Mathematics Delhi Set 2 2016-2017 Science (English Medium) Class 12 Question Paper Solution

Mathematics [Delhi Set 2]
Marks: 100 Academic Year: 2016-2017
Date & Time: 19th March 2017, 12:30 pm
Duration: 3h

[1] 1

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[2] 2

Evaluate : `int_2^3 3^x dx`

Concept: Integrals of Some Particular Functions
Chapter: [0.07] Integrals
[1] 3

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k

Concept: Invertible Matrices
Chapter: [0.03] Matrices
[4] 4

Determine the value of the constant 'k' so that function f(x) `{((kx)/|x|, ","if  x < 0),(3"," , if x >= 0):}` is continuous at x = 0

Concept: Concept of Continuity
Chapter: [0.05] Continuity and Differentiability
[2] 5

Prove that if E and F are independent events, then the events E and F' are also independent. 

Concept: Independent Events
Chapter: [0.13] Probability
[2] 6

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?

It is being given that at least one of each must be produced.

Concept: Linear Programming Problem and Its Mathematical Formulation
Chapter: [0.12] Linear Programming
[2] 7

Find  `int dx/(x^2 + 4x + 8)`

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
[2] 8

Show that all the diagonal elements of a skew symmetric matrix are zero.

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.03] Matrices
[2] 9

Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[2] 10

Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[2] 11

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Concept: Product of Two Vectors - Projection of a Vector on a Line
Chapter: [0.1] Vectors
[2] 12

For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Chapter: [0.09] Differential Equations
[4] 13 | Attempt Any One

Evaluate `int_0^pi (x sin x)/(1 + cos^2 x) dx`

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals

Evaluate `int_0^(3/2) |x sin pix|dx`

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
[4] 14

Prove that x2 – y2 = c(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2)dx = (y3 – 3x2y)dy, where C is parameter

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [0.09] Differential Equations
[4] 15

Let `veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk` then

1) Let `c_1 = 1` and `c_2 = 2`, find `c_3` which makes `veca, vecb "and" vecc`coplanar

2) if `c_2 = -1` and `c_3 = 1`, show that no value of `c_1`can make `veca, vecb and vecc` coplanar

Concept: Scalar Triple Product of Vectors
Chapter: [0.1] Vectors
[4] 16

Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

Do you also agree that the value of truthfulness leads to more respect in the society?

Concept: Bayes’ Theorem
Chapter: [0.13] Probability
[4] 17

Prove that `tan {pi/4 + 1/2 cos^(-1)  a/b} + tan {pi/4 - 1/2 cos^(-1)  a/b} = (2b)/a`

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 18 | Attempt Any One

Using properties of determinants, prove that `|(x,x+y,x+2y),(x+2y, x,x+y),(x+y, x+2y, x)| = 9y^2(x + y)`

Concept: Properties of Determinants
Chapter: [0.04] Determinants

Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O

Concept: Types of Matrices
Chapter: [0.03] Matrices
[4] 19 | Attempt Any One

Differentiate the function with respect to x.

`(sin x)^x + sin^(-1) sqrtx`

Concept: Logarithmic Differentiation
Chapter: [0.05] Continuity and Differentiability

if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`

Concept: Logarithmic Differentiation
Chapter: [0.05] Continuity and Differentiability
[4] 20

The random variable X can take only the values 0, 1, 2, 3. Given that P(2) = P(3) = p and P(0) = 2P(1). if `Sigmap_ix_i^2 = 2Sigmap_ix_i`, Find the value of p.

Concept: Variance of a Random Variable
Chapter: [0.13] Probability
[4] 21

Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).

Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[4] 22

Solve the following L.P.P. graphically Maximise Z = 4x + y 

Subject to following constraints  x + y ≤ 50

3x + y ≤ 90,

x ≥ 10

x, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[4] 23

Find `int (2x)/((x^2 + 1)(x^4 + 4))`dx

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
[6] 24 | Attempt Any One

Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).

Concept: Area Between Two Curves
Chapter: [0.08] Applications of the Integrals

Find the area bounded by the circle x2 + y2 = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.

Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals
[6] 25

Solve the differential equation `x dy/dx + y = x cos x + sin x`,  given that y = 1 when `x = pi/2`

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [0.09] Differential Equations
[6] 26 | Attempt Any One

Find the equation of the plane through the line of intersection of `vecr*(2hati-3hatj + 4hatk) = 1`and `vecr*(veci - hatj) + 4 =0`and perpendicular to the plane `vecr*(2hati - hatj + hatk) + 8 = 0`. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.

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

Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines `(x - 8)/3 = (y + 19)/(-16) = (z - 10)/7` and `(x - 15)/3 = (y - 29)/8 = (z - 5)/(-5)`

Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
[6] 27 | Attempt Any One

Consider f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`. Show that f is invertible with `f^(-1) (y) ((sqrt(y + 6)-1)/3)`

Hence Find

1) `f^(-1)(10)`

2) y if `f^(-1) (y) = 4/3`

where R+ is the set of all non-negative real numbers.

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

Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * ba − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[6] 28

A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box

Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
[6] 29

if A =  `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`

Concept: Minors and Co-factors
Chapter: [0.04] Determinants

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