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9 maths questions

Q41 Conic sections Eccentricity or Asymptote Computation View
Let $P ( 6,3 )$ be a point on the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If the normal at the point $P$ intersects the $x$-axis at $( 9,0 )$, then the eccentricity of the hyperbola is
(A) $\sqrt { \frac { 5 } { 2 } }$
(B) $\sqrt { \frac { 3 } { 2 } }$
(C) $\sqrt { 2 }$
(D) $\sqrt { 3 }$
Q42 Roots of polynomials Determine coefficients or parameters from root conditions View
A value of $b$ for which the equations $$\begin{aligned} & x ^ { 2 } + b x - 1 = 0 \\ & x ^ { 2 } + x + b = 0 \end{aligned}$$ have one root in common is
(A) $- \sqrt { 2 }$
(B) $- i \sqrt { 3 }$
(C) $i \sqrt { 5 }$
(D) $\sqrt { 2 }$
Q43 Matrices Determinant and Rank Computation View
Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $$\left[ \begin{array} { c c c } 1 & a & b \\ \omega & 1 & c \\ \omega ^ { 2 } & \omega & 1 \end{array} \right]$$ where each of $a , b$, and $c$ is either $\omega$ or $\omega ^ { 2 }$. Then the number of distinct matrices in the set $S$ is
(A) 2
(B) 6
(C) 4
(D) 8
Q44 Circles Circle Equation Derivation View
The circle passing through the point $( - 1,0 )$ and touching the $y$-axis at $( 0,2 )$ also passes through the point
(A) $\left( - \frac { 3 } { 2 } , 0 \right)$
(B) $\left( - \frac { 5 } { 2 } , 2 \right)$
(C) $\left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right)$
(D) $( - 4,0 )$
Q45 Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View
If $$\lim _ { x \rightarrow 0 } \left[ 1 + x \ln \left( 1 + b ^ { 2 } \right) \right] ^ { \frac { 1 } { x } } = 2 b \sin ^ { 2 } \theta , b > 0 \text { and } \theta \in ( - \pi , \pi ]$$ then the value of $\theta$ is
(A) $\pm \frac { \pi } { 4 }$
(B) $\pm \frac { \pi } { 3 }$
(C) $\pm \frac { \pi } { 6 }$
(D) $\pm \frac { \pi } { 2 }$
Q46 Areas by integration View
Let $f : [ - 1,2 ] \rightarrow [ 0 , \infty )$ be a continuous function such that $f ( x ) = f ( 1 - x )$ for all $x \in [ - 1,2 ]$. Let $R _ { 1 } = \int _ { - 1 } ^ { 2 } x f ( x ) d x$, and $R _ { 2 }$ be the area of the region bounded by $y = f ( x ) , x = - 1 , x = 2$, and the $x$-axis. Then
(A) $R _ { 1 } = 2 R _ { 2 }$
(B) $R _ { 1 } = 3 R _ { 2 }$
(C) $2 R _ { 1 } = R _ { 2 }$
(D) $3 R _ { 1 } = R _ { 2 }$
Q47 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Let $f ( x ) = x ^ { 2 }$ and $g ( x ) = \sin x$ for all $x \in \mathbb { R }$. Then the set of all $x$ satisfying $( f \circ g \circ g \circ f ) ( x ) = ( g \circ g \circ f ) ( x )$, where $( f \circ g ) ( x ) = f ( g ( x ) )$, is
(A) $\pm \sqrt { n \pi } , n \in \{ 0,1,2 , \ldots \}$
(B) $\pm \sqrt { n \pi } , n \in \{ 1,2 , \ldots \}$
(C) $\frac { \pi } { 2 } + 2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$
(D) $2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$
Q48 Straight Lines & Coordinate Geometry Locus Determination View
Let $( x , y )$ be any point on the parabola $y ^ { 2 } = 4 x$. Let $P$ be the point that divides the line segment from $( 0,0 )$ to $( x , y )$ in the ratio $1 : 3$. Then the locus of $P$ is
(A) $x ^ { 2 } = y$
(B) $y ^ { 2 } = 2 x$
(C) $y ^ { 2 } = x$
(D) $x ^ { 2 } = 2 y$
Q49 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
If $$f ( x ) = \begin{cases} - x - \frac { \pi } { 2 } , & x \leq - \frac { \pi } { 2 } \\ - \cos x , & - \frac { \pi } { 2 } < x \leq 0 \\ x - 1 , & 0 < x \leq 1 \\ \ln x , & x > 1 , \end{cases}$$ then
(A) $f ( x )$ is continuous at $x = -\frac{\pi}{2}$
(B) $f ( x )$ is not differentiable at $x = 0$
(C) $f ( x )$ is differentiable at $x = 1$
(D) $f ( x )$ is differentiable at $x = -\frac{3}{2}$