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Recent questions tagged isi2019-mma

3 votes
1 answer
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 $\frac{1}{2^9-1}\\$ $\frac{10}{2^9+1}\\$ $\frac{10}{2^{10}-1}\\$ $\frac{1}{2^{10}+1}$
asked May 7, 2019 in Calculus Sayan Bose 788 views
1 vote
1 answer
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 $f(0)$ $f’(0)$ $f’’(0)$ $f(100)$
asked May 7, 2019 in Calculus Sayan Bose 719 views
0 votes
1 answer
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 $1/4$ $1/6$ $1/8$ $1/24$
asked May 7, 2019 in Calculus Sayan Bose 1k views
3 votes
3 answers
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? $\begin{pmatrix} 26 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 27 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 30 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 31 \\ 5 \end{pmatrix}$
asked May 7, 2019 in Combinatory Sayan Bose 3.6k views
1 vote
1 answer
5
If $t = \begin{pmatrix} 200 \\ 100 \end{pmatrix}/4^{100} $, then $t < \frac{1}{3}$ $\frac{1}{3} < t < \frac{1}{2}$ $\frac{1}{2} < t < \frac{2}{3}$ $\frac{2}{3} < t < 1$
asked May 7, 2019 in Numerical Ability Sayan Bose 597 views
0 votes
1 answer
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 $(0,2)$
asked May 7, 2019 in Calculus Sayan Bose 520 views
1 vote
2 answers
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
asked May 7, 2019 in Calculus Sayan Bose 741 views
1 vote
1 answer
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
asked May 7, 2019 in Linear Algebra Sayan Bose 706 views
0 votes
3 answers
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 $2/7$ $1/3$ $5/7$ $2/3$
asked May 7, 2019 in Probability Sayan Bose 358 views
0 votes
1 answer
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 $x_2$
asked May 7, 2019 in Others Sayan Bose 663 views
0 votes
2 answers
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? $164025$ $190951$ $194976$ $219049$
asked May 7, 2019 in Combinatory Sayan Bose 1k views
1 vote
1 answer
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 $1$ $2$ $6$ $12$
asked May 7, 2019 in Set Theory & Algebra Sayan Bose 747 views
1 vote
1 answer
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 $\text{ln} \bigg(\frac{3}{2e} – \frac{1}{2} \bigg)$ $\text{ln} \bigg(\frac{3e}{2} – \frac{1}{4} \bigg)$ $\text{ln} \bigg(\frac{3}{e} – \frac{1}{2} \bigg)$ $\text{ln} \bigg(\frac{3}{2e} – \frac{1}{4} \bigg)$
asked May 7, 2019 in Others Sayan Bose 3.9k views
1 vote
2 answers
14
The reflection of the point $(1,2)$ with respect to the line $x + 2y =15$ is $(3,6)$ $(6,3)$ $(5,10)$ $(10,5)$
asked May 7, 2019 in Geometry Sayan Bose 401 views
1 vote
1 answer
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 $P$
asked May 7, 2019 in Geometry Sayan Bose 272 views
0 votes
2 answers
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 $t$
asked May 7, 2019 in Linear Algebra Sayan Bose 568 views
1 vote
1 answer
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 $1$ $-1$ $3$ $-3$
asked May 7, 2019 in Linear Algebra Sayan Bose 411 views
0 votes
2 answers
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 $8$ $10$ $12$ $14$
asked May 6, 2019 in Linear Algebra Sayan Bose 650 views
0 votes
3 answers
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
asked May 6, 2019 in Numerical Ability Sayan Bose 775 views
0 votes
2 answers
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
asked May 6, 2019 in Numerical Ability Sayan Bose 563 views
0 votes
1 answer
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 $A$? $3/5$ $5/7$ $10/13$ $15/19$
asked May 6, 2019 in Probability Sayan Bose 1k views
1 vote
1 answer
22
$(\cos 100^\circ + i \sin 100^\circ)(\cos 0^\circ + i \sin 110^\circ)$ is equal to $\frac{1}{2}(\sqrt3 – i)$ $\frac{1}{2}(-\sqrt3 – i)$ $\frac{1}{2}(-\sqrt3 +i)$ $\frac{1}{2}(\sqrt3 + i)$
asked May 6, 2019 in Others Sayan Bose 317 views
1 vote
1 answer
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 $1$ $2$ $3$ $4$
asked May 6, 2019 in Others Sayan Bose 421 views
1 vote
1 answer
24
The value of $\frac{1}{2\sin10^\circ}$ – $2\sin70^\circ$ is $-1/2$ $-1$ $1/2$ $1$
asked May 6, 2019 in Others Sayan Bose 289 views
0 votes
1 answer
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
asked May 6, 2019 in Calculus Sayan Bose 419 views
0 votes
1 answer
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 $-5$ $-3$ $3$ $5$
asked May 6, 2019 in Calculus Sayan Bose 360 views
1 vote
1 answer
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 $120$ $180$ $240$ $360$
asked May 6, 2019 in Combinatory Sayan Bose 800 views
2 votes
1 answer
28
The sum of all $3$ digit numbers that leave a remainder of $2$ when divided by $3$ is: $189700$ $164850$ $164750$ $149700$
asked May 6, 2019 in Numerical Ability Sayan Bose 426 views
0 votes
1 answer
29
The number of $6$ digit positive integers whose sum of the digits is at least $52$ is $21$ $22$ $27$ $28$
asked May 6, 2019 in Combinatory Sayan Bose 1.2k views
3 votes
1 answer
30
The highest power of $7$ that divides $100!$ is : $14$ $15$ $16$ $18$
asked May 6, 2019 in Numerical Ability Sayan Bose 787 views
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