Date & Time: 15th March 2016, 2:00 pm

Duration: 3h

- Candidates are required to attempt all questions from Section A and all questions either from Section B or Section C.

Find the matrix X for which:

`[(5, 4),(1,1)]` X=`[(1,-2),(1,3)]`

Chapter: [2.01] Matrices and Determinants

Solve for x, if:

tan (cos^{-1}x) = `2/sqrt5`

Chapter: [1] Relations and Functions (Section A)

Prove that the line 2x - 3y = 9 touches the conics y^{2} = -8x. Also, find the point of contact.

Chapter: [1] Relations and Functions (Section A)

Using L' Hospital's rule, evaluate:

`lim_("x"→0) (1/"x"^2 - cot"x"/"x")`

Chapter: [3.01] Continuity, Differentiability and Differentiation

Evaluate: ` int tan^3x "dx"`

Chapter: [3.03] Integrals

By using the properties of definite integrals, evaluate the integrals

`int_0^(pi/2) (sin "x" - cos "x")/(1+sin"x" cos "x") "dx"`

Chapter: [3.03] Integrals

The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.

Chapter: [9] Linear Regression (Section C)

Express `(2+"i")/((1+"i")(1-2"i")` in the form of a + ib. Find its modulus and argument

Chapter: [7] Application of Integrals (Section B)

A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of 8?

Chapter: [4] Probability (Section A)

Solve the differential equation:

`"x"("dy")/("dx")+"y"=3"x"^2-2`

Chapter: [3.04] Differential Equations

Using properties of determinants, prove that

`|[b+c , a ,a ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc

Chapter: [2.01] Matrices and Determinants

Solve the following system of linear equations using matrix method:

3x + y + z = 1

2x + 2z = 0

5x + y + 2z = 2

Chapter: [2.01] Matrices and Determinants

If `sin^-1"x" + tan^-1"x" = pi/2`, prove that `2"x"^2 + 1 = sqrt5`

Chapter: [1] Relations and Functions (Section A)

Write the Boolean function corresponding to the switching circuit given below:

A, B and C represent switches in ‘on’ position and A’, B’ and C’ represent them in ‘off position. Using Boolean algebra, simplify the function and construct an equivalent switching circuit.

Chapter: [1] Relations and Functions (Section A)

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x^{2} + 2) − log 3 on [−1, 1] ?

Chapter: [3.02] Applications of Derivatives

Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of the latus rectum is 10. Also, find its eccentricity.

Chapter: [7] Application of Integrals (Section B)

If logy = tan^{–1} x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`

Chapter: [3.02] Applications of Derivatives

A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.

Chapter: [3.02] Applications of Derivatives

Evaluate:

`int (sin"x"+cos"x")/(sqrt(9+16sin2"x")) "dx"`

Chapter: [3.03] Integrals

Find the area of the region bound by the curves y = 6x – x^{2} and y = x^{2} – 2x

Chapter: [7] Application of Integrals (Section B)

Calculate Karl Pearson’s coefficient of correlation between x and y for the following data and interpret the result:

(1, 6), (2, 5), (3, 7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)

Chapter: [9] Linear Regression (Section C)

The marks obtained by 10 candidates in English and Mathematics are given below:

Marks in English |
20 | 13 | 18 | 21 | 11 | 12 | 17 | 14 | 19 | 15 |

Marks in Mathematics |
17 | 12 | 23 | 25 | 14 | 8 | 19 | 21 | 22 | 19 |

Estimate the probable score for Mathematics if the marks obtained in English are 24.

Chapter: [9] Linear Regression (Section C)

A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.

Chapter: [4] Probability (Section A)

An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball is drawn from the second urn is a white ball.

Chapter: [4] Probability (Section A)

Find the locus of a complex number, z = x + iy, satisfying the relation `|[ z -3i}/{z +3i]| ≤ sqrt2 `. Illustrate the locus of z in the Argand plane.

Chapter: [6] Three - Dimensional Geometry (Section B)

Solve the following differential equation:

x^{2} dy + (xy + y^{2}) dx = 0, when x = 1 and y = 1

Chapter: [3.01] Continuity, Differentiability and Differentiation

For any three vectors `veca, vecb, vecc`, show that `veca - vecb, vecb - vecc, vecc - veca` are coplanar.

Chapter: [6] Three - Dimensional Geometry (Section B)

Find a unit vector perpendicular to each of the vectors `veca + vecb "and" veca - vecb "where" veca = 3hati + 2hatj + 2hatk and vecb = i + 2hatj - 2hatk`

Chapter: [5] Vectors (Section B)

Find the image of the point (2, -1, 5) in the line `(x - 11)/(10) = (y + 2)/(-4) = (z + 8)/(-11)`. Also, find the length of the perpendicular from the point (2, -1, 5) to the line.

Chapter: [6] Three - Dimensional Geometry (Section B)

Find the Cartesian equation of the plane, passing through the line of intersection of the planes `vecr. (2hati + 3hatj - 4hatk) + 5 = 0`and `vecr. (hati - 5hatj + 7hatk) + 2 = 0` intersecting the y-axis at (0, 3).

Chapter: [6] Three - Dimensional Geometry (Section B)

In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons A, B, and C carries out this task. A has a 45% chance, B has a 35% chance and C has a 20% chance of doing the task.

The probability that A, B, and C will take more than the allotted time is `(1)/(6), (1)/(10), and (1)/(20)` respectively. If it is found that the time taken is more than the allotted time, what is the probability that A has done the task?

Chapter: [4] Probability (Section A)

The difference between mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution.

Chapter: [4] Probability (Section A)

A man borrows ₹ 20,000 at 12% per annum, compounded semi-annually and agrees to pay it in 10 equal semi-annual installments. Find the value of each installment, if the first payment is due at the end of two years.

Chapter: [8] Application of Calculus (Section C)

A company manufactures two types of products A and B. Each unit of A requires 3 grams of nickel and 1 gram of chromium, while each unit of B requires 1 gram of nickel and 2 grams of chromium. The firm can produce 9 grams of nickel and 8 grams of chromium. The profit is ₹ 40 on each unit of the product of type A and ₹ 50 on each unit of type B. How many units of each type should the company manufacture so as to earn a maximum profit? Use linear programming to find the solution.

Chapter: [10] Linear Programming (Section C)

The demand function is `x = (24 - 2p)/(3)` where x is the number of units demanded and p is the price per unit. Find:**(i)** The revenue function R in terms of p.**(ii)** The price and the number of units demanded in which the revenue is maximum.

Chapter: [8] Application of Calculus (Section C)

A bill of ₹ 1,800 drawn on 10th September 2010 at 6 months was discounted for ₹ 1,782 at a bank. If the rate of interest was 5% per annum, on what date was the bill discounted?

Chapter: [8] Application of Calculus (Section C)

The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be 116. If sum of the prices in the year 2000 is ₹ 300, find the values of x and y in the data given below

Commodity |
A |
B |
C |
D |
E |
F |

Price in the year 2000 (₹) |
50 | x | 30 | 70 | 116 | 20 |

Price in the year 2010 (₹) |
60 | 24 | y | 80 | 120 | 28 |

Chapter: [1] Relations and Functions (Section A)

From the details given below, calculate the five-year moving averages of the number of students who have studied in a school. Also, plot these and original data on the same graph paper.

Year |
1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 |

Number of Students |
332 | 317 | 357 | 392 | 402 | 405 | 410 | 427 | 405 | 438 |

Chapter: [10] Linear Programming (Section C)

The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be 116. If sum of the prices in the year 2000 is ₹ 300, find the values of x and y in the data given below

Commodity |
A |
B |
C |
D |
E |
F |

Price in the year 2000 (₹) |
50 | x | 30 | 70 | 116 | 20 |

Price in the year 2010 (₹) |
60 | 24 | y | 80 | 120 | 28 |

Chapter: [1] Relations and Functions (Section A)

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