# Mathematics All India Set 1 N 2015-2016 CBSE (Commerce) Class 12 Question Paper Solution

Mathematics [All India Set 1 N]
Date & Time: 14th March 2016, 10:30 am
Duration: 3h

 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.

Concept: Operations on Matrices - Addition of Matrices
Chapter: [0.03] Matrices
 2

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

Concept: Order of a Matrix
Chapter: [0.03] Matrices
 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?

Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
 4

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

Concept: Plane - Intercept Form of the Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
 5

Find λ and μ if

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

Concept: Determinant of a Square Matrix
Chapter: [0.04] Determinants
 6

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

Concept: Components of a Vector
Chapter: [0.1] Vectors
 7 | Attempt Any One
 7.1

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

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
 7.2

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

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
 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?

Concept: Inverse of Matrix - Inverse of a Nonsingular Matrix by Elementary Transformation
Chapter: [0.03] Matrices
 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.

Concept: Continuous Function of Point
Chapter: [0.05] Continuity and Differentiability
 10 | Attempt Any One
 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

Concept: Second Order Derivative
Chapter: [0.05] Continuity and Differentiability
 10.2

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

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
 11

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

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
 12 | Attempt Any One
 12.1

Find : int((2x-5)e^(2x))/(2x-3)^3dx

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
 12.2

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

Concept: Integration as an Inverse Process of Differentiation
Chapter: [0.07] Integrals
 13

Evaluate int_(-2)^2x^2/(1+5^x)dx

Concept: Properties of Definite Integrals
Chapter: [0.07] Integrals
 14

Find : int(x+3)sqrt(3-4x-x^2dx)

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
 15

Find the particular solution of differential equation:

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

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
 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.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
 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.

Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
 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.

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
 19 | Attempt Any one
 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.

Concept: Conditional Probability
Chapter: [0.13] Probability
 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

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
 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.

Concept: Mean of a Random Variable
Chapter: [0.13] Probability
 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.

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
 22 | Attempt Any One
 22.1

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

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

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
 23

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

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

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
 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.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
 26 | Attempt Any One
 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

Concept: Properties of Determinants
Chapter: [0.04] Determinants
 26.2

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

Concept: Introduction of Operations on Matrices
Chapter: [0.03] Matrices

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