Mathematics Delhi Set 3 2016-2017 Science (English Medium) Class 12 Question Paper Solution

Mathematics [Delhi Set 3]
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
[1] 2

Evaluate : int_2^3 3^x dx

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

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
[1] 4

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
[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

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] 9

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] 10

The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm

Concept: Rate of Change of Bodies or Quantities
Chapter: [0.06] Applications of Derivatives
[2] 11

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

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.03] Matrices
[2] 12

if y = sin^(-1) (6xsqrt(1-9x^2)), 1/(3sqrt2) < x < 1/(3sqrt2) then find (dy)/(dx)

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

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" vecccoplanar

2) if c_2 = -1 and c_3 = 1, show that no value of c_1can make veca, vecb and vecc coplanar

Concept: Scalar Triple Product of Vectors
Chapter: [0.1] Vectors
[4] 14

If veca, vecb, vecc are mutually perpendicular vectors of equal magnitudes, show that the vector veca +  vecb+ vecc is equally inclined to veca, vecb and vecc.

Concept: Magnitude and Direction of a Vector
Chapter: [0.1] Vectors

If veca, vecb, vecc are mutually perpendicular vectors of equal magnitudes, find the angle which veca + vecb + veccmake with veca or vecb or vecc

Concept: Magnitude and Direction of a Vector
Chapter: [0.1] Vectors
[4] 15

The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that Sigmap_i x_i^2 = 2Sigmap_ix_i. Find the value of p

Concept: Variance of a Random Variable
Chapter: [0.13] Probability
[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 | 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] 18 | 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] 19 | 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] 20

Solve the following L.P.P graphically: Maximise Z = 20x + 10y

Subject to the following constraints x + 2y ≤ 28,

3x + y ≤ 24,

x ≥ 2,

x, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[4] 21

Show that the family of curves for which dy/dx = (x^2+y^2)/(2x^2) is given by  x2 - y2 = cx

Concept: Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Chapter: [0.09] Differential Equations
[4] 22

Find int((3 sin x - 2) cos x)/(13 - cos^2 x- 7 sin x) dx

Concept: Integration Using Trigonometric Identities
Chapter: [0.07] Integrals
[4] 23

Solve the following equation for x:  cos (tan^(-1) x) = sin (cot^(-1)  3/4)

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

Find the equation of the plane through the line of intersection of vecr*(2hati-3hatj + 4hatk) = 1and vecr*(veci - hatj) + 4 =0and 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] 26 | 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] 27

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] 28

if A = ((2,3,1),(1,2,2),(-3,1,-1)), Find A^(-1) and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8

Concept: Invertible Matrices
Chapter: [0.03] Matrices
[6] 29

Find the particular solution of the differential equation

tan x * (dy)/(dx) = 2x tan x + x^2 - y; (tan x != 0) given that y = 0 when x = pi/2

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

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