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Mathematics All India Set 1 E 2015-2016 CBSE (Science) Class 12 Question Paper Solution

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Mathematics [All India Set 1 E]
Marks: 100 Academic Year: 2015-2016
Date & Time: 14th March 2016, 10:30 am
Duration: 3h
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(i) All questions are compulsory.

(ii) Please check that this question paper contains 26 questions.

(iii) Questions 1 - 6 in Section A are very short-answer type questions carrying 1 mark each.

(iv) Questions 7 - 19 in Section B are long-answer I type questions carrying 4 marks each.

(v) Questions 20 - 26 in Section C are long-answer II type questions carrying 6 marks each.


SECTION-A
[1] 1

Write the value of `|(a-b, b- c, c-a),(b-c, c-a, a-b),(c-a, a-b, b-c)|`

Concept: Applications of Determinants and Matrices
Chapter: [2.01] Determinants
[1] 2

If A = `((1, -2, 3),(-4,2,5)) and B = ((2,3),(4,5),(2,1))` and BA = (bij), find b21 + b32.

Concept: Operations on Matrices - Properties of Scalar Multiplication of a Matrix
Chapter: [2.02] Matrices
[1] 3

Write the number of all possible matrices of order 2 x 3  with each entry 1 or 2.

Concept: Order of a Matrix
Chapter: [2.02] Matrices
[1] 4

Write the coordinates of the point which is the reflection of the point (α, β,  γ) in the XZ-plane.

Concept: Distance of a Point from a Plane
Chapter: [4.01] Three - Dimensional Geometry
[1] 5

Find the position vector of the point which divides the join of points with position vectors `vec"a" + 3vec"b" and vec"a"- vec"b"` internally in the ratio 1 : 3. 

Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
Chapter: [4.02] Vectors
[1] 6

If `|vec"a"| = 4, |vec"b"| = 3` and `vec"a".vec"b" = 6 sqrt(3)`, then find the value of `|vec"a" xx vec"b"|`.

Concept: Multiplication of a Vector by a Scalar
Chapter: [4.02] Vectors
SECTION – B
[4] 7
[4] 7.1

Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1  ("x")/(2), "x">0.`

Concept: Properties of Inverse Trigonometric Functions
Chapter: [1.01] Inverse Trigonometric Functions
OR
[4] 7.2

Prove that : `2sin^-1 (3/5) -tan^-1 (17/31) = pi/4.`

Concept: Proof Derivative X^n Sin Cos Tan
Chapter: [3.01] Continuity and Differentiability
[4] 8

On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, everyone would have got ₹ 10 more. However, if there were 16 children more, everyone would have got ₹ 10 less. Using the matrix method, find the number of children and the amount distributed by Seema. What values are reflected by Seema’s decision?

Concept: Applications of Determinants and Matrices
Chapter: [2.01] Determinants
[4] 9
[4] 9.1

If `"x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.

Concept: Exponential and Logarithmic Functions
Chapter: [3.01] Continuity and Differentiability
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OR
[4] 9.2

Verify Mean value theorem for the function f(x) = 2sin x + sin 2x on [0, π].

Concept: Mean Value Theorem
Chapter: [3.01] Continuity and Differentiability
[4] 10

Show that the function f given by:

`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`

is discontinuous at x = 0.

Concept: Concept of Continuity
Chapter: [3.01] Continuity and Differentiability
[4] 11

Find the equation of the tangent line to the curve `"y" = sqrt(5"x" -3) -5`, which is parallel to the line  `4"x" - 2"y" + 5 = 0`.

Concept: Tangents and Normals
Chapter: [3.02] Applications of Derivatives
[4] 12
[4] 12.1

Evaluate: `int_1^5{|"x"-1|+|"x"-2|+|"x"-3|}d"x"`.

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [3.05] Integrals
OR
[4] 12.2

Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.

Concept: Properties of Definite Integrals
Chapter: [3.05] Integrals
[4] 13

Find : `int_  (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.

Concept: Properties of Definite Integrals
Chapter: [3.05] Integrals
[4] 14

Find: `int_  (3"x"+ 5)sqrt(5 + 4"x"-2"x"^2)d"x"`.

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [3.05] Integrals
[4] 15

Solve the differential equation : `"x"(d"y")/(d"x") + "y" - "x" + "xy"cot"x" = 0; "x" != 0.`

Concept: Solutions of Linear Differential Equation
Chapter: [3.04] Differential Equations
[4] 16

Solve the differential equation : `("x"^2 + 3"xy" + "y"^2)d"x" - "x"^2 d"y" = 0  "given that"  "y" = 0  "when"  "x" = 1`.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [3.04] Differential Equations
[4] 17

Find the angle between the vectors `vec"a" + vec"b" and  vec"a" -vec"b" if  vec"a" = 2hat"i"-hat"j"+3hat"k" and vec"b" = 3hat"i" + hat"j"-2hat"k", and"hence find a vector perpendicular to both"  vec"a" + vec"b" and vec"a" - vec"b"`.

Concept: Vectors Examples and Solutions
Chapter: [4.02] Vectors
[4] 18

Show that the lines `("x"-1)/(3) = ("y"-1)/(-1) = ("z"+1)/(0) = λ and  ("x"-4)/(2) = ("y")/(0) = ("z"+1)/(3)` intersect. Find their point of intersection. 

Concept: Plane - Plane Passing Through the Intersection of Two Given Planes
Chapter: [4.01] Three - Dimensional Geometry
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[4] 19
[4] 19.1

A committee of 4 students is selected at random from a group consisting of 7 boys and 4 girls. Find the probability that there are exactly 2 boys in the committee, given that at least one girl must be there in the committee.

Concept: Introduction of Probability
Chapter: [6.01] Probability
OR
[4] 19.2

A random variable X has the following probability distribution :

X 0 1 2 3 4 5 6
P(X) C 2C 2C 3C C2 2C2 7C2+C

Find the value of C and also calculate the mean of this distribution.

Concept: Random Variables and Its Probability Distributions
Chapter: [6.01] Probability
SECTION – C
[6] 20

Show that the relation R defined by (a, b)R(c,d) ⇒ a + d = b + c   on the A x A  , where A =  {1, 2,3,...,10}  is an equivalence relation. Hence write the equivalence class [(3, 4)]; a, b, c,d ∈ A.

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
[6] 21
[6] 21.1

Solve for x : `|("a"+"x","a"-"x","a"-"x"),("a"-"x","a"+"x","a"-"x"),("a"-"x","a"-"x","a"+"x")| = 0`, using properties of determinants. 

Concept: Properties of Determinants
Chapter: [2.01] Determinants
OR
[6] 21.2

Using elementary row operations, find the inverse of the matrix A = `((3, 3,4),(2,-3,4),(0,-1,1))` and hence solve the following system of equations :  3x - 3y + 4z = 21, 2x -3y + 4z = 20, -y + z = 5.

Concept: Elementary Transformations
Chapter: [2.01] Determinants [2.02] Matrices
[6] 22
[6] 22.1

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.

Concept: Maximum and Minimum Values of a Function in a Closed Interval
Chapter: [3.02] Applications of Derivatives
OR
[6] 22.2

Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing. 

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
[6] 23

Using integration find the area of the triangle formed by negative x-axis and tangent and normal to the circle `"x"^2 + "y"^2 = 9  "at" (-1,2sqrt2)`.

Concept: Area Under Simple Curves
Chapter: [3.03] Applications of the Integrals
[6] 24

Find the coordinates of the foot of perpendicular and perpendicular distance from the point P(4,3,2) to the plane x + 2y + 3z = 2. Also find the image of P in the plane.

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [4.01] Three - Dimensional Geometry
[6] 25

A, B and C throw a pair of dice in that order alternatively till one of them gets a total of 9 and wins the game. Find their respective probabilities of winning, if A starts first.

Concept: Probability Examples and Solutions
Chapter: [6.01] Probability
[6] 26

A company manufactures two types of cardigans: type A and type B. It costs ₹ 360 to make a type A cardigan and ₹ 120 to make a type B cardigan. The company can make at most 300 cardigans and spend at most ₹ 72000 a day. The number of cardigans of type B cannot exceed the number of cardigans of type A by more than 200. The company makes a profit of ₹ 100 for each cardigan of type A and ₹ 50 for every cardigan of type B. 

Formulate this problem as a linear programming problem to maximize the profit to the company. Solve it graphically and find the maximum profit.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [5.01] Linear Programming
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