**CBSE Previous Year Solved Papers Class 12 Maths Delhi 2010**

**Time allowed : 3 hours Maximum Marks: 100**

**General Instructions:**

**All questions are compulsory.****Please check that this question paper contains 26 questions.****Questions 1-6 in Section A are very short-answer type questions carrying 1 mark each.****Questions 7-19 in Section B are long-answer I type questions carrying 4 marks each.****Questions 20-26 in Section C are long-answer II type questions carrying 6 marks each.****Please write down the serial number of the question before attempting it.**

**SET I**

**Note: Except for the following questions, all the remaining questions have been asked in previous sets.**

**SECTION – A**

**Question.1. What is the range of the function**

**Question.2.**

** **

** Solution:**

**Question.3.**

** **

** Solution:A = I**

**Question.4.What is the value of the determinant**

**Question.5.**

**Question.6.What is the degree of the following differential equation?**

** **

** Solution:**Degree of given differential equation is 1 as the index of highest derivative is one.

**Question.7.Write a vector of magnitude 15 units in the direction of**

**Question.8.Write the vector equation of the following line:**

**Question.9.**

** **

** Solution:**

**Question.10.What is the cosine of the angle which the vector**

** **

** Solution:** The cosine of angle which the given vector

**SECTION – B**

**Question.11.On a multiple choice examination with three possible answer (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?**

**Question.12.**

**Find the position vector of point R which ****divides the line **

**joining two points P and Q whose position vectors**

**Question.13 Find the cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1,2) and parallel to the.line**

**Question.14.Using elementary row operations, find the inverse of the following matrix:**

**Question.15.Let Z be the set of all integers and R be the relation on Z defined as R = {(a, b):a,b ∈ Z and (a – b) is divisible by 5}. Prove that R is an equivalence relation.**

**Question.16.Prove the following:**

**Solution:**

**Question.17.Show that the function/defined as follows, is continuous at x – 2, but not differential there at:**

** **

** Solution:**Test of continuity at x = 2

**Question.18.**

** **

** Solution:**

**Question.19.**

**Question.20.Find the points on the curve y = x ^{3} at which the slope of the tangent is equal to the y-coordinate of the point.**

**Solution:**Given curve is y =

**x**

^{3}**Question.21.Find the general solution of the differential equation**

** **

** Solution:**Given differential equation is

**Question.22. Find the particular solution of the differential equation satisfying the given conditions:**

** x ^{2}dy + (xy + y^{2}) dx = 0; y = 1 when x = 1.**

**Solution:**Given differential equation is

x

^{2}dy + (xy + y

^{2}) dx= 0

**SECTION – C**

**Question.23. A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on the ring is ? 300 and that on a chain is ? 190, find the number of rings and chain that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically.**

** Solution :** Let x be number of gold rings and y be the number of chain we make the following table from the given data:

**Question.24.A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to both clubs. Find the probability of the lost card being of clubs.**

**Question.25.The points A(4,5,10), B(2,3,4) and C(1, 2, -1) are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D.**

**Question.26. Using integration, find the area of the region bounded by the curve x ^{2}= 4y and the line x = 4y – 2.**

**Solution:**Given equations of curve and line are

x

^{2}= 4y

**…(1)**

**Question.27.Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is euqal to the radius of the base.**

**Question.28. Find the values of x of which f(x) = [x(x – 2)] ^{2} is an increasing function. Also, find the points on the curve, where the tangent is parallel to x-axis.**

**Question.29.Using properties of determinants show,the following:**

**SET II**

**Note: Except for the following questions, All the remaining questions have been asked in previous sets.**

**SECTION – A**

**Question.1. Find the minor of the element of second row and third column (a _{23}) in the following determinant:**

**SECTION – B**

**Question.11. Find all points of discontinuity of f, wheref is defined as follows:**

**Question.14. Let * be a binary operation on Q defined by a*b = 3ab/5. Show that * is commutative as well as associative. Also find its identity element, if it exists.**

**Question.18.**

** Solution:**

**Question.20. Find the equations of the normals to the curve y = x ^{3 } + 2x + 6 which are parallel to the line x+ 14y + 4 = 0**

**SECTION – C**

**Question.23.**

Question.29. Write the vector equations of the following lines and hence determine the distance between them:

**SET III**

**Note: Except for the following questions, all the remaining questions have been asked in previous sets**

**SECTION – A**

**Question.1.**

**Question.2.What is the degree of the following differential equation?**

** **

** Solution:**The degree of the given differential equation is

**Question.9.If A is a square matrix of order 3and |3A| =K | A |, then write the value of K.**

**SECTION – B**

**Question.11.There are two Bags, Bag I and Bag II. Bag I contains 4 white and 3 red balls while another Bag II contains 3 white and 7 red balls. One ball is drawn at random from one of the bags and it is found to be white. Find the probability that it was drawn from Bag I,**

**Question.14.**

**Question.17.Show that the relation S in the set S of real numbers, defined as S = {(a, b): a, b∈ R and a≤ b ^{3 }} is neither reflexive, nor symmetric nor transitive.**

**Solution:**We have S = {(a, b): a, b ∈ R and a ≤ b

^{3}} is

**Question.19. Find the equation of tangent to the curve**

**SECTION – C**

**Question.23. Find the intervals in which the function f given by f(x) = sin x – cos x, 0 ≤ x ≤2π is strictly increasing or strictly decreasing.**

** Solution:**

f(x) = sin x – cos x

f(x) = cosx + sinx

**Question.24.**

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