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In This Post we are  providing Chapter-6 Application of Derivatives NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON APPLICATION OF DERIVATIVES

Question 1.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:
(a) 10π
(b) 12π
(c) 8π
(d) 11π

Answer: (b) 12π

Question 2.
The total revenue received from the sale of x units of a product is given by R (x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is:
(a) 116
(b) 96
(c) 90
(d) 126.

Answer: (d) 126.

Question 3.
The interval in which y = x² e^-x is increasing with respect to x is:
(a) (-∞, ∞)
(b) (-2,0)
(c) (2, ∞)
(d) (0, 2).

Answer: (d) (0, 2).

Question 4.
The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is
(a) 3
(b) 13
(c) -3
(d) –13

Answer: (d) –13

Question 5.
The line y = x + 1 is a tangent to the curve y² = 4x at the point:
(a) (1, 2)
(b) (2, 1)
(c) (1, -2)
(d) (-1, 2).

Answer: (a) (1, 2)

Question 6.
If f(x) = 3x² + 15x + 5, then the approximate value of f(3.02) is:
(a) 47.66
(b) 57.66
(c) 67.66
(d) 77.66.

Answer: (d) 77.66.

Question 7.
The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is:
(a) 0.06 x³ m³
(b) 0.6 x³ m³
(c) 0.09 x³m³
(d) 0.9 x³ m³

Answer: (c) 0.09 x³m³

Question 8.
The point on the curve x² = 2y, which is nearest to the point (0, 5), is:
(a) (2 √2, 4)
(b) (2 √2, 0)
(c) (0, 0)
(d) (2, 2).

Answer: (a) (2 √2, 4)

Question 9.
For all real values of x, the minimum value of 1−x+x21+x+x2 is
(a) 0
(b) 1
(c) 3
(d) 13

Answer: (d) 13

Question 10.
The maximum value of [x (x – 1) + 1]^1/3, 0 ≤ x ≤ 1 is
(a) (13)¹³
(b) 12
(c) 1
(d) 0

Answer: (c) 1

Question 11.
A cylindrical tank of radius 10 m being filled with wheat at the rate of 314 cubic m per minute. Then the depth of the wheat is increasing at the rate of:
(a) 1 m/minute
(b) 0 × 1 m/minute
(c) 1 × 1 m/minute
(d) 0 × 5 m/minute.

Answer: (a) 1 m/minute

Question 12.
The slope of the tangent to the curve x = t² + 3t – 8, y = 2 t² – 2t – 5 at the point (2, -1) is:
(a) 227
(b) 67
(c) 76
(d) −67

Answer: (b) 67

Question 13.
The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is:
(a) 1
(b) 2
(c) 3
(d) 12

Answer: (a) 1

Question 14.
The normal at the point (1, 1) on the curve 2y + x² = 3 is
(a) x + y = 0
(b) x – y = 0
(c) x + y + 1 = 0
(d) x – y + 1 = 0.

Answer: (b) x – y = 0

Question 15.
The normal to the curve x² = 4y passing through (2, 1) is:
(a) x + y = 3
(b) x – y = 3
(c) x + y = 1
(d) x – y = 1.

Answer: (a) x + y = 3

In This Post we are  providing Chapter-5 Continuity And Differentiability NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON CONTINUITY AND DIFFERENTIABILITY

Question 1.
If f (x) = 2x and g (x) = x22 + 1, then which of the following can be a discontinuous function
(a) f(x) + g(x)
(b) f(x) – g(x)
(c) f(x).g(x)
(d) g(x)f(x)

Answer: (d) g(x)f(x)

Question 2.
The function f(x) = 4−x24x−x3 is
(a) discontinuous at only one point at x = 0
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) None of these

Answer: (a) discontinuous at only one point at x = 0

Question 3.
The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is
(a) R
(b) R = {12}
(c) (0, ∞)
(d) None of these

Answer: (b) R = {12}

Question 4.
The function f(x) = cot x is discontinuous on the set
(a) {x = nπ, n ∈ Z}
(b) {x = 2nπ, n ∈ Z}
(c) {x = (2n + 1) π2 n ∈ Z}
(d) {x – nπ2 n ∈ Z}

Answer: (a) {x = nπ, n ∈ Z}

Question 5.
The function f(x) = e^|x| is
(a) continuous everywhere but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) None of theseAnswer

Answer: (a) continuous everywhere but not differentiable at x = 0

Question 6.
If f(x) = x² sin1x, where x ≠ 0, then the value of the function f(x) at x = 0, so that the function is continuous at x = 0 is
(a) 0
(b) -1
(c) 1
(d) None of these

Answer: (a) 0

Question 7.
If f(x) =is continuous at x = π2, then
(a) m = 1, n = 0
(b) m = nπ2 + 1
(c) n = mπ2
(d) m = n = π2

Answer: (c) n = mπ2

Question 8.
If y = log(1−x21+x2), then dydx is equal to
(a) 4×31−x4
(b) −4×1−x4
(c) 14−x4
(d) −4×31−x4

Answer: (b) −4×1−x4

Question 9.
Let f(x) = |sin x| Then
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
(c) f is everywhere continuous but no differentiable at x = (2n + 1) π2 n ∈ Z
(d) None of these

Answer: (b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z

Question 10.
If y = sinx+y√ then dydx is equal to
(a) cosx2y−1
(b) cosx1−2y
(c) sinx1−xy
(d) sinx2y−1

Answer: (a) cosx2y−1

Question 11.
The derivative of cos^-1 (2x² – 1) w.r.t cos^-1 x is
(a) 2
(b) −121−x2√
(c) 2x
(d) 1 – x²

Answer: (a) 2

Question 12.
If x = t², y = t³, then d2ydx2
(a) 32
(b) 34t
(c) 32t
(d) 34t

Answer: (b) 34t

Question 13.
The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval [o, √3] is
(a) 1
(b) -1
(c) 32
(d) 13

Answer: (a) 1

Question 14.
For the function f(x) = x + 1x, x ∈ [1, 3] the value of c for mean value theorem is
(a) 1
(b) √3
(c) 2
(d) None of these

Answer: (b) √3

Question 15.
Let f be defined on [-5, 5] as
f(x) = {x ,if x is rational−x, if x is irrational Then f(x) is
(a) continuous at every x except x = 0
(b) discontinuous at every x except x = 0
(c) continuous everywhere
(d) discontinuous everywhere

Answer: (b) discontinuous at every x except x = 0

In This Post we are  providing Chapter-7 Integrals NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON INTEGRALS

Question 1.
The anti-derivative of (√x + 1√x) equals

Answer: (c) 23 x²³ + 2x¹² + c

Question 2.
If 1dx (f(x)) = 4x³ – 3×4 such that f(2) = 0 then f(x) is ……………

Answer: (a) x⁴ + 1×3 – 1298

Question 3.
∫10×9+10xloge10x10+10x dx equals
(a) 10^x -x¹⁰ + c
(b) 10^x + x¹⁰ + c
(c) (10^x – x¹⁰)^-1 + c
(d) log (10^x + x¹⁰) + c.

Answer: (d) log (10^x + x¹⁰) + c.

Question 4.
∫dxsin2xcos2x equals
(a) tan x + cot x + c
(b) tan x – cot x + c
(c) tan x cot x + c
(d) tan x – cot 2x + c.

Answer: (b) tan x – cot x + c

Question 5.
∫sin2x–cos2xsin2xcos2x dx is equals to
(a) tan x + cot x + c
(b) tan x + cosec x + c
(c) -tan x + cot x + c
(d) tan x + sec x + c.

Answer: (a) tan x + cot x + c

Question 6.
∫ex(1+x)cos2(xe2) dx is equals to
(a) -cot (xe^x) + c
(b) tan (xe^x) + c
(c) tan (e^x) + c
(d) cot (e^x) + c

Answer: (b) tan (xe^x) + c

Question 7.
∫dxx2+2x+2 equals
(a) x tan^-1 (x + 1) + c
(b) tan^-1 (x + 1) + c
(c) (x + 1) tan^-1 x + c
(d) tan^-1 x + c

Answer: (b) tan^-1 (x + 1) + c

Question 8.
∫dx9−25×2√ equals

Answer: (b) 15 sin^-1 (5×3) + c

Question 9.
∫xdx(x−1)(x−2) equals

(d) log |(x – 1) (x – 2)| + c.

Answer: (b) log |(x−2)2x−1| + c

Question 10.
∫dxx(x2+1) equals
(a) log |x| – 12 log (x² + 1) + c
(b) 12 log |x| + 12 log (x² + 1) + c
(c) -log |x| + 12 log (x² + 1) + c
(d) log |x| + log (x² + 1) + c

Answer: (a) log |x| – 12 log (x² + 1) + c

Question 11.
∫x² e^x³ dx equals
(a) 13 e^x³ + c
(b) 13 e^x² + c
(c) 12 e^x³ + c
(d) 12 e^x² + c

Answer: (a) 13 e^x³ + c

Question 12.
∫e^x sec x (1 + tan x) dx equals
(a) e^x cos x + c
(b) e^x sec x + c
(c) e^x sin x + c
(d) e^x tan x + c

Answer: (b) e^x sec x + c

Question 13.
∫1+x2−−−−−√ dx is equal to

Answer:

Question 14.
∫x2–8x+7−−−−−−−−√ dx is equal to

Answer:

Question 15.
∫3√1 dx1+x2 equals
(a) π3
(b) 2π3
(c) π6
(d) π112

Answer: (d) π112

In This Post we are  providing Chapter-4 Determinants NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON DETERMINANTS

Question 1.
[[[111xyzx2y2z2]]]
(a) (x – y) (y + z)(z + x)
(b) (x + y) (y – z)(z – x)
(c) (x – y) (y – z)(z + x)
(d) (x – y) (y – z) (z – x)

Answer: (d) (x – y) (y – z) (z – x)

Question 2.
The value of the determinant
[[[ 352105723]]]
(a) 124
(b) 125
(c) 134
(d) 144

Answer: (c) 134

Question 3.
If a, b, c are in A.P. then the determinant[[[x+2x+3x+4x+3x+4x+5x+2ax+2bx+2c]]]
(a) 1
(b) x
(c) 0
(d) 2x

Answer: (c) 0

Question 4.
If w is a non-real root of the equation x² – 1 = 0. then
[[[1ωω2ωω21ω21ω]]] =
(a) 0
(b) 1
(c) ω
(d) ω²

Answer: (a) 0

Question 5.
If Δ = [103026] then A =
(a) 0
(b) 10
(c) 12
(d) 60

Answer: (a) 0

Question 6.
If 7 and 2 are two roots of the equation [[[x273x672x]]] then the third root is
(a) -9
(b) 14
(c) 12
(d) None of these

Answer: (a) -9

Question 7.
If [x182x] = [61826] x is equal to
(a) 6
(b) ±6
(c) -1
(d) -6

Answer: (b) ±6

Question 8.
[[[111abca2−bcb2−cac2−ab]]] is equal to
(a) abc
(b) ab + bc + ca
(c) 0
(d) (a – b)(b – c)(c – a)

Answer: (c) 0

Question 9.
A = [αqqα] |A³| = 125 then α =
(a) ±3
(b) ±2
(c) ±5
(d) 0

Answer: (a) ±3

Question 10.
If a ≠ 0 and [[[1+a1111+a1111+a]]] = 0 then a =
(a) a = -3
(b) a = 0
(c) a = 1
(d) a = 3

Answer: (a) a = -3

Question 11.
If x > 0 and x ≠ 1. y > 0. and y ≠ 1, z > 0 and z ≠ 1 then
[[[1logyxlogzxlogxy1logzylogxzlogyz1]]] is equal to
(a) 1
(b) -1
(c) 0
(d) None of these

Answer: (c) 0

Question 12.
[[[y+zyzzz+xzxyx+y]]] is equal to
(a) 6xyz
(b) xyz
(c) 4xyz
(d) xy + yz + zx

Answer: (c) 4xyz

Question 13.
If [2541] = [2x64x] then the value of x is
(a) ±2
(b) ±13
(c) ±√3
(d) ± (0.5)

Answer: (c) ±√3

Question 14.
If [2x85x] = [67−23] then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6

Answer: (c) ±6

Question 15.
The value of determinant [[[a−bb−cc−ab+cc+aa+babc]]]
(a) a³ + b³ + c ³
(b) 3bc
(c) a³ + b³ + c³ – 3abc
(d) None of these

Answer: (c) a³ + b³ + c³ – 3abc

In This Post we are  providing Chapter-3 R NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON MATRICES

Question 1.
If A = [a_ij]_{m × n} is a square matrix, if:
(a) m < n
(b) m > n
(c) m = n
(d) None of these.

Answer: (c) m = n

Question 2.
Which of the given values of x and y make the following pair of matrices equal:
[3x+7y+152−3x] [08y−24]
(a) x = –13, y = 7
(b) Not possible to find
(c) y = 7, x = –23
(d) x = –13, y = –23

Answer: (b) Not possible to find

Question 3.
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
(a) 27
(b) 18
(c) 81
(d) 512.Answer

Answer: (d) 512.

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × 1, 2 × p, n × 3 and p × k respectively. Now answer the following (4-5):
Question 4.
The restrictions on n, k and p so that PY + WY will be defined are
(a) k = 3, p = n
(b) k is arbitrary, p = 2
(c) p is arbitrary
(d) k = 2,p = 3.

Answer: (a) k = 3, p = n

Question 5.
If n =p, then the order of the matrix 7X – 5Z is:
(a) p × 2
(b) 2 × n
(c) n × 3
(d) p × n.

Answer: (b) 2 × n

Question 6.
If A, B are symmetric matrices of same order, then AB – BA is a
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity matrix.

Answer: (a) Skew-symmetric matrix

Question 7.
If A = [cosα−sinαsinαcosα] then A + A’ = I, the value of α is
(a) π6
(b) π3
(c) π
(d) 3π2

Answer: (a) π6

Question 8.
Matrices A and B will be inverse of each other only if:
(a) AB = BA
(b) AB – BA = O
(c) AB = O, BA = I
(d) AB = BA = I.

Answer: (d) AB = BA = I.

Question 9.
If A = [αγβ−α] is such that A² = I, then
(a) 1 + α² + ßγ = 0
(b) 1 – α² + ßγ = 0
(c) 1 – α² – ßγ = 0
(d) 1 + α² – ßγ = 0

Answer: (c) 1 – α² – ßγ = 0

Question 10.
If a matrix is both symmetric and skew- symmetric matrix, then:
(a) A is a diagonal matrix
(b) A is a zero matrix
(c) A is a square matrix
(d) None of these.

Answer: (b) A is a zero matrix

Question 11.
If A is a square matrix such that A² = A, then (I + A)³ – 7A is equal to :
(a) A
(b) I – A
(c) I
(d) 3A.

Answer: (c) I

Question 12.
The matrix A = ⎡⎣⎢005050500⎤⎦⎥ is a
(a) scalar matrix
(b) diagonal matrix
(c) unit matrix
(d) square matrix.

Answer: (d) square matrix.

Question 13.
If matrix A = [a_ij]_2×2
where a_ij = 1 if i ≠ j = 0 if i = j,
(a) I
(b) A
(c) O
(d) None of these

Answer: (a) I

Question 14.

then A – B is equal to
(a) 1
(b) O
(c) 21

Answer: (c) 21

Question 15.
If A is a matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then order of matrix B is
(a) m × m
(b) n × n
(c) n × m
(d) m × n.

Answer: (d) m × n.

In This Post we are  providing Chapter-2 Inverse Trigonometric Functions NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON INVERSE TRIGONOMETRIC FUNCTIONS

Question 1.
sin^-1 (sin2π3) =
(a) 2π3
(b) π6
(c) 4π3
(d) π3

Answer: (d) π3

Question 2.
sin^-1 (1 – x) – 2 sin^-1 x = π2 then x = ?
(a) 0, 12
(b) 1, 12
(c) 12
(d) 0

Answer: (d) 0

Question 3.
tan^-1 √3 – sec^-1(-2)
(a) π
(b) –π3, 0
(c) π3
(d) 2π3

Answer: (b) –π3, 0

Question 4.
sin(sec^-1 x + cosec^-1x) =
(a) 1
(b) -1
(c) π2
(d) π3

Answer: (a) 1

Question 5.
2 tan^-1 13 + tan^-1 17 =
(a) tan^-1 4429
(b) π2
(c) 0
(d) π4

Answer: (d) π4

Question 6.
The principle value of sin^-1 √32 is
(a) 2π3
(b) π6
(c) π4
(d) π3

Answer: (d) π3

Question 7.
The value of the expression tan^-1(12cos^-12√5) is
(a) 2 + √5
(b) √5 – 2
(c) √5+24
(d) √5 + 2

Answer: (d) √5 + 2

Question 8.
Simplified form of cos^-1 (4x³ – 3x)
(a) 3 sin^-1x
(b) 3 cos^-1x
(c) π – 3 sin^-1x
(d) None of these

Answer: (b) 3 cos^-1x

Question 9.
The value of tan(tan^-1 45 + tan^-1 23) is
(a) 617
(b) 716
(c) 176
(d) None of these

Answer: (d) None of these

Question 10.
tan^-1(xy) – tan^-1(x−yx+y) is equal to
(a) π3
(b) π4
(c) π2
(d) −3π4

Answer: (b) π4

Question 11.
The value of x for which sin |cot^-1(1 – x)| = cos (tan^-1 x) is
(a) 21
(b) 1
(c) 0
(d) 12

Answer: (d) 12

Question 12.
Principal value of cos^-1(−1√2)
(a) 3π4
(b) 5π4
(c) –π4
(d) None of these

Answer: (a) 3π4

Question 13.
tan^-1 √3 – sec^-1 (-2) is equal to
(a) π
(b) –π3
(c) π3
(d) 2π3

Answer: (b) –π3

Question 14.
If y = sec^-1 x then
(a) 0 ≤ y ≤ π
(b) 0 ≤ y ≤ π2
(c) –π2 < y < π2
(d) None of these

Answer: (d) None of these

Question 15.
If x + 1x = 2 then the principal value of sin^-1 x is x
(a) π4
(b) π2
(c) π
(d) 3π2

Answer: (d) 3π2

In This Post we are  providing Chapter-1 Relation and Function NCERT MCQ for Class 12 Math which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MCQ ON RELATIONS AND FUNCTIONS

1. Let R be a relation on the set L of lines defined by l₁ R l₂ if l₁ is perpendicular to l₂, then relation R is
(a) reflexive and symmetric
(b) symmetric and transitive
(c) equivalence relation
(d) symmetric

Answer: d
Explanation: (d), not reflexive, as l₁ R l₂
⇒ l₁ ⊥ l₁ Not true
Symmetric, true as l₁ R l₂ ⇒ l2R h
Transitive, false as l₁ R l₂, l₂ R l₃
⇒ l₁ || l₃ . l₁ R l₂.

2. Given triangles with sides T₁ : 3, 4, 5; T₂ : 5, 12, 13; T₃ : 6, 8, 10; T₄ : 4, 7, 9 and a relation R in set of triangles defined as R = {(Δ₁, Δ₂) : Δ₁ is similar to Δ₂}. Which triangles belong to the same equivalence class?
(a) T₁ and T₂
(b) T₂ and T₃
(c) T₁ and T₃
(d) T₁ and T₄

Answer: c
Explanation: (c), T₁ and T₃ are similar as their sides are proportional.

3. Given set A ={1, 2, 3} and a relation R = {(1, 2), (2, 1)}, the relation R will be
(a) reflexive if (1, 1) is added
(b) symmetric if (2, 3) is added
(c) transitive if (1, 1) is added
(d) symmetric if (3, 2) is added

Answer: c
Explanation: (c), here (1,2) e R, (2,1) € R, if transitive (1,1) should belong to R.

4. Given set A = {a, b, c). An identity relation in set A is
(a) R = {(a, b), (a, c)}
(b) R = {(a, a), (b, b), (c, c)}
(c) R = {(a, a), (b, b), (c, c), (a, c)}
(d) R= {(c, a), (b, a), (a, a)}

Answer: b
Explanation: (b), A relation R is an identity relation in set A if for all a ∈ A, (a, a) ∈ R.

5. A relation S in the set of real numbers is defined as xSy ⇒  x – y+ √3 is an irrational number, then relation S is
(a) reflexive
(b) reflexive and symmetric
(c) transitive
(d) symmetric and transitive

Answer: a
Explanation:

6. Set A has 3 elements and the set B has 4 elements. Then the number of injective functions that can be defined from set A to set B is
(a) 144
(b) 12
(c) 24
(d) 64

Answer: c
Explaination: (c), total injective mappings/functions
= ⁴ P₃ = 4! = 24.

7. Given a function lf as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then
(a) g(x) = 4x + 5
(b) g(x) = 54x−5
(c) g(x) = x−45
(d) g(x) = 5x – 4

Answer: c
Explaination:

8. Let Z be the set of integers and R be a relation defined in Z such that aRb if (a – b) is divisible by 5. Then R partitions the set Z into ______ pairwise disjoint subsets.

Answer:
Explaination: Five, as remainder can be 0, 1, 2, 3, 4.

9. Consider set A = {1, 2, 3 } and the relation R= {(1, 2)}, then? is a transitive relation. State true or false.

Answer:
Explaination: True, as there is no situation
(a, b) ∈ R, (b, c) ∈ R Hence, transitive. We can also say, a relation containing only one element is transitive.

10. Every relation which is symmetric and transitive is reflexive also. State true or false.

Answer:
Explaination: False,e.g.if R is arelationinset A = {2,3,4} defined as {(2, 3), (3, 2), (2, 2)} is symmetric and transitive but not reflexive.

11. Let R be a relation in set N, given by R = {(a, b): a = b – 2, b > 6} then (3, 8) ∈ R. State true or false with reason.

Answer:
Explaination: False, as in (3, 8), b = 8
⇒ a = 8 – 2
⇒ a = 6, but here a = 3.

12. Let R be a relation defined as R = {(x, x), (y, y), (z, z), (x, z)} in set A = {x, y, z} then R is (reflexive/symmetric) relation.

Answer:
Explanation: Reflexive, as for all a ∈ A, (a, a) ∈ R.

13. Let R be a relation in the set of natural numbers N defined by R = {(a, b) ∈ N × N: a < b}. Is relation R reflexive? Give a reason.

Answer:
Explanation:
Given R = {(a, b) ∈ N × N: a < b}.
Not reflexive, as for (a, a) × R
⇒ a< a, not true.

14. Let A be any non-empty set and P(A) be the power set of A. A relation R defined on P(A) by X R Y ⇔ X ∩ Y = X, X, Y ∈ P(A). Examine whether ? is symmetric.

Answer:
Explaination: X R Y ⇔ X ∩ Y = X ⇒ Y ∩ X = X ⇒ Y R X.
Hence, symmetric.

15. State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

Answer:
Explaination: (1, 2) ∈ R, (2, 1) ∈ R, but (1, 1) ∉ R.