Date: July 2016
- All questions are compulsory.
- Figure to the right indicateJull marks.
- Answer to every question must be written on a new page.
- Graph paper is necessaryJor L.P.P.
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
The price P for demand D is given as P = 183 + 120 D – 3D^{2}.
Find D for which the price is increasing.
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
Write the Truth Value of the Negation of the Following Statement :
The Sun sets in the East.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Write the truth value of the negation of the following statement :
cos^{2} θ + sin^{2} θ = 1, for all θ ∈ R
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Simplify the following :
`{3 [(1,2,0),(0,-1,3)] - [(1,5,-2),(-3,-4,4)]} [(1),(2),(1)]`
Chapter: [0.012] Matrices [0.02] Matrices
Examine the continuity off at x = 1, if
f (x) = 5x - 3 , for 0 ≤ x ≤ 1
= x^{2} + 1 , for 1 ≤ x ≤ 2
Chapter: [0.03] Continuity
Find `"dy"/"dx"` , if `"y" = "x"^("e"^"x")`
Chapter: [0.04] Differentiation
If A = `[(1,2,3),(2,"a",2),(5,7,3)]` is a singular matrix , find the value of 'a'.
Chapter: [0.012] Matrices [0.02] Matrices
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
Show that the following statement pattern in contingency :
(~p v q) → [p ∧ (q v ~ q)]
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
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
x^{y} = e^{x-y}, then show that `"dy"/"dx" = ("log x")/("1 + log x")^2`
Chapter: [0.04] Differentiation
Evaluate : `int_0^1 "x" . "tan"^-1 "x" "dx"`
Chapter: [0.06] Indefinite Integration
If A = `[(1,-1,2),(3,0,-2),(1,0,3)]` ,
verify that A (adj A) = (adj A) A = |A| . I
Chapter: [0.012] Matrices [0.02] Matrices
A manufacturer can sell x items at a price of ₹ (280 - x) each .The cost of producing items is ₹ (x^{2} + 40x + 35) Find the number of items to be sold so that the manufacturer can make maximum profit.
Chapter: [0.05] Applications of Derivative
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
Chapter: [0.03] Continuity
Differentiate : log (1 + x^{2}) w.r.t. cot^{-1} x.
Chapter: [0.04] Differentiation
Using the Venn diagram, examine the logical equivalence of the following statements:
(a) Some politicians are actors.
(b) There are politicians who are actors.
(c) There are politicians who are not actors.
Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic
Find the volume of the solid generated by the complete revolution of the ellipse `"x"^2/36 + "y"^2/25 = 1` about Y-axis.
Chapter: [0.06] Indefinite Integration
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Chapter: [0.016] Definite Integration [0.07] Definite Integrals
The total cost of manufacturing x articles is C = (47x + 300x^{2} - x^{4}). Find x, for which average cost is increasing.
Chapter: [0.013999999999999999] Applications of Derivatives [0.05] Applications of Derivative
The total cost of manufacturing x articles C = 47x + 300x^{2} – x^{4} . Find x, for which average cost is decreasing.
Chapter: [0.013999999999999999] Applications of Derivatives
Let X = amount of time for which a book is taken out of a college library by a randomly selected student and suppose X has p.d.f.
`f(x)={(0.5x",",0≤x≤2,,),(0",","otherwise",,):}`
Calculate (a) P (X ≤ 1) (b) P (0.5 ≤ X ≤ 1.5)
Chapter: [0.07] Definite Integrals
If the correlation coefficient between X and Y is 0.8, what is the correlation coefficient between:
`"X"/2` and Y
Chapter: [0.12] Bivariate Data and Correlation
If the correlation coefficient between X and Y is 0.8, what is the correlation coefficient between:
`"X - 5"/7` and `"Y - 3"/8`
Chapter: [0.12] Bivariate Data and Correlation
Raghu, Madhu and Ramu started a business in a partnership by investing ₹ 60,000, ₹ 40,000 and ₹ 75,000 respectively. At the end of the year they found that they have incurred a loss of ₹ 24,500. Find that how much loss each one had to bear.
Chapter: [0.08] Ratio, Proportion and Partnership
If the rank correlation coefficient is 0.6 and the sum of squares of differences of ranks is 66, then find the number of pairs of observations.
Chapter: [0.12] Bivariate Data and Correlation
For an immediate annuity paid for 3 years with interest compounded at 10% p.a. its present value is ₹ 10,000. What is its accumulated value after 3 years.
[Given: (1.1)^{3} = 1.331)].
Chapter: [0.1] Insurance and Annuity
Calculate CDR for district A and B.
Age group (in years) | District A | District B | ||
Population | No.of.Deaths | Population | No.of.Deaths | |
0 - 10 | 1000 | 18 | 3000 | 70 |
10 - 55 | 3000 | 32 | 7000 | 50 |
Above 55 | 2000 | 41 | 1000 | 24 |
Chapter: [0.11] Demography
Compute Age - Specific Death rate tor the following data
Age group (in years) | Population (in thousands | No. of deaths |
Below 5 | 15 | 360 |
5 - 30 | 20 | 400 |
Above 30 | 10 | 280 |
Chapter: [0.11] Demography
The ratio of prices of two houses was 2 : 3. Two years later when price of first house has increased by 30% and that of the second by ₹ 90,000 the ratio of prices becomes 5 : 7. Find the original prices of two houses.
Chapter: [0.08] Ratio, Proportion and Partnership
The income of an agent remains unchanged though the rate of commission is increased from 5% to 6.25%. Find the percentage reduction in the value of business.
Chapter: [0.09] Commission, Brokerage and Discount
A new treatment for baldness is known to be effective in 70% of the cases treated. Four bald members from different families are treated. Find the probability that exactly two members are successfully treated.
Chapter: [0.14] Random Variable and Probability Distribution
A new treatment for baldness is known to be effective in 70% of the cases treated. Four bald members from different families are treated. Find the probability that at least one member is successfully treated.
Chapter: [0.14] Random Variable and Probability Distribution
Find the graphical solution for following system of linear inequations
`"x"_1/60 + "x"_2/90 <= 1 ; 5x_1 + 8x_2 ≤ 600 , x_1 ≥ 0 , x_2 ≥ 0`
Chapter: [0.15] Management Mathematics
Bring out the inconsistency, if any in the following :
b_{YX} + b_{XY} = 1.30 and r = 0.75
Chapter: [0.13] Regression Analysis Introduction
Bring out the inconsistency, if any in the following :
b_{YX} = b_{XY} = 1.50 and r = -0.9
Chapter: [0.13] Regression Analysis Introduction
Bring out the inconsistency, if any in the following :
b_{YX} = 1.9 and b_{XY} = -0.25
Chapter: [0.13] Regression Analysis Introduction
Bring out the inconsistency, if any in the following :
b_{YX} = 2.6 and b_{XY} = `1/2.6`
Chapter: [0.13] Regression Analysis Introduction
A pharmaceutical company has four branches, one ea.ch at city A, B, C, D. A branch manager is to be appointed one at each city, out of four candidates P, Q, R and S. The monthly business deyending upon the city and the effectiveness of the branch manager in that city is given below :
City | ||||
A | B | C | D | |
Monthly business (₹ lakh) | ||||
P | 10 | 10 | 8 | 8 |
Q | 12 | 15 | 10 | 9 |
R | 11 | 16 | 12 | 7 |
S | 15 | 13 | 15 | 11 |
Which manager should be appointed at which city so as to get the maximum total monthly business·?
Chapter: [0.02] Matrices
Fill in the blanks which are marked with a question mark in the following marked from the life table :
x | l_{x} | d_{x} | q_{x} | p_{x} | L_{x} | T_{x} |
20 | 88230 | ? | ? | ? | ? | ? |
21 | 79473 | 320552 |
Chapter: [0.11] Demography
For a bivariate data,
`bar x = 53 , bar y = 28 , "b"_"xy" = -1.5 and "b"_"xy"=- 0.2` Find
Correlation coefficient between X and Y
Chapter: [0.13] Regression Analysis Introduction
For a bivariate data,
`bar x = 53 , bar y = 28 , "b"_"xy" = - 0.2` , `"b"_"yx" = -1.5` Find
Estimate of Y , When X = 50.
Chapter: [0.13] Regression Analysis Introduction
For a bivariate data,
`bar x = 53 , bar y = 28 , "b"_"yx"=-1.5 and "b"_"xy"=- 0.2` Find Estimate of X for y = 25.
Chapter: [0.13] Regression Analysis Introduction
If X has a Poisson distribution with variance 2, find P (X = 4)
[Use e^{-2} = 0.1353]
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
If X has a Poisson distribution with variance 2, find P(X ≤ 4)
[Use e^{-2} = 0.1353]
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
If X has a Poisson distribution with variance 2, find
Mean of X [Use e^{-2} = 0.1353]
Chapter: [0.027999999999999997] Probability Distributions [0.14] Random Variable and Probability Distribution
There are 5 jobs each of which is to be processed through three machines A, B and C in the order ABC. Processing times in hours are as shown in the following table :
Job | 1 | 2 | 3 | 4 | 5 |
A | 3 | 8 | 7 | 5 | 4 |
B | 4 | 5 | 1 | 2 | 3 |
C | 7 | 9 | 5 | 6 | 10 |
Determine the optimum sequence for the five jobs and the minimum elapsed time.
Chapter: [0.15] Management Mathematics
A bill of Rs.7,500 was discounted for Rs. 7,290 at a bank on 28^{th} October 2006. If the rate of interest was 14% p.a., what is the legal due date ?
Chapter: [0.09] Commission, Brokerage and Discount
The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.
Also prepare marginal distributions :
(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )
Chapter: [0.14] Random Variable and Probability Distribution
Minimize : Z = 3x_{1} + 2x_{2}
Subject to constraints
5x_{1} + x_{2} ≥ 10
2x_{1} + 2x_{2} ≥ 12
x_{1} + 4x_{2} ≥ 12
x_{1} , x_{2} ≥ 0
Chapter: [0.15] Management Mathematics
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Maharashtra State Board previous year question papers 12th Board Exam Mathematics and Statistics with solutions 2015 - 2016
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