Board Paper of Class 12 2021 Mathematics Term-1 Set-4 CODE 065/2/4

General Instructions:

  1. This question paper comprises 50 questions out of which 40 questions are to be attempted as per instructions. All questions carry equal marks.
  2. The question paper consists of three Sections - Section A, B and C.
  3. Section - A contains 20 questions. Attempt any 16 questions from Q. No. 1 to 20.
  4. Section - B also contains 20 questions. Attempt any 16 questions from Q. No. 21 to 40.
  5. Section - C contains 10 questions including one Case Study. Attempt any 8 from Q. No. 41 to 50.
  6. There is only one correct option for every Multiple Choice Question (MCQ). Marks will not be awarded for answering more than one option.
  7. There is no negative marking.
SECTION-A
(In this section, there are 20 questions. Any 16 are to be attempted).

Question 1

Differential of  log[log(logx5)] w.r.t.x  is

  1.  5xlog(x5)log(logx5) 
  2.  5xlog(log x5) 
  3.  5x4log(x5) log(log x5) 
  4.  5x4log  x5  log(log  x5) 

Ans. (a)

Question 2

The number of all possible matrices of order 2 × 3 with each entry 1 or 2 is

  1. 16
  2. 6
  3. 64
  4. 24

Ans. (c)

Sol: Number of elements in matrix of order 2 × 3 = 6
Number of all possible matrices =  26  = 64

Question 3

A function f : R  R is defined as  f(x) = x 3 + 1  Then the function has

  1. no minimum value
  2. no maximum value
  3. both maximum and minimum values
  4. neither maximum value nor minimum value

Ans. (d)

Question 4

If  sin y = x cos (a + y),  then  dx dy  is

  1.  cos acos2(a+y)
  2.  -cos acos2(a+y)
  3.  cos asin2 y
  4.  -cos asin2y

Ans.(a)

Question 5

The points on the curve  x29+y225=1,  where tangent is parallel to x-axis are

  1. (±5, 0)
  2. (0, ±5)
  3. (0, ±3)
  4. (±3, 0)

Ans. (b)

Question 6

Three points   P(2x, x + 3), Q(0, x) and R(x + 3, x + 6)  are collinear, then x is equal to

  1. 0
  2. 2
  3. 3
  4. 1

Ans. (d)

Question 7

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

  1. π3
  2. π
  3. π3
  4. π6

Ans: (a)

Question 8

If  (x2+y2)2 = xy then dydx is

  1.  y+4xx2+y24yx2+y2-x 
  2.  y-4xx2+y2x+4x2+y2 
  3.  y-4xx2+y24yx2+y2-x 
  4.  4yx2+y2-xy-4xx2+y2 

Ans. (c)

Question 9

If a matrix A is both symmetric and skew symmetric, then A is necessarily a

  1. Diagonal matrix
  2. Zero square matrix
  3. Square matrix
  4. Identity matrix

Ans. (b)

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

  1. {(1, 1), (2, 3), (1, 2)}
  2. {(3, 3), (3, 1), (1, 2)}
  3. {(1, 1), (3, 3), (3, 1), (2, 3)}
  4. {(1, 1), (3, 3), (3, 1), (1, 2)}

Ans. (c)

Sol. To make R reflexive: Add (3, 1), (2, 3) in the relation R.
To make R symmetric: Add (1, 1), (3, 3) in the relation R.
  The ordered pairs should be added in Relation ‘R’ are {(1, 1) (3, 3) (3, 1), (2, 3)}.

Question 11

Ans. (b)

Question 12

The function f(x)={e3x-e-5xx,if x 0 k            ,if x= 0  is continuous at x = 0 for the value of k, as

  1. 3
  2. 5
  3. 2
  4. 8

Ans.(d)

Question 13

If  Cij  denotes the cofactor of element  Pij  of the matrix P = 1-1202-3324  then the value of  C31  .  C23 

  1. 5
  2. 24
  3. -5
  4. -24

Ans.(a)

Sol. Given, matrix is

P =1-1202-3324 

C31 =(-1)4 -122-3 =3-4 = -1

C23=-12+3-1-132=-2+3=-5

Then,  C31 · C23 = (1) × (5) = 5 

Question 14

The function is decreasing in the interval  y = x2 e-x  is decreasing in the interval

  1. 0 , 2
  2. 2 , 
  3. - , 0
  4. - , 0  2 ,  

Ans. (d)

Question 15

If R =(x,y);x,yZ,x2+y24  is a relation in set Z, then domain of R is

  1. {0, 1, 2}
  2. {–2, –1, 0, 1, 2}
  3. {0, –1, –2}
  4. {–1, 0, 1}

Ans. (b)

Question 16

The system of linear equations 5x + ky = 5, 3x + 3y = 5; will be consistent if

  1. k – 3
  2. k = – 5
  3. k = 5
  4. k 5

Ans. (d)

Sol.
The system of linear equation will be consistent if
  0
Now, =5k33015  3k  03k  15k  5

Question 17

The equation of the tangent to the curve  y(1 + x2 ) = 2-x,  where it crosses the x-axis is

  1. x – 5y = 2
  2. 5x – y = 2
  3. x + 5y = 2
  4. 5x + y = 2

Ans. (c)

Question 18

If  3c+6a-da+d2-3b=122-8-4  are equal, then value of ab-cd is

  1. 4
  2. 16
  3. -6
  4. -16

Ans. (a)

Question 19

The principal value of  tan-1tan9π8  is

  1. π8
  2. 3π8
  3. -π8
  4. -3π8

Ans. (a)

Question 20

For two matrices P =  34-1201  and  -121123 , P-Q is

  1.  23-300-3 
  2.  43-30-1-2 
  3.  430-3-1-2 
  4.  230-30-3 

Ans. (b)

SECTION-B
(In this section, there are 20 questions. Any 16 are to be attempted).

Question 21

The function  f(x)=2x3-15x2+36x+6  is increasing in the interval

  1. ( , 2)  (3,  ) 
  2. ( , 2) 
  3. ( , 2 ]  [ 3, )
  4. [3,)

Ans: (c);

Question 22

If  x=2cosθ-cos2θ and y=2sinθ-sin2θ, then dydx is

  1.  cosθ+cos2θsinθ-sin2θ 
  2.  cosθ-cos2θsin2θ-sinθ 
  3.  cosθ-cos2θsinθ-sin2θ 
  4.  cos2θ-cosθsin2θ-sinθ 

Ans. (b)

Question 23

What is the domain of the function  cos-12x-3?

  1. -1, 1
  2. 1,2
  3. -1,1
  4. 1, 2

Ans: (d);

Question 24

Ans: (b);

Question 25

If a function f defined by  f(x)=kcosxπ-2x,if xπ23          ,if x=π2 , then the value of k is

  1. 2
  2. 3
  3. 6
  4. -6

Ans. (c)

Question 26

For the matrix  X=011101110, X2-X  is

  1. 2I
  2. 3I
  3. I
  4. 5I

Ans. (a)

Question 27

Let X=x2:xN  and the function f: NX is defined by fx=x2,xN. Then this function is

  1. injective only
  2. not bijective
  3. surjective only
  4. bijective

Ans.(d)

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

  1. P
  2. Q
  3. R
  4. S

Sol. (c);

Given, points of the feasible region: P (0, 5), Q (1, 5), R (4, 2) and S (12, 0) and objective function Z = 2x + 5y

Here, ZP = 2 × 0 + 5 × 5 = 25 
ZQ = 2 × 1 + 5 × 5 = 27
ZR = 2 × 4 + 5 × 2 = 18  Minimum value
ZS = 2 × 12 + 5 × 0 = 24 

Hence, the minimum value of Z is at the point R.

Question 29

The equation of the normal to the curve  ay2=x3  at the point  (am2 , am3 ) is

  1. 2y  3mx + am3  = 0 
  2. 2x + 3my  3am4   am2  = 0
  3. 2x + 3my + 3am4   2am2  = 0 
  4. 2x + 3my  3am4   2am2  = 0

Ans. (d)

Question 30

If A is a square matrix of order 3 and  A = 5  ,then  adj A  is

  1. 125
  2. –25
  3. 25
  4. ±25

Ans. (c)

Question 31

The simplest form of  tan-11+x-1-x1+x+1-x 

  1. π4-π2 
  2. π4+π2 
  3. π4-12cos-1x
  4. π4+12cos-1x

Ans.(c)

Question 32

If for the matrix  A=α-2-2α, A3=125,  then the value of  α  is

  1. ±3
  2. -3
  3. ±1
  4. 1

Ans : (a);

Question 33

If  y = sin (m sin-1x),  then which one of the following equations is true?

  1. 1-x2d2ydx2+xdydx+m2y=0
  2. 1-x2d2ydx2-xdydx+m2y=0
  3. 1+x2d2ydx2-xdydx-m2y=0
  4. 1+x2d2ydx2+xdydx-m2x=0

Ans. (b)

Question 34

The principal value of  tan-13-cot-1-3  is

  1. π
  2. -π2
  3. 0
  4. 23

Ans. (b)

Question 35

The maximum value of  1xx  is

  1. e1/e
  2. e
  3. 1e1/e
  4. ee

Ans. (a)

Question 36

Let matrix  X=xij is given by  X=11-234-521-3. Then the matrix  Y=mij  where  mij = Minor of xij ,is

  1. 7-5-3191-11-1117
  2. 7-19-115-1-13117
  3. 719-11-3117-5-1-1
  4. 719-11-1-17-3-117

Ans. (d)

Question 37

A function  f:RR  defined by  f(x) = 2 + x2 is 

  1. not one-one
  2. one-one
  3. not onto
  4. neither one-one nor onto

Ans. (d)

Question 38

Ans. (b)

Sol: Objective function Z = 2x – y + 5

Points of the feasible region are
A(0, 10), B(12, 6), C(20, 0) and O(0, 0)
 ZA = 2 × 0  10 + 5 =  5  Min. value    ZB = 2 × 12  6 + 5 = 24  6 + 5 = 23    ZC = 2 × 20  0 + 5 = 40 + 5 = 45     ZO = 0  0 + 5 = 5

Question 39

If x = – 4 is a root of  x231x132x=0,  then the sum of the other two roots is

  1. 4
  2. -3
  3. 2
  4. 5

Ans. (a)

Question 40

The absolute maximum value of the function  f(x) = 4x12x2  in the interval  2,92  is

  1. 8
  2. 9
  3. 6
  4. 10

Ans. (a)

SECTION-C
(Attempt any 8 questions out of the Questions 41–50. Each question is of one mark).

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

  1. 2π2rh (2rh + h2 )
  2. π2hr (2rh + h2 )
  3. 2π2r (2rh2 - h3)
  4. 2π2r2 (2rh - h2)

Ans. (c)

Question 42

The corner points of the feasible region determined by a set of constraints (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

  1. a – 5b = 0
  2. a – 3b = 0
  3. a – 2b = 0
  4. a – 8b = 0

Ans. (d)

Question 43

If curves  y2  = 4x  and xy = c cut at right angles, then the value of c is

  1. 42
  2. 8
  3. 22
  4. -42

Ans. (a)

Question 44

The inverse of the matrix  X=200030004  is

  1. 24120001300014 
  2. 124100010001 
  3. 124200030004 
  4. 120001300014 

Ans. (d)

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.

Ans. (b)

Sol. Z = 4x + 3y

Corner points of feasible region are P(0, 40), O(0, 0), R(40, 0), and Q(30, 20)
ZP = 4 × 0 + 3 × 40 = 120ZO = 0ZR = 4 × 40 + 3 × 0 = 160ZQ = 4 × 30 + 3 × 20 = 120 + 60 = 180  Maximum value 

Question 46

Let side of square plot is x m and its depth is h metres, then cost c for the pit is

  1. 50h+400h2
  2. 12500h+400h2
  3. 250h+h2
  4. 250h+400h2

Ans. (b)

Question 47

Value of h (in m) for which  dcdh=0  is

  1. 1.5
  2. 2
  3. 2.5
  4. 3

Ans. (c)

Question 48

 d2cdh2   is given by

  1. 25000h3+800
  2. 500h3+800
  3. 100h3+800
  4. 500h3+2

Ans. (a)

Question 49

Value of x (in m) for minimum cost is

  1. 5
  2. 1053
  3. 55
  4. 10

Ans. (d)

Question 50

Total minimum cost of digging the pit (in ₹) is

  1. 4,100
  2. 7,500
  3. 7,850
  4. 3,220

Ans. (b)

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