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# Question Paper Solutions - Mathematics 2015 - 2016-CBSE 12th-Class 12 CBSE (Central Board of Secondary Education)

SubjectMathematics
Year2015 - 2016 (March)

#### Alternate Sets

Marks: 100
[1]1

If A =  ([cos alpha, sin alpha],[-sinalpha, cos alpha]) , find α satisfying 0 < α < pi/rwhen A+A^T=sqrt2I_2 where AT is transpose of A.

Chapter: [2.01] Matrices and Determinants
Concept: Operations on Matrices - Addition of Matrices
[1]2

If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k.

Chapter: [11] Matrices
Concept: Order of a Matrix
[1]3

For what values of k, the system of linear equations

x + y + z = 2
2x + y – z = 3
3x + 2y + kz = 4

has a unique solution?

Chapter: [2.02] Determinants
Concept: Elementary Transformations
[1]4

Write the sum of intercepts cut off by the plane vecr.(2hati+hatj-k)-5=0 on the three axes

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Plane - Intercept Form of the Equation of a Plane
[1]5

Find λ and μ if

(hati+3hatj+9k)xx(3hati-lambdahatj+muk)=0

Chapter: [2.02] Determinants
Concept: Determinant of a Square Matrix
[1]6

If veca=4hati-hatj+hatk then find a unit vector parallel to the vector veca+vecb

Chapter: [4.01] Vectors
Concept: Components of a Vector
[4]7 | Attempt Any One
[4]7.1

Solve for x : tan-1 (x - 1) + tan-1x + tan-1 (x + 1) = tan-1 3x

Chapter: [3] Trigonometric Functions
Concept: Properties of Inverse Trigonometric Functions
[4]7.2

Prove that tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3

Chapter: [3] Trigonometric Functions
Concept: Properties of Inverse Trigonometric Functions
[4]8

A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

Chapter: [2.01] Matrices
Concept: Inverse of a Matrix by Elementary Operations
[4]9

If f(x)= {((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}

is continuous at x = 0, then find the values of a and b.

Chapter: [3] Continuity
Concept: Continuous Function of Point
[4]10 | Attempt Any One
[4]10.1

If x cos(a+y)= cosy then prove that dy/dx=(cos^2(a+y)/sina)

Hence show that sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0

Chapter: [4] Differentiation
Concept: Second Order Derivative
[4]10.2

if y = sin^(-1)[(6x-4sqrt(1-4x^2))/5] Find dy/dx .

Chapter: [13] Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[4]11

Find the equation of tangents to the curve y= x3 + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.

Chapter: [14] Applications of Derivative
Concept: Tangents and Normals
[4]12 | Attempt Any One
[4]12.1

Find : int((2x-5)e^(2x))/(2x-3)^3dx

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[4]12.2

Find :int(x^2+x+1)/((x^2+1)(x+2))dx

Chapter: [3.03] Integrals
Concept: Integration as an Inverse Process of Differentiation
[4]13

Evaluate int_(-2)^2x^2/(1+5^x)dx

Chapter: [7] Definite Integrals
Concept: Properties of Definite Integrals
[4]14

Find : int(x+3)sqrt(3-4x-x^2dx)

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[4]15

Find the particular solution of differential equation:

dy/dx=-(x+ycosx)/(1+sinx) " given that " y= 1 " when "x = 0

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[4]16

Find the particular solution of the differential equation

2y ex/y dx + (y - 2x ex/y) dy = 0

given that x = 0 when y = 1.

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[4]17

Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4) are coplanar.

Chapter: [4.02] Three - Dimensional Geometry
Concept: Shortest Distance Between Two Lines
[4]18

Find the coordinates of the foot of perpendicular drawn from the point A (-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

Chapter: [4.02] Three - Dimensional Geometry
Concept: Three - Dimensional Geometry Examples and Solutions
[4]19 | Attempt Any one
[4]19.1

A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y.

Chapter: [22] Probability
Concept: Conditional Probability
[4]19.2

A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first

Chapter: [6.01] Probability
Concept: Probability Examples and Solutions
[6]20

Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X.Also, find the mean and variance of the distribution.

Chapter: [4] Probability (Section A)
Concept: Mean of a Random Variable
[6]21

LetA= R × R and * be a binary operation on A defined by (a, b) * (c, d) = (a+c, b+d)

Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element (a, b) ε A.

Chapter: [12.03] Functions
Concept: Concept of Binary Operations
[6]22 | Attempt Any One
[6]22.1

Prove that y=(4sintheta)/(2+costheta)-theta

Chapter: [3.02] Applications of Derivatives
Concept: Simple Problems on Applications of Derivatives
[6]22.2

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is cos^(-1)(1/sqrt3)

Chapter: [3.02] Applications of Derivatives
Concept: Simple Problems on Applications of Derivatives
[6]23

Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).

Chapter: [3.04] Applications of the Integrals
Concept: Area of the Region Bounded by a Curve and a Line
[6]24

Find the equation of the plane which contains the line of intersection of the planes

vecr.(hati-2hatj+3hatk)-4=0" and"

vecr.(-2hati+hatj+hatk)+5=0

and whose intercept on x-axis is equal to that of on y-axis.

Chapter: [6] Three - Dimensional Geometry (Section B)
Concept: Vector and Cartesian Equation of a Plane
[6]25

A retired person wants to invest an amount of Rs. 50, 000. His broker recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximise his returns.

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
[6]26 | Attempt Any One
[6]26.1

Using properties of determinants, prove that

|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3

Chapter: [2.02] Determinants
Concept: Properties of Determinants
[6]26.2

If A= ((1,0,2),(0,2,1),(2,0,3)) and A3 - 6A2 +7A + kI3 = O find k.

Chapter: [2.01] Matrices
Concept: Introduction of Operations on Matrices
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