QB1
Permutations & ArrangementsForming Numbers with Digit ConstraintsView
Find the sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4,5$, each digit appearing at most once.
QB2
ProofDirect Proof of a Stated Identity or EqualityView
Consider the squares of an $8 \times 8$ chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260.
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QB3
CirclesCircles Tangent to Each Other or to AxesView
An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.
QB4
Number TheoryDivisibility and Divisor AnalysisView
Let $a _ { n } = 1 \ldots 1$ with $3 ^ { n }$ digits. Prove that $a _ { n }$ is divisible by $3 a _ { n - 1 }$.
QB5
CirclesIntersection of Circles or Circle with ConicView
If a circle intersects the hyperbola $y = 1 / x$ at four distinct points $\left( x _ { i } , y _ { i } \right) , i = 1,2,3,4$, then prove that $x _ { 1 } x _ { 2 } = y _ { 3 } y _ { 4 }$.
QB6
Stationary points and optimisationFind critical points and classify extrema of a given functionView
Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.
QB7
Combinations & SelectionSubset Counting with Set-Theoretic ConditionsView
Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.
QB8
Stationary points and optimisationFind absolute extrema on a closed interval or domainView
Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.
QB9
Curve SketchingNumber of Solutions / Roots via Curve AnalysisView
How many real roots does $x ^ { 4 } + 12 x - 5$ have?
QB10
Stationary points and optimisationFind critical points and classify extrema of a given functionView
Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.
QB11
Modulus functionProof involving modulus in iterative or analytic settingView
For real numbers $x , y$ and $z$, show that $$| x | + | y | + | z | \leq | x + y - z | + | y + z - x | + | z + x - y |$$
Q1
4 marksSequences and SeriesProof of Inequalities Involving Series or Sequence TermsView
Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then, (a) $a _ { n } < b _ { n }$ for all $n > 1$ (b) $a _ { n } > b _ { n }$ for all $n > 1$ (c) $a _ { n } = b _ { n }$ for infinitely many $n$ (d) none of the above.
Q2
4 marksNumber TheoryProperties of Integer Sequences and Digit AnalysisView
The last digit of $( 2004 ) ^ { 5 }$ is: (a) 4 (b) 8 (c) 6 (d) 2
Q3
4 marksNumber TheoryQuadratic Diophantine Equations and Perfect SquaresView
If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then (a) $n$ must be odd (b) $n$ cannot be a perfect square (c) $2 n$ cannot be a perfect square (d) none of the above.
Q4
4 marksBinomial Theorem (positive integer n)Find a Specific Coefficient in a Single Binomial ExpansionView
The coefficient of $a ^ { 3 } b ^ { 4 } c ^ { 5 }$ in the expansion of $( b c + c a + a b ) ^ { 6 }$ is: (a) $\frac { 12 ! } { 3 ! 4 ! 5 ! }$ (b) $\frac { 6 ! } { 3 ! }$ (c) 33 (d) $3 \cdot \left( \frac { 6 ! } { 3 ! 3 ! } \right)$
Q5
4 marksLaws of LogarithmsSolve a Logarithmic EquationView
If $\log _ { 10 } x = 10 ^ { \log _ { 100 } 4 }$, then $x$ equals (a) $4 ^ { 10 }$ (b) 100 (c) $\log _ { 10 } 4$ (d) none of the above.
Q6
4 marksComplex Numbers ArithmeticTrue/False or Property Verification StatementsView
Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned}
& A = \{ ( z ,w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\
& B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\}
\end{aligned}$$ Then, (a) $A = B$ (b) $A \subset B$ and $A \neq B$ (c) $B \subset A$ and $B \neq A$ (d) none of the above.
Q7
4 marksInequalitiesSolve Polynomial/Rational Inequality for Solution SetView
The set of all real numbers $x$ such that $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is: (a) the interval $2 \leq x < \infty$ (b) the interval $0 \leq x < \infty$ (c) the interval $- 1 \leq x < \infty$ (d) none of the above.
Q8
4 marksComplex Numbers ArithmeticIdentifying Real/Imaginary Parts or ComponentsView
Let $z$ be a non-zero complex number such that $\frac { z } { 1 + z }$ is purely imaginary. Then (a) $z$ is neither real nor purely imaginary (b) $z$ is real (c) $z$ is purely imaginary (d) none of the above.
Q9
4 marksStandard trigonometric equationsMonotonicity, symmetry, or parity analysis of a trigonometric functionView
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.
Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is: (a) $f _ { n } ( x )$ (b) $f _ { n } ( x ) f _ { n - 1 } ( x )$ (c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$ (d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.
Q12
4 marksModulus functionDifferentiability of functions involving modulusView
Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then (a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$ (d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.
Q13
4 marksStationary points and optimisationDetermine intervals of increase/decrease or monotonicity conditionsView
If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then (a) $f ( x )$ is an increasing function on the real line (b) $f ( x )$ is a decreasing function on the real line (c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$ (d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.
Q14
4 marksAreas Between CurvesSelect Correct Integral ExpressionView
The area of the region bounded by the straight lines $x = \frac { 1 } { 2 }$ and $x = 2$, and the curves given by the equations $y = \log _ { e } x$ and $y = 2 ^ { x }$ is (a) $\frac { 1 } { \log _ { e } 2 } ( 4 + \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$ (b) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2$ (c) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$ (d) none of the above.
The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is (a) 0 (b) 1 (c) 2 (d) 3
Q16
4 marksCombinations & SelectionCounting Functions or Mappings with ConstraintsView
The number of maps $f$ from the set $\{ 1,2,3 \}$ into the set $\{ 1,2,3,4,5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is (a) 60 (b) 50 (c) 35 (d) 30
Q17
4 marksCirclesIntersection of Circles or Circle with ConicView
Let $a$ be a real number. The number of distinct solutions $( x , y )$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be (a) $0,1,2,3,4$ or 5 (b) 0, 1 or 3 (c) 0, 1, 2 or 4 (d) $0,2,3$, or 4
Q18
4 marksDiscriminant and conditions for rootsParameter range for no real roots (positive definite)View
The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is (a) $m < \frac { 1 } { 4 }$ (b) $m \geq 0$ (c) $0 \leq m \leq \frac { 1 } { 4 }$ (d) $0 \leq m < \frac { 1 } { 4 }$.
Q19
4 marksConnected Rates of ChangeShadow Rate of Change ProblemView
A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is (a) $3.6 \mathrm { ft } . / \mathrm { sec }$. (b) $2.4 \mathrm { ft } . / \mathrm { sec }$. (c) $3 \mathrm { ft } . / \mathrm { sec }$. (d) $12 \mathrm { ft } . / \mathrm { sec }$.
Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is: (a) $r \operatorname { cosec } \frac { \pi } { n }$ (b) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$ (c) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$ (d) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.
Q21
4 marksNumber TheoryProperties of Integer Sequences and Digit AnalysisView
The digit in the units' place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is (a) 3 (b) 0 (c) 1 (d) 7.
Q22
4 marksSequences and SeriesLimit Evaluation Involving SequencesView
The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is: (a) $\frac { 3 } { 4 }$ (b) $\frac { 1 } { 4 }$ (c) 1 (d) 4.
Q23
4 marksStationary points and optimisationDetermine intervals of increase/decrease or monotonicity conditionsView
The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if (a) $\alpha \geq 2$ (b) $\alpha < 2$ (c) $\alpha < - 1$ (d) $\alpha > 2$.
Q24
4 marksSequences and series, recurrence and convergenceMultiple-choice on sequence propertiesView
For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$ (a) does not have a maximum (b) attains maximum at exactly one value of $n$ (c) attains maximum at exactly two values of $n$ (d) attains maximum for infinitely many values of $n$.
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.
Q26
4 marksCirclesCircle-Related Locus ProblemsView
Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is (a) a circle (b) an ellipse (c) a line segment (d) segment of a parabola.
Q27
4 marksNumber TheoryArithmetic Functions and Multiplicative Number TheoryView
Let $d _ { 1 } , d _ { 2 } , \ldots , d _ { k }$ be all the factors of a positive integer $n$ including 1 and $n$. If $d _ { 1 } + d _ { 2 } + \ldots + d _ { k } = 72$, then $\frac { 1 } { d _ { 1 } } + \frac { 1 } { d _ { 2 } } + \cdots + \frac { 1 } { d _ { k } }$ is: (a) $\frac { k ^ { 2 } } { 72 }$ (b) $\frac { 72 } { k }$ (c) $\frac { 72 } { n }$ (d) none of the above.
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then: (a) neither $S$ nor $T$ is a ring (b) $S$ is a ring, $T$ is not a ring. (c) $T$ is a ring, $S$ is not a ring. (d) both $S$ and $T$ are rings.