ssl=1\")
<\/p>\nCBSE Previous Year Question Papers Class 12 Maths 2012 Outside Delhi Set III<\/h3>\n
\nWrite the value of \\((\\hat{k} \\times \\hat{i}) \\cdot \\hat{j}+\\hat{i} \\cdot \\hat{k}\\) [1]
\nSolution:
\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-87.png?resize=284%2C130&ssl=1\")
<\/p>\n
\nFind the value of x + y from the following equations:
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-88.png?resize=208%2C55&ssl=1\")
[1]
\nSolution:
\nGiven,
\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-89.png?resize=329%2C338&ssl=1\")
<\/p>\n
![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-90.png?resize=375%2C573&ssl=1\")
![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-91.png?resize=373%2C347&ssl=1\")
![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-92.png?resize=365%2C414&ssl=1\")
<\/p>\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-93.png?resize=373%2C544&ssl=1\")
![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-94.png?resize=367%2C523&ssl=1\")
![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-95.png?resize=376%2C436&ssl=1\")
![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-96.png?resize=272%2C54&ssl=1\")
[4]![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-97.png?resize=367%2C522&ssl=1\")
![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-98.png?resize=368%2C172&ssl=1\")
<\/p>\n![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-99.png?resize=366%2C160&ssl=1\")
![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-100.png?resize=371%2C516&ssl=1\")
![\"CBSE](\"https:\/\/i2.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-101.png?resize=384%2C478&ssl=1\")
<\/p>\n![\"CBSE](\"https:\/\/i0.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-102.png?resize=369%2C574&ssl=1\")
![\"CBSE](\"https:\/\/i1.wp.com\/www.cbselabs.com\/wp-content\/uploads\/2019\/08\/CBSE-Previous-Year-Question-Papers-Class-12-Maths-2012-Outside-Delhi-103.png?resize=365%2C584&ssl=1\")