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Board Paper of Class 12-Science Term-I 2021 Math Delhi(Set 4) - Solutions

General Instructions:
(i) This question paper comprises 50 questions out of which 40 questions are to be attempted as per instructions. All questions carry equal marks.
(ii) The question paper consists three Sections – Section A, B and C.
(iii) Section – A contains 20 questions. Attempt any 16 questions from Q.No. 1 to 20.
(iv) Section – B also contains 20 questions. Attempt any 16 questions from Q.No. 21 to 40.
(v) Section – C contains 10 questions including one Case Study. Attempt any 8 from Q.No. 41 to 50.
(vi) There is only one correct option for every Multiple Choice Question (MCQ). Marks will not be awarded for answering more than one option.
(vii) There is no negative marking.


  • Question 1
    Differential of log log log x5 w.r.t x is
    (a) 5x log x5 log log x5

    (b) 5x log log x5

    (c) 5x4log x5 log log x5

    (d) 5x4log x5 log log x5 VIEW SOLUTION


  • Question 2
    The number of all possible matrices of order 2 × 3 with each entry 1 or 2 is
    (a) 16
    (b) 6
    (c) 64
    (d) 24 VIEW SOLUTION


  • Question 3
    A function f : RR is defined as f(x) = x3+1. Then the function has
    (a) no minimum value
    (b) no maximum value
    (c) both maximum and minimum values
    (d) neither maximum value nor minimum value VIEW SOLUTION


  • Question 4
    If sin y = x cos (a + y), then dxdy is
    (a) cos acos2 (a+y)

    (b) -cos acos2 (a+y)

    (c) cos asin2 y

    (d) -cos asin2 y VIEW SOLUTION


  • Question 5
    The points on the curve x29+y225=1, where tangent is parallel to x-axis are
    (a) ±5, 0
    (b) 0, ±5
    (c) 0, ±3
    (d) ±3, 0 VIEW SOLUTION


  • Question 6
    Three points P(2x, x + 3), Q(0, x) and R(x + 3, x + 6) are collinear, then x is equal to
    (a) 0
    (b) 2
    (c) 3
    (d) 1 VIEW SOLUTION


  • Question 7
    The principal value of cos-112+sin-1-12 is

    (a) π12

    (b) π3

    (c) π

    (d) π6 VIEW SOLUTION


  • Question 8
    If (x2 + y2)2 = xy, then dydx is

    (a) y+4x(x2+y2)4y(x2+y2)-x

    (b) y-4x(x2+y2)x+4(x2+y2)

    (c) y-4x(x2+y2)4y(x2+y2)-x

    (d) 4y(x2+y2)-xy-4x(x2+y2) VIEW SOLUTION


  • Question 9
    If a matrix A is both symmetric and skew symmetric, then A is necessarily a
    (a) Diagonal matrix
    (b) Zero square matrix
    (c) Square matrix
    (d) Identity matrix VIEW SOLUTION


  • Question 10
    Let set X = {1, 2, 3} and a relation R is defined in X as : R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are
    (a) {(1, 1), (2, 3), (1, 2)} 
    (b) {(3, 3), (3, 1), (1, 2)}
    (c) {(1, 1), (3, 3), (3, 1), (2, 3)} 
    (d) {(1, 1), (3, 3), (3, 1), (1, 2)}  
      VIEW SOLUTION


  • Question 11
    A Linear Programming Problem is as follows :
    Minimise                               z = 2x + y
    subject to the constraints       x ≥ 3, x ≤ 9, y ≥ 0
                                               xy ≥ 0, x + y ≤ 14
    The feasible region has
    (a) 5 corner points including (0, 0) and (9, 5)
    (b) 5 corner points including (7, 7) and (3, 3)
    (c) 5 corner points including (14, 0) and (9, 0)
    (d) 5 corner points including (3, 6) and (9, 5)  VIEW SOLUTION


  • Question 12
    The function fx=e3x-e-5xx, if x0k              , if x=0 is continuous at = 0 for the value of k, as
    (a) 3
    (b) 5
    (c) 2
    (d) 8 VIEW SOLUTION


  • Question 13
    If Cij denotes the cofactor of element pij of the matrix P=1-1202-3324, then the value of C31 ∙ C23 is
    (a) 5
    (b) 24
    (c) –24
    (d) –5 VIEW SOLUTION


  • Question 14
    The function y = x2eis decreasing in the interval
    (a) (0, 2)
    (b) (2, ∞)
    (c) (–∞, 0)
    (d) (–∞, 0) ∪ (2, ∞) VIEW SOLUTION


  • Question 15
    If R = {(x, y); x, y ∊ Z, x+ y2 ≤ 4} is a relation in set Z, then domain of R is
    (a) {0, 1, 2}
    (b) {–2, –1, 0, 1, 2}
    (c) {0, –1, –2} 
    (d) {–1, 0, 1} VIEW SOLUTION


  • Question 16
    The system of linear equations
    5x + ky = 5
    3x + 3y = 5
    will be consistent if 
    (a) k  –3
    (b) k = –5
    (c) k = 5 
    (d) k 5  VIEW SOLUTION


  • Question 17
    The equation of the tangent to the curve y (1 + x2) = 2 – x, where it crosses the x-axis is
    (a) x – 5y = 2
    (b) 5x – y = 2
    (c) + 5y = 2
    (d) 5+ = 2  VIEW SOLUTION


  • Question 18
    If 3c+6a-da+d2-3b=122-8-4 are equal, then value of abcd is 
    (a) 4 
    (b) 16
    (c) –4
    (d) –16 VIEW SOLUTION


  • Question 19
    The principal value of tan-1tan9π8 is
    (a) π8

    (b) 3π8

    (c) -π8

    (d) -3π8
      VIEW SOLUTION


  • Question 20
    For two matrices P = 34-1201 and QT-121123 P – Q is 
    (a) 23-300-3

    (b) 43-30-1-2

    (c) 430-3-1-2

    (d) 230-30-3 VIEW SOLUTION


  • Question 21
    The function f(x) = 2x3 – 15x2 + 36x + 6 is increasing in the interval
    (a) (–∞, 2) ∪ (3, ∞)
    (b) (–∞, 2)
    (c) (–∞, 2] ∪ [3, ∞)
    (d) [3, ∞) VIEW SOLUTION


  • Question 22
    If x = 2 cosθ – cos 2θ and y = 2 sinθ – 2θ, then dydx is

    (a) cosθ+cos 2θsinθ-sin 2θ

    (b) cosθ-cos 2θsin 2θ-sinθ

    (c) cosθ-cos 2θsinθ-sin 2θ

    (d) cos 2θ-cosθsin 2θ+sinθ VIEW SOLUTION


  • Question 23
    What is the domain of the function cos–1 (2x – 3)?
    (a) [–1, 1]
    (b) (1, 2)
    (c) (–1, 1)
    (d) [1, 2] VIEW SOLUTION


  • Question 24
    A matrix A = [aij]3 × 3 is defined by

    aij=2i+3j,i<j5,i=j3i-2j,i>j

    The number of elements in A which are more than 5, is
    (a) 3
    (b) 4
    (c) 5
    (d) 6 VIEW SOLUTION


  • Question 25
    If a function f defined by

    fx=k cosxπ-2x,if xπ23,if x=π2
    is continuous at x=π2, then the value of k is
    (a) 2
    (b) 3
    (c) 6
    (d) –6 VIEW SOLUTION


  • Question 26
    For the matrix X=011101110, (X2 – X) is
    (a) 2 I
    (b) 3 I
    (c) I
    (a) 5 I VIEW SOLUTION


  • Question 27
    Let X = {x2 : xN} and the function f : N → X is defined by f(x) = x2, xN. Then this function is
    (a) injective only
    (b) not bijective
    (c) surjective only
    (d) bijective VIEW SOLUTION


  • Question 28
    The corner points of the feasible region for a Linear Programming problem are P(0, 5), Q(1, 5), R(4, 2) and S(12, 0). The minimum value of the objective function Z = 2x + 5y is at the point
    (a) P
    (b) Q
    (c) R
    (d) S VIEW SOLUTION


  • Question 29

    The equation of the normal to the curve ay2 = x3 at the point (am2, am3) is
    (a) 2y – 3mx + am3 = 0
    (b) 2x + 3my – 3am4am2 = 0
    (c) 2x + 3my + 3am4 – 2am2 = 0
    (d) 2x + 3my – 3am4 – 2am2 = 0

    VIEW SOLUTION


  • Question 30
    If A is a square matrix of order 3 and |A| = –5, then |adj A| is
    (a) 125
    (b) –25
    (c) 25
    (d) ±25 VIEW SOLUTION


  • Question 31
    The simplest form of  tan–1 1+x-1-x1+x+1-x is
    (a) π4-x2

    (b) π4+x2

    (c) π4-12cos-1x

    (d) ​π4+12cos-1x VIEW SOLUTION


  • Question 32
    If for the matric A =α-2-2αA3=125, then the value of α is
    (a) ± 3
    (b) –3
    (c) ± 1
    (d) 1 VIEW SOLUTION


  • Question 33
    If y = sin(m sin–1 x), then which one of the following equations is true?
    (a) 1-x2d2ydx2+xdydx+m2y=0

    (b) 1-x2d2ydx2-xdydx+m2y=0

    (c) 1+x2d2ydx2-xdydx-m2y=0

    (d) ​1+x2d2ydx2+xdydx-m2x=0 VIEW SOLUTION


  • Question 34
    The principal value of tan-13-cot-1-3 is
    (a) π
    (b) -π2
    (c) 0
    (d) 23 VIEW SOLUTION


  • Question 35
    The maximum value of 1xx is
    (a) e1e
     
    (b) e

    (c) 1e1e

    (d) ee VIEW SOLUTION


  • Question 36
    Let matrix X = [xij] is given by X = 112345213. Then the matrix Y = [mij], where mij = Minor of xij, is
    (a) X = 753191111117
    (b) X = 719115113117
    (c) X = 719113117511
    (d) X = 719111113117 VIEW SOLUTION


  • Question 37
    A function f: RR defined by f(x) = 2 + x2 is
    (a) not one-one
    (b) one-one
    (c) not onto
    (d) neither one-one nor onto VIEW SOLUTION


  • Question 38
    A Linear Programming Problem is as follows: 
    Maximise / Minimise objective function Z = 2x – y +5 
    Subject to the constraints
    3x + 4y ≤ 60
    x + 3y ≤ 30
    x ≥ 0, ≥ 0
    If the corner points of the feasible region are A (0, 10), B(12, 6), C(20, 0) and O(0, 0), then which of the following is true? 
    (a) Maximum value of Z is 40 
    (b) Minimum value of Z is – 5 
    (c) Difference of maximum and minimum values of Z is 35 
    (d) At two corner points, value of Z are equal VIEW SOLUTION


  • Question 39
    If x = –4 is a root of x231x132x =0, then the sum of the other two roots is
    (a) 4
    (b) –3
    (c) 2
    (d) 5 VIEW SOLUTION


  • Question 40
    The absolute maximum value of the function fx=4x12x2 in the interval 2,92 is
    (a) 8
    (b) 9
    (c) 6
    (d) 10 VIEW SOLUTION


  • Question 41
    In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is
    (a) 2π2rh (2rh + h2)
    (b) π2hr (2rh + h2)
    (c) 2π2r(2rh2h3)
    (d) 2π2r2 (2rhh2) VIEW SOLUTION


  • Question 42
    The corner points of the feasible region determined by a set of constrains (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and the 
    objective function is Z = ax + 2by where a, b > 0. The condition on a and b such that the maximum Z occurs at Q and S is
    (a) a – 5b = 0
    (b) a – 3b = 0
    (c) a – 2b = 0
    (d) a – 8b = 0 VIEW SOLUTION


  • Question 43
    If curves y2 = 4x and xy = c cut at right angles, then the value of c is
    (a) 42
    (b) 8
    (c) 22
    (d) -42 VIEW SOLUTION


  • Question 44
    The inverse of the matrix X=200030004 is
    (a) 241/20001/30001/4

    (b) 124100010001

    (c) 124200030004

    (d) 1/20001/30001/4 VIEW SOLUTION


  • Question 45
    For an L.P.P. the objective function is Z = 4x + 3y, and the feasible region determined by a set of constraints (linear inequations) is shown in the  graph.



    Which one of the following statements is true ?
    (a) Maximum value of Z is at R.
    (b) Maximum value of Z is at Q.
    (c) Value of Z at R is less than the value at P.
    (d) Value of Z at Q is less than the value at R. VIEW SOLUTION


  • Question 46

    In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
    Based on this information, answer the any 4 of the following questions.

    Let side of square plot is x m and its depth is h metres, then cost c for the pit is
    (a) 50h+400 h2

    (b) 12500h+400 h2

    (c) 250h+h2

    (d) 250h+400 h2 VIEW SOLUTION


  • Question 47

    In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
    Based on this information, answer the any 4 of the following questions.

    Value of h (in m) for which dcdh=0 is
    (a) 1.5
    (b) 2
    (c) 2.5
    (d) 3 VIEW SOLUTION


  • Question 48

    In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
    Based on this information, answer the any 4 of the following questions.

    d2cdh2 is given by

    (a) 25000h3+800

    (b) 500h3+800

    (c) 100h3+800

    (b) 500h3+2
      VIEW SOLUTION


  • Question 49

    In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
    Based on this information, answer the any 4 of the following questions.

    Value of x (in m) for minimum cost is
    (a) 5
    (b) 1053
    (c) 55
    (d) 10 VIEW SOLUTION


  • Question 50

    In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents welfare association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of ₹50 per square metre for space so that there is no misuse of the space and Resident welfare association takes it seriously. Association hired a labourer for digging out 250 m3 and he charged ₹400 × (depth)2. Association will like to have minimum cost.
    Based on this information, answer the any 4 of the following questions.

    Total minimum cost of digging the pit (in ₹) is
    (a) 4,100
    (b) 7,500
    (c) 7,850
    (d) 3,220 VIEW SOLUTION
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