Mathematics Delhi Set 2 2015-2016 Science (English Medium) Class 12 Question Paper Solution

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

[1] 1

Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b

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

Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1

Concept: Basic Concepts of Vector Algebra
Chapter: [0.1] Vectors
[1] 3

The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A

Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
Chapter: [0.1] Vectors
[1] 4

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is `2hati-3hatj+6hatk`

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
[1] 5

Find the maximum value of `|(1,1,1),(1,1+sintheta,1),(1,1,1+costheta)|`

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

 If A is a square matrix such that A2 = I, then find the simplified value of (A – I)3 + (A + I)3 – 7A.

Concept: Types of Matrices
Chapter: [0.03] Matrices
[4] 7

Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[4] 8 | Attempt Any One
[4] 8.1

Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 8.2

Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`

Concept: Methods of Integration: Integration by Parts
Chapter: [0.07] Integrals
[4] 9

Find `intsqrtx/sqrt(a^3-x^3)dx`

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 10

Evaluate `int_(-1)^2|x^3-x|dx`

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[4] 11

Find the particular solution of the differential equation

(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[4] 12

Find the general solution of the following differential equation : 

`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[4] 13

Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[4] 14

Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines. 

`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " and "vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk)`



Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
[4] 15 | Attempt Any One
[4] 15.1

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C

Concept: Bayes’ Theorem
Chapter: [0.13] Probability
[4] 15.2

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
[4] 16 | Attempt Any One
[4] 16.1

Prove that:


Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 16.2

Solve for x:

`2tan^(-1)(cosx)=tan^(-1)(2"cosec" x)`

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 17

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
Chapter: [0.04] Determinants
[4] 18 | Attempt Any One
[4] 18.1

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  `dy/dx `at t = `pi/4`

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.05] Continuity and Differentiability
[4] 18.2

If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
[4] 19

Find the values of p and q for which

f(x) = `{((1-sin^3x)/(3cos^2x),`

is continuous at x = π/2.

Concept: Concept of Continuity
Chapter: [0.05] Continuity and Differentiability
[6] 20 | Attempt Any One
[6] 20.1

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
[6] 20.2

Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[6] 21

Using integration find the area of the region {(x, y) : x2+y2 2ax, y2 ax, x, y  0}.

Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals
[6] 22

Find the coordinate of the point P where the line through A(3, –4, –5) and B(2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0).
Also, find the ratio in which P divides the line segment AB.

Concept: Section Formula
Chapter: [0.1] Vectors
[6] 23

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

Concept: Mean of a Random Variable
Chapter: [0.13] Probability
[6] 24

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 25

Let f : N→N be a function defined as f(x)=`9x^2`+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find `f^-1`(43) and` f^−1`(163).

Concept: Inverse of a Function
Chapter: [0.01] Relations and Functions
[6] 26 | Attempt Any One
[6] 26.1

Prove that  `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.

Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
[6] 26.2

Using elementary transformations, find the inverse of the matrix A =  `((8,4,3),(2,1,1),(1,2,2))`and use it to solve the following system of linear equations :

8x + 4y + 3z = 19

2xyz = 5

x + 2y + 2z = 7

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

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