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Board Paper of Class 12-Commerce 2023 Math Delhi(Set 3) - Solutions

General Instructions:
Read the following instructions very carefully and follow them:
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) Question paper is divided into FIVE Sections-Section A, B, C, D and E.
(ii) In Section A - Question Number 1 to 18 are Multiple Choice Questions (MCQ) type and Question Number 19 & 20 are Assertion-Reason based questions of 1 mark each.
(iv) In Section B - Question Number 21 to 25 are Very Short Answer (VSA) type questions of 2 marks each.
(v) In Section C - Question Number 26 to 31 are Short Answer (SA) type questions, carrying 3 marks each.
(vi) In Section D - Question Number 32 to 35 are Long Answer (LA) type questions carrying 5 marks each.
(vii) In Section E - Question Number 36 to 38 are case study based questions carrying 4 marks each where 2 VSA type questions are of 1 mark each and 1 SA type question is of 2 marks. Internal choice is provided in 2 marks question in each case-study.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B - 3 questions in Section C, 2 questions in
Section-D and 2 questions in Section - E.
(ix) Use of calculators is NOT allowed.

  • Question 1
    Let R be a relation in the set N given by
    R = {a, b): a = b − 2, b > 6}.
    (a) (8, 7) ∊ R
    (b) (6, 8) ∊ R
    (c) (3, 8) ∊ R
    (d) (2, 4) ∊ R VIEW SOLUTION

  • Question 2
    If A=5xy0 and A = AT, where AT is the transpose of the matrix A, then
    (a) x = 0, y = 5
    (b) x = y
    (c) x + y = 5
    (d) x = 5, y = 0 VIEW SOLUTION

  • Question 3
    sinπ3+sin-112 is equal to
    (a) 1
    (b) 12
    (c) 13
    (d) 14 VIEW SOLUTION

  • Question 4
    If for a square matrix, A, A2 − A + I = O, then A−1 equals
    (a) A
    (b) A + I
    (c) I − A

  • Question 5
    If α34121141=0, then the value of α is
    (a) 1
    (b) 2
    (c) 3

  • Question 6
    If f(x) = | cos |, then f3π4 is
    (a) 1
    (b) −1
    (c) -12
    (d) 12 VIEW SOLUTION

  • Question 7
    If x = A cos 4t + B sin 4t, then d2xdt2 is equal to
    (a) x
    (b) −x
    (c) 16x
    (d) −16x  VIEW SOLUTION

  • Question 8
    The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x, is continuous at
    (a) x = 1
    (b) x = 1.5
    (c) x = −2
    (d) x = 4 VIEW SOLUTION

  • Question 9
    The function f(x) = x3 + 3x is increasing in interval
    (a) (−∞, 0)
    (b) (0, ∞)
    (d) (0, 1) VIEW SOLUTION

  • Question 10
    -11x-2x-2dx, x2 is equal to
    (a) 1
    (b) −1
    (c) 2
    (d) −2 VIEW SOLUTION

  • Question 11
    sec xsec x-tan x dx equals
    (a) sec x − tan x + c
    (b) sec x + tan x + c
    (c) tan x − sec x + c
    (d) −(sec x + tan x) + c VIEW SOLUTION

  • Question 12
    The order and the degree of the differential equation 1+3dydx2=4d3ydx3 respectively are:
    (a) 1, 23
    (b) 3, 1
    (c) 3, 3
    (d) 1, 2 VIEW SOLUTION

  • Question 13
    If a·i^=a · (i^+j^)=a·(i^+j^+k^)=1, then a is
    (a) k^(b) i^(c) j^(d) i^+j˙+k^ VIEW SOLUTION

  • Question 14
    Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is
    (a) 2732
    (b) 532
    (c) 3132
    (d) 132 VIEW SOLUTION

  • Question 15
    If for any two events A and B, P(A)=45 and P(A ∩ B)=710, then P(B/A) is equal to
    (a) 110 
    (b) 18
    (c) 78
    (d) ​1720 VIEW SOLUTION

  • Question 16
    The angle between the lines 2x = 3y = −z and 6x = −y = −4z is 
    (a) 0°​
    (b) 30°​
    (c) 45°​
    (d) 90°​ VIEW SOLUTION

  • Question 17
    If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are
    (a) 0,-12, 12
    (b) -12, 0, 12
    (c) 12, 0, -12
    (d) 0, 12, 12 VIEW SOLUTION

  • Question 18
    The magnitude of the vector 6i^-2j^+3k^ is
    (a) 1
    (b) 5
    (c) 7
    ​(d) 12 VIEW SOLUTION

  • Question 19
    Assertion (A): 2810-xx+10-xdx=3
    Reason (R): abf(x)dx=abf(a+b-x)dx

    (a) Both (A) and (R) are true and (R) is the correct explanation of (A).
    (b) Both (A) and (R) are true, but (R) is not the correct explanation of (A).
    (c) (A) is true and (R) is false.
    (d) (A) is false, but (R) is true.  VIEW SOLUTION

  • Question 20
    Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is 13.
    Reason (R): Let E and F be two events with a random experiment, then P(F/E)=P(EF)P(E).
    ​(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
    (b) Both (A) and (R) are true, but (R) is not the correct explanation of (A).
    (c) (A) is true and (R) is false.
    (d) (A) is false, but (R) is true.  VIEW SOLUTION

  • Question 21
    If x = a cos t and y = b sin t, then find d2ydx2.


    Find the value of k for which the function f given as
    fx=1-cos x2x2 ,  if x0       k       ,  if x0 is countinuous at x = 0 VIEW SOLUTION

  • Question 22
    Find the value of tan-12cos2sin-112+tan-1 1. VIEW SOLUTION

  • Question 23
    Find the vector and the cartesian equations of a line that passes through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7= 35z. VIEW SOLUTION

  • Question 24
    Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis. Hence, obtain its area using integration. VIEW SOLUTION

  • Question 25
    If the vectors a and b are such that a=3, b =23 and a×b is a unit vector, then  find the angle between a and b.


    Find the area of a parallelogram whose adjacent sides are determined by the vectors a=i^j^+3k^ and b=2i^-7j^+k^. VIEW SOLUTION

  • Question 26
    Show that the determinant xsin θcos θ-sin θ-x1cos θ1x is independent of θ. VIEW SOLUTION

  • Question 27
    Using integration, find the area of the region bounded by y = mx (m > 0), x = 1, x = 2 and the x-axis. VIEW SOLUTION

  • Question 28
    Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line x-153=y-298=z-5-5


    If a=i^+j^+k^ and b=i^+2j^+3k^ then find a unit vector perpendicular to both a+b and a-b. VIEW SOLUTION

  • Question 29
    Find the distance between the lines : 
    r=(i^+2j^-4k^)+λ(2i^+3j^+6k^) ;r=(3i^+3j^-5k^)+μ(4i^+6j^+12k^) VIEW SOLUTION

  • Question 30
    If y = tan x + sec x, then prove that d2ydx2=cosx1-sinx2.


    Differentiate sec–1 11-x2 w.r.t. sin-12x1-x2. VIEW SOLUTION

  • Question 31
    Evaluate : 02π11+esinxdx


    Find : x4(x1)(x2+1)dx VIEW SOLUTION

  • Question 32
    Solve the following Linear Programming Problem graphically:
    Minimize : Z = 60x + 80y
    Subject o constraints:
    3x + 4y ≥ 8
    5x + 2y ≥ 11
    x, y ≥ 0

  • Question 33
    The median of an equilateral triangle is increasing at the rate of 23 cm/s. Find the rate at which its side is increasing.


    Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers. VIEW SOLUTION

  • Question 34
    Evaluate: 0π2 sin 2x tan-1sinxdx VIEW SOLUTION

  • Question 35
    In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 35 be the probability that he knows the answer and 25 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 13 . What is the probability that the student knows the answer, given that he answered it correctly ?


    A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize. VIEW SOLUTION

  • Question 36
    Case Study
    An organization conducted bike race under two different categories - Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.
    Let B = {b1, b2, b3} and G = {g1, g2}, where B represents the set of Boys selected and G the set of Girls selected for the final race.

    Based on the above information, answer the following questions
    (I) How many relations are possible from B to G ?
    (II) Among all the possible relations from B to G, how many functions can be formed from B to G ?
    (III) Let R : B → B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation.


    (III) A function f : B → G be defined by f = {b1, g1). (b2, g2), (b3, g1)}. Check if f is bijective. Justify your answer. VIEW SOLUTION

  • Question 37
    Case Study
    Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250.

    Based on the above information, answer the following questions:
    (I) Convert the given above situation into a matrix equation of the form AX = B.
    (II) Find |A|.
    (III) Find A–1.


    (III) Determine P = A2 – 5A. VIEW SOLUTION

  • Question 38
    Case Study
    An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form dydx=Fx, y is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if Fλx, λy=λn Fx, y. To solve a homogeneous differential equation of the type dydx=Fx, y= gyx , we make the substitution = vx and then separate the variables.

    Based on the above, answer the following questions:
    (I) Show that (x2y2) dx + 2xy dy = 0 is a differential equation of the type dydx=gyx.
    (II) Solve the above equation to find its general solution. VIEW SOLUTION
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