For a real number $x$, $$x ^ { 3 } - 7 x + 6 > 0$$ if and only if (A) $x > 2$. (B) $- 3 < x < 1$. (C) $x < - 3$ or $1 < x < 2$. (D) $- 3 < x < 1$ or $x > 2$.
Define a polynomial $f ( x )$ by $$f ( x ) = \left| \begin{array} { l l l }
1 & x & x \\
x & 1 & x \\
x & x & 1
\end{array} \right|$$ for all $x \in \mathbb { R }$, where the right hand side above is a determinant. Then the roots of $f ( x )$ are of the form (A) $\alpha , \beta \pm i \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R } , \gamma \neq 0$ and $i$ is a square root of $- 1$. (B) $\alpha , \alpha , \beta$ where $\alpha , \beta \in \mathbb { R }$ are distinct. (C) $\alpha , \beta , \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R }$ are all distinct. (D) $\alpha , \alpha , \alpha$ for some $\alpha \in \mathbb { R }$.
Let $S$ be the set of those real numbers $x$ for which the identity $$\sum _ { n = 2 } ^ { \infty } \cos ^ { n } x = ( 1 + \cos x ) \cot ^ { 2 } x$$ is valid, and the quantities on both sides are finite. Then (A) $S$ is the empty set. (B) $S = \{ x \in \mathbb { R } : x \neq n \pi$ for all $n \in \mathbb { Z } \}$. (C) $S = \{ x \in \mathbb { R } : x \neq 2 n \pi$ for all $n \in \mathbb { Z } \}$. (D) $S = \{ x \in \mathbb { R } : x \neq ( 2 n + 1 ) \pi$ for all $n \in \mathbb { Z } \}$.
Consider a right angled triangle $\triangle A B C$ whose hypotenuse $A C$ is of length 1. The bisector of $\angle A C B$ intersects $A B$ at $D$. If $B C$ is of length $x$, then what is the length of $CD$? (A) $\sqrt { \frac { 2 x ^ { 2 } } { 1 + x } }$ (B) $\frac { 1 } { \sqrt { 2 + 2 x } }$ (C) $\sqrt { \frac { x } { 1 + x } }$ (D) $\frac { x } { \sqrt { 1 - x ^ { 2 } } }$
Consider a triangle with vertices $( 0,0 ) , ( 1,2 )$ and $( - 4,2 )$. Let $A$ be the area of the triangle and $B$ be the area of the circumcircle of the triangle. Then $\frac { B } { A }$ equals (A) $\frac { \pi } { 2 }$. (B) $\frac { 5 \pi } { 4 }$. (C) $\frac { 3 } { \sqrt { 2 } } \pi$. (D) $2 \pi$.
Let $f , g$ be continuous functions from $[ 0 , \infty )$ to itself, $$h ( x ) = \int _ { 2 ^ { x } } ^ { 3 ^ { x } } f ( t ) d t , x > 0$$ and $$F ( x ) = \int _ { 0 } ^ { h ( x ) } g ( t ) d t , x > 0$$ If $F ^ { \prime }$ is the derivative of $F$, then for $x > 0$, (A) $F ^ { \prime } ( x ) = g ( h ( x ) )$. (B) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ f \left( 3 ^ { x } \right) - f \left( 2 ^ { x } \right) \right]$. (C) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ x 3 ^ { x - 1 } f \left( 3 ^ { x } \right) - x 2 ^ { x - 1 } f \left( 2 ^ { x } \right) \right]$. (D) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ 3 ^ { x } f \left( 3 ^ { x } \right) \ln 3 - 2 ^ { x } f \left( 2 ^ { x } \right) \ln 2 \right]$.
For real numbers $a , b , c , d , a ^ { \prime } , b ^ { \prime } , c ^ { \prime } , d ^ { \prime }$, consider the system of equations $$\begin{aligned}
a x ^ { 2 } + a y ^ { 2 } + b x + c y + d & = 0 \\
a ^ { \prime } x ^ { 2 } + a ^ { \prime } y ^ { 2 } + b ^ { \prime } x + c ^ { \prime } y + d ^ { \prime } & = 0
\end{aligned}$$ If $S$ denotes the set of all real solutions $( x , y )$ of the above system of equations, then the number of elements in $S$ can never be (A) 0. (B) 1. (C) 2. (D) 3.
Suppose $z \in \mathbb { C }$ is such that the imaginary part of $z$ is non-zero and $z ^ { 25 } = 1$. Then $$\sum _ { k = 0 } ^ { 2023 } z ^ { k }$$ equals (A) 0. (B) 1. (C) $- 1 - z ^ { 24 }$. (D) $- z ^ { 24 }$.
If $f : [ 0 , \infty ) \rightarrow \mathbb { R }$ is a continuous function such that $$f ( x ) + \ln 2 \int _ { 0 } ^ { x } f ( t ) d t = 1 , x \geq 0$$ then for all $x \geq 0$, (A) $f ( x ) = e ^ { x } \ln 2$. (B) $f ( x ) = e ^ { - x } \ln 2$. (C) $f ( x ) = 2 ^ { x }$. (D) $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$.
Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done? (A) 3 (B) 4 (C) 5 (D) 6
The straight line $O A$ lies in the second quadrant of the $(x, y)$-plane and makes an angle $\theta$ with the negative half of the $x$-axis, where $0 < \theta < \frac { \pi } { 2 }$. The line segment $C D$ of length 1 slides on the $(x, y)$-plane in such a way that $C$ is always on $O A$ and $D$ on the positive side of the $x$-axis. The locus of the mid-point of $C D$ is (A) $x ^ { 2 } + 4 x y \cot \theta + y ^ { 2 } \left( 1 + 4 \cot ^ { 2 } \theta \right) = \frac { 1 } { 4 }$. (B) $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 } + \cot ^ { 2 } \theta$. (C) $x ^ { 2 } + 4 x y \cot \theta + y ^ { 2 } = \frac { 1 } { 4 }$. (D) $x ^ { 2 } + y ^ { 2 } \left( 1 + 4 \cot ^ { 2 } \theta \right) = \frac { 1 } { 4 }$.
Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$? (A), (B), (C), (D) [graphs as provided in the figure]
Consider the function $f : \mathbb { C } \rightarrow \mathbb { C }$ defined by $$f ( a + i b ) = e ^ { a } ( \cos b + i \sin b ) , a , b \in \mathbb { R }$$ where $i$ is a square root of $-1$. Then (A) $f$ is one-to-one and onto. (B) $f$ is one-to-one but not onto. (C) $f$ is onto but not one-to-one. (D) $f$ is neither one-to-one nor onto.