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Mathematics 2015-2016 ISC (Commerce) Class 12 Question Paper Solution

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Mathematics
Marks: 100Academic Year: 2015-2016
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.

Section A (80 Marks)
[30]1
[3]1.i

Find the matrix X for which:
`[(5, 4),(1,1)]` X=`[(1,-2),(1,3)]`

Concept: Order of a Matrix
Chapter: [2.01] Matrices and Determinants
[3]1.ii

Solve for x, if:

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

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [1] Relations and Functions (Section A)
[3]1.iii

Prove that the line 2x - 3y = 9 touches the conics y2 = -8x. Also, find the point of contact.

Concept: Inverse of a Function
Chapter: [1] Relations and Functions (Section A)
[3]1.iv

Using L' Hospital's rule, evaluate:

`lim_("x"→0) (1/"x"^2 - cot"x"/"x")`

Concept: L' Hospital'S Theorem
Chapter: [3.01] Continuity, Differentiability and Differentiation
[3]1.v

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

Concept: Introduction of Integrals
Chapter: [3.03] Integrals
[3]1.vi

By using the properties of definite integrals, evaluate the integrals

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

Concept: Properties of Definite Integrals
Chapter: [3.03] Integrals
[3]1.vii

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.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [9] Linear Regression (Section C)
[3]1.viii

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

Concept: Applications of the Integrations - Application of Integrals - Modulus Function
Chapter: [7] Application of Integrals (Section B)
[3]1.ix

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

Concept: Introduction of Probability
Chapter: [4] Probability (Section A)
[3]1.x

Solve the differential equation:

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

Concept: Basic Concepts of Differential Equation
Chapter: [3.04] Differential Equations
[10]2
[5]2.i

Using properties of determinants, prove that

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

Concept: Properties of Determinants
Chapter: [2.01] Matrices and Determinants
[5]2.ii

Solve the following system of linear equations using matrix method: 
3x + y + z = 1
2x + 2z = 0
5x + y + 2z = 2

Concept: Determinant of a Matrix of Order 3 × 3
Chapter: [2.01] Matrices and Determinants
[10]3
[5]3.i

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

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [1] Relations and Functions (Section A)
[5]3.ii

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.

Concept: Concept of Relations - Types of Relations - Identity Relation
Chapter: [1] Relations and Functions (Section A)
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[10]4
[5]4.i

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x2 + 2) − log 3 on [−1, 1] ?

Concept: Maximum and Minimum Values of a Function in a Closed Interval
Chapter: [3.02] Applications of Derivatives
[5]4.ii

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. 

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [7] Application of Integrals (Section B)
[10]5
[5]5.i

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

Concept: Simple Problems on Applications of Derivatives
Chapter: [3.02] Applications of Derivatives
[5]5.ii

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. 

Concept: Maxima and Minima
Chapter: [3.02] Applications of Derivatives
[10]6
[5]6.i

Evaluate:

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

Concept: Definite Integral as the Limit of a Sum
Chapter: [3.03] Integrals
[5]6.ii

Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x 

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [7] Application of Integrals (Section B)
[10]7
[5]7.i

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)

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [9] Linear Regression (Section C)
[5]7.ii

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.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [9] Linear Regression (Section C)
[10]8
[5]8.i

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. 

Concept: Introduction of Probability
Chapter: [4] Probability (Section A)
[5]8.ii

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. 

Concept: Laws of Probability
Chapter: [4] Probability (Section A)
[10]9
[5]9.i

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.

Concept: Plane - Intercept Form of the Equation of a Plane
Chapter: [6] Three - Dimensional Geometry (Section B)
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[5]9.ii

Solve the following differential equation: 
x2 dy + (xy + y2) dx = 0, when x = 1 and y = 1

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [3.01] Continuity, Differentiability and Differentiation
Section – B
[10]10
[5]10.i

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

Concept: Concept of Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
Chapter: [6] Three - Dimensional Geometry (Section B)
[5]10.ii

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` 

Concept: Types of Vectors
Chapter: [5] Vectors (Section B)
[10]11
[5]11.i

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. 

Concept: Concept of Line - Equation of Line Passing Through Given Point and Parallel to Given Vector
Chapter: [6] Three - Dimensional Geometry (Section B)
[5]11.ii

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).

Concept: Vector and Cartesian Equation of a Plane
Chapter: [6] Three - Dimensional Geometry (Section B)
[10]12
[5]12.i

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?

Concept: Introduction of Probability
Chapter: [4] Probability (Section A)
[5]12.ii

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

Concept: Variance of Binomial Distribution (P.M.F.)
Chapter: [4] Probability (Section A)
Section – C
[10]13
[5]13.i

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.

Concept: Application of Calculus in Commerce and Economics in the Cost Function
Chapter: [8] Application of Calculus (Section C)
[5]13.ii

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.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [10] Linear Programming (Section C)
[10]14
[5]14.i

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. 

Concept: Application of Calculus in Commerce and Economics in the Demand Function
Chapter: [8] Application of Calculus (Section C)
[5]14.ii

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?

Concept: Application of Calculus in Commerce and Economics in the Cost Function
Chapter: [8] Application of Calculus (Section C)
[10]15
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[5]15.i

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 80  120 28
Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [1] Relations and Functions (Section A)
[5]15.ii

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
Concept: Graphical Method of Solving Linear Programming Problems
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 80  120 28
Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [1] Relations and Functions (Section A)
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CISCE previous year question papers Class 12 Mathematics with solutions 2015 - 2016

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