Date & Time: 15th March 2017, 2:00 pm

Duration: 3h

- Attempt all questions from section A.
- Answer any TWO questions from either section B or Section C.

If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.

Chapter: [0.021] Matrices and Determinants

If y – 2x – k = 0 touches the conic 3x^{2} – 5y^{2} = 15, find the value of k.

Chapter: [0.06] Three - Dimensional Geometry (Section B)

Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`

Chapter: [0.031] Continuity, Differentiability and Differentiation

Using L ‘Hospital’s Rule, evaluate:

`lim_("x"->pi/2) ("x" "tan""x" - pi/4 . "sec" "x")`

Chapter: [0.031] Continuity, Differentiability and Differentiation

Evaluate: `int 1/"x"^2 "sin"^2 (1/"x") "dx"`

Chapter: [0.033] Integrals

Evaluate: `int_0^(pi/4) "log" (1 + "tan" theta) "d" theta`

Chapter: [0.033] Integrals

By using the data `bar"x"` = 25 , `bar"y" = 30 ; "b"_"yx" = 1.6` and `"b"_"xy" = 0.4` find,

(a) The regression equation y on x.

(b) What is the most likely value of y when x = 60?

(c) What is the coefficient of correlation between x and y?

Chapter: [0.09] Linear Regression (Section C)

A problem is given to three students whose chances of solving it are `1/4, 1/5` and `1/3` respectively. Find the probability that the problem is solved.

Chapter: [0.04] Probability (Section A)

If a + ib = `("x" + "iy")/("x" - "iy"),` prove that `"a"^2 +"b"^2 = 1` and `"b"/"a" = (2"xy")/("x"^2 - "y"^2)`

Chapter: [0.034] Differential Equations

Solve : `"dy"/"dx" = 1 - "xy" + "y" - "x"`

Chapter: [0.031] Continuity, Differentiability and Differentiation

Using properties of determinants, prove that:

`|(a,b,b+c),(c,a,c+a),(b,c,a+b)|` = (a+b+c)(a-c)^{2}

Chapter: [0.021] Matrices and Determinants

Given that: A = `((1,-1,0),(2,3,4),(0,1,2))` and B = `((2,2,-4),(-4,2,-4),(2,-1,5))` , find AB. Using this result, solve the following system of equation: x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.

Chapter: [0.021] Matrices and Determinants

Solve the equation for x:

sin^{-1}x + sin^{-1}(1 - x) = cos^{-1}x, x ≠ 0

Chapter: [0.034] Differential Equations

If A, B and C are the elements of Boolean algebra, simplify the expression (A’ + B’) (A + C’) + B’ (B + C). Draw the simplified circuit.

Chapter: [0.01] Relations and Functions (Section A)

Verify Langrange’s mean value theorem for the function:

f(x) = x (1 – log x) and find the value of c in the interval [1, 2].

Chapter: [0.031] Continuity, Differentiability and Differentiation

Find the coordinates of the centre, foci and equation of directrix of the hyperbola x^{2} – 3y^{2} – 4x = 8.

Chapter: [0.034] Differential Equations

If y = cos (sin x), show that: `("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y" "cos"^2"x" = 0`

Chapter: [0.031] Continuity, Differentiability and Differentiation

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

Chapter: [0.032] Applications of Derivatives

Evaluate: `int ("sin 2x")/((1 + "sin x")(2 + "sin x")) "dx"`

Chapter: [0.033] Integrals

Draw a rough sketch of the curve y^{2} = 4x and find the area of region enclosed by the curve and the line y = x.

Chapter: [0.07] Application of Integrals (Section B)

Calculate the Spearman’s rank correlation coefficient for the following data and interpret the result:

X | 35 | 54 | 80 | 95 | 73 | 73 | 35 | 91 | 83 | 81 |

Y | 40 | 60 | 75 | 90 | 70 | 75 | 38 | 95 | 75 | 70 |

Chapter: [0.09] Linear Regression (Section C)

Find the line of best fit for the following data, treating x as the dependent variable (Regression equation x on y):

X | 14 | 12 | 13 | 14 | 16 | 10 | 13 | 12 |

Y | 14 | 23 | 17 | 24 | 18 | 25 | 23 | 24 |

Hence, estimate the value of x when y = 16.

Chapter: [0.09] Linear Regression (Section C)

In a class of 60 students, 30 opted for Mathematics, 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random, find the probability that:

(i) The student opted for Mathematics or Biology.

(ii) The student has opted neither Mathematics nor Biology.

(iii) The student has opted in Mathematics but not Biology.

Chapter: [0.04] Probability (Section A)

Bag A contains 1 white, 2 blue and 3 red balls. Bag B contains 3 white, 3 blue and 2 red balls. Bag C contains 2 white, 3 blue and 4 red balls. One bag is selected at random and then two balls are drawn from the selected bag. Find the probability that the balls draw n are white and red.

Chapter: [0.04] Probability (Section A)

Prove that locus of z is circle and find its centre and radius if is purely imaginary.

Chapter: [0.01] Relations and Functions (Section A)

Solve: (x^{2} – yx^{2}) dy + (y^{2} + xy^{2}) dx = 0

Chapter: [0.033] Integrals

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, show that the vector `veca + vecb+ vecc` is equally inclined to `veca, vecb` and `vecc`.

Chapter: [0.05] Vectors (Section B)

Find the value of λ for which the four points with position vectors `6hat"i" - 7hat"j", 16hat"i" - 19hat"j" - 4hat"k" , lambdahat"j" - 6hat"k" "and" 2hat"i" - 5hat"j" + 10hat"k"` are coplanar.

Chapter: [0.05] Vectors (Section B)

Show that the lines `("x" - 4)/1 = ("y" + 3)/-4 = ("z" + 1)/7` and `("x" - 1)/2 = ("y" + 1)/-3 = ("z" + 10)/8` intersect. Find the coordinates of their point of intersection.

Chapter: [0.034] Differential Equations

Find the equation of the plane passing through the point (1, -2, 1) and perpendicular to the line joining the points A(3, 2, 1) and B(1, 4, 2).

Chapter: [0.034] Differential Equations

A fair die is rolled. If face 1 turns up, a ball is drawn from Bag A. If face 2 or 3 turns up, a ball is drawn from Bag B. If face 4 or 5 or 6 turns up, a ball is drawn from Bag C. Bag A contains 3 red and 2 white balls, Bag B contains 3 red and 4 white balls and Bag C contains 4 red and 5 white balls. The die is rolled, a Bag is picked up and a ball is drawn. If the drawn ball is red; what is the probability that it is drawn from Bag B?

Chapter: [0.04] Probability (Section A)

An urn contains 25 balls of which 10 balls are red and the remaining green. A ball is drawn at random from the urn, the colour is noted and the ball is replaced. If 6 balls are drawn in this way, find the probability that:

(i) All the balls are red.

(ii) Not more than 2 balls are green.

(iii) The number of red balls and green balls is equal.

Chapter: [0.04] Probability (Section A)

A machine costs ₹ 60,000 and its effective life is estimated to be 25 years. A sinking fund is to be created for replacing the machine at the end of its life when its scrap value is estimated as ₹ 5000. The price of the new machine is estimated to be 100% more than the price of the present one. Find the amount that should be set aside at the end of each year, out of the profits, for the sinking fund it accumulates at an interest of 6% per annum compounded annually.

Chapter: [0.08] Application of Calculus (Section C)

A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 6% phosphoric acid and of type B which contains 5% nitrogen and 10% phosphoric acid. After the soil test, it is found that at least 7 kg of nitrogen and the same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs ₹ 5.00 per kg and the type B costs ₹ 8.00 per kg. Using Linear programming, find how many kilograms of each type of fertilizer should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph.

Chapter: [0.1] Linear Programming (Section C)

The demand for a certain product is represented by the equation p = 500 + 25x - `(x^2)/(3)` in rupees, where x is the number of units and p is the price per unit. Find:**(i)** Marginal revenue function.**(ii)** The marginal revenue when 10 units are sold.

Chapter: [0.08] Application of Calculus (Section C)

A bill of ₹ 60000 payable 10 months after the date was discounted for ₹ 57300 on 30th June 2007. If the rate of interest was 11`(1)/(4)` % per annum, on what date was the bill drawn?

Chapter: [0.08] Application of Calculus (Section C)

The price of relatives and weights of a set of commodities are given below:

Commodity |
A |
B |
C |
D |

Price relatives | 125 | 120 | 127 | 119 |

Weights | x | 2x | y | y + 3 |

If the sum of the weights is 40 and the weighted average of price relatives index number is 122, find the numerical values of x and y.

Chapter: [0.08] Application of Calculus (Section C)

Construct 3 yearly moving averages from the following data and show on a graph against the original data:

Year |
2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 |

Annual sales in lakhs |
18 | 22 | 20 | 26 | 30 | 22 | 24 | 28 | 32 | 35 |

Chapter: [0.08] Application of Calculus (Section C)

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