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Questions by Arjun

1 votes
2 answers
661
ISI2015-MMA-27
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is $0$ $1/32$ $15/32$ $10/32$
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is$0$$1/32$$...
1 votes
1 answer
662
ISI2015-MMA-28
In a triangle $ABC$, $AD$ is the median. If length of $AB$ is $7$, length of $AC$ is $15$ and length of $BC$ is $10$ then length of $AD$ equals $\sqrt{125}$ $69/5$ $\sqrt{112}$ $\sqrt{864}/5$
In a triangle $ABC$, $AD$ is the median. If length of $AB$ is $7$, length of $AC$ is $15$ and length of $BC$ is $10$ then length of $AD$ equals$\sqrt{125}$$69/5$$\sqrt{11...
0 votes
1 answer
665
ISI2015-MMA-31
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above
Consider the sets defined by the real solutions of the inequalities$$A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$$Then$B \subseteq A$$A \...
0 votes
1 answer
666
ISI2015-MMA-32
If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals $\sqrt{2/3}$ $\sqrt{3/2}$ $3/ \sqrt{2}$ $\sqrt{2}/3$
If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals$\sqrt{2/3}$$\sqrt{3/2}$$3/ \sqrt{2}$$\sqrt{2}/3$
1 votes
1 answer
667
ISI2015-MMA-33
If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x,$ for every $x \in \mathbb{R}$, then $f(2)$ is $-15$ $22$ $11$ $0$
If $f(x)$ is a real valued function such that $$2f(x)+3f(-x)=15-4x,$$ for every $x \in \mathbb{R}$, then $f(2)$ is$-15$$22$$11$$0$
0 votes
1 answer
668
ISI2015-MMA-34
If $f(x) = \dfrac{\sqrt{3}\sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[-1, \sqrt{3}/2]$ the interval $[- \sqrt{3}/2, 1]$ the interval $[-1, 1]$ none of the above
If $f(x) = \dfrac{\sqrt{3}\sin x}{2+\cos x}$, then the range of $f(x)$ isthe interval $[-1, \sqrt{3}/2]$the interval $[- \sqrt{3}/2, 1]$the interval $[-1, 1]$none of the ...
0 votes
0 answers
669
ISI2015-MMA-35
If $f(x)=x^2$ and $g(x)= x \sin x + \cos x$ then $f$ and $g$ agree at no points $f$ and $g$ agree at exactly one point $f$ and $g$ agree at exactly two points $f$ and $g$ agree at more than two points
If $f(x)=x^2$ and $g(x)= x \sin x + \cos x$ then$f$ and $g$ agree at no points$f$ and $g$ agree at exactly one point$f$ and $g$ agree at exactly two points$f$ and $g$ agr...
1 votes
1 answer
670
ISI2015-MMA-36
For non-negative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m-1, 1) & \text{ if } m \neq 0, n=0 \\ f(m-1, f(m,n-1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$
For non-negative integers $m$, $n$ define a function as follows$$f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m-1, 1) & \text{ if } m \neq 0, n=0 \\ f(m-1, f(m,n-1))...
1 votes
1 answer
671
ISI2015-MMA-37
Let $a$ be a non-zero real number. Define $f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$
Let $a$ be a non-zero real number. Define$$f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$$ for $x \in \mathbb{R}$....
0 votes
0 answers
672
ISI2015-MMA-38
A real $2 \times 2$ matrix $M$ such that $M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1- \varepsilon \end{pmatrix}$ exists for all $\varepsilon > 0$ does not exist for any $\varepsilon > 0$ exists for some $\varepsilon > 0$ none of the above is true
A real $2 \times 2$ matrix $M$ such that $$M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1- \varepsilon \end{pmatrix}$$exists for all $\varepsilon 0$does not exist for any $\vare...
7 votes
3 answers
673
ISI2015-MMA-39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are$1,1,4$$1,4,4$$0,1,4$$0,4,4$
0 votes
0 answers
674
ISI2015-MMA-40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ ... $(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ by$$\textbf{A} = \begin{...
1 votes
1 answer
675
ISI2015-MMA-41
Let $k$ and $n$ be integers greater than $1$. Then $(kn)!$ is not necessarily divisible by $(n!)^k$ $(k!)^n$ $n! \cdot k! \cdot$ $2^{kn}$
Let $k$ and $n$ be integers greater than $1$. Then $(kn)!$ is not necessarily divisible by$(n!)^k$$(k!)^n$$n! \cdot k! \cdot$$2^{kn}$
2 votes
1 answer
677
ISI2015-MMA-43
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta = 1, -2$ $\eta = -1, -2$ $\eta = 3, -3$ $\eta = 1, 2$
The values of $\eta$ for which the following system of equations$$\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & =...
0 votes
1 answer
679
ISI2015-MMA-45
Angles between any pair of $4$ main diagonals of a cube are $\cos^{-1} 1/\sqrt{3}, \pi – \cos ^{-1} 1/\sqrt{3}$ $\cos^{-1} 1/3, \pi – \cos ^{-1} 1/3$ $\pi/2$ none of the above
Angles between any pair of $4$ main diagonals of a cube are$\cos^{-1} 1/\sqrt{3}, \pi – \cos ^{-1} 1/\sqrt{3}$$\cos^{-1} 1/3, \pi – \cos ^{-1} 1/3$$\pi/2$none of the ...
0 votes
1 answer
680
ISI2015-MMA-46
If the tangent at the point $P$ with coordinates $(h,k)$ on the curve $y^2=2x^3$ is perpendicular to the straight line $4x=3y$, then $(h,k) = (0,0)$ $(h,k) = (1/8, -1/16)$ $(h,k) = (0,0) \text{ or } (h,k) = (1/8, -1/16)$ no such point $(h,k)$ exists
If the tangent at the point $P$ with coordinates $(h,k)$ on the curve $y^2=2x^3$ is perpendicular to the straight line $4x=3y$, then$(h,k) = (0,0)$$(h,k) = (1/8, -1/16)$$...
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