Mathematics Patna Set 2 2014-2015 Science (English Medium) Class 12 Question Paper Solution

Mathematics [Patna Set 2]
Marks: 100 Academic Year: 2014-2015
Date: March 2015

[1] 1

Write the value of `vec a .(vecb xxveca)`

Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[1] 2

If `veca=hati+2hatj-hatk, vecb=2hati+hatj+hatk and vecc=5hati-4hatj+3hatk` then find the value of `(veca+vecb).vec c`

Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[1] 3

Write the direction ratios of the following line :

`x = −3, (y−4)/3 =( 2 −z)/1`

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [0.11] Three - Dimensional Geometry
[1] 4

If `A=[[2,3],[5,-2]]` then write A-1

Concept: Invertible Matrices
Chapter: [0.03] Matrices
[1] 5

Find the differential equation representing the curve y = cx + c2.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[1] 6

Write the integrating factor of the following differential equation:

(1+y2) dx(tan1 yx) dy=0

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
[4] 7

Using the properties of determinants, prove the following:


Concept: Properties of Determinants
Chapter: [0.04] Determinants
[4] 8

If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1cos 2t), show that `dy/dx=β/αtan t`

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.05] Continuity and Differentiability
[4] 9

Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
[4] 10

Find the derivative of the following function f(x) w.r.t. x, at x = 1 : 

`f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x`

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
[4] 11 | Attempt any one :
[4] 11.1

Evaluate :`int_0^(pi/2)(2^(sinx))/(2^(sinx)+2^(cosx))dx`

Concept: Fundamental Theorem of Calculus
Chapter: [0.07] Integrals
[4] 11.2

Evaluate `∫_0^(3/2)|x cosπx|dx`

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[4] 12

To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs 20, Rs 15 and Rs 5 per unit respectively. School A sold 25 paper bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper bags, 15 scrap-books and 28 pastel sheets while School C sold 26 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.

By such exhibition, which values are generated in the students?

Concept: Multiplication of Two Matrices
Chapter: [0.03] Matrices
[4] 13 | Attempt any one:
[4] 13.1

Prove that :

`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 13.2

Solve the following for x :


Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 14

If `A=[[2,0,1],[2,1,3],[1,-1,0]]` , find A2 − 5 A + 16 I.

Concept: Introduction of Operations on Matrices
Chapter: [0.03] Matrices
[4] 15

Show that four points A, B, C and D whose position vectors are 

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Concept: Coplanarity of Two Lines
Chapter: [0.11] Three - Dimensional Geometry
[4] 16 | Attempt any one
[4] 16.1

Show that the following two lines are coplanar:

`(x−a+d)/(α−δ)= (y−a)/α=(z−a−d)/(α+δ) and (x−b+c)/(β−γ)=(y−b)/β=(z−b−c)/(β+γ)`

Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
[4] 16.2

Find the acute angle between the plane 5x − 4y + 7z − 13 = 0 and the y-axis.

Concept: Angle Between Line and a Plane
Chapter: [0.11] Three - Dimensional Geometry
[4] 17 | Attempt any one
[4] 17.1

A and B throw a die alternatively till one of them gets a number greater than four and wins the game. If A starts the game, what is the probability of B winning?

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
[4] 17.2

A die is thrown three times. Events A and B are defined as below:
A : 5 on the first and 6 on the second throw.
B: 3 or 4 on the third throw.

Find the probability of B, given that A has already occurred.

Concept: Conditional Probability
Chapter: [0.13] Probability
[4] 18

Evaluate :


Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 19



Concept: Integrals of Some Particular Functions
Chapter: [0.07] Integrals
[6] 20

Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[6] 21 | Attempt any one :
[6] 21.1

Find the the differential equation for all the straight lines, which are at a unit distance from the origin.

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [0.09] Differential Equations
[6] 21.2

Show that the differential  equation `2xydy/dx=x^2+3y^2`  is homogeneous and solve it.

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [0.09] Differential Equations
[6] 22

Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle π/4 with the plane x + y = 3. Also find the equation of the plane

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
[6] 23 | Attempt any one :
[6] 23.1

If the function f : R → R be defined by f(x) = 2x − 3 and g : R → R by g(x) = x3 + 5, then find the value of (fog)−1 (x).

Concept: Composition of Functions and Invertible Function
Chapter: [0.01] Relations and Functions
[6] 23.2

Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) =  (ac, b + ad), for all (a, b) (c, d) ∈ A.
(i) the identity element in A
(ii) the invertible element of A.

(iii)and hence write the inverse of elements (5, 3) and (1/2,4)

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[6] 24

If the function f(x)=2x39mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.


Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
[6] 25

The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10. According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively. A man receives Rs 225 a day and a woman receives Rs 200 a day. How many men and women helpers should be hired to keep the pay-roll at a minimum ? Formulate an LPP and solve it graphically.

Concept: Mathematical Formulation of Linear Programming Problem
Chapter: [0.12] Linear Programming
[6] 26

40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ?

Concept: Conditional Probability
Chapter: [0.13] Probability

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