2022 NDA -1 Math -834

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NDA Exam (abhipedia)
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2022 NDA -1 Math (Test Code: 834)
Qs: 1

If ?I = |  1    p    q |                       |  1    1    1 |

          |  1    q    r  |   and ?2 =      |  q    r     p |

          |  1   r     p |                        |  r    p     q |

Where p  ≠  q  ≠ r,  then ?1 +?2 is

         

  1. 0

  2. always positive

  3. always negative

  4. positive.if p, q, rare positive else negative

Qs: 2

lf(a-b)(b-c)(c-a)=2 and abc =  6, then  what  is the value of 
|a    b      c|
|a2   b2   c2| ?
|a3   b3   c3|
 

  1. 3

  2. 12

  3. 14

  4. 15

Qs: 3

Under which of the following condi- tions does the determinant
|a    b   c|
|b     c  a| vanish?
|c   a   b|
1. a+b+c=0
2. a3 + b3 + c3 = 3abc
3. a2 + b2 + c2- ab - bc - ca= 0
Select the correct answer using the code given below :

  1. 1  and  2 only

  2. 2 and 3  only

  3. 1 and 3  only

  4. 1, 2 and 3

Qs: 4

Consider the following in respect of the matrices :
A=[m n], B=[-n -m] and C=[ m]
                                          [-m]
1. CA=CB
2. AC=BC
3. C(A + B) = CA + CB
Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. 2 and 3

  4. 1 and 2

Qs: 5

         [|2 sin θ   cos θ   0]
If A = [-2 cos θ  sin θ   0], then
         [-1                 1          1 ]
what is A(adjA) equal to?
where / is the identity matrix.

  1. Null matrix
  2. -I

  3. I

  4. 2I

Qs: 6

For what value of  k is the matrix singular ?

         [|2cos2θ     2cos2θ    6]
         [1-2 sin2θ  2cos2θ-1  3]
         [ k                 2k         1]

  1. 0 only

  2. 1 only

  3. 2 only

  4. Any real value

Qs: 7

Let A be a non-singular matrix and B = adjA.   Which    of   the   following statements is/are correct?
1. AB= BA
2. AB is a scalar matrix
3. AB can be a null matrix
Select the correct answer using the code given below :

  1. 1 only

  2. 1 and 2 only

  3. 2 only

  4. 1, 2 and 3

Qs: 8

Consider the following statements in respect of square matrices A arid B of same order:
l. If AB is a null matrix, then a; least one of A and B is a null matrix.
2. If AB is. an identity matrix, then BA=AB.
Which of the above statements is/are correct?
 

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

Qs: 9

If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C=A +B?

  1. 1

  2. 2

  3. 4

  4. 8

Qs: 10

Let  A and  B be  non-singular  matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct?
1. A2= A
2. AB2= A2B
Select the correct answer using the code given below :

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

Qs: 11

How many terms are there in the expansion of 
[1+2/x]9 [1-2/x]9 ?

  1. 9

  2. 10

  3. 19

  4. 20

Qs: 12

Consider the following statements in respect of the expansion of (x +y)10 :
1. Among all the coefficients of the terms, the coefficient of the 6th  term has the highest value
2. The coefficient of the 3rd term is equal to coefficient of the 9th term
Which of the above statements is/are correct. 

  1. 1 only

  2. 2 only .

  3. Both 1 and 2

  4. Neither 1 nor 2

Qs: 13

If C(3n, 2n) = C(3n, 2n-7), then what is the value of C(n, n-5)?

  1. 42

  2. 35

  3. 28

  4. 21

Qs: 14

What is the value of

C(51, 21)- C(51, 22) + C(51, 23)­ -

C(51, 24) + C(51, 25)- C(51, 26) +

C(51, 27)- C(51, 28) + C(51, 29)­ -

C(51, 30) ?

  1. C(51, 25)

  2. C(51, 27)

  3. C(51, 51)- C(51, 0)

  4. C(51,,25)- C(51, 27)

Qs: 15

How many odd numbers between 300 and 400 are there in which none of the digits is repeated?

  1. 32

  2. 36

  3. 40

  4. 25

Qs: 16

How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even positions ?

  1. 72

  2. 36

  3. 18

  4. 12

Qs: 17

Let a and /3 be the roots of  the equation x2 + px + q = 0. If a3 and β3  are  the  roots  of   the   equation x2+ mx + n = 0, then what is the value of m +n?

  1. p3+q3+pq
     

  2. p3+q3-pq

  3. p3 + q3 + 3pq

  4. P3+ q3 - 3pq

Qs: 18

Let a and /3 be the roots of the equa­tion x2- ax - bx + ab - c = 0. What is the quadratic equation whose roots are a and b?

  1. x2- ax - βx +  aβ + c = 0

  2. x- ax - βx +  β + c = 0

  3. x2- ax - βx +  c = 0

  4. x2- ax - βx +  aβ + c 

Qs: 19

If the roots of the equation x2 - ax - bx -q + be +•ca = 0 

are equal, then which one of the following •is correct ?

  1. a + b + c = 0

  2. a -  b  + c = 0

  3. a + b -  c = 0

  4. -a + b + c = 0

Qs: 20

Let  a  and  β (a > β)  be  the  roots of the equation x2 - 8x + q = 0. If a2 -β2 = 16,  then  what  is the value of q? 

  1. -15

  2. -10

  3. 10

  4. 15

Qs: 21

What is the maximum value of n such that 5n divides (30! + 35!), where n is a natural number ?

  1. 4

  2. 6

  3. 7

  4. 8

Qs: 22

What is the value of
2(_2x 1)  + 3(3  x2x 1)+4(4x3x2x1)+
5(5x4x3x2x 1) +    ......    ......    ......+
9(9x8 x7x6x5 x 4 x 3 x2 x 1)+2 ?

  1. 11!

  2. 10!

  3. 10+10!

  4. 11+10!

Qs: 23

If A= {{l, 2, 3}}, then how many elements are there in  the  power  set of A?

  1. 1

  2. 2

  3. 4

  4. 8

Qs: 24

If a, b, c are in GP where a > 0, b > 0, c > 0, then which of  the  following are correct ?

1. a2, b2, c2 are in GP

2. 1/a, 1/b' 1/care m GP

3. √a, √b , √c are in GP

Select the correct answer using the code given below :

  1. 1 and  2 only

  2. 2 and  3 only

  3. 1 and  3 only

  4. 1, 2 and 3

Qs: 25

If a+b/2,  b,  b+c/2 are in HP. then which one of the following is correct?

  1. a, b, care in AP

  2. a, b, c are  in GP

  3. a + b, b + c, c + a are in GP

  4. a+ b, b + c, c + a are in AP

Qs: 26

What is value of cot215° + tan215° ?

  1. 12

  2. 14

  3. 16

  4. 4

Qs: 27

In a triangle ABC,

sinA - cosB - cosC = 0.

What is angle B equal to ?

  1. π/6

  2. π/4

  3. π/3

  4. π/2

Qs: 28

If a +β = π/4  and 2tanα = 1, then what is tan 2f3 equal to ?

  1. \frac{1}{3}

  2. \frac{2}{3}

  3. \frac{3}{4}

  4. \frac{3}{5}

Qs: 29

If tan(45° + θ)  = 1 + sin2θ, where _π/4 <  θ < π/4 then what is the value of cos2 θ ?

  1. 0

  2. -1

  3. 1

  4. 2

Qs: 30

Let sin2θ= cos3θ,  where θ is acute angle. What is the value of 1 + 4sinθ ?
(given that sin18° = √5-1/4)

  1. √3

  2. 2

  3. √5

  4. 3

Qs: 31

If tanθ = - 5/12  then what can be the value of sinθ?

  1. \frac{5}{13}but can not be \frac{-5}{13}

  2. \frac{-5}{13}but can not be \frac{5}{13}

  3. \frac{5}{13}but can not be -\frac{5}{13}

  4. None of the above

Qs: 32

What is the value of

cos4  7π/8 + cos4  5π/8 ?

  1. \frac{3}{2}

  2. \frac{3}{4}

  3. \frac{3}{8}

  4. \frac{3}{16}

Qs: 33

What is sin2  (π/4 + θ) - sin2  (π/4 - θ) equal to?

  1. sin2θ

  2. cos2θ

  3. 2sinθ

  4. 2cosθ

Qs: 34

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are 0 and 20 respectively. What is the height of the tower ?

  1. hcosθ

  2. hsinθ

  3. hcos2θ

  4. hsin2θ

Qs: 35

The shadow of a tower becomes x metre longer, when the angle of elevation of sun changes from 60° to θ. If the height of the tower is √3 x metre, then which one of the following is correct?

  1. 0 < θ < 30°

  2. 30°  < θ < 45°

  3. 45°  < θ < 60°

  4. 60°  < θ < 90°

Qs: 36

If tan-1(1/2 )+tan-1(x/3 )= π/4, where 0 < x < 6, then what is x equal to ?

  1. 1

  2. 2

  3. 3

  4. 5

Qs: 37

If  3sin-1x + cos-1x = π,  then  what  is x equal to?

  1. 0

  2. 1/2

  3. 1/√2

  4. 1/√3

Qs: 38

If tanα + tanβ = 1-tanα.tanβ, where tanα.tanβ ≠ 1, then which of the following is one of the values of (α + β?

  1. \frac{\pi }{6}

  2. \frac{\pi }{4}

  3. \frac{\pi }{3}

  4. \frac{\pi }{2}

Qs: 39

If (1 + tanθ)(1+tan9θ) = 2, then what is the value of tan(10θ)?

  1. 0

  2. 1

  3. 2

  4. Infinite

Qs: 40

What is the value of sin0° + sin10° + sin2O° + sin3O° + ... + sin360°?

  1. -1

  2. 0

  3. 1

  4. 2

Qs: 41

Consider all the subsets  of  the  set  A= {1, 2, 3, 4}. How many of them are supersets of the set {4} ?

  1. 6

  2. 7

  3. 8

  4. 9

Qs: 42

Consider the following statements in respect of two non-empty sets A and B :
I. x∉ (AUB) → x∉ or x A or x∉ B
2.x∉((A ∩ B)) → x∉ or x A and x∉ B
Which of the above statements is/are correct?
 

  1. 1 only

  2. 2 only
     

  3. Both 1 and 2

  4. Neither 1 nor 2

Qs: 43

Consider the following statements in respect of two non-empty sets A and B :

I. AUB = A ∩ B iff  A = B

2.A?B = φ iff  A = B

Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither1nor 2

Qs: 44

Consider the following statements in respect of the relation R in the set IN  of natural numbers defined by xRy if x2-5xy +4y2=0:
1. R is reflexive
2. R is symmetric
3. R is transitive
Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. 1 and 2 only

  4. 1, 2 and 3

Qs: 45

Consider the following statements in respect of any relation R on a set A :
1. If R is reflexive, then R-1 is also reflexive
2. If R is symmetric, then R-1 is also symmetric
3. If R is transitive, thenR-1 is also transitive
Which of the above statements are correct?

  1. 1 and 2 only

  2. 2 and  3 only

  3. 1 and 3  only

  4. 1, 2 and 3

Qs: 46

What is the principal argument of  1/1i where i = √-1?

  1. -\frac{3\pi }{4}

  2. -\frac{\pi }{4}

  3. \frac{\pi }{4}

  4. \frac{3\pi }{4}

Qs: 47

What is the modulus of  [√-3/2 - 1/2]200?

  1. \frac{1}{4}

  2. \frac{1}{2}

  3. 1

  4. \frac{1}{3}

Qs: 48

Consider the following statements :

1. n!/3!   is divisible  by 6, where n > 3

2. n!/3! + 3  is  divisible  by  7, where n > 3

Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

Qs: 49

In how many ways can a team of 5 players be selected out of 9 players so as to exclude two particular players ?

  1. 14

  2. 21

  3. 35

  4. 42

Qs: 50

In  the expa•nsion  of (x + 1/x)2n , what is the (n + 1)th term for the end (when arranged in decending powers of x)?

  1. C(2n, n)x

  2. C(2n, n- l)x

  3. C(2n, n)

  4. C(2n, n - 1)

Qs: 51

If the sum of the first 9 terms of an AP is equal to sum of e •first 11 terms, then what is the sum of the first 20 terms?

  1. 20

  2. 10

  3. 2

  4. 0

Qs: 52

If   the  5th  term  of  an AP  is 1/10 and  its 1oth term is 1/5, then what is the sum of first 50 terms?

  1. 25

  2. 25.5

  3. 26

  4. 26.5

Qs: 53

What is (1110011)2 + (10111)2 equal to?

  1. (101)2

  2. (1001)2

  3. (111)2

  4. (1011)2

Qs: 54

If  x3+ y3 = (100010111)2  and  x+y = (11111)2,  then . what   is  (x - y )2 + xy equal to? 

  1. (1101)2

  2. (1001)2

  3. (1011)2

  4. (1111)2

Qs: 55

Consider the inequations 5x-4y+ 12 < 0,  x+y<2, x<0 and y > 0.  Which   one  of  the following points lies in the common region ? 

  1. (0,0)

  2. (-2,4)

  3. (-1,4)

  4. (-1,2)

Qs: 56

Consider the following statements  in respect of the function y = [x], x∈(-1, 1) where [.] is the greatest integer function :

1. Its derivative is 0 at x = 0.5
2. It is continuous at x = 0
Which of the above statements is/are correct?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

Qs: 57

What is the degree of the differential 

equation 1+ (dy/dx)2 = (d2y/dx2)4/3?

  1. \frac{4}{3}

  2. 2

  3. 3

  4. 4

Qs: 58

A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant .(propor­ tionality constant) ?

  1. \frac{In2}{100}

  2. \frac{In5}{100}

  3. \frac{In10}{100}

  4. \frac{2In2}{100}

Qs: 59

What is the domain of the function
f (x) = √1- (x - 1)2   ?

  1. (0,1)

  2. [-1,1]

  3. (0,2)

  4. [0,2]

Qs: 60

The area of the region bounded by the parabola y2 = 4kx, where k > 0 and its latus . rectum is 24 square units. What is the value of k ?

  1. 1

  2. 2

  3. 3

  4. 4

Qs: 61

Calculate Interest coverage ratio from the following data: NP after tax = 60,000; Tax = 40%; 15% debt = 10,00,000

  1. 1.50

  2. 1.67

  3. 1.75

  4. 2.50

Qs: 62

What is \int(sin x)-1/2 (cos x)-3/2 dx equal to ?

  1. √tanx + c

  2. 2√tanx + c

  3. √cot x + c

  4. √2tanx + c

Qs: 63

If I1 = \intex/dx / ex+e-x and I2 = \intdx/e2x +1,

then what is I1 + I2 is equal to ?

  1. \frac{x}{2}+c

  2. x + c

  3. \frac{x}{2}

  4. \frac{x}{1}

Qs: 64

What is \int-1 -2    x/|x|   dx equal to ?

  1. -2

  2. -1

  3. 1

  4. 2

Qs: 65

How many extreme values does sin4x + 2x, where O < x <  π/2 have ?

  1. 1

  2. 2

  3. 4

  4. 8

Qs: 66

What is the maximum value of the funciton f(x) =  1/tan x + cot x, where 0 < x <  π/2 ?

  1. \frac{1}{4}

  2. \frac{1}{2}

  3. 1

  4. 2

Qs: 67

if 4f(x)-f(\frac{1}{x})=(2x+\frac{1}{x})(2x-\frac{1}{x}),

then what is f (2) equal to ?

  1. 0

  2. 1

  3. 2

  4. 4

Qs: 68

If  f(x) = 4x + 3, then what is fo fo  f (- 1) equal to?

  1. -1

  2. 0

  3. 1

  4. 2

Qs: 69

If xYyx = 1, then what is dy/dx at (1, 1) equal to?

  1. -1

  2. 0

  3. 1

  4. 4

Qs: 70

If y = (xx)x, then what is the value of dy/dx at x= 1?

  1. \frac{1}{2}

  2. 1

  3. 2

  4. 4

Qs: 71

Let y = [x + l]-,  - 4 < x < - 3  where[.] is the greatest integer function. What is the derivative of y with respect to x at x= -3.5?

  1. -4

  2. -3.5

  3. -3

  4. 0

Qs: 72

If dy/dx = (ln5)y with y(0) = ln5, then what is y(1) equal to?

  1. 0

  2. 5

  3. 2In5

  4. 5In5

Qs: 73

Consider the following in respect of the function f (x) = 10x :
1. Its domain is (-∞, ∞) 
2. It is a continuous function
3. It is differentiable at x = 0
Which of the above statements are correct?

  1. 1 and 2 only

  2. 2 and 3 only

  3. 1 and 3 only

  4. 1, 2 and 3

Qs: 74

What is lim x3 ( cosecx)2 equal to?

             x→0

  1. 0

  2. \frac{1}{2}

  3. 1

  4. Limit does not exixt

Qs: 75

What  is   lim  x3-1/√x-1 equal to ?

               x→1

  1. 0

  2. 3

  3. 6

  4. Limit does not exixt

Qs: 76

In which one of the following intervals is the function
f(x)=  x3/3 - 7x2/2 + 6x+5 decreasing ?

  1. (-∞,1) only

  2. (1,6)

  3. (6∞) only

  4. (-∞,1) U (6∞)

Qs: 77

If  the  derivative  of  the   function f(x) =m/x +2nx+1 vanishes at x = 2, then what is the value of m + 8n ?

  1. -2

  2. 0

  3. 2

  4. Cannot be determined due to insufficient data

Qs: 78

What is the area included in the first quadrant  between  the curves y = x and y=x3?

  1. \frac{1}{8} square unit

  2. \frac{1}{4} square unit

  3. \frac{1}{2} square unit

  4. 1 square unit

Qs: 79

If xy = 4225 where x, y are natural numbers, then what is the minimum value  of x + y ?

  1. 130

  2. 260

  3. 2113

  4. 4226

Qs: 80

What  does  th•e equation  x dy/dx -2y = 0 represent?

  1. A family of straight lines

  2. A family of circles

  3. A family of parabolas

  4. A family of ellipses

Qs: 81

If the points with coordinates (-5, 0), (5p2, I0p) and (5q2, l0q) are co,linear, then what is the value of pq where p ≠ q?

  1. -2

  2. -1

  3. 1

  4. 2

Qs: 82

What is the equation of the straight line which passes through the point (1, -2) and cuts off equal intercepts from the axes ?

  1. x+y-1=0

  2. x-y-1=0

  3. x+y+1=0

  4. x-y-2=0

Qs: 83

What is the equation  of  the  circle which touches both the axes in the first quadrant and the line y.- 2 = 0 ?

  1. x2 + y2x- 2y- 1 = 0

  2. x2 + y2 + 2x + 2y + 1 = 0

  3. x2 + y2  -2x- 2y + 1 = 0 

  4. x2 + y2 - 4x - 4y + 4 = 0

Qs: 84

What is the equation of  the  para­ bola with focus (-3, 0) and directrix x-3=0?

  1. y2 = 3x

  2. x2 = 12y

  3. y2 = 12x

  4. y2 = -12x

Qs: 85

What is the distance between the foci of the ellipse x2 + 2y2 = 1 ?

  1. 1

  2. √2

  3. 2

  4. 2√2

Qs: 86

Let a, b, c be the lengths of sides BC, CA, AB respectively of a triangle ABC. If p is the perimeter and q is the area of the triangle, then what is 

p(p-2a) tan {A/2} equal to ?

  1. q

  2. 2q

  3. 3q

  4. 4q

Qs: 87

A straight line passes through the point of intersection of x + 2y + 2 =.0 and 2x - 3y - 3 = 0. It cuts equal inter­ cepts in the fourth quadrant. What is the sum of the absolute values of the intercepts ?

  1. 2

  2. 3

  3. 4

  4. 6

Qs: 88

Under which one of the following conditions are the lines ax + by + c = 0 and bx + ay + c = 0 parallel (a ≠ 0, b ≠ 0)?

  1. a - b = 0 only

     

  2. a + b = 0 only

  3. a2 -b2 = 0

  4. ab + 1 = 0

Qs: 89

What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y =p , .(where p is a real number) by the coordinate axes?
 

  1. x - y = 0

  2. x+y=0

  3. x-y=p

  4. x+y=p

Qs: 90

If the point (x, y) is equidistant from the points (2a, 0)  and  (0,  3a)  wher > 0, then which one of the following is correct?

  1. 2x-3y=0

  2. 3x-2y=0

  3. 4x - 6y + 5a = 0

  4. 4x-6y-5a = 0

Qs: 91

Consider the following for the next three (03) items that follow :
The plane 6x + ky + 3z-12 = 0 where k:≠0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x2 + y2 + z2-2x-3y-4z= 0. 

Question:

What is the value of k ?

  1. 3

  2. 4

  3. 6

  4. 12

Qs: 92

Consider the following for the next three (03) items that follow :
The plane 6x + ky + 3z-12 = 0 where k:≠0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x2 + y2 + z2-2x-3y-4z= 0. 

Question:

If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correct?

  1. 0 < p < 0•5
     

  2. 0.5 < p < 1

  3. l<p<1.5
     

  4. > 1.5

Qs: 93

Consider the following for the next three (03) items that follow :
The plane 6x + ky + 3z-12 = 0 where k:≠0 meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x2 + y2 + z2-2x-3y-4z= 0. 

Question:

What is the equation of the line  ough the origin and the centre of the sphere?

  1. x=y=z

  2. 2x =  3y = 4z

  3. 6x=3y=4z

  4. 6x = 4y = 3z

Qs: 94

Consider the following for the next two (02) items that follow :
Let the plane 2x/k+ 2y/3 +z/3 = 2 pass through the point (2, 3, -6).

Question:

What are the direction ratios of a normal to the plane ?

  1. <3, 2, 1>

  2. <2, 3,6>

  3. <6, 3, 2>

  4. <1, 2, 3>

Qs: 95

Consider the following for the next two (02) items that follow :
Let the plane 2x/k+ 2y/3 +z/3 = 2 pass through the point (2, 3, -6).

Question:

If p, q and r are the . intercepts made  by the plane on the coordinate axes respectively, then what is (p + q + r) equal to?

  1. 10

  2. 11

  3. 12

  4. 13

Qs: 96

If 4i+j - 3k and pi+qj - 2k are collinear vectors, then what. are the possibJe values of p and q respec­tively?

  1. 4, 1

  2. 1, 4

  3. \frac{8}{3},\frac{2}{3}

  4. \frac{2}{3},\frac{8}{3}

Qs: 97

  1. a

  2. b

  3. c

  4. d

Qs: 98

Consider the following statements :
1. Dot product over vector addition is distributive
2. Cross product over vector addition is distributive
3. Cross product of vectors is asso­ ciative
Which of the above statements is/ are correct? :

  1. 1 only

  2. 2 only

  3. 1 and 2 only

  4. 1, 2 and 

Qs: 99

A company has taken a Rs 70 crore loan from a foreign company. The contract is denominated in dollars. Exchange rate as on 31-03-2020 was $1 = Rs 70. Exchange rate as on 31-03-2021 was $1 = Rs 72.5. Payment of loan has to be made in 3 equal instalments. What is the proft or loss as per AS-11? 

  1. Proft of Rs 2.5 crore

  2. Loss of Rs 2.5 crore

  3. Loss of Rs 1 crore

  4. Profit of Rs 1 crore

Qs: 100

  1. a

  2. b

  3. c

  4. d

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