# Recent questions tagged isi2016-mmamma

1
How many complex numbers $z$ are there such that $\mid z+1 \mid = \mid z+i \mid$ and $\mid z \mid =5$? $0$ $1$ $2$ $3$
1 vote
The number of real roots of the equation $2 \cos \big(\frac{x^2+x}{6}\big)=2^x+2^{-x}$ is $0$ $1$ $2$ $\infty$
The $a, b, c$ and $d$ satisfy the equations$\begin{matrix} a & + & 7b & + & 3c & + & 5d & = &16 \\ 8a & + & 4b & + & 6c & + & 2d & = &-16 \\ 2a & + & 6b & + & 4c & + & 8d & = &16 \\ 5a & + & 7b & + & 3c & + & 5d & = &-16 \end{matrix}$Then $(a+d)(b+c)$ equals $-4$ $0$ $16$ $-16$
Let $f(x, y) = \begin{cases} \dfrac{x^2y}{x^4+y^2}, & \text{ if } (x, y) \neq (0, 0) \\ 0 & \text{ if } (x, y) = (0, 0) \end{cases}$ Then $\lim_{(x, y) \rightarrow (0,0)}$f(x,y)$equals$0$equals$1$equals$2$does not exist 0 votes 1 answer 5 Find the centroid of the triangle whose sides are given by the following equations:$\begin{matrix} 4x & - & y & = &19 \\ x &- & y & = & 4 \\ x& + & 2y & = & -11 \end{matrix}\left(\frac{11}{3}, -\frac{7}{3}\right)\left(\frac{5}{3}, -\frac{7}{3}\right)\left(-\frac{11}{3}, -\frac{7}{3}\right)\left(\frac{7}{3}, -\frac{11}{3}\right)$0 votes 0 answers 6 The set of value(s) of$\alpha$for which$y(t)=t^{\alpha}$is a solution to the differential equation$t^2 \frac{d^2y}{dx^2}-2t \frac{dy}{dx}+2y =0 \: \text{ for } t>0$is$\{1\}\{1, -1\}\{1, 2\}\{-1, 2\}$2 votes 1 answer 7 Let$g: \mathbb{R} \rightarrow \mathbb{R}$be differentiable with$g'(x^2)=x^3$for all$x>0$and$g(1) =1$. Then$g(4)$equals$64/532/537/567/5$0 votes 2 answers 8 Suppose$X$and$Y$are two independent random variables both following Poisson distribution with parameter$\lambda$. What is the value of$E(X-Y)^2$?$\lambda2 \lambda\lambda^24 \lambda^2$0 votes 1 answer 9 If$A_1, A_2, \dots , A_n$are independent events with probabilities$p_1, p_2, \dots , p_n$respectively, then$P( \cup_{i=1}^n A_i)$equals$\Sigma_{i=1}^n \: \: p_i\Pi_{i=1}^n \: \: p_i\Pi_{i=1}^n \: \: (1-p_i)1-\Pi_{i=1}^n \: \: (1-p_i)$0 votes 1 answer 10 Ravi asked his neighbor to water a delicate plant while he is away. Without water, the plant would die with probability 4/5 and with water it would die with probability 3/20. The probability that Ravi's neighbor would remember to water the plant is 9/10. If the plant actually died, what is the probability that Ravi's neighbor forgot to water the plant? 4/5 27/43 16/43 2/25 0 votes 1 answer 11 Suppose there are$n$positive real numbers such that their sum is 20 and the product is strictly greater than 1. What is the maximum possible value of n? 18 19 20 21 0 votes 2 answers 12 Which one of the following statements is correct regarding the elements and subsets of the set$\{1, 2, \{1, 2, 3\}\}$?$\{1, 2\} \in \{1, 2, \{1, 2, 3\} \}\{1, 2\} \subseteq \{1, 2, \{1, 2, 3\} \}\{1, 2, 3\} \subseteq \{1, 2, \{1, 2, 3\} \}3 \in \{1, 2, \{1, 2, 3\} \}$0 votes 1 answer 13 The number of positive integers$n$for which$n^2 +96$is a perfect square$0124$0 votes 0 answers 14 Suppose a 6 digit number$N$is formed by rearranging the digits of the number 123456. If$N$is divisible by 5, then the set of all possible remainders when$N$is divided by 45 is$\{30\}\{15, 30\}\{0, 15, 30\}\{0, 5, 15, 30\}$0 votes 0 answers 15 The number of positive integers$n$for which$n^3 +(n+1)^3 +(n+2)^3 = (n+3)^3$is$0123$2 votes 1 answer 16 Let$A=\begin{pmatrix} -1 & 2 \\ 0 & -1 \end{pmatrix}$, and$B=A+A^2+A^3+ \dots +A^{50}$. Then$B^2 =1B^2 =0B^2 =AB^2 =B$1 vote 1 answer 17 Let$A$be a real$2 \times 2$matrix. If$5+3i$is an eigenvalue of$A$, then$det(A)$equals 4 equals 8 equals 16 cannot be determined from the given information 2 votes 0 answers 18 Let$f : (0, \infty) \rightarrow (0, \infty)$be a strictly decreasing function. Consider$h(x) = \dfrac{f(\frac{x}{1+x})}{1+f(\frac{x}{1+x})}$. Which one of the following is always true?$h$is strictly decreasing$h$is strictly increasing$h$is strictly decreasing at first and then strictly increasing$h$is strictly increasing at first and then strictly decreasing 1 vote 0 answers 19 Let$A=\{1, 2, 3, 4, 5, 6, 7, 8 \}$. How many functions$f: A \rightarrow A$can be defined such that$f(1)< f(2) < f(3)$?$\begin{pmatrix} 8 \\ 3 \end{pmatrix}\begin{pmatrix} 8 \\ 3 \end{pmatrix} 5^8\begin{pmatrix} 8 \\ 3 \end{pmatrix} 8^5\frac{8!}{3!}$0 votes 0 answers 20 The infinite series$\Sigma_{n=1}^{\infty} \frac{a^n \log n}{n^2}$converges if and only if$a \in [-1, 1)a \in (-1, 1]a \in [-1, 1]a \in (-\infty, \infty)$0 votes 0 answers 21 Given that$\int_{-\infty}^{\infty} e^{-x^2/2} dx = \sqrt{2 \pi}$, what is the value of$\int_{- \infty}^{\infty} \mid x \mid ^{-1/2} e^{- \mid x \mid} dx$?$0\sqrt{\pi}2 \sqrt{\pi}\infty$0 votes 0 answers 22 Let$f: \mathbb{R} \rightarrow \mathbb{R}$be a strictly increasing function. Then which one the following is always true? The limits$\lim_{x \rightarrow a+} f(x) $and$\lim_{x \rightarrow a-} f(x)$exist for all real numbers$a$If$f$is differentiable at$a$... number$B$such that$f(x)<B$for all real$x$There cannot be any real number$L$such that$f(x)>L$for all real$x$0 votes 1 answer 23 A integer is said to be a$\textbf{palindrome}$if it reads the same forward or backward. For example, the integer$14541$is a$5$-digit palindrome and$12345$is not a palindrome. How many$8$-digit palindromes are prime?$011119$2 votes 2 answers 24 Consider the function$f(x) = \dfrac{e^{- \mid x \mid}}{\text{max}\{e^x, e^{-x}\}}, \: \: x \in \mathbb{R}$. Then$f$is not continuous at some points$f$is continuous everywhere, but not differentiable anywhere$f$is continuous everywhere, but not differentiable at exactly one point$f$is differentiable everywhere 1 vote 0 answers 25 Let$A$be a square matrix such that$A^3 =0$, but$A^2 \neq 0$. Then which of the following statements is not necessarily true?$A \neq A^2$Eigenvalues of$A^2$are all zero rank($A$) > rank($A^2$) rank($A$) > trace($A$) 0 votes 0 answers 26 Suppose$a$is a real number for which all the roots of the equation$x^4 -2ax^2+x+a^2-a=0$are real. Then$a<-\frac{2}{3}a=00<a<\frac{3}{4}a \geq \frac{3}{4}$0 votes 0 answers 27 A club with$n$members is organized into four committees so that each member belongs to exactly two committees and each pair of committees has exactly one member in common. Then$n=4n=6n=8n\$ cannot be determined from the given information