Date & Time: 26th February 2016, 11:00 am
Duration: 3h
- All questions are compulsory.
- Answer to every question must be written on a new page.
- Log lable will be provided on demand.
- Answer to Section-I and Section-II should be written in two separate answer books.
- Questions from Section-I attempted in the answer book of Section-II and vice versa will not be assessed/not given any credit.
- Graph paper is necessary for L.P.P.
- Figure to the right indicate full marks.
If y = ( sin x )^{x} , Find `dy/dx`
Chapter: [0.03] Continuity
If A = `[(1, 3), (3, 1)]`, Show that A^{2} - 2A is a scalar matrix.
Chapter: [0.012] Matrices [0.02] Matrices
Write the negation of the Following Statement :
∀ y ∈ N, y^{2} + 3 ≤ 7
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Write the negation of the following statement :
If the lines are parallel then their slopes are equal.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
The total revenue R = 720 - 3x^{2} where x is number of items sold. Find x for which total revenue R is increasing.
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
Evaluate : `int (sec^2 x)/(tan^2 x + 4)` dx
Chapter: [0.04] Differentiation
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
Discuss the continuity of function f at x = 0.
Where f(X) = `[ [sqrt ( 4 + x ) - 2 ]/ ( 3x )]`, For x ≠ 0
= `1/12`, For x = 0
Chapter: [0.03] Continuity
State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
State if the following sentence is a statement. In case of a statement, write down the truth value :
√-4 is a rational number.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Solve the following equations by the inversion method :
2x + 3y = - 5 and 3x + y = 3.
Chapter: [0.012] Matrices [0.02] Matrices
Find x and y, if
`{ 3[(1, 2, 0), (0, -1, 3)] - [(1, 5, -2), (-3, -4, 4)]}[(1), (2), (1)] = [(x), (y)]`
Chapter: [0.02] Matrices
Express the truth of each of the following statements using Venn diagram.
(1) All teachers are scholars and scholars are teachers.
(2) If a quadrilateral is a rhombus then it is a parallelogram..
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Write converse and inverse of the following statement :
"If Ravi is good in logic then Ravi is good in Mathematics."
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Find the area of the region bounded by the lines 2y + x = 8, x = 2 and x = 4.
Chapter: [0.07] Definite Integrals
Evaluate : `int_3^9 [root(3)(12-x)]/[ root(3)(x) + root(3)(12 - x)]`
Chapter: [0.07] Definite Integrals
If f(x) = `("e"^(2"x") - 1)/"ax"` , for x < 0 , a ≠ 0
= 1 for x = 0
= `("log" (1 + 7"x"))/"bx"` , for x > 0 , b ≠ 0
is continuous at x = 0, then find a and b.
Chapter: [0.07] Definite Integrals
If the function f is continuous at x = 0 then find f(0),
where f(x) = `[ cos 3x - cos x ]/x^2`, `x!=0`
Chapter: [0.03] Continuity
If f '( x ) = 4x^{3} - 3x^{2} + 2x + k and f(0) = 1, f(1) = 4, find f(x).
Chapter: [0.06] Indefinite Integration
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure E_{c} of a person with income I is given as E_{c} = ( 0.0003 ) I^{2} + ( 0.075 ) I when I = 1000.
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
Cost of assembling x wallclocks is `( x^3/3 - 40x^2)` and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.
Chapter: [0.05] Applications of Derivative
If `cos^-1[(x^2 - y^2)/(x^2 + y^2)] = 2k "show that y" dy/dx = xtan^2k`
Chapter: [0.06] Indefinite Integration
Anandi and Rutuja invested Rs. 10,000 each in a business. Anandi withdrew her capital after 7 months. Rutuja continued for the year. After one year, the profit earned by them was Rs. 5,700. Find the profit earned by each person.
Chapter: [0.08] Ratio, Proportion and Partnership
Calculate age specific death (A-SDR) rates for the following data :
Age group ( in years ) | Population (in '000) | Number of Deaths |
Below 10 | 25 | 50 |
10 - 30 | 30 | 90 |
30 - 45 | 40 | 160 |
45 - 70 | 20 | 100 |
Chapter: [0.11] Demography
For a bivariate data b_{yx} = -1.2 and b_{xy} = -0.3. Find the correlation coefficient between x and y.
Chapter: [0.12] Bivariate Data and Correlation
A random variable x has the following probability distribution :
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
P( X = x) | k | 3k | 5k | 7k | 9k | 11k | 13k |
Find k.
Chapter: [0.14] Random Variable and Probability Distribution
The probability distribution function of continuous random variable X is given by
f( x ) = `x/4`, 0 < x < 2
= 0, Otherwise
Find P( x ≤ 1)
Chapter: [0.03] Continuity
From the two regression equations y = 4x - 5 and 3x = 2y + 5, find `barx and bary`
Chapter: [0.13] Regression Analysis Introduction
Draw scatter diagram for the following data and interpret it :
x | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
y | 32 | 20 | 24 | 36 | 40 | 28 | 38 |
Chapter: [0.11] Demography
If Σd^{2} = 66 and n = 10 then find the rank correlation coefficient.
Chapter: [0.12] Bivariate Data and Correlation
Determine `l_92 and l_93, "given that" l_91 = 97, d_91 = 38 and q_92 = 27/59`
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
Calculate CDR for districts A and B and compare them. Also state which district is more healthy..
Age group ( in years ) |
District A | District B | ||
No. of persons ('000) | No of Deaths | No. of persons ('000) | No. of Deaths | |
0 - 15 | 1 | 20 | 2 | 50 |
15 - 60 | 3 | 30 | 7 | 70 |
60 and above | 2 | 40 | 1 | 25 |
Chapter: [0.11] Demography
If for a bivariate data `barx = 10, bary = 12, V(X) = 9, σ_y = 4` and r = 0.6, estimate y when x = 5.
Chapter: [0.12] Bivariate Data and Correlation
Calculate the coefficient of correlation between X and Y series from the following data :
`n = 15 ,bar x = 25, bary = 18, σ_x = 3.01, σ_y = 3.03,`
`sum ("x"_i - bar x) ("y"_i - bar y) = 122`
Chapter: [0.12] Bivariate Data and Correlation
Solve the following minimal assignment problem and hence find minimum time where '- ' indicates that job cannot be assigned to the machine :
Machines | Processing time in hours | ||||
A | B | C | D | E | |
M_{1} | 9 | 11 | 15 | 10 | 11 |
M_{2} | 12 | 9 | - | 10 | 9 |
M_{3} | - | 11 | 14 | 11 | 7 |
M_{4} | 14 | 8 | 12 | 7 | 8 |
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
Solve the following maximal assignment problem :
Branch Manager | Monthly Business ( Rs. lakh) | |||
A | B | C | D | |
P | 11 | 11 | 9 | 9 |
Q | 13 | 16 | 11 | 10 |
R | 12 | 17 | 13 | 8 |
S | 16 | 14 | 16 | 12 |
Chapter: [0.027000000000000003] Assignment Problem and Sequencing [0.15] Management Mathematics
Find the true discount, banker's discount and banker's gain on a bill of Rs. 36,600 due 4 months hence at 5% p.a.
Chapter: [0.09] Commission, Brokerage and Discount
Mr. Anil wants to invest at most Rs. 60,000 in Fixed Deposit (F.D.) and Public Provident Fund ( P.P.F. ). He wants to invest at least Rs. 20,000 in F.D. and at least Rs. 15,000 in P.P.F. The rate of interest on F.D. is 8% p.a. and that on P.P.F. is 10% p.a. Formulate the above problem as L.P.P. to determine maximum yearly income.
Chapter: [0.1] Insurance and Annuity
Find graphical solution for following system of linear inequations :
3x + 2y ≤ 180; x+ 2y ≤ 120, x ≥ 0, y ≥ 0
Hence find co-ordinates of corner points of the common region.
Chapter: [0.023] Linear Regression [0.13] Regression Analysis Introduction
Mrs. Menon plans to save for her daughter's marriage. She wants to accumulate a sum of ₹ 4,00,000 at the end of 4 years. How much should she invest at the end of each year from now, if she can get interest compounded at 10% p.a. ?
[Given (1.1)^{4} = 1.4641]
Chapter: [0.1] Insurance and Annuity
A car valued at Rs. 4,00,000 is insured for Rs. 2,50,000. The rate of premium is 5% less 20%. How much loss does the owner bear including the premium if value of the car is reduced to 60% of its original value ?
Chapter: [0.08] Ratio, Proportion and Partnership
If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
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