Board Paper of Class 12-Science 2013 Maths (SET 3) - Solutions
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted.
- Question 1
Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line.VIEW SOLUTION
- Question 2
Write unit vector in the direction of the sum of vectors andVIEW SOLUTION
- Question 3
If and are two equal vectors, then write the value of
x + y + z.VIEW SOLUTION
- Question 4
Write the degree of the differential equation.VIEW SOLUTION
- Question 5
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.VIEW SOLUTION
- Question 6
If , then find the matrix A.VIEW SOLUTION
- Question 7
If , then write the value of x.VIEW SOLUTION
- Question 8
Find the value of a ifVIEW SOLUTION
- Question 9
Write the value of.VIEW SOLUTION
- Question 10
Write the principal value of.VIEW SOLUTION
- Question 11
A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are the likely to contradict each other in stating the same fact?
Do you think that statement of B is true?VIEW SOLUTION
- Question 12
Find the coordinates of the point, where the line intersects the plane x − y + z − 5 = 0. Also find the angle between the line and the plane.
Find the vector equation of the plane which contains the line of intersection of the planes and which is perpendicular to the plane.VIEW SOLUTION
- Question 13
Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).VIEW SOLUTION
- Question 14
Evaluate: .VIEW SOLUTION
- Question 15
Evaluate: .VIEW SOLUTION
- Question 16
- Question 17
Show that the function is continuous but not differentiable at x = 3.
If x = a sin t and y = a, findVIEW SOLUTION
- Question 18
If, show that.VIEW SOLUTION
- Question 19
Differentiate the following function with respect to x:VIEW SOLUTION
- Question 20
Using properties of determinants prove the following:
- Question 21
Find the value of the following:
Prove that:VIEW SOLUTION
- Question 22
Show that the function f in defined as is one-one and onto.
Hence find f −1.VIEW SOLUTION
- Question 23
Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.VIEW SOLUTION
- Question 24
A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?VIEW SOLUTION
- Question 25
Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane, passing through the points (2, 2, 1), (3, 0, 1) and (4, −1, 0).VIEW SOLUTION
- Question 26
Show that the differential equation is homogeneous. Find the particular solution of this differential equation, given that x = 1 when y = 1.VIEW SOLUTION
- Question 27
Using integration, find the area bounded by the curve x2 = 4y and the line x = 4y − 2.
Using integration, find the area of the region enclosed between the two circles x2 + y2 = 4 and (x − 2)2 + y2 = 4.VIEW SOLUTION
- Question 28
Show that the height of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is. Also find the maximum volume.
Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.VIEW SOLUTION
- Question 29
A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs 6,000. Three times the award money for Hard work added to that given for honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.VIEW SOLUTION