List of top Mathematics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

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)

The volume of the region 𝑅={(𝑥, 𝑦, 𝑧) ∈ ℝ × ℝ × ℝ ∶ 𝑥 ² + 𝑦 ² ≤ 4, 0 ≤ 𝑧 ≤ 4 − 𝑦} is

For 𝑥∈ℝ, the curve 𝑦=𝑥² intersects the curve 𝑦=𝑥 sin 𝑥+cos 𝑥 at exactly 𝑛 points. Then, 𝑛 equals

Let M= \(\begin{pmatrix} 1 & -1 & 0\\ -a & 2 & -1 \\0&-1&1\end{pmatrix}\)If a non-zero vector 𝑋=(𝑥, 𝑦, 𝑧)^T∈ℝ³ satisfies 𝑀⁶𝑋=𝑋, then a subspace of ℝ3 that contains the vector 𝑋 is

Let 𝑀=𝑀₁𝑀₂, where 𝑀₁ and 𝑀₂ are two 3×3 distinct matrices. Consider the following two statements:
(I) The rows of 𝑀 are linear combinations of rows of 𝑀₂. 
(II) The columns of 𝑀 are linear combinations of columns of 𝑀₁. 
Then,

For real constants 𝛼 and 𝛽, suppose that the system of linear equations
𝑥+2𝑦+3𝑧=6; x+ 𝑦 +𝛼𝑧 = 3; 2𝑦+𝑧=𝛽, 
has infinitely many solutions. Then, the value of 4𝛼 + 3𝛽 equals

Which of the following is/are TRUE ?

Among all the points on the sphere 𝑥²+𝑦²+𝑧²=24, the point (𝛼, 𝛽, 𝛾) is closest to the point (1, 2,−1). Then, the value of 𝛼+𝛽+𝛾 equals

Let 𝛼 and 𝛽 be real constants such that\(lim_{x-0^+}\frac{∫^x_0(\frac{αt^2}{1+t^4})dt}{βx-sin\,x}\)=1 .
Then, the value of 𝛼 + 𝛽 equals ________

Let 𝑓:ℝ×ℝ→ℝ be a function defined by
\(f(x,y)=\begin{cases} \frac{xy(x+y}{x^2+y^2} & \quad \text{if }(x,y)≠(0,0),\\  0, & \quad if\,(x,y)=(0,0). \end{cases}\)
Then, which of the following statements is/are TRUE?

Let 𝑓: ℝ→ ℝ be a function defined by 𝑓(𝑥)=𝑥²−𝑥, 𝑥 ∈ ℝ. Let 𝑔: ℝ→ℝ be a twice differentiable function such that 𝑔(𝑥)=0 has exactly three distinct roots in the open interval (0, 1). Let ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥), 𝑥 ∈ ℝ, and ℎ′′ be the second order derivative of the function ℎ. If 𝑛 is the number of roots of ℎ′′(𝑥)=0 in (0, 1), then the minimum possible value of 𝑛 equals ________

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 __________

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

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

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

Let 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 \(A=\begin{pmatrix} 1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 1 \end{pmatrix}\) 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 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 ?

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 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 : [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