LFM Stats And Pure

iran-konkur 2020 Q118 Asymptote Determination View

118. The graph of $f(x) = \dfrac{-2x^2 + 3x}{ax^2 + bx + c}$ has asymptotes $y = -1$, $y = -2$, $x = -2$, and $x = 1$. What is $f(-1)$?
(1) $1.25$ (2) $1.5$ (3) $1.75$ (4) $-1.5$

iran-konkur 2021 Q118 Continuity and Discontinuity Analysis of Piecewise Functions View

118. The function $f(x) = \dfrac{ax^3 - bx^2 + 2}{ax^3 - bx + 2}$ is discontinuous at exactly two points and has exactly two asymptotes parallel to the coordinate axes. What are the values of $a$ and $b$?
(1) $a = 0,\ b = 2$ (2) $a = 8,\ b = 10$
(3) $a = -2,\ b = 0$ (4) $a = -8,\ b = -6$

iran-konkur 2022 Q124 Asymptote Determination View

124-- The locus of the intersection of the asymptotes of the hyperbola $y=\dfrac{ax+3}{(a+1)x+(a-1)}$ is $y=\dfrac{3}{2}x^{2}+x+\dfrac{5}{6}$. The graph of this hyperbola intersects the $x$-axis at which length?
(1) $3$ (2) $-3$ (3) $\dfrac{3}{2}$ (4) $-\dfrac{3}{2}$

iran-konkur 2023 Q16 Continuity and Discontinuity Analysis of Piecewise Functions View

16. For a specific value of $k$, the function $$f(x) = \begin{cases} |x - [-x]| & x \in [x] \text{ even} \\ x - [x] + k & x \in [x] \text{ odd} \end{cases}$$ is continuous at $x = n$ and $x = -n$. Which case is correct regarding $n$ specifically? $(k, n \in \mathbb{N})$

[(1)] $n$ even
[(2)] $n$ odd
[(3)] $f$ is continuous for all values of $n$.
[(4)] $f$ is not continuous for any value of $n$.

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isi-entrance None Q7 Sketching a Curve from Analytical Properties View

Draw the graph (on plain paper) of $f(x) = \min\{|x|-1, |x-1|-1, |x-2|-1\}$.

isi-entrance 2005 Q6 Sketching a Curve from Analytical Properties View

Let $h(x) = \frac{x^4}{(1-x)^4}$ and $g(x) = f(h(x)) = h(x) + 1/h(x)$. Sketch the graph of $g(x)$ and show that $g(x)$ has a root between $0$ and $1$.

isi-entrance 2006 Q8 Exponential Inequality Solving View

Find all values of $c$ for which the equation $\log_2 x = cx$ has solutions.

isi-entrance 2011 Q15 Solve trigonometric equation for solutions in an interval View

The number of solutions of the equation $\sin ( \cos \theta ) = \theta$, $- 1 \leq \theta \leq 1$, is
(a) 0
(b) 1
(c) 2
(d) 3

isi-entrance 2012 Q3 Number of Solutions / Roots via Curve Analysis View

Find the number of intersection points of $y = \log x$ and $y = x^2$.

isi-entrance 2012 Q28 Set Operations Using Inequality-Defined Sets View

Let $A = \{(x,y) : x^4 + y^2 \leq 1\}$ and $B = \{(x,y) : x^6 + y^4 \leq 1\}$. Which of the following is true?
(A) $A = B$
(B) $A \subset B$ (A is a proper subset of B)
(C) $B \subset A$ (B is a proper subset of A)
(D) Neither $A \subset B$ nor $B \subset A$

isi-entrance 2013 Q14 4 marks Monotonicity, symmetry, or parity analysis of a trigonometric function View

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

isi-entrance 2013 Q36 4 marks Accumulation Function Analysis View

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as given by the respective graphs.

isi-entrance 2013 Q70 4 marks Reflection and Image in a Line View

The equation $x^3 y + x y^3 + x y = 0$ represents
(A) a circle
(B) a circle and a pair of straight lines
(C) a rectangular hyperbola
(D) a pair of straight lines

isi-entrance 2015 Q9 4 marks Monotonicity, symmetry, or parity analysis of a trigonometric function View

In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(a) never changes sign
(b) changes sign only once
(c) changes sign more than once, but finitely many times
(d) changes sign infinitely many times.

isi-entrance 2015 Q9 4 marks Monotonicity, symmetry, or parity analysis of a trigonometric function View

In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(a) never changes sign
(b) changes sign only once
(c) changes sign more than once, but finitely many times
(d) changes sign infinitely many times.

isi-entrance 2015 Q25 4 marks Line Equation and Parametric Representation View

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

isi-entrance 2016 Q14 4 marks Continuity and Differentiability of Special Functions View

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

isi-entrance 2017 Q27 Full function study (variation table, limits, asymptotes) View

Let the function $f : [0,1] \rightarrow \mathbb{R}$ be defined as $$f(x) = \max\left\{\frac{|x - y|}{x + y + 1} : 0 \leq y \leq 1\right\} \text{ for } 0 \leq x \leq 1$$ Then which of the following statements is correct?
(A) $f$ is strictly increasing on $\left[0, \frac{1}{2}\right]$ and strictly decreasing on $\left[\frac{1}{2}, 1\right]$.
(B) $f$ is strictly decreasing on $\left[0, \frac{1}{2}\right]$ and strictly increasing on $\left[\frac{1}{2}, 1\right]$.
(C) $f$ is strictly increasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly decreasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.
(D) $f$ is strictly decreasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly increasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.

isi-entrance 2018 Q5 Find critical points and classify extrema of a given function View

Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair?
(A) $( 4,4 )$
(B) $( 3,3 )$
(C) $( 2,2 )$
(D) $( 0,0 )$

isi-entrance 2019 Q2 Sequence of functions convergence View

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be defined by $$f(x) = \lim_{n \rightarrow \infty} \cos^{n}\left(\frac{1}{n^{x}}\right)$$
(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.

isi-entrance 2019 Q25 Locus or solution set characterization of a trigonometric relation View

The locus of points ( $x , y$ ) in the plane satisfying $\sin ^ { 2 } ( x ) + \sin ^ { 2 } ( y ) = 1$ consists of
(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope $\pm 1$.
(D) finitely many lines with slope $\pm 1$.

isi-entrance 2019 Q29 Number of Solutions / Roots via Curve Analysis View

The number of real solutions of the equation $x ^ { 2 } = e ^ { x }$ is:
(A) 0
(B) 1
(C) 2
(D) 3 .

isi-entrance 2019 Q30 Number of Solutions / Roots via Curve Analysis View

The number of distinct real roots of the equation $x \sin ( x ) + \cos ( x ) = x ^ { 2 }$ is
(A) 0
(B) 2
(C) 24
(D) none of the above.

isi-entrance 2020 Q4 Number of Solutions / Roots via Curve Analysis View

The number of real solutions of $e ^ { x } = \sin ( x )$ is
(A) 0
(B) 1
(C) 2
(D) infinite.

isi-entrance 2022 Q30 Optimization Subject to an Algebraic Constraint View

The range of values that the function $$f ( x ) = \frac { x ^ { 2 } + 2 x + 4 } { 2 x ^ { 2 } + 4 x + 9 }$$ takes as $x$ varies over all real numbers in the domain of $f$ is:
(A) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 1 } { 2 }$
(B) $\frac { 3 } { 7 } \leq f ( x ) < \frac { 1 } { 2 }$
(C) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 4 } { 9 }$
(D) $\frac { 3 } { 7 } \leq f ( x ) \leq \frac { 1 } { 2 }$

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12