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Mathematics Foreign Set 2 2015-2016 CBSE (Commerce) Class 12 Question Paper Solution

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Mathematics [Foreign Set 2]
Marks: 100 Academic Year: 2015-2016
Date & Time: 14th March 2016, 10:30 am
Duration: 3h
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[1] 1

If A`((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.

 

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.03] Matrices
[1] 2

If `veca","vecb","vecc`are unit vectors such that `veca+vecb+vecc=0`, then write the value of  `veca.vecb+vecb.vecc+vecc.veca`

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[1] 3

if `|vecaxxvecb|^2+|veca.vecb|^2=400 ` and `|vec a| = 5` , then write the value of `|vecb|`

Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[1] 4

Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.

Concept: Distance of a Point from a Plane
Chapter: [0.11] Three - Dimensional Geometry
[1] 5

If `[2     1       3]([-1,0,-1],[-1,1,0],[0,1,1])([1],[0],[-1])=A` , then write the order of matrix A.

Concept: Operations on Matrices - Multiplication of Matrices
Chapter: [0.03] Matrices
[1] 6

If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.

Concept: Applications of Determinants and Matrices
Chapter: [0.04] Determinants
[4] 7

Find the values of a and b, if the function f defined by 

\[f\left( x \right) = \begin{cases}x^2 + 3x + a & , & x \leqslant 1 \\ bx + 2 & , & x > 1\end{cases}\] is differentiable at = 1.
Concept: Algebra of Continuous Functions
Chapter: [0.05] Continuity and Differentiability
[4] 8 | Attempt any one of the following
[4] 8.1

Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
[4] 8.2

If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
[4] 9

Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .

 
Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[4] 10

Evaluate : \[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\] .

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
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[4] 11 | Attempt any one of the following
[4] 11.1

Find : \[\int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\] .

Concept: Integrals of Some Particular Functions
Chapter: [0.07] Integrals
[4] 11.2

Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .

Concept: Integration as an Inverse Process of Differentiation
Chapter: [0.07] Integrals
[4] 12

Find : \[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\] .

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[4] 13

Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .

Concept: Basic Concepts of Differential Equation
Chapter: [0.09] Differential Equations
[4] 14

Solve the following differential equation: \[\left( \cot^{- 1} y + x \right) dy = \left( 1 + y^2 \right) dx\] .

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [0.09] Differential Equations
[4] 15

If \[\vec{a}  \times  \vec{b}  =  \vec{c}  \times  \vec{d}   \text { and }   \vec{a}  \times  \vec{c}  =  \vec{b}  \times  \vec{d}\] , show that \[\vec{a}  -  \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .

Concept: Basic Concepts of Vector Algebra
Chapter: [0.1] Vectors
[4] 16

Prove that the lines through A (0, −1, −1) and B (4, 5, 1) intersects the line through C (3, 9, 4) and D (−4, 4, 4). Also, find their point of intersection. 

Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
[4] 17 | Attempt any one of the following
[4] 17.1

A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.

Concept: Conditional Probability
Chapter: [0.13] Probability
[4] 17.2

Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in number of colleges. It is given that

\[P\left( X = x \right) = \begin{cases}kx & , & if x = 0 or 1 \\ 2 kx & , & if x = 2 \\ k\left( 5 - x \right) & , & if x = 3 or 4 \\ 0 & , & if x > 4\end{cases}\]

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.13] Probability
[4] 18 | Attempt any one of the following
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[4] 18.1

Prove that : \[\cot^{- 1} \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} = \frac{x}{2}, 0 < x < \frac{\pi}{2}\] .

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 18.2

Solve for x : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 19

A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?

Concept: Types of Matrices
Chapter: [0.03] Matrices
[6] 20

Using integration find the area of the region bounded by the curves \[y = \sqrt{4 - x^2}, x^2 + y^2 - 4x = 0\] and the x-axis.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[6] 21

Find the equation of the plane which contains the line of intersection of the planes \[x + 2y + 3z - 4 = 0 \text { and } 2x + y - z + 5 = 0\] and whose x-intercept is twice its z-intercept.

Concept: Plane - Equation of a Plane in Normal Form
Chapter: [0.11] Three - Dimensional Geometry
[6] 22

Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B.

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
[6] 23

In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :

Tablets  Iron Calcium Vitamin
x 6 3 2
y 2 3 4

The person needs to supplement at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 24

 If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

Concept: Types of Functions
Chapter: [0.01] Relations and Functions
[6] 25 | Attempt any one of the following
[6] 25.1

If \[a, b\] and c  are all non-zero and 

\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c\end{vmatrix} =\] 0, then prove that 
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} +\]1
= 0

 

Concept: Applications of Determinants and Matrices
Chapter: [0.04] Determinants
[6] 25.2

If  \[A = \begin{pmatrix}\cos\alpha & - \sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix},\] ,find adj·A and verify that A(adj·A) = (adj·A)A = |A| I3.

Concept: Operations on Matrices - Multiplication of Matrices
Chapter: [0.03] Matrices
[6] 26 | Attempt any one of the following
[6] 26.1

The sum of the surface areas of a cuboid with sides x, 2x and \[\frac{x}{3}\] and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of  the sum of their volumes.

Concept: Graph of Maxima and Minima
Chapter: [0.06] Applications of Derivatives
[6] 26.2

Find the equation of tangents to the curve y = cos(+ y), –2π ≤ x ≤ 2π that are parallel to the line + 2y = 0.

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives

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