List of practice Questions

For t ∈ \(\R\), let [𝑡] denote the greatest integer less than or equal to t.
Let D = {(x, y) ∈ \(\R^2\) ∶ x2 + y2 < 4}. Let f : D → \(\R\) and g : D → \(\R\) be defined by f(0, 0) = g(0, 0) = 0 and
\(f(x,y)=[x^2+y^2]\frac{x^2y^2}{x^4+y^4},\ \ \ g(x,y)=[y^2]\frac{xy}{x^2+y^2}\)
for (x, y) ≠ (0, 0). Let E be the set of points of D at which both f and g are discontinuous. The number of elements in the set E is _________.

Given that y(x) is a solution of the deferential equation
\(x^2y''+xy'-4y-x^2\) 
on the interval (0,∞) such that the \(lim_{x→0^+}\) y(x) exists and y(1)=1. the value y'(1) is qual to____.(Rounded off to two decimal places).

If G is the region in \(\R^2\) given by
\(G=\left\{(x,y)\in \R^2:x^2+y^2<1,\frac{x}{\sqrt3}< y < \sqrt3x,x>0,y>0\right\}\)
then the value of
\(\frac{200}{\pi}\iint_Gx^2dx\ dy\)
is equal to _________. (Rounded off to two decimal places)

Let \(A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{pmatrix}\)and let A^T denote the transpose of A. Let \(u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}\) and \(v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)be column vectors with entries in \(\R\) such that \(u^2_1+u_2^2=1\) and \(v_1^2+v_2^2+v^2_3=1\). Suppose
\(Au=\sqrt2v\) and \(A^Tv=\sqrt2u.\)
Then |\(𝑢1 + 2 \sqrt2 𝑣_1\)| is equal to _________. (Rounded off to two decimal places)

Let a triangle be bounded by the lines L₁ : 2x + 5y = 10; L₂ : –4x + 3y = 12 and the line L₃, which passes through the point P(2, 3), intersects L₂ at A and L₁ at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to

Let f : [0, π] → \(\R\) be the function defined by
\(f(x)=\begin{cases} (x-\pi)e^{\sin x} & \text{if } 0 \le x \le \frac{\pi}{2},\\ xe^{\sin x}+\frac{4}{\pi} & \text{if }\frac{\pi}{2}\lt x \le \pi. \end{cases}\)
Then the value of
\(\int\limits_0^{\pi}f(x)dx\)
is equal to _________. (Rounded off to two decimal places)

Let r be the radius of convergence of the power series
\(\frac{1}{3}+\frac{x}{5}+\frac{x^2}{3^2}+\frac{x^3}{5^2}+\frac{x^4}{3^3}+\frac{x^5}{5^3}+\frac{x^6}{3^4}+\frac{x^7}{5^4}+...\)
Then the value of r² is equal to _________. (Rounded off to two decimal places)

Consider the 4 × 4 matrix \(M=\begin{pmatrix} 11 & 10 & 10 & 10 \\ 10 & 11 & 10 & 10 \\ 10 & 10 & 11 & 10 \\ 10 & 10 & 10 & 11 \end{pmatrix}\). Then the value of the determinant of M is equal to _________.

Let f : (−1, 1) → \(\R\) and g : (−1, 1) → \(\R\) be thrice continuously differentiable functions such that f(x) ≠ g(x) for every nonzero x ∈ (−1, 1). Suppose
f(0) = ln 2, f'(0) = π, f"(0) = π² , and f"'(0) = π⁹
and
g(0) = ln 2, g'(0) = π, g′′(0) = π², and g'"(0) = π³.
Then the value of the limit
\(\lim\limits_{x\rightarrow0}\frac{e^{f(x)}-e^{g(x)}}{f(x)-g(x)}\)
is equal to _________. (Rounded off to two decimal places)

Let σ be the permutation in the symmetric group S₅ given by
σ(1) = 2, σ(2) = 3, σ(3) = 1, σ(4) = 5, σ(5) = 4.
Define
N(σ) = {𝜏 ∈ S₅ ∶ σ ∘ 𝜏 = 𝜏 ∘ σ}.
Then the number of elements in N(σ) is equal to _________.

Define f : \(\R^2 → \R\) by
f(x, y) = x² + 2y² - x for (x, y) ∈ \(\R^2\).
Let D = {(x, y) ∈ \(\R^2\) ∶ x² + y² ≤ 1} and \(E=\left\{(x,y)\in \R^2:\frac{x^2}{4}+\frac{y^2}{9} \le 1\right\}.\)
Consider the sets
D_max = {(a, b) ∈ D ∶ f has absolute maximum on D at (a, b)}, 
D_min = {(a, b) ∈ D ∶ f has absolute minimum on D at (a, b)}, 
E_max = {(c, d) ∈ E ∶ f has absolute maximum on E at (c, d)}, 
E_min = {(c, d) ∈ E ∶ f has absolute minimum on E at (c, d)}.
Then the total number of elements in the set
D_max ∪ D_min ∪ E_max ∪ E_min
is equal to _________.

Let V be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 5, together with the zero polynomial. Let T: V→ \(\R\) be the linear map defined by 7(1) = 1 and T(x(x-1)... (x-k+1)) = 1 for 1 ≤ k ≤ 5.
Then which of the following is/are true?

If f : [0, ∞) → \(\R\) and g : [0, ∞) → [0, ∞) are continuous functions such that
\(\int^{x^3+x^2}_0f(t)dt=x^2\) and \(\int_0^{g(x)}t^2dt=9(x+1)^3\) for all x ∈ [0, ∞),
then the value of
f(2) + g(2) + 16 f(12)
is equal to _________. (Rounded off to two decimal places)

Two isomers ‘A’ and ‘B’ with molecular formula C₄H₈ give different products on oxidation with KMnO₄/H⁺ results in effervescence of a gas and gives ketone. The compound ‘A’ is

Incorrect statement for Tyndall effect is:

Let a circle C in complex plane pass through the points \(z_1 = 3 + 4i\), \(z_2 = 4 + 3i\) and \(z_3 = 5i\). If \(z(≠z_1)\) is a point on C such that the line through \(z\) and \(z_1\) is perpendicular to the line through \(z_2\) and \(z_3\), then \(arg\ (z)\) is equal to:

Haemoglobin contains \(0.34\%\) of iron by mass. The number of Fe atoms in \(3.3\) g of haemoglobin is(Given: Atomic mass of Fe is \(56\) u, \(N_A\) = \(6.022 × 1023\) \(mol^{–1}\))

If the electric potential at any point (x, y, z)m in space is given by V = 3x² volt. The electric field at the point (1, 0, 3)m will be

The mean and variance of a binomial distribution are $\alpha$ and $\frac{\alpha}{3}$ respectively If $P(X=1)=\frac{4}{243}$, then $P(X=4$ or 5$)$ is equal to :

The spin-only magnetic moment value of an octahedral complex among \(CoCl_3⋅4NH_3\), \(NiCl_2⋅6H_2O\) and \(PtCl_4⋅2HCl\), which upon reaction with excess of \(AgNO_3\) gives 2 moles of \(AgCl\) is _____ B.M. (Nearest Integer)

A vessel contains 16 g of hydrogen and 128 g of oxygen at standard temperature and pressure. The volume of the vessel in cm³ is: