**CBSE Sample Paper Maths Class 12**** **includes 29 questions according to 12 CBSE Board maths exam pattern. Download complete sample papers in PDF format by using link below to last question.

**1 MARK QUESTIONS:**

**Question:** Find the slope of tangent to the curve y = x^{2} – 2 at the point whose abscissa is 2.

**Question:** Differentiate: sin^{o} x with respect to *x *.

**Question:** Let Z be the set of all integers and let R be a relation in Z, defined by R= {(a, b) : ( a-b) is even} show that R is an equivalence relation in Z.

**Question:** If A is a square matrix of order 3 such that |Adj. A| = 64. Find |A’|.

**Question:** Find the domain of the function f(x) = log x^{2}.

**Question:** Find the slope of the tangent to the curve y = 3x^{2} + 4x at the point whose abscissa is (–2).

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**Question:** Find the direction cosines of a passing through origin and lying in the first octant, making equal angles with the three co-ordinates axes.

**Question:** Find the angle between the planes 2x – 3y + 4z = 1 and –x + y = 4.

**Question:** If A is a square matrix satisfying A^{2} = 1, then what is the inverse of A?

**Question:** Give example of a function which is continuous at *x* = 1 but not differentiable at *x* = 1.

**Question:** Write the range of the principal branch of sec^{–1} (x) defined on the domain R – (–1, 1).

**Question:** If A= [1 2 3] then evaluate AA^{T}.

**Question:** If A is an invertible matrix of order 3 and |A| = 5, then find |adj A|.

**Question:** Find the point on the curve y^{2} = 8x for which the abscissa and ordinate change at the same rate.

### 3 MARKS QUESTIONS:

**Question:** Find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5). Also find the distance of the plane from the origin.

**Question:** Find the interval in which the following function is (a) increasing, (b) decreasing: f(x) = x^{4} – 8x^{3} + 22x^{2} – 24x + 21

**4 MARK QUESTIONS:**

**Question:** Find the coordinates of the foot of the perpendicular drawn from the point A (1, 8, 4) to the line joining the points B (0, −1, 3) and C (2, −3, −1).

**Question:** Show that the relation R in the set A = {x ; x Î Z, 0 < x < 12} given by R = {(a, b) : |a – b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.

**Question:** If the papers of 4 students can be checked by any of the 7 teachers, then show that the probability that all the 4 papers are checked by exactly 2 teachers is 6/42.

**Question:** A biased die is twice as likely to show even number and odd number. The die is rolled three times. If occurrence of an even number is considered a success, then write the probability distribution of number of successes. Also find the mean number of successes.

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**Question:** A die is thrown twice and the sum of the numbers appearing is observed to be 8 . what is the conditional probability that the number 5 has appeared at least once.

**Question:** Find the equation of plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5). Also, find the perpendicular distance of the plane from the origin.

**Question:** There are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

**Question:** Find all the points of discontinuity of the function ¦ (x) = [x^{2}] jon [1, 2), where [.] denotes the greatest integer function.

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**Question:** Let N be the set of all natural numbers and let R be a Relation in N, defined by R= {(a, b): a is a factor of b} then, show that R is reflexive and transitive but not symmetric.

**Question:** Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

**Question:** Two cards are drawn successively with replacement from well shuffled deck of 52 cards. Find the probability distribution of the number of aces.

**6 MARK QUESTIONS:**

**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 4 times as many passengers prefer to travel by second class than by the first class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. Form an L.P.P. and solve it graphically.

**Question:** Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius R.

**Question:** Find the equation of the plane passing through the points (–1, –1, 2) and perpendicular to each of the planes whose equations are 2x + 3y – 3z = 2 and 5x – 4y + z = 6.

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**Question:** A man known to speak _{ }the 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.

**Question:** Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units /kg of vitamin A and 5 units /kg of vitamin B while food Q contains 4 units /kg of vitamin A and 2 units /kg of vitamin B. Determine the minimum cost of the mixture?

**Question:** In an examination, 10 questions of true false type are asked. A student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’ and if it falls tails, he answers ‘false’. Show that the probability that the answers at most 7 questions correctly is 121/128.

**Question:** A window is in the form of a rectangle surmounted by a semi circle. Total perimeter of the window is 10 metres. Find the dimensions of the rectangle so as to admit maximum light through the whole opening.

**Question:** Using the method of integration, find the area of the region bounded by the lines 2x + y = 4, 3x – 2y = 6, x – 3y + 5 = 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 co-ordinates of the foot of the perpendicular drawn from the point (1, 1, 1) to the plane.

**Question:** Two bags A and B contain 4 white and 3 black balls respectively. From bag A, two balls are drawn at random and then transferred to bag B. A ball is then drawn from bag b and is found to be a black ball. What is the probability that the transferred balls were 1 white and 1 black?

**Question:** Show that right circular cylinder of given volume, open at the top has minimum total surface area if its height is equal to the radius of base.

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**Question:** Find the equation of plane passing through the point (1, 2, 1)and perpendicular to the line joining points (1, 4, 2) and (2, 3, 5).Also find the co-ordinates of foot of perpendicular of the point (4, 0, 3) from the above found plane.

**Question:** A dietician wishes to mix two types of foods in such a way that the vitamin contents of mixture contains at least 8 units of vitamin A and 10 units of units of vitamin C. Food I contains 2 units per k.g. of vitamin A and one unit of per k.g of vitamin C. Food II contains 1 unit per k.g. of vitamin A and two unit per k.g. of vitamin C. It costs Rs 50 per k.g. to purchase food I and Rs 70 per k.g. to purchase food II .Formulate the problem as L.P.P to minimise the cost of such mixture and find the minimum cost graphically.

**Question:** A wire of length36 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

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**Question:** Given three identical boxes A, B and C each containing two coins. In box A, both the coins are gold coins, in box B both are silver coins and in box C there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold.

**Question:** A dietitian has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to maximise the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

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**Question:** If a Young man rides his motorcycle at 25 km/hr he has to spend Rs. 2 per km on petrol ; if he rides it at a faster speed of 40km/hr, the petrol cost increases to 5 Rs per km .he has Rs 100 to spend on petrol and wishes to find the maximum distance he can travel within one hour. Express this as a linear problem and then solve it.

**Question:** There are two types of fertilizers F_{1} and F_{2}. F_{1} consists of 10% nitrogen and 6% phosphoric acid and F_{2} consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F_{1} cost Rs 6/kg and F_{2} costs Rs 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

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**Paper 3:** https://examsegg.com/downloads/12/cbse-class-12-maths-sample-paper-download.pdf

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