Important Questions For You!

Maths is considered one of the most difficult subjects in CBSE Class 12 Commerce. One should be attentive and practice Maths regularly. In CBSE Class 12 Maths, the level of difficulty is high, as you get introduced to complex topics like integration and differentiation. Our Maths experts simplify complex Maths problems by assisting you with the right methods to solve them so that you can score full marks. You may still have doubts while referring to the Maths revision notes or Maths NCERT solutions. Get these doubts resolved by Maths experts through the UnDoubt section of our website.

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Chapter 1: Relations and Functions

1. Let A = Q × Q, Q being the set of rationals. Let ‘*’ be a binary operation on A, defined by (a, b) * (c, d) = (ac, ad + b).  Then ‘*’ is

  1. Commutative
  2. Associative
  3. Transitive
  4. All of the above                                                                                                                 [1M]

2. Let N be the set of natural numbers and R be the relation in N defined as R = {(a, b) : a = b + 2, b < 4}. Then

  1. (4, 2) ∈ R
  2. (5, 4) ∈ R
  3. (2, 1) ∈ R
  4. (4, 6) ∈ R                                                                                                                          [1M]

3. Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by begin mathsize 12px style straight f left parenthesis straight x right parenthesis equals fraction numerator straight x minus 2 over denominator straight x minus 3 end fraction end style.Show that f is one - one and onto. [4M]

4. begin mathsize 12px style straight f colon straight R rightwards arrow straight A comma space straight A equals left curly bracket straight x colon straight x element of straight R comma space minus 1 less than straight x 1 right curly bracket comma space straight f left parenthesis straight x right parenthesis equals fraction numerator straight x over denominator 1 plus open vertical bar straight x close vertical bar end fraction comma space straight x element of straight R end style. Show that the function f is a bijective function.  [4M]

5. A relation R on the set of complex numbers is defined by begin mathsize 12px style straight z subscript 1 Rz subscript 2 equals fraction numerator straight z subscript 1 minus straight z subscript 2 over denominator straight z subscript 1 plus straight z subscript 2 end fraction end style  Show that R is an equivalence relation. [6M]

6. Let A = Q × Q, where Q is the set of all rational numbers and * is a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ε A. Then find:

  1. The identity element of * in A.
  2. Invertible elements of A, and hence write the inverse of elements (5, 3) and begin mathsize 12px style open parentheses 1 half comma space 4 close parentheses end style.  [6M]

Chapter 2: Inverse Trigonometric Functions

1. The value of begin mathsize 12px style tan to the power of negative 1 end exponent open square brackets 2 sin open parentheses 2 cos to the power of negative 1 end exponent fraction numerator square root of 3 over denominator 2 end fraction close parentheses close square brackets end style is.

  1. begin mathsize 12px style straight pi over 2 end style
  2. begin mathsize 12px style straight pi over 6 end style
  3. begin mathsize 12px style straight pi over 3 end style
  4. begin mathsize 12px style straight pi over 4 end style                                                                                                                            [1M]
2. If begin mathsize 12px style open parentheses tan to the power of negative 1 end exponent straight x close parentheses squared plus open parentheses cot to the power of negative 1 end exponent straight x close parentheses squared equals fraction numerator 5 straight pi squared over denominator 8 end fraction end style then find x.
  1. 1
  2. –1
  3. 0
  4. 2                                                                                                                                [1M]
3. Solve the following for x:
begin mathsize 12px style sin to the power of negative 1 end exponent open parentheses 1 minus straight x close parentheses minus 2 x space sin to the power of negative 1 end exponent space straight x equals straight pi over 2 end style                                                                                              [4M]
4. Prove that begin mathsize 12px style tan 1 half open curly brackets sin to the power of negative 1 end exponent fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction plus cos to the power of negative 1 end exponent fraction numerator 1 minus straight y squared over denominator 1 plus straight y squared end fraction close curly brackets equals fraction numerator straight x plus straight y over denominator 1 minus xy end fraction comma space if space open vertical bar straight x close vertical bar less than 1 comma space straight y greater than 0 space and space xy less than 1. end style     [4M]
5. Prove that begin mathsize 12px style cot to the power of negative 1 end exponent open curly brackets fraction numerator square root of 1 plus sinx end root plus square root of 1 minus sinx end root over denominator square root of 1 plus sinx end root minus square root of 1 minus sinx end root end fraction close curly brackets equals straight x over 2 comma space 0 less than straight x less than straight x over 2 end style                                               [4M]

6. begin mathsize 12px style If space cos to the power of negative 1 end exponent straight x over straight a plus cos to the power of negative 1 end exponent straight y over straight b equals straight a comma space prove space that space straight x squared over straight a squared minus 2 xy over ab cos space straight a plus straight y squared over straight b squared equals sin squared straight a. end style                                [4M]
 

Chapter 3: Matrices

1. Find the matrix X such that 2A + B + X = 0, where begin mathsize 12px style straight A equals open square brackets table row cell negative 1 end cell 2 row 3 4 end table close square brackets space and space straight B equals open square brackets table row 3 cell negative 2 end cell row 1 5 end table close square brackets end style

  1. begin mathsize 12px style open square brackets table row 13 7 row 1 2 end table close square brackets end style
  2. begin mathsize 12px style open square brackets table row cell negative 1 end cell cell negative 2 end cell row cell negative 7 end cell cell negative 13 end cell end table close square brackets end style
  3. begin mathsize 12px style open square brackets table row 1 cell negative 2 end cell row 7 cell negative 13 end cell end table close square brackets end style
  4. begin mathsize 12px style open square brackets table row cell negative 1 end cell 2 row cell negative 7 end cell 13 end table close square brackets end style                                                                                                                 [1M]

2. Find the value(s) of x such that begin mathsize 12px style open square brackets 1 cross times 1 close square brackets open square brackets table row 1 3 2 row 2 5 1 row 15 3 2 end table close square brackets open square brackets table row 1 row 2 row straight x end table close square brackets equals 0 end style

  1. –2 or –14
  2. –1 or –13
  3. 2 or 14
  4. 1 or 13                                                                                                                        [1M]
3. Using matrices solve the following system of linear equations:

x – y + 2x = 7

3x + 4y – 5z = –5

2x – y + 3z = 12                                                                                                                    [6M]

4. Using elementary operations, find the inverse of the following matrix:

begin mathsize 12px style open square brackets table row cell negative 1 end cell 1 2 row 1 2 3 row 3 1 1 end table close square brackets end style                                                                                                                          [6M]


Chapter 4: Determinants

1. The value of the determinant begin mathsize 12px style open square brackets table row cell straight b minus straight c end cell cell straight c minus straight a end cell cell straight a minus straight b end cell row cell straight c minus straight a end cell cell straight a minus straight b end cell cell straight b minus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell straight c minus straight a end cell end table close square brackets end style is

  1. 0
  2. 1
  3. –1
  4. a                                                                                                                               [1M]
2. Without expanding, find the value of begin mathsize 12px style open square brackets table row 0 cell straight b minus straight a end cell cell straight c minus straight a end cell row cell straight a minus straight b end cell 0 cell straight c minus straight b end cell row cell straight a minus straight c end cell cell straight b minus straight c end cell 0 end table close square brackets end style.
  1. 0
  2. 1
  3. –1
  4. a                                                                                                                               [1M]
3. If a, b, c are all positive and are pth, qth and rth terms of a G.P., then we have begin mathsize 12px style increment equals open square brackets table row cell log space a end cell p 1 row cell log space b end cell q 1 row cell log space c end cell straight r 1 end table close square brackets equals 0 end style.        [4M]
4. If begin mathsize 12px style straight f open parentheses straight x close parentheses equals open square brackets table row straight a cell negative 1 end cell 0 row ax straight a cell negative 1 end cell row cell ax squared end cell ax straight a end table close square brackets end style , find the value of f(2x) – f(x).                                                            [4M]
5. Show that begin mathsize 12px style open square brackets table row straight a straight b straight c row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row bc ca ab end table close square brackets equals open square brackets table row 1 1 1 row cell straight a squared end cell cell straight b squared end cell cell straight c squared end cell row cell straight a cubed end cell cell straight b cubed end cell cell straight c cubed end cell end table close square brackets equals open parentheses straight a minus straight b close parentheses open parentheses straight b minus straight c close parentheses open parentheses straight c minus straight a close parentheses open parentheses ab plus bc plus ca close parentheses end style.                          [6M]

6. An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investments is Rs. 70 more than the income from the third. Find the amount of each investment by matrix method.                                                                          [6M]

 

Chapter 5: Continuity and Differentiability

1. Let begin mathsize 12px style straight f open parentheses straight x close parentheses equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell fraction numerator 1 minus cos 4 straight x over denominator straight x squared end fraction comma end cell cell if space straight x less than 0 end cell row cell straight a comma end cell cell if space straight x equals 0 end cell row cell fraction numerator square root of straight x over denominator square root of 16 plus square root of straight x end root minus 4 end fraction comma end cell cell if space straight x greater than 0 end cell end table close end style

Determine the value of a so that f(x) is continuous at x = 0.                                                      [4M]

2. Differentiate begin mathsize 12px style tan to the power of negative 1 end exponent open square brackets fraction numerator square root of 1 plus straight x squared end root minus 1 over denominator straight x end fraction close square brackets end style with respect to x.                                                          [4M]

3. If x = a sin t and y = 0begin mathsize 12px style open parentheses cos space straight t plus log space tan 1 half close parentheses comma space find space fraction numerator straight d squared straight y over denominator dx squared end fraction end style.                                                       [4M]

4. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 - cos 2t), find the values of begin mathsize 12px style dy over dx space at space straight t equals straight pi over 4 space and space straight t equals straight pi over 3 end style.       [4M]

Chapter 6: Application of Derivatives

1. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).  [4M]

2. Show that of all the rectangles inscribed in a given circle, the square has the maximum area.   [6M]

3. Find the values of ‘a’ for which the function f(x) = (a + 2)x3 – 3ax2 + 9ax – 1 decreases for all real values of x. [6M]


Chapter 7: Integrals

1. Evaluate: begin mathsize 12px style open curly brackets fraction numerator straight x squared over denominator open parentheses straight x squared plus 4 close parentheses open parentheses straight x squared plus 9 close parentheses end fraction dx close end style                                                                                        [4M]

2. Evaluate: begin mathsize 12px style open curly brackets fraction numerator sin open parentheses straight x minus straight a close parentheses over denominator sin open parentheses straight x plus straight a close parentheses end fraction dx close end style                                                                                                 [4M]

3. Evaluate: begin mathsize 12px style integral subscript 0 superscript straight pi fraction numerator 4 straight x space sin space straight x over denominator 1 plus cos squared straight x end fraction dx end style                                                                                             [4M]

4. Prove: begin mathsize 12px style integral subscript 0 superscript straight pi over 4 end superscript square root of tan space straight x end root plus square root of cot space straight x end root space dx equals square root of 2. straight pi over 2 end style                                                                        [6M]

5. Evaluate begin mathsize 12px style integral subscript 0 superscript 3 open parentheses 2 straight x squared plus 5 straight x close parentheses dx end style as a limit of sum.                                                                     [6M]


Chapter 8: Application of Integrals

1. Using integration, find the area bounded by the curve x2 = 4y which passes through the point (1, 2). Also, find the equation of the corresponding tangent.                                                                                                           [6M]

2. Sketch the region bounded by the curves begin mathsize 12px style straight y equals square root of 5 minus straight x squared end root space and space straight y equals open vertical bar straight x minus 1 close vertical bar. end style Find its area using integration.    [6M]


Chapter 9: Differential Equations

1. Form the DE of the family of circles in the second quadrant and touching the coordinate axes.   [4M]

2. Solve: x(1 + y2)dx – y(1 + x2)dy = 0 given that when y = 0, x = 1.                                        [4M]

3. Find the particular solution of the differential equation begin mathsize 12px style log space open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y end style, given that y = 0, when x = 0.   [6M]

4. Solve the differential equation x2dy + y(x + y)dx = 0 given that y = 1 when x = 1.                  [6M]

5. Solve: begin mathsize 12px style cos squared straight x dy over dx plus straight y equals tan space straight x end style                                                                                               [6M]

 

Chapter 10 and 11: Vector Algebra and Three Dimensional Geometry

1. If begin mathsize 12px style straight a with rightwards harpoon with barb upwards on top space and space straight b with rightwards harpoon with barb upwards on top end style are two vectors such that begin mathsize 12px style open vertical bar straight a with rightwards harpoon with barb upwards on top plus straight b with rightwards harpoon with barb upwards on top close vertical bar equals open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar end style then prove that vector begin mathsize 12px style 2 straight a with rightwards harpoon with barb upwards on top plus straight b with rightwards harpoon with barb upwards on top end style is perpendicular to vector begin mathsize 12px style straight b with rightwards harpoon with barb upwards on top end style.   [4M]

2. The scalar product of the vector begin mathsize 12px style straight a with rightwards harpoon with barb upwards on top equals straight i with hat on top plus straight j with hat on top plus straight k with hat on top end style with a unit vector along the sum of vectors  begin mathsize 12px style straight b with rightwards harpoon with barb upwards on top equals 2 straight i with hat on top plus 4 straight j with hat on top minus 5 straight k with hat on top space and space straight c with rightwards harpoon with barb upwards on top equals straight lambda straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top end style is equal to one. Find the value of λ and hence find the unit vector along begin mathsize 12px style straight b with rightwards harpoon with barb upwards on top plus straight c with rightwards harpoon with barb upwards on top end style.  [4M]

3. 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.          [6M]

4. Find the equation of the line passing through the point (–1, 3, –2) and perpendicular to the lines  begin mathsize 12px style straight x over 1 equals straight y over 2 equals straight z over 3 space and space fraction numerator straight x plus 2 over denominator negative 3 end fraction equals fraction numerator straight y minus 1 over denominator 2 end fraction equals fraction numerator straight z plus 1 over denominator 5 end fraction. end style                                                                      [6M]

5. Find the equation of the plane which contains the line of intersection of the planes begin mathsize 12px style straight r with rightwards harpoon with barb upwards on top. open parentheses straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top close parentheses minus 4 equals 0 space and space straight r with rightwards harpoon with barb upwards on top. open parentheses negative 2 straight i with hat on top plus straight j with hat on top plus straight k with hat on top close parentheses plus 5 equals 0 end style and whose intercept on x-axis is equal to the y-axis.   [6M]

Chapter 12: Linear Programming

1. A company manufactures three kinds of calculators: A, B and C in its two factories I and II. The company has got an order for manufacturing at least 6400 calculators of kind A, 4000 of kind B and 4800 of kind C. The daily output of factory I is of 50 calculators of kind A, 50 calculators of kind B, and 30 calculators of kind C. The daily output of factory II is 40 calculators of kind A, 20 of kind B and 40 of kind C. The cost per day to run factory I is Rs.12, 000 and of factory II is Rs.15, 000. How many days to the two factories have to be in operation to produce the order with the minimum cost? Solve it graphically.               [6M]

2. 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? Solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?                                                                                                                              [6M]

3. A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units /kg of vitamin A and 1 unit /kg of vitamin C while food II contains 1 unit /kg of vitamin A and 2 units / kg of vitamin 1 unit /kg of vitamin C. It costs Rs 5 per kg to purchases food I and Rs 7 per kg to purchases Food II. Determine the minimum cost of such a mixture by solving it graphically.               [6M]


Chapter 13: Probability

1. How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%?   [4M]

2. In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.              [4M]

3. A speaks truth in 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A?                                                                                                      [4M]

4. 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?                                    [6M]

5. Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation.

[6M]

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Our Maths experts give you the best solutions for Maths textbook questions and sample paper questions. Chapter-wise NCERT solutions for CBSE Class 12 Commerce Maths are easily accessible at TopperLearning. Use these solutions to practise problems based on concepts such as direction ratios, probability, area between lines, inverse trigonometric functions and more.

To prepare for your Maths exam, you need to attempt solving different kinds of Maths questions. One of the best ways to assess your problem-solving abilities is to attempt solving previous years’ papers with a timer set. Our Maths solutions for previous years’ papers will come in handy to help you with checking your answers. So, to score more marks in your Class CBSE 12 board exams, make use of our Maths study resources.

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