List of top Mathematics Questions asked in IIT JAM MS

Let
\(𝑆_𝑛 =∑_{k=1}^n\frac{1+k2^k}{4^{k-1}},n=1,2,...\)
Then, lim𝑛_→∞ 𝑆_𝑛 equals (round off to two decimal places)

If, for some 𝛼∈(0, ∞),
\(∫^∞_02^{-x^2}dx=a\sqrt\pi,\) 
then 𝛼 equals (round off to two decimal places)

Let {an}_n≥1 be a sequence of non-zero real numbers. Then which one of the following statements is true ?

Let f : \(\R → \R\) be the function defined by
\(f(x)=\text{det}\begin{pmatrix}1+x & 9 & 9 \\ 9 & 1+x & 9 \\ 9 & 9 & 1+x \end{pmatrix}\)
Then the maximum value of f on the interval [9, 10] equals

Let f : \(\R^2 → \R\) be the function defined by
\(f(x,y) = \begin{cases}   x^2\sin\frac{1}{x}+y^2\cos y, & x \ne 0 \\   0, & x=0. \end{cases}\)
Then which one of the following statements is NOT true ?

Let f : [1, 2] → \(\R\) be the function defined by
\(f(t)=\int^t_1\sqrt{x^2e^{x^2}-1}\ dx.\)
Then the arc length of the graph of f over the interval [1, 2] equals

Let F : [0, 2] → \(\R\) be the function defined by
\(F(x)=\int_{x^2}^{x+2}e^{x[t]}dt,\)
where [t] denotes the greatest integer less than or equal to t. Then the value of the derivative of F at x = 1 equals

Let the system of equations
x + ay + z = 1
2x + 4y + z = -b
3x + y + 2z = b + 2
have infinitely many solutions, where a and b are real constants. Then the value of 2a + 8b equals

Let \(A=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}\). Then the sum of all the elements of A¹⁰⁰ equals

Let {a_n}_n≥1 be a sequence of real numbers such that \(a_n=\frac{1}{3^n}\) for all n ≥ 1. Then which of the following statements is/are true ?

Let f : \(\R^2 → \R\) be the function defined by
f(x, y) = 8(x² - y²) - x⁴ + y⁴.
Then which of the following statements is/are true ?

If n ≥ 2, then which of the following statements is/are true ?

Let {a_n}_n≥1 be a sequence of real numbers such that a_1+5m = 2, a_2+5m = 3, a_3+5m = 4, a_4+5m = 5, a_5+5m = 6, m = 0, 1, 2, … . Then limsup_n→∞ a_n + liminf_n→∞ a_n equals __________

Let f : \(\R → \R\) be a function such that
20(x - y) ≤ f(x) - f(y) ≤ 20(x - y) + 2(x - y)²
for all x, y ∈ \(\R\) and f(0) = 2. Then f(101) equals __________

Let A be a 3 × 3 real matrix such that det(A) = 6 and
\(adj\ A=\begin{pmatrix} 1 & -1 & 2 \\ 5 & 7 & 1 \\ -1 & 1 & 1 \end{pmatrix}\)
where adj A denotes the adjoint of A.
Then the trace of A equals __________ (round off to 2 decimal places)

Let f ∶ \(\R^2 → \R\) be the function defined by f(x, y) = x² - 12y. If M and m be the maximum value and the minimum value, respectively, of the function f on the circle x² + y² = 49, then |M| + |m| equals __________

The value of
\(\int^2_0 \int_0^{2-x}(x+y)^2e^{\frac{2y}{x+y}}\ dy\ dx\)
equals __________ (round off to 2 decimal places)

Let \(A=\begin{pmatrix} 1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 1 \end{pmatrix}\) and let \(\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\)be an eigenvector corresponding to the smallest eigenvalue of A, satisfying \(x^2_1+x^2_2+x^2_3=1\). Then the value of |x₁| + |x₂| + |x₃| equals __________ (round off to 2 decimal places)

Let \( \alpha, \beta \) and \( \gamma \) be the eigenvalues of \[ M = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 3 \\ -1 & 2 & 2 \end{bmatrix}. \] If \( \gamma = 1 \) and \( \alpha > \beta \), then the value of \( 2\alpha + 3\beta \) is .............

Let \[ M = \begin{bmatrix} 5 & -6 \\ 3 & -4 \end{bmatrix} \] be a \(2 \times 2\) matrix. If \(\alpha = \det(M^4 - 6I_2)\), then the value of \(\alpha^2\) is ............

Let \( M \) be a \(3 \times 3\) real matrix. Let \[ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ -1 \\ \alpha \end{pmatrix} \] be eigenvectors of \(M\) corresponding to three distinct eigenvalues. Then, which of the following is NOT a possible value of \(\alpha\)?

For \( a \in \mathbb{R} \), consider the system of linear equations \[ \begin{cases} a x + a y = a + 2,
x + a y + (a - 1)z = a - 4,
a x + a y + (a - 2)z = -8, \end{cases} \] in the unknowns \(x, y, z\). Then, which of the following statements is TRUE?

Consider the linear system \( A x = b \), where \(A\) is an \(m \times n\) matrix, \(x\) is an \(n \times 1\) vector of unknowns and \(b\) is an \(m \times 1\) vector. Further, suppose there exists an \(m \times 1\) vector \(c\) such that the linear system \(A x = c\) has NO solution. Then, which of the following statements is/are necessarily TRUE?

Let \(A\) be a \(3 \times 3\) real matrix such that \(A \ne I_3\) and the sum of the entries in each row of \(A\) is 1. Then, which of the following statements is/are necessarily TRUE?

Consider the function \[ f(x,y) = 3x^2 + 4xy + y^2, \quad (x,y) \in \mathbb{R}^2. \] If \( S = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\} \), then which of the following statements is/are TRUE?