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

Mathematics [Foreign Set 2]
Date & Time: 14th March 2016, 10:30 am
Duration: 3h

 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
 2

If veca","vecb","veccare 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
 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
 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
 5

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

Concept: Operations on Matrices - Multiplication of Matrices
Chapter: [0.03] Matrices
 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
 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
 8 | Attempt any one of the following
 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
 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
 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
 10

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

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
 11 | Attempt any one of the following
 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
 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
 12

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

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
 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
 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
 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
 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
 17 | Attempt any one of the following
 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
 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
 18 | Attempt any one of the following
 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
 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
 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
 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
 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
 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
 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
 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
 25 | Attempt any one of the following
 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
 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
 26 | Attempt any one of the following
 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
 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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