CBSE Class 12 Maths board exam 2018 has been scheduled to conduct on 21st March (as per CBSE Date Sheet 2018). This is the last time when all the students are searching the important questions and specially the long answer questions carrying 6 Marks.
We have done the analysis of last 7 years’ question papers of cbse class 12th mathematics along with CBSE Mathematics sample papers. We found some questions in the 6 marks category which have been repeatedly asked by bit of manipulations.
A list of 6 marks important questions class 12 Maths exam are available here. According to the latest blueprint of CBSE Class 12 Maths, students will get six long-answer-II type questions in the Mathematics questions paper.
Before knowing about important questions, first we will learn about important topics from which these types of questions are generally asked in the board exam:
• Relation functions
• Matrices (Algebra)
• Area of the curve
• Definite integrals
• Application of the Integrals (Finding the area under simple curves, especially lines, circles/parabolas/ellipses)
• Definite Integral
• Three - dimensional Geometry (Angle between two lines, two planes, a line and a plane, distance of a point from a plane)
After knowing about these topics, let’s learn about important 6 mark questions which are expected in CBSE Class 12 Maths board exam 2018
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Evaluate:
Important 4 marks questions for Class 12 Maths board exam 2018
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Question:
A company produces soft drink that has a contract which requires that a minimum of 80 units of the chemical A and 60 units of the chemical B go into each bottle of the drink. The chemicals are available in prepared mix packets from two different suppliers. Supplier S had a packet of mix of 4 units of A and 2 units of B that costs Rs. 10. The supplier T has a packet of mix of 1 unit of A and 1 unit of B costs Rs. 4. How many packets of mixed from S and T should the company purchase to honour the contract requirement and yet minimize cost? Make a LPP and solve graphically.
Question:
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?
Question:
An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 500 is made on each executive class ticket out of which 20% will go to the welfare fund of the employees. Similarly a profit of Rs. 400 is made on each economy ticket out of which 25% will go for the improvement of facilities provided to economy class passengers. In both cases, the remaining profit goes to the airline’s fund. The airline reserves at least 20 seats for executive class. However at least four times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the net profit of the airline. Make the above as an LPP and solve graphically. Do you think, more passengers would prefer to travel by such an airline than by others?
Question:
10 students were selected from a school on the basis of values for giving awards and were divided into three groups. The first group comprises hard workers, the second group has honest and law abiding students and the third group contains vigilant and obedient students. Double the number of students of the first group added to the number in the second group gives 13, while the combined strength of first and second group is four times that of the third group. Using matrix method, find the number of students in each group. Apart from the values, hard work, honesty and respect for law, vigilance and obedience, suggest one more value, which in your opinion, the school should consider for awards.
Important 2 marks questions for Class 12 Maths board exam 2018
Question:
A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of ` 80 on each piece of type A and Rs. 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?
Question:
A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs.25 and that from a shade is Rs.15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit. Formulate an LPP and solve it graphically.
Question:
One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other integredients used in making the cakes. Formulate the above as a linear programming problem and solve graphically.
Question:
A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has a space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize the profit? Formulate this as a linear programming problem and solve it graphically.
Question:
An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs. 300 is made on each second class ticket. The airline reserves at least 20 seats for first class. However, at least four times as many passengers prefer to travel by second class then by first class. Determine how many tickets of each type must be sold to maximise profit for the airline. Form an LPP and solve it graphically.
Question:
A manufacturer can sell x items at a price of Rs. (5 ‒ 100/x) each. The cost price of x items is Rs. (x/5 + 500). Find the number of items he should sell to earn maximum profit.
Question:
Find the equation of a plane which is at a distance of 3√3 units from origin and the normal to which is equally inclined to the coordinate axes.
Question:
Find the area enclosed by the parabola 4y = 3x^{2} and the line 2y = 3x + 12.
Question:
Find the distance of the point (2, 12, 5) from the point of intersection of the line r = (2i ‒ 4j + 2k) + λ (3i + 4j +2k) and plane r (i ‒ 2j + k) = 0.
Question:
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.
Question:
Find the distance of the point P(–1, –5, –10) from the point of intersection of the line joining the points A(2, –1, 2) and B(5, 3, 4) with the plane x – y + z = 5.
Question:
Find the equation of the plane that contains the point (1, – 1, 2) and is perpendicular to both the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8. Hence find the distance of point P (–2, 5, 5) from the plane obtained above.
Question:
Show that the lines (x + 3)/‒3 = (y ‒ 1)/1 = (z ‒ 5)/5; (x + 1)/‒1 = (y ‒ 2)/2 = (z ‒ 5)/5 are coplanar. Also find the equation of the plane containing the lines.
Question:
Find the equation of the plane passing through the point (‒1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
Question:
Find the equation of the plane determined by the points A (3, - 1, 2), B (5, 2, 4) and C (–1, –1, 6). Also find the distance of the point P (6, 5, 9) from the plane.
Question: Find the equation of the plane passing through the point (‒ 1, ‒ 1, 2) and perpendicular to each of the following planes:
2x + 3y ‒ 3z = 2 and 5x ‒ 4y + z = 6
Question: Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line
(x + 3)/2 = (y ‒ 3)/2 = (z ‒ 2)/5.
Question:
Using integration, find the area of the region:
{(x, y): 9x^{2} + y^{2} ≤ 36 and 3x + y ≥ 6}
Question:
Find the area of the region included between the parabola y^{2 }= x and the line x + y = 2.
Question. Using integration find the area of the region bounded by the parabola y^{2 }= 4x and the circle
Question:
Using the method of integration, find the area of the region bounded by the lines
2x + y = 4, 3x ‒ 2y = 6 and x ‒ 3y + 5 = 0.
Question:
If the sum of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is π/3.
Question:
Find the area of the smaller region bounded by the ellipse x^{2}/9 + y^{2}/4 = 1 and the line x/3 + y/2 = 1
Question:
Obtain the inverse of the following matrix, using elementary operations
Question:
Using properties of determinants, prove the following:
Question.
Using elementary transformations, find the inverse of the following matrix:
Question:
Using matrices, solve the following system of equations:
2x ‒ 3y + 5z = 11
3x + 2y ‒ 4z = ‒5
x + y ‒ 2z = ‒3
Question:
Using matrices, solve the following system of linear equations:
3x - 2y + 3z = 8
2x + y ‒ z = 1
4x - 3y + 2z = 4
Note: As per latest examination pattern, questions based on probability are expected in 2 marks and 4 marks. But these questions (each carrying 6 marks) have been asked in previous years’ papers. Due to this reason we are also including questions based on probability here.
Question:
A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.
Question:
From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.
Question:
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.
Question:
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?
Question:
Five cards are drawn one by one, with replacement, from a well shuffled deck of 52 cards,
Find the probability that
(i) all the five cards are diamonds.
(ii) only 3 cards are diamonds.
(iii) none is a diamond.
Question:
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?
Question:
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.
Question:
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident for them are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver or a car driver?
Question:
Two cards are drawn simultaneously (or successively without replacement) from a well suffled pack of 52 cards. Find the mean and variance of the number of red cards.
Question:
25. Three bags contain balls as shown in the table below:
Number of White balls |
Number of Black balls |
Number of Red balls |
(Bag I): 1 |
2 |
3 |
(Bag II): 2 |
1 |
1 |
(Bag III): 4 |
3 |
2 |
A bag is chosen at random and two balls are drawn from it. They happen to be white and red. What is the probability that they came from the III bag?
Question:
An insurance company insured 3000 scooter drivers, 5000 car drivers and 7000 truck drivers. The probabilities of their meeting with an accident respectively are 0.04, 0.05 and 0.15 one of the insured persons meets with an accident. Find the probability that he is a car driver.
Question:
A man is known to speak truth 3 out of 4 times. He throws a die and report that it is a 6. Find the probability that it is actually 6.
Question.
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver.
Important 1 mark questions for Class 12 Maths board exam 2018