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Mathematics Set 1 2018-2019 ISC (Commerce) Class 12 Question Paper Solution

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Mathematics [Set 1]
Marks: 100Academic Year: 2018-2019
Date: March 2019

section A (80 marks)
[20]1
[2]1.i

If f : R → R, f(x) = x and g: R → R , g(x) =  2x+ 1, and R is the set of real numbers, then find fog(x) and gof (x)

Concept: Composition of Functions and Invertible Function
Chapter: [1] Relations and Functions (Section A)
[2]1.ii

Solve:  sin(2 tan -1 x)=1

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

 Using determinants, find the values of k, if the area of triangle with vertices (–2, 0), (0, 4) and (0, k) is 4 square units. 

Concept: Area of a Triangle
Chapter: [2.01] Matrices and Determinants
[1]1.iv

Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`

Concept: Types of Matrices
Chapter: [2.01] Matrices and Determinants
[1]1.v

`f(x)=(x^2-9)/(x - 3)` is not defined at x = 3. what value should be assigned to f(3) for continuity of f(x) at = 3?

Concept: Continuous Function of Point
Chapter: [3.01] Continuity, Differentiability and Differentiation
[1]1.vi

 Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R. 

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
[2]1.vii

 Evaluate :  `int sec^2x/(cosec^2x)dx`

Concept: Introduction of Integrals
Chapter: [3.03] Integrals
[2]1.viii

Using L’Hospital Rule, evaluate: `lim_(x->0)  (8^x - 4^x)/(4x
)`

Concept: Definite Integral as the Limit of a Sum
Chapter: [3.03] Integrals
[2]1.ix

 Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?

Concept: Conditional Probability
Chapter: [4] Probability (Section A)
[2]1.x

If events A and B are independent, such that `P(A)= 3/5`,  `P(B)=2/3` 'find P(A ∪ B).

Concept: Conditional Probability
Chapter: [4] Probability (Section A)
[4]2

If f A→ A and A=R - `{8/5}` , show that the function `f (x) = (8x + 3)/(5x - 8)` is one-one onto. Hence,find `f^-1`.

Concept: Inverse of a Function
Chapter: [1] Relations and Functions (Section A)
[4]3
[4]3.i

Solve for x:
`tan^-1 [(x-1),(x-2)] + tan^-1 [(x+1),(x+2)] = x/4`

Concept: Basic Concepts of Trigonometric Functions
Chapter: [1] Relations and Functions (Section A)
OR
[4]3.ii

if sec-1  x = cosec-1  v. show that `1/x^2 + 1/y^2 = 1`

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [1] Relations and Functions (Section A)
[4]4

Using propertiesof determinants prove that:
`|(x , x(x^2), x+1), (y, y(y^2 + 1), y+1),( z, z(z^2 + 1) , z+1) | = (x-y) (y - z)(z - x)(x + y+ z)`

Concept: Properties of Determinants
Chapter: [2.01] Matrices and Determinants
[4]5
[4]5.i

 Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4. 

Concept: Concept of Continuity
Chapter: [3.01] Continuity, Differentiability and Differentiation
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OR
[4]5.ii

Verify the Lagrange’s mean value theorem for the function: 
`f(x)=x + 1/x ` in the interval [1, 3]

Concept: Mean Value Theorem
Chapter: [3.01] Continuity, Differentiability and Differentiation
[4]6

IF `y = e^(sin-1x)   and  z =e^(-cos-1x),` prove that `dy/dz = e^x//2`

Concept: Derivatives of Functions in Parametric Forms
Chapter: [3.01] Continuity, Differentiability and Differentiation
[4]7

A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

Concept: Rate of Change of Bodies Or Quantities
Chapter: [3.02] Applications of Derivatives
[4]8
[4]8.i

Solve the differential equation  `dy/dx = (x + y+2)/(2(x+y)-1)`

Concept: Introduction of Relations and Functions
Chapter: [1] Relations and Functions (Section A)
OR
[4]8.ii

 Evaluate: `int_-6^3 |x+3|dx`

Concept: Introduction of Relations and Functions
Chapter: [1] Relations and Functions (Section A)
[4]9

Solve the differential equation  `dy/dx = (x + y+2)/(2(x+y)-1)`

Concept: Introduction of Relations and Functions
Chapter: [1] Relations and Functions (Section A)
[4]10

Bag A contains 4 white balls and 3 black balls. While Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B?

Concept: Conditional Probability
Chapter: [4] Probability (Section A)
[6]11

Solve the following system of linear equation using matrix method: 
`1/x + 1/y +1/z = 9`

`2/x + 5/y+7/z = 52`

`2/x+1/y-1/z=0`

Concept: Introduction of Matrices
Chapter: [2.01] Matrices and Determinants
[6]12
[6]12.i

 The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it. 

Concept: Maxima and Minima
Chapter: [3.02] Applications of Derivatives
OR
[6]12.ii

 Find the point on the straight line 2x+3y = 6,  which is closest to the origin. 

Concept: Maxima and Minima
Chapter: [3.02] Applications of Derivatives
[6]13

Evaluate: `int_0^x (xtan x)/(sec x + tan x) dx`

Concept: Introduction of Integrals
Chapter: [3.03] Integrals
[6]14
[6]14.i

Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person choose a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver. 

Concept: Introduction of Integrals
Chapter: [3.03] Integrals
OR
[6]14.ii

 Determine the binomial distribution where mean is 9 and standard deviation is `3/2` Also, find the probability of obtaining at most one success. 

Concept: Bernoulli Trials and Binomial Distribution
Chapter: [4] Probability (Section A)
Section - B(20 Marks)
[20]15
[2]15.i

If \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors, \[\left| \vec{a} + \vec{b} \right| = 13\] and \[\left| \vec{a} \right| = 5\] find the value of \[\left| \vec{b} \right|\]

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [5] Vectors (Section B)
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[2]15.ii

 Find the length of the perpendicular from origin to the plane `vecr. (3i - 4j-12hatk)+39 = 0`

Concept: Plane - Intercept Form of the Equation of a Plane
Chapter: [6] Three - Dimensional Geometry (Section B)
[2]15.iii

 Find the angle between the two lines `2x = 3y = -z and 6x =-y = -4z`

Concept: Angle Between Two Lines
Chapter: [6] Three - Dimensional Geometry (Section B)
[4]16
[4]16.i

(a)  If `veca  =  hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb`  are perpendicular.

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [5] Vectors (Section B)
OR
[4]16.ii

 Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [6] Three - Dimensional Geometry (Section B)
[4]17
[4]17.i

 Find the equation of the plane passing through the interesection of the planes 2x + 2y -3z -7 =0 and 2x +2y - 3z -7=0 such that the intercepts made by the resulting plane on the x - axis and the z - axis are equal. 

Concept: Intersection of the Line and Plane
Chapter: [6] Three - Dimensional Geometry (Section B)
OR
[4]17.ii

 Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [6] Three - Dimensional Geometry (Section B)
[6]18

Draw a rough sketch and find the area bounded by the curve x2 = y and x + y = 2.

Concept: Area Under Simple Curves
Chapter: [7] Application of Integrals (Section B)
[6]19
[2]19.i

A company produces a commodity with Rs. 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue received on selling the products, is at the rate of Rs. 8 per unit. Find the break-even point. 

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
Chapter: [8] Application of Calculus (Section C)
[2]19.ii

 The total cost function for a production is given by `C(x) = 3/4 x^2 - 7x +  27`
Find the number of units produced for which M.C. = A.C 
(M.C. = Marginal Cost and A. C. = Average Cost.) 

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
Chapter: [8] Application of Calculus (Section C)
[2]19.iii

If `vecx = 18, vecy=100,σ=20` , bary` and correlation coefficient xyr 0.8,  find the regression equation of y on x. 

Concept: Identification of Regression Equations
Chapter: [9] Linear Regression (Section C)
[4]20
[4]20.i

The following results were obtained with respect to two variables x and y. 
∑ x = 15 , ∑y = 25, ∑xy = 83, ∑xy = 55, ∑y=135 and n =5
(i) Find the regression coefficient xy b . 
(ii) Find the regression equation of x on y. 

Concept: Identification of Regression Equations
Chapter: [9] Linear Regression (Section C)
OR
[4]20.ii

Find the equation of the regression line of y on x, if the observations (x, y) are as follows : 
(1,4),(2,8),(3,2),(4,12),(5,10),(6,14),(7,16),(8,6),(9,18)
Also, find the estimated value of y when x = 14.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [9] Linear Regression (Section C)
[4]21
[4]21.i

The cost function of a product is given by C(x) =`x^3/3 - 45x^2 -  900x + 36` where x is the number of units produced. How many units should be produced to minimise the marginal cost?

Concept: Application of Calculus in Commerce and Economics in the Marginal Cost and Its Interpretation
Chapter: [8] Application of Calculus (Section C)
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OR
[4]21.ii

The marginal cost function of x units of a product is given by 2MC= 3x2 -10x +3x2 The cost of producing one unit is Rs. 7. Find the cost function and average cost function. 

Concept: Application of Calculus in Commerce and Economics in the Marginal Cost and Its Interpretation
Chapter: [8] Application of Calculus (Section C)
[6]22

A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for Rs. 48 per unit and product B is sold for Rs. 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum gross income? Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph.

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

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