Mathematics  – Important Question Bank for Bihar Class 12 Intermediate (HSC) Board Exam 2017 

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Bihar Math question

1. A factory has two machines M₁ and M₂. Machine M₁ produced. 60% of the items and machine M₂ produced 40% of the items. Further, 2% of the items produced by machine M₁ and 1% of the items produced by machine M₂ were defective. All the items are collected in one group and then one item is chosen at random from this group. It is found to be defective. What is the probability that it was produced by machine M₁ ?
2. India plays two matches each with Sri Lanka and South Africa. In any match the probability of India getting points 0, 1, 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, find the probability of India getting at least 7 points
3. A and B are two independent events, The probability that both A and B occur is 1/6 and probability that neither of them occur is 1/3. Find the probability of happening of A and B respectively.
4. Let S = {s1, s2, s3, s4, s5, s6} represent the set of observable symptoms of diseases D₁, D₂ & D₃. A random sample of 1000 patients contains 320 patients with disease D₁, 350 patients with disease D₂ and 330 patients with disease D₃. Also, 310 patients with disease D₁, 330 patients with disease D₂ and 300 patients with disease D₃ show symptoms S. Knowing that the patient has symptoms S, the doctor wants to determine the patient’s illness. On the basis of this information, what should the doctor conclude?
5. A boy throws a die. If he gets a 5 or 6, he tosses a coin 3 times and notes the number of heads. If he gets 1, 2, 3 or 4 he tosses a coin once and notes whether a head or a tail is obtained. If he obtained exactly one head, what is the probability that he threw 1, 2, 3 or 4 with the die?
6. A man speaks truth 3 out of 4 times. He throws a die and reports that it is, a six. Find the probability that it is actually a six.
7. If each element of a second order determinant is either 0 or 1, what is the probability that the value of the determinant is positive?
8. Solve the following LPP graphically

Maximize   z = 5x₁ + 7x₂

Subject to x₁ + x₂ ≤ 4, 3x₁ = 8x₂ ≤ 24, 10x₁ + 7x₂ ≤ 35; x₁, x₂, ≥ 0

9. One kind of cake requires 300 gm of flour and 15g. of fat another kind of cake requires 150 g. of flour and 30 g. of fat. Find the maximum number of cakes which can be made from 7.5 kg. of flour and 600g. of fat, assuming that there is no shortage of ingredients used in making the cakes. Write this as a L.P.P and solve it graphically
10. A person wants to invest at most Rs. 12000 in Bonds P and Q. According to rules he has to invest at least Rs. 2000 in Bond P and at least Rs. 4000 in Bond Q. If the rate of interest on Bond P is 8% per annum and on Bond Q is 10% per annum, how much should he invest his money for maximum interest? Solve the problem graphically
11. Solve the following transportation problem

Requirements    5           5                4

12. Find the co-ordinates of the point where the line through the points (3, -4, -5) and (2, -3, 1) cuts the plane determined by the points A (1, 2, 3), B (2, 2, 1) and C (-1, 3, 6).
13. Find the perpendicular distance of the point P (3, 2, 1) from the plane 2x – y+ z + 1 = 0 and the foot of this perpendicular. Also, find the image of the point P in the plane
14. Using vectors prove that angle in a semi-circle is a right circle
15. Form the differential equation of family of curves y² – 2ay + x² = a² where a is an arbitrary constant
16. Find the area of the region enclosed between the circles x² + y² = 1 and (x-1)² + y² = 1
17. Find the area of that part of the circle x² + y² = 16 which is exterior to the parabola y² = 6x
18. A wire of length 36 cm. is cut into two pieces. One of the pieces is turned in the form of a square and other in the form of an equilateral triangle, find the length of each piece so that the sum of the areas of the two shapes be minimum.
19. If the lengths of three sides of a trapezium other than base are each equal to 10 cm, then find the maximum area of the trapezium
20. Using integration find the area of region bounded by the lines 4x-y + 5 = 0, x + y = 5 and x – 4y + 5 =0
21. A square plate of metal is expanding and each of its side is increasing at the rate of 2 cm per minute. At what rate is the area of the plate increasing when the side is 20 cm long?
22. The area between the x-axis and the curve y = sin x from x = 0 to x = 2π is ……………………..
23. Find the acute angle between two lines that have the direction numbers 1, 1, 0 and 2, 1, 2
24. Find the direction numbers of a line that is perpendicular to each of two lines whose direction numbers are 2, -1, 2 and 3, 0 ,1.
25. Find the equations of the plane parallel to the plane 6x – 3y – 2z + 9 =0 and at a distance 2 from the origin
26. Find the length of the normal from the origin to the plane x + 2y – 2 = 9
27. Find the equation of the plane through the point (α,β,γ) and parallel to the plane ax + by + cz = 0
28. Find the equation of the straight line through (2,1,-2) and equally inclined to axes
29. Find the direction cosines of the line equally inclined to the axes
30. Find the probability that when a two digit number’s units and tens place is interchanged one will get the same number
31. One mapping is selected at random from the mappings from set {a,b,c,d} to {x,y}. Find the probability that the selected mapping onto
32. Two numbers are selected at random from the numbers 1,2,3,4……24. Find the probability that their sum will be divisible by
33. A bag A contains 2 white and 2 red balls and another bag B contains 4 white and 5 red balls. A ball is drawn and is found to be red. Find the probability that it was drawn from the bag B
34. A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, what is the probability that all letters are not placed in the right envelopes?
35. The probability of two events A and B are 0.25 and 0.40 respectively. The probability that both A and B occur is 0.15. Find the probability that neither A nor B occurs.
36. Let S=N x N and * be a binary operation on S defined by (a,b)*(c,d)=(a+c,b+d). Show that * is commutative and associative. Also find the identity element for * on S
37. If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, Find the approximate error in calculating its surface area.
38. A rectangle is inscribed in a semi-circle of radius r with one of its sides on a diameter of the semi-circle. Find the dimensions of the rectangle so that its area is maximum. Also, find the maximum area
39. Form the differential equation representing the family of curves y = a Sin(x +b) where a and b are arbitrary constants
40. Form the differential equation representing the given family of curves, by eliminating arbitrary constants a and b , b² x² +a² y² = a²b²