Maths can be a nightmare to many students as the subject requires conceptual knowledge and a good understanding. It is not a child’s play to excel in the subject without having a masterly skill and good grip on the concepts. As the subject requires a lot of practice to learn theorems and their proofs, it also requires timely revision, or else it’s easy to forget the concepts within a matter of time.

Maths can be a nightmare to many students as the subject requires conceptual knowledge and a good understanding. It is not a child’s play to excel in the subject without having a masterly skill and good grip on the concepts. As the subject requires a lot of practice to learn theorems and their proofs, it also requires timely revision, or else it’s easy to forget the concepts within a matter of time.

As the Board examination are very near for the students, it is prime important to score well in their board, especially in the subject of mathematics. As most of the colleges require students with high reasoning skills and knowledge of mathematics, excelling in the subject can improve the chances to get into a reputed college with ease.

So here are a few important 6 marks questions for students of class 12th who aims to secure good percentage:

**Important 6 Marks Questions for Class 12 Maths Board are as follows-**

**Question 1- **In answering a question on a multiple choice test, a student either knows the answer or guesses. Let \(\frac{3}{5}\) be the probability that he knows the answer and \(\frac{2}{5}\) be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability \(\frac{1}{3}\), what is the probability that the student knows the answer given that he answered it correctly ?

**Question 2- **Using integration, find the area enclosed by the parabola 4y = 3×2 and the line 2y = 3x + 12.

**Question 3- **Find the vector and cartesian equations of the plane passing through the line of intersection of the planes

and \(\vec{r} .(2\hat{i} + 5\hat{j} + 3\hat{k}) = 9\)

such that the intercept made by the plane on x-axis and z-axis are equal.

**Question 4- **Solve the differential equation \(\frac{dy}{dx}-3y \cot x = \sin 2x\) given \(y = 2\) when \(x = \frac{\pi}{2}\).

**Question 5- **The sum of surface areas of a sphere and a cuboid with sides \(\frac{x}{3}\), x and 2x, is constant. Show that the sum of their volumes is minimum if x is equal to three times the radius of sphere.

**Question 6- **A manufacturer produces nuts and bolts. It takes 2 hours work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 2 hours on machine B to produce a package of bolts. He earns a profit of 24 per

package on nuts and 18 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 10 hours a day. Make an L.P.P. from above and solve it graphically ?

**Question 7- **Prove that the binary operation * on A = R – {-1} defined as a*b = a + b + ab for all a, b \(\in\) A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.

**Question 8- **Using the properties of determinants, show that \(\Delta ABC\) is isosceles if,

**Question 9- **A shopkeeper has 3 varieties of pens ‘A’, ‘B’ and ‘C’. Meenu purchased 1 pen of each variety for a total of Rs. 21. Jeevan purchased 4 pens of ‘A’ variety, 3 pens of ‘B’ variety and 2 pens of ‘C’ variety for Rs. 60. While Shikha purchased 6 pens of ‘A’ variety, 2 pens of ‘B’ variety and 3 pens of ‘C’ variety for Rs. 70. Using matrix method, find the cost of each variety of pen.

**Question 10- **Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector \(2\hat{i} + 3\hat{j} + 4 \hat{k}\) to the plane \(\vec{r}.(2\hat{i} + \hat{j} +3\hat{k})- 26 = 0\). Also find the image of P in the plane.

**Question 11- **If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is \(\frac{\pi}{3}\).

**Question 12- **Five bad oranges are accidentally mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

**Question 13- **Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \(6 \sqrt{3}\)r.

**Question 14- **Find:

**Question 15- **Solve the differential equation \((\tan^{-1}x – y)dx = (1 + x^{2})dy\).

**Question 16- **xy + yx = 1, then show that \(\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}} = \left (\frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}\)

**Question 17- **A variable plane \(\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}} = \frac{1}{p^{2}}\).

**Question 18- **Find the coordinates of the point where the line through the points \((3, -4, -5)\) and \((2, -3, 1)\), crosses the plane determined by the points \((1, 2, 3)\), \((4, 2, -3)\) and \((0, 4, 3)\).

**Question 19- **Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

**Question 20- **A company manufactures two types of sweaters, type A and type B. It cost Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make atmost 300 sweaters and can spend atmost Rs. 72,000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs. 200 on each unit of type A but considering the difficulties of a common man the company charges a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve the problem for maximum profit.