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ISC Class XII Board Examination

ISC Mathematics — Board Examination 2025

Board2025Subject code 860
Open source paper at oswaal360.com (opens in a new tab)
Board
ISC Class XII Board Examination
Class
Class 12
Subject
Mathematics
Year
2025
Total marks
Not verified
Duration
Not verified

Verification and rights

Follows the published ICSE syllabus structure and links back to the source, but has not been reconciled line by line against a Council document.

Imported board paper from Oswaal Books (reproduction of CISCE board paper) on 2026-08-29.

Questions in this paper

  1. 1((i))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    If A =   , then A is: (iv) ∫ 2 dx is equal to:
    0 0  0 x +1    1 mark
  2. 1((ii))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    Which of the following is a homogenous differential -2
    equation? (c) (d) 0
    (a) (4x2 + 6y + 5) dy - (3y2 + 2x + 4) dx = 0
  3. 1((v))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    Assertion: Consider the two events A and B such
    (b) (xy) dx - (x3 + y3) dy = 0
    A B
    (c) (x3 + 2y2) dx + 2xy dy = 0 that n(A) = n(B) and P   = P   .
    B A
    (d) y2 dx + (x2 - xy - y2) dy = 0
    Reason: The events A and B are mutually exclusive.    1 mark
  4. 1((iii))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    Consider the graph of the function f(x) shown below:
  5. 1((vi))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    The existence of unique solution of the system of
    equations x + y = λ and 5x + ky = 2 depends on:
    1 
    Statement 1: The function f(x) is increasing in  , 2  . λ
    2     1 mark
  6. 1((viii))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    If f ( x ) =- x 2 - 2 0 ≤ x < 1 Question 4
     x x ≥1 dy log x
     (i) If xy = ex-y, prove that =
    dx ( 1 + log x )2
    then the number of point(s) of discontinuity of f(x),
    is / are: OR
    (a) 1 (b) 3 1    2 marks
  7. 1((ii) [cont.])(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    If f(x) = log(1 + x) + , show that f(x) attains its
    (c) 2 (d) 0 1+x
  8. 1((ix))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    Assertion: If Set A has m elements, Set B has n minimum value at x = 0 .
    elements and n < m, then the number of one-one Question 5
    function(s) from A → B is zero. Three shopkeepers Gaurav, Rizwan and Jacob use
    Reason: A function f: A → B is defined only if all carry bags made of polythene, handmade paper
    elements in Set A have an image in Set B. and newspaper. The number of polythene bags,    2 marks
  9. 1((x))(a) Statement 1 is true and Statement 2 is false. In subparts (i) to (xi) choose the correct options and (b) Statement 2 is true and Statement 1 is false. in subparts (xii) to (xv), answer the questions as (c) Both the statements are true. instructed. (d) Both the statements are false. 0 a  16 1x -1
    Let X be a discrete random variable. The probability Using the concepts of matrices and determinants,
    distribution of X is given below: answer the following questions:
    X 30 10 -10 (i) Represent the above information in matrix form.
    P(X) (ii) Find the values of ` A, ` B and ` C.
    1 3 1
  10. 6((i))2] 5 10 2  2 x +1 3x 
    Differentiate sin-1   with respect to x.
    Then E(X) will be:  1 + ( 36 )x 
     
  11. 6((ii))2] 5 10 2  2 x +1 3x 
    Show that tan-1x + tan-1y = C is the general solution
  12. 6((xi))2] 5 10 2  2 x +1 3x 
    Statement 1: If ‘A’ is an invertible matrix, then (A2)-1
    of the differential equation (1 + x2) dy + (1 + y2)
    = (A-1)2
    dx = 0
    Statement 2: If ‘A’ is an invertible matrix, then |A-1|
  13. 8((xii))Write the smallest equivalence relation from the set
    cos x
    A to A, where A = {1, 2, 3}. (i) Evaluate: ∫ dx
    3cos x - 5
     0 1 -2 
     OR
  14. 8((xiii))For what value of x, is A =  -1 0 3  a skew
  15. 8((ii))Evaluate: ∫ ( log x ) dx
     x -3 0 
    symmetric matrix? Question 9    4 marks
  16. 8((xiv))Three critics review a book. Odds in favour of the 1  d 2
    y
    book are 5:2, 4:3 and 3:4, respectively for the three (i) If x = tan  log y  then show that (1 + x2) +
    a  dx 2
    critics. Find the probability that all critics are in
    favour of the book. dy
    (2x - a) =0
    5 dx
    OR
  17. 8((xv))Evaluate: ∫ dx
    2x + 7 (ii) The graph of f(x) = -x3 + 27x - 2 is given below:    1 mark
  18. 10((i))under the responsibility of the security agency X? Pia, Sia and Dia displayed their paintings in an art exhibition. The three artists displayed 15, 5 and 10 of SECTION-B (15 MARKS) their paintings, respectively. A person bought three Question 15 paintings from the exhibition. In subparts (i) and (ii) choose the correct options
    Find the probability that he bought one painting and in subparts (iii) and (iv), answer the questions as
    from each of them. instructed.
  19. 10((ii))under the responsibility of the security agency X? Pia, Sia and Dia displayed their paintings in an art exhibition. The three artists displayed 15, 5 and 10 of SECTION-B (15 MARKS) their paintings, respectively. A person bought three Question 15 paintings from the exhibition. In subparts (i) and (ii) choose the correct options
    Find the probability that he bought all the three   2   2
    paintings from the same person. ( ) ( )
  20. 10((i) [cont.])under the responsibility of the security agency X? Pia, Sia and Dia displayed their paintings in an art exhibition. The three artists displayed 15, 5 and 10 of SECTION-B (15 MARKS) their paintings, respectively. A person bought three Question 15 paintings from the exhibition. In subparts (i) and (ii) choose the correct options
    Assertion: a + b + b - a = 2 (a2 + b2)    1 mark
  21. 11((ii))6] Reason: Dot product of any two vectors is 3π commutative.
    Evaluate: ∫ dx is not the correct explanation for Assertion.
    ( x - 1) ( x 2 + 1)
    (c) Assertion is true and Reason is false.
  22. 12((i))6] (d) Assertion is false and Reason is true.
    Solve the differential equation: (ii) The angle between the two planes x + y + 2z = 9
    dy and 2x - y + z = 15 is:
    (x + 5y2) = y when x = 2 and y = 1 π π
    dx (a) (b)
    OR 2 3    1 mark
  23. 12((ii))6] (d) Assertion is false and Reason is true.
    Find the particular solution of the differential
    3π
    equation: (c) π (d)
    (x2 - 2y2) dx + 2xydy = 0, when x = 1 and y = 1
  24. 13((i))(iii) Show that points P(-2, 3, 5) Q(1, 2, 3) and R(7, 0, -1) are collinear. [1] Observe the two graphs, Graph 1 and Graph 2 given below and answer the questions that follow. (iv) Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by Y Y  ( ) ( ) the bees is given r = i + 2 j + 3k + λ 2i + 3 j + 4 k . X (0, 0) X (0, 0) Graph 1 Graph 2
    Which one of the graphs represents y = sin-1x?    1 mark
  25. 13((ii))(iii) Show that points P(-2, 3, 5) Q(1, 2, 3) and R(7, 0, -1) are collinear. [1] Observe the two graphs, Graph 1 and Graph 2 given below and answer the questions that follow. (iv) Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by Y Y  ( ) ( ) the bees is given r = i + 2 j + 3k + λ 2i + 3 j + 4 k . X (0, 0) X (0, 0) Graph 1 Graph 2
    Write the domain and range of y = sin-1x.
    1 2 π    1 mark
  26. 13((iii))(iii) Show that points P(-2, 3, 5) Q(1, 2, 3) and R(7, 0, -1) are collinear. [1] Observe the two graphs, Graph 1 and Graph 2 given below and answer the questions that follow. (iv) Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by Y Y  ( ) ( ) the bees is given r = i + 2 j + 3k + λ 2i + 3 j + 4 k . X (0, 0) X (0, 0) Graph 1 Graph 2
    Prove that sin -1 + sin -1 =
    5 5 2
      3     2 marks
  27. 13((iv))(iii) Show that points P(-2, 3, 5) Q(1, 2, 3) and R(7, 0, -1) are collinear. [1] Observe the two graphs, Graph 1 and Graph 2 given below and answer the questions that follow. (iv) Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by Y Y  ( ) ( ) the bees is given r = i + 2 j + 3k + λ 2i + 3 j + 4 k . X (0, 0) X (0, 0) Graph 1 Graph 2
    Find the value of tan -1  2 sin  2 cos-1   (a) Write the above-mentioned equation in Cartesian
      2
       form.
    (b) Find the equation of the path traced by the other
    honeybee passing through the point (2, 4, 5).    1 mark
  28. 16((i))2] (iii) Consider the following data:
    Find the equation of the plane passing through the
    x 1 2 3 6
    points (2, 2, -1), (3, 4, 2) and (7, 0, 6).
    OR y 6 5 4 1
  29. 16((ii))2] (iii) Consider the following data:
    Find the equation of the plane passing through the x-x
    points (2, 3, 1), (4, -5, 3) and parallel to x-axis.
  30. 17((i))y-y Consider the position vectors of A, B and C as   OA =2i - 2 j + k , OB =+ i 2 j - 2 k and (a) Calculate x and y [1]  OC = 2i - j + 4 k (b) Complete the table. [1]   (c) Calculate bxy [1]
    Calculate AB and BC.
      Question 20    1 mark
  31. 17((ii))y-y Consider the position vectors of A, B and C as   OA =2i - 2 j + k , OB =+ i 2 j - 2 k and (a) Calculate x and y [1]  OC = 2i - j + 4 k (b) Complete the table. [1]   (c) Calculate bxy [1]
    Find the projection of AB on BC. (i) Find the regression line of best fit from the following
  32. 17((iii))y-y Consider the position vectors of A, B and C as   OA =2i - 2 j + k , OB =+ i 2 j - 2 k and (a) Calculate x and y [1]  OC = 2i - j + 4 k (b) Complete the table. [1]   (c) Calculate bxy [1]
    Find the area of the triangle ABC whose sides are data.
     
    AB and BC. ∑x = 24, ∑y = 44, ∑xy = 306, ∑x2 = 164, ∑y2 = 576,
  33. 18((i))The equation y = 4 - x2 represents a parabola. OR
    (a) Make a rough sketch of the graph of the given (ii) Two lines of regression are given as 4x + 3y + 7 = 0
    function. and 3x + 4y + 8 = 0. Identify the line of regression
    of x on y.
    (b) Determine the area enclosed between the curve,
    the x-axis, the lines x = 0 and x = 2 Question 21
    (c) Hence, find the area bounded by the parabola (i) A utensil manufacturer produces ‘x’ dinner sets per
    600 - p
    and the x-axis. week and sells each set at ` p, where x = .
    OR
  34. 18((ii))A farmer has a field bounded by three lines x + 2y = The cost of production of ‘x’ sets is ` x2 + 78x + 2000
    2, y - x = 1, 2x + y = 7. Using integration, find the (a) Write the revenue function.
    area of the region bounded by these lines. (b) Write the profit function.
    SECTION-C (15 MARKS) (c) Calculate the number of dinner sets to be produced
    and sold per week to ensure maximum profit.    1 mark
  35. 19((i))OR In subparts (i) and (ii) choose the correct options and (ii) The Average Cost of producing ‘x’ units of in subpart (iii), answer the questions as instructed. commodity is given by:
    The total revenue received from the sale of ‘x’ units
    x2 x 5000
    of a product is R(x) = 36x + 3x2 + 5. Then, the actual AC = - - 30 +
    revenue for selling the 10th item will be: 200 50 x
    (a) 27 (b) 90 (a) Find the Cost function.
    (c) 93 (d) 33 (b) Find the Marginal Cost function.    1 mark
  36. 19((ii))OR In subparts (i) and (ii) choose the correct options and (ii) The Average Cost of producing ‘x’ units of in subpart (iii), answer the questions as instructed. commodity is given by:
    Read the following statements and choose the (c) Find the Marginal Average Cost function.
    correct option. d MC - AC
    (I) The correlation coefficient and the regression
    (d) Verify that ( AC ) =
    dx x
    coefficients are of the same sign Question 22
    (II) The correlation coefficient is the arithmetic mean Two different types of books have to be stacked in
    between the regression coefficients. the shelf of a library. The first type of book weighs
    (III) The product of two regression coefficients is always 1 kg and has a thickness of 6 cm. The second type
    equal to 1. of book weighs 1.5 kg and has a thickness of 4 cm.
    (IV) Both the regression coefficients cannot be The shelf is 96 cm long and can support a maximum
    numerically greater than unity. weight of 21 kg.    1 mark