# Recent questions tagged isi2019-mma

1
Consider the function $h$ defined on $\{0,1,…….10\}$ with $h(0)=0, \: h(10)=10$ and $2[h(i)-h(i-1)] = h(i+1) – h(i) \: \text{ for } i = 1,2, \dots ,9.$ Then the value of $h(1)$ is $\frac{1}{2^9-1}\\$ $\frac{10}{2^9+1}\\$ $\frac{10}{2^{10}-1}\\$ $\frac{1}{2^{10}+1}$
1 vote
2
Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\psi(y) =0$ for all $y \notin [0,1]$ and $\int_{0}^{1} \psi(y) dy=1$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function. Then the value of $\lim _{n\rightarrow \infty}n \int_{0}^{100} f(x)\psi(nx)dx$ is $f(0)$ $f’(0)$ $f’’(0)$ $f(100)$
3
Consider the functions $f,g:[0,1] \rightarrow [0,1]$ given by $f(x)=\frac{1}{2}x(x+1) \text{ and } g(x)=\frac{1}{2}x^2(x+1).$ Then the area enclosed between the graphs of $f^{-1}$ and $g^{-1}$ is $1/4$ $1/6$ $1/8$ $1/24$
4
A general election is to be scheduled on $5$ days in May such that it is not scheduled on two consecutive days. In how many ways can the $5$ days be chosen to hold the election? $\begin{pmatrix} 26 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 27 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 30 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 31 \\ 5 \end{pmatrix}$
1 vote
5
If $t = \begin{pmatrix} 200 \\ 100 \end{pmatrix}/4^{100}$, then $t < \frac{1}{3}$ $\frac{1}{3} < t < \frac{1}{2}$ $\frac{1}{2} < t < \frac{2}{3}$ $\frac{2}{3} < t < 1$
6
Let $a,b,c$ be non-zero real numbers such that $\int_{0}^{1} (1 + \cos^8x)(ax^2 + bx +c)dx = \int_{0}^{2}(1+ \cos^8x)(ax^2 + bx + c) dx =0$ Then the quadratic equation $ax^2 + bx +c=0$ has no roots in $(0,2)$ one root in $(0,2)$ and one root outside this interval one repeated root in $(0,2)$ two distinct real roots in $(0,2)$
1 vote
7
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f''(x)$ exists for every $x \in \mathbb{R}$, where $f''(x) = f \circ f^{n-1}(x)$ for $n \geq 2$ ... $S \subset T$ $T \subset S$ $S = T$ None of the above
1 vote
8
Let $A$ be $2 \times 2$ matrix with real entries. Now consider the function $f_A(x)$ = $Ax$ . If the image of every circle under $f_A$ is a circle of the same radius, then A must be an orthogonal matrix A must be a symmetric matrix A must be a skew-symmetric matrix None of the above must necessarily hold
9
A coin with probability $p (0 < p < 1)$ of getting head, is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $2/5$, then the value of $p$ is $2/7$ $1/3$ $5/7$ $2/3$
10
A function $f:\mathbb{R^2} \rightarrow \mathbb{R}$ is called degenerate on $x_i$, if $f(x_1,x_2)$ remains constant when $x_i$ varies $(i=1,2)$. Define $f(x_1,x_2) = \mid 2^{\pi _i/x_1} \mid ^{x_2} \text{ for } x_1 \neq 0$, where $i = \sqrt {-1}$. Then which ... $f$ is degenerate on $x_1$ but not on $x_2$ $f$ is degenerate on $x_2$ but not on $x_1$ $f$ is neither degenerate on $x_1$ nor on $x_2$
11
Suppose that the number plate of a vehicle contains two vowels followed by four digits. However, to avoid confusion, the letter $‘O’$ and the digit $‘0’$ are not used in the same number plate. How many such number plates can be formed? $164025$ $190951$ $194976$ $219049$
1 vote
12
Let $G =\{a_1,a_2, \dots ,a_{12}\}$ be an Abelian group of order $12$ . Then the order of the element $( \prod_{i=1}^{12} a_i)$ is $1$ $2$ $6$ $12$
1 vote
13
For the differential equation $\frac{dy}{dx} + xe^{-y}+2x=0$ It is given that $y=0$ when $x=0$. When $x=1$, $\:y$ is given by $\text{ln} \bigg(\frac{3}{2e} – \frac{1}{2} \bigg)$ $\text{ln} \bigg(\frac{3e}{2} – \frac{1}{4} \bigg)$ $\text{ln} \bigg(\frac{3}{e} – \frac{1}{2} \bigg)$ $\text{ln} \bigg(\frac{3}{2e} – \frac{1}{4} \bigg)$
1 vote
14
The reflection of the point $(1,2)$ with respect to the line $x + 2y =15$ is $(3,6)$ $(6,3)$ $(5,10)$ $(10,5)$
1 vote
15
If $S$ and $S’$ are the foci of the ellipse $3x^2 + 4y^2=12$ and $P$ is a point on the ellipse, then the perimeter of the triangle $PSS’$ is $4$ $6$ $8$ dependent on the coordinates of $P$
16
The rank of the matrix $\begin{bmatrix} 0 &1 &t \\ 2& t & -1\\ 2& 2 & 0 \end{bmatrix}$ equals $3$ for any real number $t$ $2$ for any real number $t$ $2$ or $3$ depending on the value of $t$ $1,2$ or $3$ depending on the value of $t$
1 vote
17
If the system of equations $\begin{array} \\ax +y+z= 0 \\ x+by +z = 0 \\ x+y + cz = 0 \end{array}$ with $a,b,c \neq 1$ has a non trivial solutions, the value of $\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}$ is $1$ $-1$ $3$ $-3$
18
Let $V$ be the vector space of all $4 \times 4$ matrices such that the sum of the elements in any row or any column is the same. Then the dimension of $V$ is $8$ $10$ $12$ $14$
19
Given a positive integer $m$, we define $f(m)$ as the highest power of $2$ that divides $m$. If $n$ is a prime number greater than $3$, then $f(n^3-1) = f(n-1)$ $f(n^3-1) = f(n-1) +1$ $f(n^3-1) = 2f(n-1)$ None of the above is necessarily true
20
How many triplets of real numbers $(x,y,z)$ are simultaneous solutions of the equations $x+y=2$ and $xy-z^2=1$? $0$ $1$ $2$ infinitely many
21
The chance of a student getting admitted to colleges $A$ and $B$ are $60\%$ and $40\%$, respectively. Assume that the colleges admit students independently. If the student is told that he has been admitted to at least one of these colleges, what is the probability that he has got admitted to college $A$? $3/5$ $5/7$ $10/13$ $15/19$
1 vote
22
$(\cos 100^\circ + i \sin 100^\circ)(\cos 0^\circ + i \sin 110^\circ)$ is equal to $\frac{1}{2}(\sqrt3 – i)$ $\frac{1}{2}(-\sqrt3 – i)$ $\frac{1}{2}(-\sqrt3 +i)$ $\frac{1}{2}(\sqrt3 + i)$
1 vote
23
For $0 \leq x < 2 \pi$, the number of solutions of the equation $\sin^2x + 2 \cos^2x + 3\sin x \cos x = 0$ is $1$ $2$ $3$ $4$
1 vote
24
The value of $\frac{1}{2\sin10^\circ}$ – $2\sin70^\circ$ is $-1/2$ $-1$ $1/2$ $1$
25
The solution of the differential equation $\frac{dy}{dx} = \frac{2xy}{x^2-y^2}$ is $x^2 + y^2 = cy$, where $c$ is a constant $x^2 + y^2 = cx$, where $c$ is a constant $x^2 – y^2 = cy$ , where $c$ is a constant $x^2 - y^2 = cx$, where $c$ is a constant
26
If $f(a)=2, \: f’(a) = 1, \: g(a) =-1$ and $g’(a) =2$, then the value of $\lim _{x\rightarrow a}\frac{g(x) f(a) – f(x) g(a)}{x-a}$ is $-5$ $-3$ $3$ $5$
1 vote
27
Suppose that $6$-digit numbers are formed using each of the digits $1, 2, 3, 7, 8, 9$ exactly once. The number of such $6$-digit numbers that are divisible by $6$ but not divisible by $9$ is equal to $120$ $180$ $240$ $360$
The sum of all $3$ digit numbers that leave a remainder of $2$ when divided by $3$ is: $189700$ $164850$ $164750$ $149700$
The number of $6$ digit positive integers whose sum of the digits is at least $52$ is $21$ $22$ $27$ $28$
The highest power of $7$ that divides $100!$ is : $14$ $15$ $16$ $18$