# Mathematics Set 1 2016-2017 ISC (Commerce) Class 12 Question Paper Solution

Mathematics [Set 1]
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.

Section A( 80 Marks )
 1
 1.a

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

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.021] Matrices and Determinants
 1.b

If y – 2x – k = 0 touches the conic 3x2 – 5y2 = 15, find the value of k.

Concept: Equation of a Line in Space
Chapter: [0.06] Three - Dimensional Geometry (Section B)
 1.c

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

Concept: Concept of Continuity
Chapter: [0.031] Continuity, Differentiability and Differentiation
 1.d

Using L ‘Hospital’s Rule, evaluate:

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

Concept: L' Hospital'S Theorem
Chapter: [0.031] Continuity, Differentiability and Differentiation
 1.e

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

Concept: Introduction of Integrals
Chapter: [0.033] Integrals
 1.f

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

Concept: Introduction of Integrals
Chapter: [0.033] Integrals
 1.g

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?

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
 1.h

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.

Concept: Introduction of Probability
Chapter: [0.04] Probability (Section A)
 1.i

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

Concept: Basic Concepts of Differential Equation
Chapter: [0.034] Differential Equations
 1.j

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

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [0.031] Continuity, Differentiability and Differentiation
 2
 2.a

Using properties of determinants, prove that:

|(a,b,b+c),(c,a,c+a),(b,c,a+b)| = (a+b+c)(a-c)2

Concept: Properties of Determinants
Chapter: [0.021] Matrices and Determinants
 2.b

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.

Concept: Introduction of Determinant
Chapter: [0.021] Matrices and Determinants
 3
 3.a

Solve the equation for x:

sin-1x + sin-1(1 - x) = cos-1x, x ≠ 0

Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.034] Differential Equations
 3.b

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.

Concept: Introduction of Relations and Functions
Chapter: [0.01] Relations and Functions (Section A)
 4
 4.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].

Concept: Mean Value Theorem
Chapter: [0.031] Continuity, Differentiability and Differentiation
 4.b

Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.

Concept: Basic Concepts of Differential Equation
Chapter: [0.034] Differential Equations
 5
 5.a

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

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.031] Continuity, Differentiability and Differentiation
 5.b

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

Concept: Maxima and Minima
Chapter: [0.032] Applications of Derivatives
 6
 6.a

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

Concept: Some Properties of Indefinite Integral
Chapter: [0.033] Integrals
 6.b

Draw a rough sketch of the curve y2 = 4x and find the area of region enclosed by the curve and the line y = x.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.07] Application of Integrals (Section B)
 7
 7.a

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
Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.09] Linear Regression (Section C)
 7.b

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.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
 8
 8.a

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.

Concept: Probability Distribution Function
Chapter: [0.04] Probability (Section A)
 8.b

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.

Concept: Introduction of Probability
Chapter: [0.04] Probability (Section A)
 9
 9.a

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

Concept: Introduction of Relations and Functions
Chapter: [0.01] Relations and Functions (Section A)
 9.b

Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0

Concept: Definite Integral as the Limit of a Sum
Chapter: [0.033] Integrals
Section B
 10
 10.a

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.

Concept: Magnitude and Direction of a Vector
Chapter: [0.05] Vectors (Section B)
 10.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.

Concept: Vectors and Their Types
Chapter: [0.05] Vectors (Section B)
 11
 11.a

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.

Concept: Application on Coordinate Geometry
Chapter: [0.034] Differential Equations
 11.b

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).

Concept: Basic Concepts of Differential Equation
Chapter: [0.034] Differential Equations
 12
 12.a

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?

Concept: Independent Events
Chapter: [0.04] Probability (Section A)
 12.b

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.

Concept: Introduction of Probability
Chapter: [0.04] Probability (Section A)
Section – C ( 20 Marks )
 13
 13.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.

Concept: Application of Calculus in Commerce and Economics in the Cost Function
Chapter: [0.08] Application of Calculus (Section C)
 13.b

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.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.1] Linear Programming (Section C)
 14
 14.a

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.

Concept: Application of Calculus in Commerce and Economics in the Marginal Revenue Function and Its Interpretation
Chapter: [0.08] Application of Calculus (Section C)
 14.b

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?

Concept: Application of Calculus in Commerce and Economics in the Marginal Revenue Function and Its Interpretation
Chapter: [0.08] Application of Calculus (Section C)
 15
 15.a

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.

Concept: Application of Calculus in Commerce and Economics in the Average Cost
Chapter: [0.08] Application of Calculus (Section C)
 15.b

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
Concept: Application of Calculus in Commerce and Economics in the Average Cost
Chapter: [0.08] Application of Calculus (Section C)

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