Mathematics All India Set 1 N 2015-2016 Science (English Medium) Class 12 Question Paper Solution

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

[1] 1

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

Concept: Algebraic Operations on Matrices - Addition of Matrices
Chapter: [0.03] Matrices
[1] 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
[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?


Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
[1] 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
[1] 5

Find λ and μ if


Concept: Determinant of a Square Matrix
Chapter: [0.04] Determinants
[1] 6

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

Concept: Components of Vector
Chapter: [0.1] Vectors
[4] 7 | Attempt Any One
[4] 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
[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`

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

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

Concept: Continuous Function of Point
Chapter: [0.05] Continuity and Differentiability
[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`

Concept: Second Order Derivative
Chapter: [0.05] Continuity and Differentiability
[4] 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
[4] 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
[4] 12 | Attempt Any One
[4] 12.1

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 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
[4] 13

Evaluate `int_(-2)^2x^2/(1+5^x)dx`

Concept: Properties of Definite Integrals
Chapter: [0.07] Integrals
[4] 14

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 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
[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.

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

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

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
[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.

Concept: Conditional Probability
Chapter: [0.13] 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

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

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

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[6] 22 | Attempt Any One
[6] 22.1

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

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

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

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[6] 24

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

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


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
[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.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 26 | Attempt Any One
[6] 26.1

Using properties of determinants, prove that



Concept: Properties of Determinants
Chapter: [0.04] Determinants
[6] 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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