# Important Questions For You!

Chapter 1: Relations and Functions

1. Let A = Q × Q, Q being the set of rationals. Let ‘*’ be a binary operation on A, defined by (a, b) * (c, d) = (ac, ad + b).  Then ‘*’ is

1. Commutative
2. Associative
3. Transitive
4. All of the above                                                                                                                 [1M]

2. Let N be the set of natural numbers and R be the relation in N defined as R = {(a, b) : a = b + 2, b < 4}. Then

1. (4, 2) ∈ R
2. (5, 4) ∈ R
3. (2, 1) ∈ R
4. (4, 6) ∈ R                                                                                                                          [1M]

3. Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by .Show that f is one - one and onto. [4M]

4. . Show that the function f is a bijective function.  [4M]

5. A relation R on the set of complex numbers is defined by   Show that R is an equivalence relation. [6M]

6. Let A = Q × Q, where Q is the set of all rational numbers and * is a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ε A. Then find:

1. The identity element of * in A.
2. Invertible elements of A, and hence write the inverse of elements (5, 3) and .  [6M]

Chapter 2: Inverse Trigonometric Functions

1. The value of  is.

1.                                                                                                                             [1M]
2. If  then find x.
1. 1
2. –1
3. 0
4. 2                                                                                                                                [1M]
3. Solve the following for x:
[4M]
4. Prove that      [4M]
5. Prove that                                                [4M]

6.                                 [4M]

Chapter 3: Matrices

1. Find the matrix X such that 2A + B + X = 0, where

1.                                                                                                                  [1M]

2. Find the value(s) of x such that

1. –2 or –14
2. –1 or –13
3. 2 or 14
4. 1 or 13                                                                                                                        [1M]
3. Using matrices solve the following system of linear equations:

x – y + 2x = 7

3x + 4y – 5z = –5

2x – y + 3z = 12                                                                                                                    [6M]

4. Using elementary operations, find the inverse of the following matrix:

[6M]

Chapter 4: Determinants

1. The value of the determinant  is

1. 0
2. 1
3. –1
4. a                                                                                                                               [1M]
2. Without expanding, find the value of .
1. 0
2. 1
3. –1
4. a                                                                                                                               [1M]
3. If a, b, c are all positive and are pth, qth and rth terms of a G.P., then we have .        [4M]
4. If  , find the value of f(2x) – f(x).                                                            [4M]
5. Show that .                          [6M]

6. An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investments is Rs. 70 more than the income from the third. Find the amount of each investment by matrix method.                                                                          [6M]

Chapter 5: Continuity and Differentiability

1. Let

Determine the value of a so that f(x) is continuous at x = 0.                                                      [4M]

2. Differentiate  with respect to x.                                                          [4M]

3. If x = a sin t and y = 0.                                                       [4M]

4. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 - cos 2t), find the values of .       [4M]

Chapter 6: Application of Derivatives

1. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).  [4M]

2. Show that of all the rectangles inscribed in a given circle, the square has the maximum area.   [6M]

3. Find the values of ‘a’ for which the function f(x) = (a + 2)x3 – 3ax2 + 9ax – 1 decreases for all real values of x. [6M]

Chapter 7: Integrals

1. Evaluate:                                                                                         [4M]

2. Evaluate:                                                                                                  [4M]

3. Evaluate:                                                                                              [4M]

4. Prove:                                                                         [6M]

5. Evaluate  as a limit of sum.                                                                     [6M]

Chapter 8: Application of Integrals

1. Using integration, find the area bounded by the curve x2 = 4y which passes through the point (1, 2). Also, find the equation of the corresponding tangent.                                                                                                           [6M]

2. Sketch the region bounded by the curves  Find its area using integration.    [6M]

Chapter 9: Differential Equations

1. Form the DE of the family of circles in the second quadrant and touching the coordinate axes.   [4M]

2. Solve: x(1 + y2)dx – y(1 + x2)dy = 0 given that when y = 0, x = 1.                                        [4M]

3. Find the particular solution of the differential equation , given that y = 0, when x = 0.   [6M]

4. Solve the differential equation x2dy + y(x + y)dx = 0 given that y = 1 when x = 1.                  [6M]

5. Solve:                                                                                                [6M]

Chapter 10 and 11: Vector Algebra and Three Dimensional Geometry

1. If  are two vectors such that  then prove that vector  is perpendicular to vector .   [4M]

2. The scalar product of the vector  with a unit vector along the sum of vectors   is equal to one. Find the value of λ and hence find the unit vector along .  [4M]

3. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Also find the distance of the plane obtained above, from the origin.          [6M]

4. Find the equation of the line passing through the point (–1, 3, –2) and perpendicular to the lines                                                                        [6M]

5. Find the equation of the plane which contains the line of intersection of the planes  and whose intercept on x-axis is equal to the y-axis.   [6M]

Chapter 12: Linear Programming

1. A company manufactures three kinds of calculators: A, B and C in its two factories I and II. The company has got an order for manufacturing at least 6400 calculators of kind A, 4000 of kind B and 4800 of kind C. The daily output of factory I is of 50 calculators of kind A, 50 calculators of kind B, and 30 calculators of kind C. The daily output of factory II is 40 calculators of kind A, 20 of kind B and 40 of kind C. The cost per day to run factory I is Rs.12, 000 and of factory II is Rs.15, 000. How many days to the two factories have to be in operation to produce the order with the minimum cost? Solve it graphically.               [6M]

2. 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? Solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?                                                                                                                              [6M]

3. A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units /kg of vitamin A and 1 unit /kg of vitamin C while food II contains 1 unit /kg of vitamin A and 2 units / kg of vitamin 1 unit /kg of vitamin C. It costs Rs 5 per kg to purchases food I and Rs 7 per kg to purchases Food II. Determine the minimum cost of such a mixture by solving it graphically.               [6M]

Chapter 13: Probability

1. How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%?   [4M]

2. In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.              [4M]

3. A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A?                                                                                                      [4M]

4. There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of The three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?                                    [6M]

5. 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.

[6M]

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