Subject: Mathematics
Class XII
Time Allowed: 3 hours
Max. Marks: 80
General Instructions:
1. All questions are compulsory
2. Section A contains 20 questions each carries 1 mark
3. Section B contains 5 questions each carries 2 marks
4. Section C contains 6 questions each carries 3 marks
5. Section D contains 3 case based questions of 4 marks each.
6. Section E contains 4 questions each carries 5 mark.
7. There are some internal choice questions in Sections B, C and E.
Section A
1. Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is:
a) Transitive and symmetric
b) Reflexive and symmetric
c) Reflexive, transitive but not symmetric
d) Equivalence
2. f : R — > R: f(x) = x3 → is
a) many one and into b) one one and onto c) many one and onto d) one one and into
3. Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into:
a) intersecting sets b) two sets c) mutually disjoint subsets d) three sets
4. Let θ = sin–1 (sin (– 600°)), then value of θ is:
a) −2π/3 b) 2π/3 c) π/2 d) π/3
5. sin (cot-1 x) is equal to:
a) None of these b) x/√(1 + x2) c) 1/√(1 + x2) d) √(1 + x2)
6. Which of the following is the principal value branch of cos–1x?
a) (0, π) − {π/2} b) [-π/2, π/2] c) (0, π) d) [0, π]
a) none of these b) 1 x 1 matrix c) not defined d) 3 x 3 matrix
9. If A is 3 4 matrix and B is a matrix such that ATB and BAT are both defined. Then, B is of the type:
a) 4 x 4 b) 4 x 3 c) 3 x 3 d) 3 x 4
10. If A is singular matrix, then adj A is:
a) symmetric b) non-singular c) not defined d) singular
11. If A is singular then A(adj A) = ?
a) None of these b) A null matrix c) A unit matrix d) A symmetric matrix
12. If B is non-singular matrix and A is a square matrix, then det (B-1AB) is equal to:
a) Det (A-1) b) Det (A) c) Det (B) d) Det (B-1)
15. If y = a sin mx + b cos m x, then d2y/dx2 is equal to
a) my1 b) None of these c) -m2y d) m2y
16. How many non-decreasing functions f : A — > B where A = {1, 2, 3}, B = {1, 2, 3, 4, 5} can be defined?
a) 60 b) 35 c) 25 d) 20
17. AB is a vertical lamp post (height = 600 cm) and CD represents a man (height = 150 cm). The man walks towards the lamp post at 180 cm/s. How fast is the shadow diminishing?
a) 420 cm/s b) -420 cm/s c) -60 cm/s d) 60 cm/s
18. The function/(x) = 3x + cos 3x is:
a) Decreasing on R
b) Strictly decreasing on R
c) Strictly increasing on R
d) Increasing on R
19. Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q > 0. Condition on p and q so that the minimum of Z occurs at (3, 0) and (1, 1) is:
a) p = 3q b) p = q/2 c) p = 2q d) p = q
20. The feasible region for a LPP is shown in Figure. Find the minimum value of Z = 11x + 7y.
a) 22 b) 21 c) 19 d) 20
21. Find the principal value of tan (-1)
22. Construct a 3 x 2 matrix whose elements are given by aij = (2i – j).
23. If the points (x, – 2), (5, 2) and (8, 8) are collinear, find x using determinants.
24. Find the derivative of the function given by f(x) = sin (x2).
25. Find the points of local maxima or local minima, if any, using the first derivative test. Also find the local maximum or local minimum values, as the case may be: f(x) = sin2x, 0 < x < π
OR
Find the intervals in function f(x) = 2x3 – 24x + 107 is increasing or decreasing.
Section C
26. Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a +1} is reflexive, symmetric or transitive.
27. Find the value of tan–1(1) + cos–1 (−1/2) + sin-1 (−1/2)
29. Write minors and cofactors of the element of:
30. Differentiating the function w.r.t. x:
OR
31. Determine the maximum value of Z = 11x + 7y subject to the constraints:
2x + y ⩽ 6,x ⩽ 2,x ⩾ 0,y ⩾ 0.
Section D
32. Read the text carefully and answer the questions that follow:
Sherlin and Danju were playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time it belonged to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
(i) Let R ∶ B → B be defined by R = {(x, y): y is divisible by x} is:
a) Not reflexive but symmetric and transitive
b) Equivalence
c) Reflexive and transitive but not symmetric
d) Reflexive and symmetric and not transitive
(ii) Raji wants to know the number of functions from A to B. How many number of functions are possible?
a) 6! b) 62 c) 212 d) 26
(iii) Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R is:
a) Transitive b) Reflexive c) None of these d) Symmetric
(iv) Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
a) 212 b) 6! c) 62 d) 26
33. Read the text carefully and answer the questions that follow:
While working with excel, we need to switch or rotate cells. You can do this by copying, pasting, and using the Transpose option. But doing that creates duplicated data. If you don’t want that, you can type a formula instead using the TRANSPOSE function. For example, in the following picture the formula =TRANSPOSE (A1:B4) takes the cells A1 through B4 and arranges them horizontally.
(i) A square matrix A is expressed as sum of symmetric and skew-symmetric matrices and then symmetric part of A is:
a) ½(AT – A) b) ½(A + AT) c) c) None of these d) ½(A – AT)
(ii) A square matrix A is expressed as sum of symmetric and skew-symmetric matrices and then skew-symmetric part of A is:
a) ½(A – AT) b) ½(A + AT) c) None of these d) ½(AT – A)
34. Read the text carefully and answer the questions that follow:
Sofia wants to prepare a handmade gift box for her friend’s birthday at home. For making the lower part of the box, she takes a square piece of cardboard of side 20 cm.
i) If x cm be the length of each side of the square cardboard which is to be cut off from corners of the square piece of side 20 cm, then a possible value of x will be given by the interval
a) None of these b) (0, 3) c) [0, 20] d) (0, 10)
ii) Volume of the open box formed by folding up the cutting corner can be expressed as
a) V = x/3(20 – 2x)(20 + 2x)
b) V = x(20 – 2x)(20 – 2x)
c) V = x(20 – 2x)(20 – x)
d) V = x/2(20 + 2x)(20 – x)
iii) The values of x for which dV/dx = 0, are:
a) 3, 4 b) 0, 10/3 c) 0, 10 d) 10, 10/3
iv) Sofia is interested in maximising the volume of the box. So, what should be the side of the square to be cut off so that the volume of the box is maximum?
a) 2 cm b) 10/3 cm c) 12 cm d) 8 cm
Section – E
35. Using matrix method, solve the following system of linear equations:
x + y = 5
y + z = 3
x + z = 4
36. Find the values of p and q so that
differentiable at x = 1
OR
Find dy/dx when y = xsin x + (sin x) x
37. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
OR
A window is in the form of a rectangle surmounted by a semicircular opening the total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
38. Minimise and maximise z = 5x + 2y subject to the following constraints:
x – 2y ≤ 2,
3x + 2y ≤ 12,
– 3x + 2y ≤ 3,
x ≥ 0, y ≥ 0.