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Mathematics for Class 12

Application of Derivatives -01

Time Left 00:48

Maximum Marks :20

Instructions

1. All questions are compulsory. 2. There is no overall choice 3. Use of calculators is not permitted

4. Please give the explanation for the answer where applicable. 5. To view the answer, click on the Answer link alongside the Question.

*To view the answer(s), click View Answer when the link gets highlighted.

Q1 - Find the equation of the tangent to the curve which is

parallel to the line (4 Marks)

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Q2 - A square piece of tin of side 18 cm is to be made into a box without top, by cutting a

square from each corner and folding up the flaps to form the box. What should be the

side of the square to be cut off so that the volume of the box is the maximum possible.

(6 Marks)

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Q3 - A wire of length 28 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?

(6 Marks)

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Q4 - Find the rate of change of the area of a circle with respect to its radius when = 1 cm.

(1 Mark)

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Q5 - Find the rate of change of volume of a spherical body with respect to radius, when

the radius is 10 cm.

(1 Mark)

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Q6 - Find the slope of the tangent to the curve at (1 Mark)

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Q7 - Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 2%.

(1 Mark)

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Application of Derivatives -02

Time Left 00:52

Maximum Marks :20

Instructions




Q1 - Prove that the curves and cut at right angles if . (4 Marks)

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Q2 - Show that the right circular cone of least curved surface and given

volume has an altitude equal to times the radius of the base. (6 Marks)

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Q3 - A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is

. (6 Marks)

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Q4 - An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume

of the cube increasing when the edge is 10 cm long?

(1 Mark)

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Q5 - Show that the function given by f(x) = 5x - 2 is strictly increasing on R.(1 Mark)

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Q6 - Show that the function given by is strictly increasing on R.

(1 Mark)

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Q7 - If , then give the equation of tangent at .(1 Mark)

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Application of Integrals -01

Time Left 00:47

Maximum Marks :20

Instructions




Q1 - Find the area of the region bounded by , , and the -axis.

(4 Marks)

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Q2 - Find the area of the region bounded by and .(4 Marks)

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Q3 - Using integration, find the area bounded by the curves and

. (6 Marks)

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Q4 - Find the area of the region bounded by the line , the -axis

and the ordinates and .(6 Marks)

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Application of Integrals -02

Time Left 00:49

Maximum Marks :20

Instructions




Q1 - Find the area of the region bounded by and .(4 Marks)

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Q2 - Find the area of the region bounded by ellipse . (4 Marks)

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Q3 - Using the method of integration find the area bounded by the curve

.(6 Marks)

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Q4 - Find the area of the circle exterior to the parabola .(6 Marks)

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Continuity and Differentiability -01

Time Left 00:48

Maximum Marks :20

Instructions




Q1 - Find , if . (3 Marks)

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Q2 - If , find in terms of alone. (3 Marks)

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Q3 - If , then prove that . (4 Marks)

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Q4 - Differentiate w.r.t. .(4 Marks)

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Q5 -

(4 Marks)

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Q6 - Examine the continuity of function given by at .

(1 Mark)

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Q7 - Differentiate with respect to x.(1 Mark)

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Continuity and Differentiability -02

Time Left 00:46

Maximum Marks :20

Instructions




Q1 - Find , if . (3 Marks)

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Q2 - Find , if (3 Marks)

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Q3 - If , for some prove that is a

constant, independent of and . (4 Marks)

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Q4 - If , show that .(4 Marks)

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Q5 - If find at .(4 Marks)

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Q6 - Find the points where the constant function f(x) = k is continuous.(1 Mark)

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Q7 - Prove that the identity function on real numbers given by is continuous at every real number.

(1 Mark)

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Determinants -01

Time Left 00:15

Maximum Marks :20

Instructions




Q1 - Prove that: (4 Marks)

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Q2 - Show that: (3 Marks)

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Q3 - If are in A.P then show that:

(3 Marks)

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Q4 - If , find AB. Using AB solve the following system of equations:

(6 Marks)

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Q5 - Evaluate: . (1 Mark)

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Q6 - Show that .

(2 Marks)

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Q7 -

(1 Mark)

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Q8 - Find the area of a triangle whose vertices are given

by (1 Mark)

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Determinants -02

Time Left 00:44

Maximum Marks :20

Instructions




Q1 - If , verify . (4 Marks)

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Q2 - If , then show that is

factor of .(3 Marks)

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Q3 - Prove that:

(3 Marks)

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Q4 - The sum of three numbers is 6. If we multiply third number by 3 and add second number to it we get 11. By adding first and third numbers we get double of the second number. Represent it algebraically and find the numbers using matrix method.

(6 Marks)

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Q5 - Find values of for which . (1 Mark)

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Q6 - Evaluate: .(1 Mark)

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Q7 - The value of determinant will _______ if its rows and columns are interchanged.

(1 Mark)

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Q8 - If P, Q, R are three non-null square matrices of the same order, write the condition on P such that PQ = PR Q = R.

(1 Mark)

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Differential Equation -01

Time Left 00:44

Maximum Marks :20

Instructions




Q1 - Determine the order and degree of differential equation:

(1 Mark)

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Q2 - Solve: (1 Mark)

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Q4 - Show that is a solution of the differential equation .

(1 Mark)

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Q5 - Find the equation of the curve that passes through the point (1, 2) and

satisfies the differential equation . (4 Marks)

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Q6 - Find the particular solution of the differential equation

at .(6 Marks)

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Q7 - Solve: (6 Marks)

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Practice PapersBack




Differential Equation -02

Time Left 00:46

Maximum Marks :20

Instructions




Q1 - Determine the order of differential equation: (1 Mark)

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Q2 - Determine the degree of differential equation: (1 Mark)

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Q3 - Verify that the given function is a solution of the corresponding differential equation: y = ex + 1 : y'' - y' = 0

(1 Mark)

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Q4 - Verify that the given function is a solution of the corresponding differential equation:

(1 Mark)

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Q5 - Form the differential equation representing the family of curves given

by: (1 Mark)

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Q8 - Form the differential equation of the curve where

are arbitrary constants.(4 Marks)

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Q9 - Solve the differential equation: (6 Marks)

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Integrals -01

Time Left 00:44

Maximum Marks :20

Instructions




Q1 - Evaluate:

(1 Mark)

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Q2 - Evaluate:

(1 Mark)

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Q3 - Evaluate:

(2 Marks)

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Q4 - Evaluate:

(2 Marks)

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Q5 - Evaluate:

(4 Marks)

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Q6 - Evaluate:

(6 Marks)

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Q7 - Evaluate:

(6 Marks)

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Integrals -02

Time Left 00:44

Maximum Marks :20

Instructions




Q1 - Evaluate:

(1 Mark)

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Q2 - Evaluate:

(1 Mark)

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Q3 - Evaluate:

(2 Marks)

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Q4 - Evaluate:

(1 Mark)

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Q5 - Evaluate:

(4 Marks)

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Q6 - Evaluate:

(6 Marks)

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Q7 - Evaluate:

(6 Marks)

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Inverse Trigonometric Functions -01

Time Left 00:45

Maximum Marks :20

Instructions




Q1 - Prove that (3 Marks)

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Q2 - Find the value of (3 Marks)

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Q3 - Show that (6 Marks)

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Q5 - Find the principal value of .(1 Mark)

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Q6 - Show that .(2 Marks)

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Q7 - Find the value of . (2 Marks)

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Q8 - If sin-1(3/5) = x, find the value of cos x . (1 Mark)

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Inverse Trigonometric Functions -02

Time Left 00:45

Maximum Marks :20

Instructions




Q1 - Write in the simplest form.

(2 Marks)

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Q2 - Express in the simplest form.(3 Marks)

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Q5 - Is the following statement true?

(1 Mark)

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Q6 - If sin-1 x = y, then what will be the range of y?(1 Mark)

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Q7 - Find the value of . (1 Mark)

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Q8 - Find the value of . (2 Marks)

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Linear Programming -01

Time Left 00:32

Maximum Marks :20

Instructions

1. All questions are compulsory.

2. There is no overall choice 3. Use of calculators is not permitted



Q1 - Solve the following linear programing problem graphically:

Maximise Subject to constraints

(4 Marks)

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Minimise Subject to constraints

(4 Marks)

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Q3 - A manufacturing company makes two models A and B of a product. Each piece of

Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each

piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing.

For fabricating and finishing, the maximum labour hours available are 180 and 30

respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs

12000 on each piece of Model B. How many pieces of Model A and Model B should be

manufactured per week to realise a maximum profit? What is the maximum profit per

week?

(6 Marks)

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Q4 - A retired person has Rs 70,000 to invest and two types of bonds are available in the

market for investment. First type of bond yields an annual income of 8% on the amount

invested and the second type of bonds yields 10% per annum. As per norms, he has to

invest a minimum of Rs 10,000 in the first type and not more than Rs 30,000 in the

second type. How should he plan his investment so as to get the maximum return, after

one year of investment?

(6 Marks)

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Linear Programming -02

Time Left 00:08

Maximum Marks :20

Instructions





Maximise Subject to constraints

(4 Marks)

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Q2 - A dietician 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 atleast 240 units of calcium, atleast 460 units of iron and at most 300 units

of cholesterol. How many packets of each food should be used to minimise the amount of

vitamin A in the diet? What is the minimum amount of vitamin A?

(6 Marks)

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Q3 - If a young man rides his motorcycle at 25 km/hour, he had to spend Rs 2 per km on

petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs 5 per

km. He has Rs 100 to spend on petrol and wishes to find what is the maximum distance

he can travel within one hour. Express this as an LPP and solve it graphically.

(6 Marks)

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Q4 - A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. Formulate the LPP.

(4 Marks)

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Matrices -01

Time Left 00:46

Maximum Marks :20

Instructions




Q1 - If .(3 Marks)

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Q2 -

(6 Marks)

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Q3 - Find the matrix so that .

(4 Marks)

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Q4 - Show that for any square matrix A, A + A’ is a symmetric matrix and A - A’ is a skew symmetric matrix.

(3 Marks)

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Q5 - Find the order of matrix A = [aij]m x n.(1 Mark)

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Q6 - If A and B are two equal square matrices such that A = and

B = , then find the value of .

(1 Mark)

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Q7 - Construct matrix whose elements are given by .(1 Mark)

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Q8 - Define identity matrix. (1 Mark)

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Matrices -02

Time Left 00:48

Maximum Marks :20

Instructions




Q1 -

(3 Marks)

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Q2 - If A, B are symmetric matrix of same order, then show that AB - BA is a skew- symmetric matrix.

(3 Marks)

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Q3 -

(4 Marks)

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Q4 -

(4 Marks)

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Q5 - A matrix has 24 elements. What are the possible orders it can have?(1 Mark)

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Q6 - If A and B are two square matrices and K is a scalar quantity then K(A+B) = ______.

(1 Mark)

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Q7 - A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an ________.

(1 Mark)

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Q8 - If and , find .(1 Mark)

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Probability -01

Time Left 00:39

Maximum Marks :20

Instructions




Q1 - A spinner has 4 equal sectors coloured yellow, blue, green and red. What is the probability of landing on red after spinning the spinner?

(1 Mark)

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Q2 - A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble?

(1 Mark)

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Q3 - Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4’.

(2 Marks)

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Q4 - Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?

(1 Mark)

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Q5 - A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

(4 Marks)

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Q6 - A and B throw a dice alternatively till one of them gets a ‘6’ and wins the game. Find

their respective probabilities of winning, if A starts first.

(4 Marks)

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Q7 - In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear

each hurdle is 5/6 . What is the probability that he will knock down fewer than 2 hurdles?

(4 Marks)

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Probability -02

Time Left 00:39

Maximum Marks :20

Instructions




Q1 - A single 6-sided dice is rolled. What is the probability of rolling an even number?

(1 Mark)

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Q2 - Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E|F) and P(F|E).

(1 Mark)

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Q3 - If P(A) = 0.8, P(B) = 0.5 and P(B|A) = 0.4, find P(A B).(1 Mark)

View Answer

Q4 - If 2P(A) = P(B) = 5/13 and P(A|B) = 2/5, evaluate P(A B). (1 Mark)

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Q5 - If and , find , where A and B are independent events.

(1 Mark)

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Q6 - A dice is thrown. If E is the event ‘the number appearing is a multiple of 3’ and F be the event ‘the number appearing is even’ then find whether E and F are independent?

(1 Mark)

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Q7 - Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that(i) the youngest is a girl, (ii) at least one is a girl?

(4 Marks)

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Q8 - A person has undertaken a construction job. The probabilities are 0.65 that there

will be strike, 0.80 that the construction job will be completed on time if there is no

strike, and 0.32 that the construction job will be completed on time if there is a strike.

Determine the probability that the construction job will be completed on time.

(4 Marks)

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Q9 - The probability of a shooter hitting a target is 3/4. How many minimum number of

times must he/she fire so that the probability of hitting the target at least once is more

than 0.99?

(4 Marks)

View Answer



Relations and Functions -01

Time Left 00:43

Maximum Marks :20

Instructions


4. Please give the explanation for the answer where applicable.

5. To view the answer, click on the Answer link alongside the Question. *To view the answer(s), click View Answer when the link gets highlighted.

Q1 - Check the injectivity and surjectivity of the following:(i) f: N N given by f(x) = x2

(ii) f: R R given by f(x) = x2 (4 Marks)

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Q2 - Let A = R - {3} and B = R - {1}. Consider the function f: A B defined

by . Is f one- one and onto? Justify your answer.(4 Marks)

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Q3 - Consider the binary operation on the set {1, 2, 3, 4, 5} defined by a b = min{a, b}. Write the composition table of the operation .

(3 Marks)

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Q4 - If , Show that for all . What is the

inverse of ?(4 Marks)

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Q5 - A relation R in a set A is called …, if no element of A is related to any element of A.

(1 Mark)

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Q6 - A relation R in a set A is called ..., if (a,a) R, for every a A.(1 Mark)

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Q7 - R is a relation in a set {1, 2, 4} given by R = {(1, 1), (2, 2), (4, 4)}. State whether R

is reflexive or symmetric.

(1 Mark)

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Q8 - Show that is the inverse of x 0 for the multiplication operation on R.

(1 Mark)

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Relations and Functions -02

Time Left 00:44

Maximum Marks :20

Instructions




Q1 - Show that the Relation R on the set A = {x Z : 0 x 12}, given by

R = {(a, b) : |a - b| is a multiple of 4}

is an equivalence relation.(3 Marks)

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Q2 - Show that the function f: R R given by is neither

one-one nor onto.

(2 Marks)

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Q3 - Let f: N R be a function defined as f(x) = 4x2 + 12x + 15. Show that f: N S where, S is the range of f(x). Find the inverse of f

(4 Marks)

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Q4 - Define a binary operation * on the set as

Show that 0 is the identity for this operation and each element of the set is invertible with 6-a being the inverse of a .

(4 Marks)

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Q5 - A relation R in a set A is called ..., if each element of A is related to every element of

A.

(1 Mark)

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Q6 - A relation R in a set A is called ... if (a1, a2) R (a2, a1) R for all (a1, a2) A.

(1 Mark)

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Q7 - A function f : X Y is defined to be one-one (or injective), if for all a1, a2

X, f (a1) = f (a2) , implies ... Otherwise f is ...(1 Mark)

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Q8 - Let f = {(3, 1), (2, 3), (1, 2)}, find f -1 if f is one-one and onto. (1 Mark)

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Three Dimensional Geometry -01

Time Left 00:39

Maximum Marks :20

Instructions




Q1 - Find the direction cosines of the line passing through the two points (– 2, 4, – 5) and

(1, 2, 3).

(1 Mark)

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Q2 - Write the equation of a line that passes through a given point A and

parallel to a given vector .

(1 Mark)

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Q3 - Find the distance of the point (3, -5, 12) from X-axis.(1 Mark)

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Q4 - Find the vector equation for the line passing through the points (–1, 0, 2) and (3, 4,

6).

(1 Mark)

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Q5 - Find the angle between the pair of lines given by

and .(1 Mark)

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Q6 - Find the distance of the plane from the origin.(1 Mark)

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Q7 - Find the equation of the plane passing through the points (1, 2, 3) and

(0, –1, 0) and parallel to the line .(4 Marks)

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Q8 - 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).

(4 Marks)

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Q9 - Find the coordinates of the point where the line through the points A (3, 4, 1) and

B(5, 1, 6) crosses the XY-plane.

(6 Marks)

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Three Dimensional Geometry -02

Time Left 00:35

Maximum Marks :20

Instructions




Q1 - If a line makes angle 90°, 60° and 30° with the positive direction of x, y and z-axis

respectively, find its direction cosines.

(1 Mark)

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Q2 - If a line has direction ratios 2, – 1, – 2, determine its direction cosines.

(1 Mark)

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Q3 - Show that the points A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11) are collinear.

(1 Mark)

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Q4 - Write the Cartesian equation of the line, if the coordinates of the given

point A are and the direction ratios of the line are (1 Mark)

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Q5 - The Cartesian equation of a line is . Find the vector equation for the line.

(1 Mark)

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Q6 -

(3 Marks)

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Q7 - Find the angle between the planes x + y + 2z = 9 and 2x - y + z = 15.(1 Mark)

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Q8 - Find the equation of the plane through the point (1, 4, -2) and parallel to the plane -

2x + y - 3z = 7.

(1 Mark)

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Q9 - If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane

, then find the value of p.(6 Marks)

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Vector Algebra -01

Time Left 00:43

Maximum Marks :20

Instructions




Q1 - If is a unit vector and , then find .

(1 Mark)

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Q2 - Find angle between the vectors and .(1 Mark)

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Q3 - Find , if and .(1 Mark)

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Q4 -

(3 Marks)

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Q5 - If and are any vectors, then prove that .(4 Marks)

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Q6 - If A, B, C have position vectors (2, 0, 0), (0, 1, 0) and (0, 0, 2), show that ABC is isosceles.

(2 Marks)

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Q8 - If are unit vectors such that , then find the value of

.

(4 Marks)

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Vector Algebra -02

Time Left 00:42

Maximum Marks :20

Instructions


4. Please give the explanation for the answer where applicable.

5. To view the answer, click on the Answer link alongside the Question. *To view the answer(s), click View Answer when the link gets highlighted.

Q1 - Find the scalar projection of the vector on the vector

.(1 Mark)

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Q2 -

(1 Mark)

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Q3 - Prove that the angle in a semi-circle is a right angle.(4 Marks)

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Q4 -

(2 Marks)

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Q5 - Find the area of a triangle having the points A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1) as its vertices.

(4 Marks)

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Q6 - Express the vector as sum of two vectors such that one is parallel

to the vector and other is perpendicular to .

(6 Marks)

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mathe question

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