IIT JAM 2024 Mathematics (MA) Exam Paper PDF

## JAM 2024 ### Mathematics (MA) ### Special Instructions / Useful Data - **Zn = {0, 1, ..., n-1}** denotes the additive group of integers modulo n. - **IR** the set of all real numbers. - **N** = the set of all positive integers. - **Z** = the set of all integers. - **C** = the set of all complex...

## JAM 2024 ### Mathematics (MA) ### Special Instructions / Useful Data - **Zn = {0, 1, ..., n-1}** denotes the additive group of integers modulo n. - **IR** the set of all real numbers. - **N** = the set of all positive integers. - **Z** = the set of all integers. - **C** = the set of all complex numbers. - **Q** = the set of all rational numbers. - **gcd(r, n)** = the greatest common divisor of the integers r and n. - **Sn** = the symmetric group of all permutations of {1,2, ..., n}. - **An** = the group of all even permutations in Sn. - **Mn(C)** = the set of all n x n matrices with entries from C. - **Mn(R)** = the set of all n x n matrices with entries from IR. - **MT** = the transpose of the matrix M. - **In** = the n x n identity matrix. - **Pn(x)** = the real vector space of polynomials, in the variable x with real coefficients and having degree at most n, together with the zero polynomial. These polynomials are regarded as functions from R to R. - **(n)** **k)** **=** the binomial coefficient defined as **(n)** **k)** **= n! / (k! * (n-k)!)**. - **f°g** = the composite function defined by **(f°g)(x) = f(g(x))**. - **A \ B** = the complement of the set B in the set A, that is, **{x ∈ A:x & B}**. - **log x** = the logarithm of x to the base e for a positive number x. - **IR"** = the n-dimensional Euclidean space. - **A × B** = the Cartesian product of the sets A and B. - **M-1** = the inverse of an invertible matrix M. ### Section A: Q.1 - Q.10 Carry ONE mark each. **Q.1** Let *yc: R→ (0,∞)* be the solution of the Bernoulli's equation *dy/dx - y + y³ = 0, y(0) = c > 0*. Then, for every *c > 0*, which one of the following is true? **(A)** lim *yc(x)* = 0 as *x→∞* **(B)** lim *y(x)* = 1 as *x→∞* **(C)** lim *y(x)* = *e* as *x→∞* **(D)** lim *y(x)* does not exist as *x→∞* **Q.2** For a twice continuously differentiable function *g: R→ R*, define *ug(x, y) = (1/y) ∫(-y) ^0 g(x + t)dt* for *(x, y) ∈ R², y > 0*. Which one of the following holds for all such *g*? **(A)** (∂²ug/∂x²) = (2 ∂ug/y ∂y) + (∂²ug/∂y²) **(B)** (∂²ug/∂x²) = (1 ∂ug/y ∂y) + (∂²ug/∂y²) **(C)** (∂²ug/∂x²) = (2 ∂ug/y ∂y) - (∂²ug/∂y²) **(D)** (∂²ug/∂x²) = (1 ∂ug/y ∂y) - (∂²ug/∂y²) **Q.3** Let *y(x)* be the solution of the differential equation *dy/dx = 1 + y secx for x ∈ (-π/2, π/2)* that satisfies *y(0) = 0*. Then, the value of *y(π/3)* equals: **(A)** √3 log(√3/2) **(B)** (√2/2)log(√3/2) **(C)** (√2/2) log 3 **(D)** √3 log 3 **Q.4** Let F be the family of curves given by *x² + 2hxy + y² = 1, -1 < h < 1*. Then, the differential equation for the family of orthogonal trajectories to F is: **(A)** (x²y - y³ + y) *dy/dx* - (xy² - x³ + x) = 0 **(B)** (x²y - y³ + y) *dy/dx* + (xy² – x³ + x) = 0 **(C)** (x²y + y³ + y) *dy/dx* - (xy² + x³ + x) = 0 **(D)** (x²y + y³ + y) *dy/dx* + (xy² + x³ + x) = 0 **Q.5** Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE? **(A)** G is necessarily cyclic. **(B)** G is abelian but need not be cyclic. **(C)** G need not be abelian. **(D)** G has 13 elements of order 13. **Q.6** For a positive integer n, let *U(n) = {r∈Zn: gcd(r, n) = 1}* be the group under multiplication modulo n. Then, which one of the following statements is TRUE? **(A)** U(5) is isomorphic to U(8) **(B)** U(10) is isomorphic to U(12) **(C)** U(8) is isomorphic to U(10) **(D)** U(8) is isomorphic to U(12) **Q.7** Which one of the following is TRUE for the symmetric group S12? **(A)** S12 has an element of order 42. **(B)** S12 has an element of order 35. **(C)** S12 has an element of order 27. **(D)** S12 has no element of order 60. **Q.8** Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE? **(A)** The order of G is necessarily even. **(B)** The order of G is not necessarily even. **(C)** G is necessarily cyclic. **(D)** G is necessarily abelian but need not be cyclic **Q.9** Consider the following statements. P: If a system of linear equations *Ax = b* has a unique solution, where A is an *m x n* matrix and *b* is an *m × 1* matrix, then *m = n*. Q: For a subspace W of a nonzero vector space V, whenever *u∈V\W* and *v∈V \ W*, then *u + v∈V\W*. Which one of the following holds? **(A)** Both P and Q are true. **(B)** P is true but Q is false. **(C)** P is false but Q is true. **(D)** Both P and Q are false **Q.10** Let *g: R→ R* be a continuous function. Which one of the following is the solution of the differential equation *d²y/dx² + y = g(x)* for *x ∈ R*, satisfying the conditions *y(0) = 0, y'(0) = 1*? **(A)** *y(x) = sin x - ∫(0) ^x sin(x - t) g(t)dt* **(B)** *y(x) = sin x + ∫(0) ^x sin(x - t) g(t)dt* **(C)** *y(x) = sin x - ∫(0) ^x cos(x - t) g(t)dt* **(D)** *y(x) = sin x + ∫(0) ^x cos(x - t) g(t)dt* ### Section A: Q.11 - Q.30 Carry TWO marks each. **Q.11** Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6? **(A)** The alternating group A6 **(B)** The alternating group A5 **(C)** S6 **(D)** S5 **Q.12** Consider the group *G = {A∈ M2(R): AAT = 12}* with respect to matrix multiplication. Let *Z(G) = {AEG: AB = BA, for all BEG}*. Then, the cardinality of *Z(G)* is: **(A)** 1 **(B)** 2 **(C)** 4 **(D)** Infinite **Q.13** Let V be a nonzero subspace of the complex vector space *M₂(C)* such that every nonzero matrix in V is invertible. Then, the dimension of V over C is: **(A)** 1 **(B)** 2 **(C)** 7 **(D)** 49 **Q.14** For *n ∈ N*, let *an = 1 / ((3n+2)(3n + 4))* and *bn = (n³ + cos(3^n)) / (3^n + n³)*. Then, which one of the following is TRUE? **(A)** ∑(n=1) ^∞ *an* is convergent but ∑(n=1) ^∞ *bn* is divergent **(B)** ∑(n=1) ^∞ *an* is divergent but ∑(n=1) ^∞ *bn* is convergent **(C)** Both ∑(n=1) ^∞ *an* and ∑(n=1) ^∞ *bn* are divergent **(D)** Both ∑(n=1) ^∞ *an* and ∑(n=1) ^∞ *bn* are convergent **Q.15** Let *a = [[1,√3],[-1,1],[√3,0]]*. Consider the following two statements. P: The matrix *14 - aaT* is invertible. Q: The matrix *14 – 2aaT* is invertible. Then, which one of the following holds? **(A)** P is false but Q is true **(B)** P is true but Q is false **(C)** Both P and Q are true **(D)** Both P and Q are false **Q.16** Let A be a 6 x 5 matrix with entries in R and B be a 5 x 4 matrix with entries in R. Consider the following two statements. P: For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 x 4 zero matrix. Q: For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 x 5 zero matrix. Which one of the following holds? **(A)** Both P and Q are true **(B)** P is true but Q is false **(C)** P is false but Q is true **(D)** Both P and Q are false **Q.17** Let *P11(x)* be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 11, together with the zero polynomial. Let *E = {So(x), S₁(X), ..., S11 (X)}, F = {ro(x), r₁(x), ..., r₁₁ (x)}* be subsets of *P11(x)* having 12 elements each and satisfying *So(3) = S₁(3) = . = $11(3) = 0, ro(4) = r₁ (4) = … = r₁₁(4) = 1*. Then, which one of the following is TRUE? **(A)** Any such E is not necessarily linearly dependent and any such F is not necessarily linearly dependent. **(B)** Any such E is necessarily linearly dependent but any such F is not necessarily linearly dependent. **(C)** Any such E is not necessarily linearly dependent but any such F is necessarily linearly dependent. **(D)** Any such E is necessarily linearly dependent and any such F is necessarily linearly dependent. **Q.18** For the differential equation *y(8x - 9y)dx + 2x(x-3y)dy = 0*, which one of the following statements is TRUE? **(A)** The differential equation is not exact and has x² as an integrating factor **(B)** The differential equation is exact and homogeneous **(C)** The differential equation is not exact and does not have x² as an integrating factor **(D)** The differential equation is not homogeneous and has x² as an integrating factor **Q.19** For *x ∈ R*, let *[x]* denote the greatest integer less than or equal to x. For *x, y ∈ R*, define *min{x,y} = {x if x ≤ y, y otherwise}*. Let *f: [-2π, 2π] → R* be defined by *f(x) = sin(min{x, x - [x]})* for *x ∈ [−2π, 2π]*. Consider the set *S = {x ∈ [−2π, 2π]: f is discontinuous at x}*. Which one of the following statements is TRUE? **(A)** S has 13 elements **(B)** S has 7 elements **(C)** S is an infinite set **(D)** S has 6 elements **Q.20** Define the sequences *{an}n=3* and *{bn}n=3* as *an = (logn + log log n)logn* and *bn=n (1+logn)*. Which one of the following is TRUE? **(A)** ∑(n=3) ^∞ 1/an is convergent but ∑(n=3) ^∞ 1/bn is divergent **(B)** ∑(n=3) ^∞ 1/an is divergent but ∑(n=3) ^∞ 1/bn is convergent **(C)** Both ∑(n=3) ^∞ 1/an and ∑(n=3) ^∞ 1/bn are divergent **(D)** Both ∑(n=3) ^∞ 1/an and ∑(n=3) ^∞ 1/bn are convergent **Q.21** For *p,q,r ∈ R, r≠ 0* and *n∈ N*, let *an = pr^n (n/n+2)²* and *bn = (n^n / n! * r^n) (n/n+2)*. Then, which one of the following statements is TRUE? **(A)** If 1 <p<e² and q > 1, then ∑(n=1) ^∞ *an* is convergent **(B)** If e² <p <e and q > 1, then ∑(n=1) ^∞ *an* is convergent **(C)** If 1 <r <e, then ∑(n=1) ^∞ *bn* is convergent **(D)** If *e/e < r < 1*, then ∑(n=1) ^∞ *bn* is convergent **Q.22** Let *P7(x)* be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 7, together with the zero polynomial. Let *T: P(x) → P(x)* be the linear transformation defined by *T(f(x)) = f(x) + df(x)/dx*. Then, which one of the following is TRUE? **(A)** T is not a surjective linear transformation **(B)** There exists k ∈ N such that Tk is the zero linear transformation **(C)** 1 and 2 are the eigenvalues of T **(D)** There exists r∈ N such that *(T – I)”* is the zero linear transformation, where I is the identity map on *P7(x)*. **Q.23** For *a ∈ R*, let *ya(x)* be the solution of the differential equation *dy/dx + 2y = 1/(1 + x²) for x EIR* satisfying *y(0) = a*. Then, which one of the following is TRUE? **(A)** lim *ya (x)* = 0 for every *a ∈R* as *x→∞* **(B)** lim *ya(x)* = 1 for every *a∈R* as *x→∞* **(C)** There exists an *a ∈ R* such that lim *ya(x)* exists but its value is different from 0 and 1 as *x→∞*. **(D)** There is an *a∈ R* for which lim *ya(x)* does not exist as *x→∞*. **Q.24** Consider the following two statements. P: There exist functions *f: R→R, g: RR* such that *f* is continuous at *x = 1* and *g* is discontinuous at *x = 1* but *g o f* is continuous at *x = 1*. Q: There exist functions *f: R→R, g: RR* such that both *f* and *g* are discontinuous at *x = 1* but *g o f* is continuous at *x = 1*. Which one of the following holds? **(A)** Both P and Q are true **(B)** Both P and Q are false **(C)** P is true but Q is false **(D)** P is false but Q is true **Q.25** Let *f: R→ R* be defined by *f(x) = (x² + 1)² / (x⁴ + x² + 1)* for *x ∈ R*. Then, which one of the following is TRUE? **(A)** f has exactly two points of local maxima and exactly three points of local minima **(B)** f has exactly three points of local maxima and exactly two points of local minima **(C)** f has exactly one point of local maximum and exactly two points of local minima **(D)** f has exactly two points of local maxima and exactly one point of local minimum **Q.26** Let *f: IR → R* be a solution of the differential equation *d²y/dx² - 2 dy/dx + y = 2ex* for *x∈R*. Consider the following statements. P: If *f(x) > 0* for all *x ∈ R*, then *f'(x) > 0* for all *x ∈ R*. Q: If *f'(x) > 0* for all *x ∈ IR*, then *f(x) > 0* for all *x ∈ R*. Then, which one of the following holds? **(A)** P is true but Q is false **(B)** P is false but Q is true **(C)** Both P and Q are true **(D)** Both P and Q are false **Q.27** For *a > b > 0*, consider *D = {(x,y,z) ∈ R3: x² + y² + z² ≤ a² and x² + y² ≥ b²}*. Then, the surface area of the boundary of the solid D is: **(A)** 4π(a + b)√a² – b² **(B)** 4π (α² – b√a²-b²) **(C)** 4π(a – b)√a² – b² **(D)** 4π (a² + b√a²-b²) **Q.28** For *n ≥ 3*, let a regular *n*-sided polygon *Pn* be circumscribed by a circle of radius *Rn* and let *rn* be the radius of the circle inscribed in *Pn*. Then *lim(n→∞) (Rn / rn)^2* equals: **(A)** e(π²) **(B)** e(π²/2) **(C)** e(π²/3) **(D)** e(2π²) **Q.29** Let L₁ denote the line *y = 3x + 2* and L2 denote the line *y = 4x + 3*. Suppose that *f: IR → IR* is a four times continuously differentiable function such that the line L₁ intersects the curve *y = f(x)* at exactly three distinct points and the line L2 intersects the curve *y = f(x)* at exactly four distinct points. Then, which one of the following is TRUE? **(A)** *df/dx* does not attain the value 3 on R. **(B)** *d²f/dx²* vanishes at most once on R. **(C)** *d³f/dx³* vanishes at least once on R. **(D)** *df/dx* does not attain the value 7/2 on R. **Q.30** Define the function *f: R² → R* by *f(x, y) = 12xy e-(2x+3y-2)*. If *(a, b)* is the point of local maximum of *f*, then *f (a, b)* equals: **(A)** 2 **(B)** 6 **(C)** 12 **(D)** 0 ### Section B: Q.31 - Q.40 Carry TWO marks each. **Q.31** Let *{an}n=1* be a sequence of real numbers. Then, which of the following statements is/are always TRUE? **(A)** If ∑(n=1) ^∞ *an* converges absolutely, then ∑(n=1) ^∞ *an* converges absolutely **(B)** If ∑(n=1) ^∞ *an* converges absolutely, then ∑(n=1) ^∞ *a^2n* converges absolutely **(C)** If ∑(n=1) ^∞ *an* converges, then ∑(n=1) ^∞ *a^2n* converges **(D)** If ∑(n=1) ^∞ *an* converges, then ∑(n=1) ^∞ *an* converges **Q.32** Which of the following statements is/are TRUE? **(A)** ∑(n=1) ^∞ n log(1+1/n) is convergent **(B)** ∑(n=1) ^∞ (1-cos(1/√n)) log n is convergent **(C)** ∑(n=1) ^∞ n² log (1+1/n) is convergent **(D)** ∑(n=1) ^∞ (1-cos(1/√n)) log n is convergent **Q.33** Which of the following statements is/are TRUE? **(A)** The additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. **(B)** The multiplicative group of nonzero real numbers is isomorphic to the multiplicative group of nonzero complex numbers. **(C)** The additive group of real numbers is isomorphic to the multiplicative group of nonzero complex numbers. **(D)** The additive group of real numbers is isomorphic to the additive group of rational numbers. **Q.34** Let *f: (1,∞) → (0,∞)* be a continuous function such that for every *n ∈ N*, *f(n)* is the smallest prime factor of *n*. Then, which of the following options is/are CORRECT? **(A)** lim f(x) exists as *x→∞* **(B)** lim f(x) does not exist as *x→∞* **(C)** The set of solutions to the equation *f(x) = 2024* is finite **(D)** The set of solutions to the equation *f(x) = 2024* is infinite **Q.35** Let *S = {(x, y) ∈ R²: x > 0, y > 0}* and *f: S→ R* be given by *f(x,y) = 2x² + 3y² – logx - (1/6) logy*. Then, which of the following statements is/are TRUE? **(A)** There is a unique point in S at which f (x, y) attains a local maximum. **(B)** There is a unique point in S at which *f (x, y)* attains a local minimum. **(C)** For each point *(xo, yo) ES*, the set *{(x,y) ∈ S: f(x,y) = f (xo, yo) }* is bounded. **(D)** For each point *(x0,yo) ES*, the set *{(x, y) ∈ S: f(x, y) = f (xo, yo) }* is unbounded. **Q.36** The center *Z(G)* of a group G is defined as *Z(G) = {x ∈ G:xg = gx for all g∈G}*. Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G? **(A)** If G is non-abelian and *Z(G)* contains more than one element, then the center of the quotient group *G/Z(G)* contains only one element. **(B)** If |G| ≥ 2, then there exists a non-trivial homomorphism from Z to G. **(C)** If |G| ≥ 2 and G is non-abelian, then there exists a non-identity isomorphism from G to itself. **(D)** If |G| = p³, where p is a prime number, then G is necessarily abelian. **Q.37** For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, CE M10 (R) satisfying A = BC ? **(A)** Rowspace(A)⊆ Rowspace(B) **(B)** Rowspace(A)⊆ Rowspace(C) **(C)** Colspace(A) ⊆ Colspace(B) **(D)** Colspace(A)⊆ Colspace(C) **Q.38** Define *f: IR → R* and *g: R→R* as follows *f(x) = ∑(m=0) ^∞ (-1)^m x^(2m) / (2^(2m) (m!)²)* and *g(x) = ∑(m=0) ^∞ (-1)^m x^(2m) / (2^(2m) (m + 1)! m!)* for *x ∈ R*. Let *x1, x2, x3, x4 ∈ R* be such that *0 < x1 < x2, 0< X3 < X4*, *f(x1) = f(x2) = 0, f(x) ≠ 0* when *x₁ < x < X2*, *g(x3) = g(x4) = 0* and *g(x) ≠ 0* when *x3 < x < X4*. Then, which of the following statements is/are TRUE? **(A)** The function *f* does not vanish anywhere in the interval *(X3, X4)* **(B)** The function *f* vanishes exactly once in the interval *(X3, X4)* **(C)** The function *g* does not vanish anywhere in the interval *(x1,X2)* **(D)** The function *g* vanishes exactly once in the interval *(x1, x2)* **Q.39** For *0 < a < 4*, define the sequence *{Xn}n=1* of real numbers as follows: *X₁ = a* and *Xn+1 +2 = -xn(xn-4)* for *n∈N*. Which of the following is/are TRUE? **(A)** {Xn}n=1 converges for at least three distinct values of *a ∈ (0,1)* **(B)** {Xn}n=1 converges for at least three distinct values of *a ∈ (1,2)* **(C)** {Xn}n=1 converges for at least three distinct values of *a ∈ (2,3)* **(D)** {Xn}n=1 converges for at least three distinct values of *a ∈ (3,4)* **Q.40** Consider *G = {m + n√2: m, n ∈ Z}* as a subgroup of the additive group IR. Which of the following statements is/are TRUE? **(A)** G is a cyclic subgroup of R under addition **(B)** G∩I is non-empty for every non-empty open interval *I⊆R* **(C)** G is a closed subset of IR **(D)** G is isomorphic to the group Z × Z, where the group operation in *ZX Z* is defined by *(m₁, n₁) + (m2, n₂) = (m₁ + m2, n₁ + n₂)* ### Section C: Q.41 - Q.50 Carry ONE mark each. **Q.41** The area of the region *R = {(x, y) ∈ R²: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 1/4 ≤xy ≤ 1/2}* is ____ (rounded off to two decimal places). **Q.42** Let *y: IR→ R* be the solution to the differential equation *d²y/dx² + 2 dy/dx + 5y = 1* satisfying *y(0) = 0* and *y'(0) = 1*. Then, lim *y(x)* equals ____ (rounded off to two decimal places). as *x→∞* **Q.43** For *a > 0*, let *ya (x)* be the solution to the differential equation *2 d²y/dx² - dy/dx - y = 0* satisfying the conditions *y(0) = 1, y'(0) = α*. Then, the smallest value of *a* for which *ya(x)* has no critical points in R equals ____ (rounded off to the nearest integer). **Q.44** Consider the 4 x 4 matrix *M = [[0,1,2,3],[1,0,1,2],[2,1,0,1],[3,2,1,0]]*. If *ai,j* denotes the *(i, j)*th entry of *M-1*, then *a4,1* equals ____ (rounded off to two decimal places). **Q.45** Let *P12(x)* be the real vector space of polynomials in the variable x with real coefficients and having degree at most 12, together with the zero polynomial. Define *V = {f ∈ P₁₂(x): f(-x) = f(x) for all x ∈ R and f(2024) = 0}*. Then, the dimension of V is ____. **Q.46** Let *S = {f: IR→R: f is a polynomial and f(f(x)) = (f(x))2024 for x ER}*. Then, the number of elements in S is ____. **Q.47** Let a₁ = 1, b₁ = 2 and c₁ = 3. Consider the convergent sequences {an}n=1, {bn}n=1 and {Cn}n=1 defined as follows: *an+1 = (an + bn)/2, bn+1 = (bn + cn)/2, and cn+1 = (cn + an)/2* for *n ≥ 1*. Then, ∑(n=1) ^∞ bnCn(an+1 - an) + ∑(n=1) ^∞ (bn+1Cn+1-bnCn)an+1 equals ____ (rounded off to two decimal places). **Q.48** Let *S = {(x, y, z) ∈ R³: x² + y² + z² = 4, (x − 1)² + y² ≤ 1, z ≥ 0}*. Then, the surface area of S equals ____ (rounded off to two decimal places). **Q.49** Let P7(x) be the real vector space of polynomials in x with degree at most 7, together with the zero polynomial. For r = 1, 2, ..., 7, define *Sr(x) = x(x − 1) ... (x − (r-1))* and *so(x) = 1*. Consider the fact that B = {so(x), S₁ (X), ..., S7 (x)} is a basis of P7(x). If *x^5 = ∑(k=0) ^7 a5,k Sk(X)*, where *a5,k ∈ R*, then *a5,2* equals ____ (rounded off to two decimal places) **Q.50** Let *M = [[0,0,0,-1],[0,-4,0,0],[0,0,0,3],[2,2,2,0]]*. If *p(x)* is the characteristic polynomial of M, then *p(2) - 1* equals ____. ### Section C: Q.51 - Q.60 Carry TWO marks each. **Q.51** For *a ∈ (−2π, 0)*, consider the differential equation *x^2 d²y/dx² + αχ dy/dx + y = 0* for *x > 0*. Let D be the set of all *αε (−2π, 0)* for which all corresponding real solutions to the above differential equation approach zero as *x → 0+*. Then, the number of elements in *D ∩ Z* equals ____. **Q.52** The value of *∫(1) ^π (log(t + 1/t)^(-1) / x * sin^2(5x) dx)* equals ____ (rounded off to two decimal places). **Q.53** Let T be the planar region enclosed by the square with vertices at the points (0,1), (1,0), (0,-1) and (-1,0). Then, the value of *∫∫T (cos(π(x - y)) - cos(π(x + y)))^2 dx dy* equals ____ (rounded off to two decimal places). **Q.54** Let *S = {(x, y, z) ∈ R3: x2 + y2 + z² < 1}*. Then, the value of *1/π ∫∫∫S ((x - 2y + z)² + (2x − y − z)² + (x − y + 2z)²)dxdydz* equals ____ (rounded off to two decimal places). **Q.55** For *n ∈ N*, if *an = 1/(n³ + 1) + 2²/(n³ + 2) + ... + n²/(n³ + n)*,