List of practice Questions

The number of points of intersection of $|z - (4 + 3i)| = 2$ and $|z| + |z - 4| = 6$, $z \in \mathbb{C}$, is

Sum of squares of modulus of all the complex numbers z satisfying
$\overline{z}=iz^2+z^2–z $
is equal to ________.

Let S be the set of (α,β),π<α,β<2π,for which the complex number
$\frac{1-i\sinα}{1+2i\sinα}$ is purely imaginary and $\frac{1+i\cosβ}{1-2i\cosβ}$ is purely real,
Let $Zαβ = \sin2α+i\cos2β, (α,β) ∈ S$. Then
$\sum_{(\alpha, \beta) \in S} \left(iZ_{\alpha\beta} + \frac{1}{iZ_{\alpha\beta}}\right)$
is equal to

If z = x + iy satisfies | z | – 2 = 0 and |z – i| – | z + 5i| = 0, then

Let
$S = \left\{z∈C : z^2+\overline{z} = 0 \right\}$. Then $∑_{z∈S}(Re(z)+Im(z))$
is equal to____.

Let a function ƒ : N →N be defined by
$f(n) = \left\{ \begin{array}{ll} 2n & n = 2,4,6,8,\ldots \\ n - 1 & n = 3,7,11,15,\ldots \\ \frac{n+1}{2} & n = 1,5,9,13 \end{array} \right.$
then, ƒ is

Let f: ℝ → ℝ satisfy f(x + y) = 2^x f(y) + 4^y f(x), ∀ x, y∈ ℝ. If f(2) = 3, then 14. f'(4)/f'(2) is equal to ___.

The number of real solutions of x⁷ + 5x³ + 3x + 1 = 0 is equal to ______.

The number of real solutions of the equation e^4x + 4e^3x - 58e^2x + 4ex + 1 = 0 is _____.

g :R→R be two real valued functions defined as
$f(x) = \begin{cases} -|x + 3| & x < 0 \\ e^x, & x \geq 0 \end{cases}$
and
$g(x) = \begin{cases} x^2 + k_1x ,& x < 0 \\ 4x + k_2 ,& x \geq 0 \end{cases}$
where k₁ and k₂ are real constants. If (goƒ) is differentiable at x = 0, then (goƒ) (–4) + (goƒ) (4) is
equal to:

$\alpha = \sin 36^\circ$ is a root of which of the following equation?

Let $\alpha, \beta$ be the roots of the equation $x^2 - 4\lambda x + 5 = 0$ and α, γ be the roots of the equation $x^2 - (3\sqrt{2} + 2\sqrt{3})x + 7 + 3\lambda\sqrt{3} = 0$. If $\beta + \gamma = 3\sqrt{2}$ then $(\alpha + 2\beta + \gamma)^2$ is equal to _______.

Let
$f(x)=max\left\{|x+1|,|x+2|,……,|x+5|\right\} $
Then $\int_{-6}^{0} f(x) \, dx$
is equal to_______

Let f(x) be a quadratic polynomial such that f(–2) + f(3) = 0. If one of the roots of f(x) = 0 is –1, then the sum of the roots of f(x) = 0 is equal to:

Let f,g : R → R be functions defined by*
$f(x) = \begin{cases} [x], & x < 0 \\ |1 - x|, & x \geq 0 \end{cases}$
and $g(x) = \begin{cases} e^x - x, & x < 0 \\ {(x - 1)^2 - 1}, & x \geq 0 \end{cases}$
Where [x] denotes the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly:

If
$f(x) = \begin{cases} x + a, & x \leq 0 \\ |x - 4|, & x > 0 \end{cases}$ and $g(x) = \begin{cases} x + 1, & x < 0 \\ (x - 4)^2 + b, & x \geq 0 \end{cases}$
are continuous on R, then (gof) (2) + (fog) (–2) is equal to

Let
$f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \leq 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$
Then the set of all values of b, for which f(x) has maximum value at x = 1, is

The number of functions f, from the set
$A = {x∈N: x^2-10x+9≤0} $to the set $B = {n62:n∈N}$
such that
$f(x)≤(x-3)^2+1$, for every $x∈A,$
is ______.

Let a function f: ℝ → ℝ be defined as :
$\begin{array}{l}f(x) = \left\{\begin{matrix}\int_{0}^{x}(5-|t-3|)dt, & x>4 \\x^2 + bx, & x \le4 \\\end{matrix}\right.\end{array}$
where b ∈ ℝ. If f is continuous at x = 4 then which of the following statements is NOT true?

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) – f(c) = f(d) is :

Let A and B be two events such that

$\begin{array}{l} P\left(B/A\right)\frac{2}{5},\ P\left(A/B\right)=\frac{1}{7}\ \text{and}\ P\left(A\cap B\right)=\frac{1}{9}.\end{array} Consider \begin{array}{l} \left(S1\right)P\left(A’\cup B\right)=\frac{5}{6},\end{array}$

$\begin{array}{l} \left(S2\right)P\left(A’\cap B’\right)=\frac{1}{18}.\end{array}$Then

Let E₁, E₂, E₃be three mutually exclusive events such that
$P(E_1)=\frac{2+3p}{6}, P(E_2)=\frac{2−p}{8} and\ P(E_3)=\frac{1−p}{2}.$
If the maximum and minimum values of p are p₁ and p₂, then (p₁ + p₂) is equal to :

A six faced die is biased such that
3 × P (a prime number) = 6 × P (a composite number) = 2 × P (1).
Let X be a random variable that counts the number of times one gets a perfect square on some
throws of this die. If the die is thrown twice, then the mean of X is :

Let S = {1, 2, 3, …, 2022}. Then the probability that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :