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

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

[1] 1

Write the number of vectors of unit length perpendicular to both the vectors `veca=2hati+hatj+2hatk and vecb=hatj+hatk`

Concept: Components of Vector
Chapter: [0.1] Vectors
[1] 2

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

Concept: Order of a Matrix
Chapter: [0.03] Matrices
[1] 3

If ` x in N and |[x+3,-2],[-3x,2x]|=8` , then find the value of x.

Concept: Determinants of Matrix of Order One and Two
Chapter: [0.04] Determinants
[1] 4

Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.

Concept: Basic Concepts of Vector Algebra
Chapter: [0.1] Vectors
[1] 5

Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
[1] 6

Use elementary column operation C2 → C2 + 2C1 in the following matrix equation :

`[[2,1],[2,0]] = [[3,1],[2,0]] [[1,0],[-1,1]]`

Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
[4] 7

The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[4] 8

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
[4] 9

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

Concept: Integrals of Some Particular Functions
Chapter: [0.07] Integrals
[4] 10

The two adjacent sides of a parallelogram are `2hati-4hatj-5hatk and 2 hati+2hatj+3hatj` . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Concept: Geometrical Interpretation of Scalar
Chapter: [0.1] Vectors
[4] 11

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[4] 12 | Attempt any one
[4] 12.1

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses

Concept: Conditional Probability
Chapter: [0.13] Probability
[4] 12.2

A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?

Concept: Independent Events
Chapter: [0.13] Probability
[4] 13

A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question?

Concept: Invertible Matrices
Chapter: [0.03] Matrices
[4] 14 | Attempt any one
[4] 14.1

Differentiate xsinx+(sinx)cosx with respect to x.

Concept: Derivative - Exponential and Log
Chapter: [0.05] Continuity and Differentiability
[4] 14.2

If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`

Concept: Second Order Derivative
Chapter: [0.05] Continuity and Differentiability
[4] 15 | Attempt any one
[4] 15.1

Solve the equation for x:sin1x+sin1(1x)=cos1x

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 15.2

If `cos^-1( x/a) +cos^-1 (y/b)=alpha` , prove that `x^2/a^2-2(xy)/(ab) cos alpha +y^2/b^2=sin^2alpha`

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 16

 If x=a sin 2t(1+cos 2t) and y=b cos 2t(1cos 2t), find `dy/dx `

Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.05] Continuity and Differentiability
[4] 17

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [0.09] Differential Equations
[4] 18 | Attempt any one
[4] 18.1

Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`

Concept: Fundamental Theorem of Calculus
Chapter: [0.07] Integrals
[4] 18.2

Evaluate `∫_0^(3/2)|x cosπx|dx`

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[4] 19

Find : `int x^2/(x^4+x^2-2) dx`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.07] Integrals
[6] 20 | Attempt any one
[6] 20.1

Using properties of determinants, show that ΔABC is isosceles if:`|[1,1,1],[1+cosA,1+cosB,1+cosC],[cos^2A+cosA,cos^B+cosB,cos^2C+cosC]|=0​`

Concept: Properties of Determinants
Chapter: [0.04] Determinants
[6] 20.2

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.

Concept: Inverse of Matrix - Inverse of a Matrix by Elementary Transformation
Chapter: [0.03] Matrices
[6] 21

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs Rs 10 per kg and 'B' cost Rs 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 22 | Attempt any one:
[6] 22.1

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[6] 22.2

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.

Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
[6] 23

Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

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

Prove that the curves y2 = 4x and x2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[6] 25

Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[6] 26

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector

`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.

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

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