isi-entrance

Q1 4 marks Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

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 marks Permutations & Arrangements Forming Numbers with Digit Constraints View

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

Q3 4 marks Number Theory Properties of Integer Sequences and Digit Analysis View

The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2

Q4 4 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View

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) $\binom { 6 } { 3 } 3 !$
(C) 33
(D) $3 \binom { 6 } { 3 }$

Q5 4 marks Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View

Let $A B C D$ be a unit square. Four points $E , F , G$ and $H$ are chosen on the sides $A B , B C , C D$ and $D A$ respectively. The lengths of the sides of the quadrilateral $E F G H$ are $\alpha , \beta , \gamma$ and $\delta$. Which of the following is always true?
(A) $1 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 \sqrt { 2 }$
(B) $2 \sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4 \sqrt { 2 }$
(C) $2 \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 4$
(D) $\sqrt { 2 } \leq \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } \leq 2 + \sqrt { 2 }$

Q6 4 marks Laws of Logarithms Solve a Logarithmic Equation View

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

Q7 4 marks Complex Numbers Arithmetic Modulus Computation View

$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is
(A) real and negative
(B) real and positive
(C) purely imaginary
(D) none of the above need to be true always

Q8 4 marks Inequalities Solve Polynomial/Rational Inequality for Solution Set View

The set of all real numbers $x$ satisfying the inequality $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is
(A) the interval $[ 2 , \infty )$
(B) the interval $[ 0 , \infty )$
(C) the interval $[ - 1 , \infty )$
(D) none of the above

Q9 4 marks Stationary points and optimisation Geometric or applied optimisation problem View

The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is
(A) $a b$
(B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$
(C) $\frac { ( a + b ) ^ { 2 } } { 2 }$
(D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$

Q10 4 marks Straight Lines & Coordinate Geometry Locus Determination View

Let $A$ be the fixed point $(0,4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $A B$ and let the perpendicular bisector of $A B$ meet the $y$-axis at $R$. The locus of the mid-point $P$ of $M R$ is
(A) $y + x ^ { 2 } = 2$
(B) $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 1 / 4$
(C) $( y - 2 ) ^ { 2 } - x ^ { 2 } = 1 / 4$
(D) none of the above

Q11 4 marks Sine and Cosine Rules Find an angle using the cosine rule View

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

Q12 4 marks Sine and Cosine Rules Heights and distances / angle of elevation problem View

Two poles, $A B$ of length two metres and $C D$ of length twenty metres are erected vertically with bases at $B$ and $D$. The two poles are at a distance not less than twenty metres. It is observed that $\tan \angle A C B = 2 / 77$. The distance between the two poles is
(A) $72 m$
(B) 68 m
(C) 24 m
(D) 24.27 m

Q13 4 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

Q14 4 marks Trig Graphs & Exact Values 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

Q15 4 marks Taylor series Limit evaluation using series expansion or exponential asymptotics View

The limit $$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$$ (A) does not exist
(B) exists and equals 0
(C) exists and equals $2 / 3$
(D) exists and equals 1

Q16 4 marks Chain Rule Iterated/Nested Exponential Differentiation View

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 equal to
(A) $f _ { n } ( x )$
(B) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$
(D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$

Q17 4 marks Differentiating Transcendental Functions Determine parameters from function or curve conditions View

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

Q18 4 marks Stationary points and optimisation Geometric or applied optimisation problem View

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

Q19 4 marks Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View

If $b = \int _ { 0 } ^ { 1 } \frac { e ^ { t } } { t + 1 } d t$ then $\int _ { a - 1 } ^ { a } \frac { e ^ { - t } } { t - a - 1 } d t$ is
(A) $b e ^ { a }$
(B) $b e ^ { - a }$
(C) $- b e ^ { - a }$
(D) $- b e ^ { a }$

Q20 4 marks Trigonometric equations in context View

In the triangle $A B C$, the angle $\angle B A C$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $A B C$ is
(A) obtuse angled
(B) right angled
(C) acute angled but not equilateral
(D) equilateral

Q21 4 marks Circles Area and Geometric Measurement Involving Circles View

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4 n ^ { 2 }$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4 n - 2$
(B) $4 n$
(C) $8 n - 4$
(D) $8 n - 2$

Q22 4 marks Connected Rates of Change Shadow Rate of Change Problem View

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 ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

Q23 4 marks Circles Circles Tangent to Each Other or to Axes View

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3 \sqrt { 3 } / 2$
(B) $\sqrt { 3 } / 2$
(C) $\sqrt { 3 }$
(D) $4 / \sqrt { 3 }$

Q24 4 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

Q25 4 marks Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1, 2, \ldots , n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

Q26 4 marks Combinations & Selection Counting Functions or Mappings with Constraints View

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

Q27 4 marks Combinations & Selection Counting Functions or Mappings with Constraints View

Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n _ { i }$, the label of the ball drawn from the $i$-th box, $i = 1, 2, 3$. Then the number of ways in which the balls can be chosen such that $n _ { 1 } < n _ { 2 } < n _ { 3 }$ is
(A) 120
(B) 130
(C) 150
(D) 160

Q28 4 marks Simultaneous equations View

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

Q29 4 marks Stationary points and optimisation Geometric or applied optimisation problem View

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e ^ { - x }$ is:
(A) $1 / e$
(B) 1
(C) $1 / 2$
(D) $e$

Q30 4 marks Number Theory Prime Number Properties and Identification View

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a ^ { n } + b ^ { n }$ is prime, what can you say about $n$?
(A) The integer $n$ must be 2
(B) The integer $n$ need not be 2, but must be a power of 2
(C) The integer $n$ need not be a power of 2, but must be even
(D) None of the above is necessarily true

Q31 4 marks Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

Q32 4 marks Integration by Substitution Substitution Combined with Symmetry or Companion Integral View

The value of the integral $$\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$$ equals
(A) 1
(B) $\pi$
(C) $e$
(D) none of these

Q33 4 marks Quadratic trigonometric equations View

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer
(C) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer
(D) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer

Q34 4 marks Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View

The number $$\left( \frac { 2 ^ { 10 } } { 11 } \right) ^ { 11 }$$ is
(A) strictly larger than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$
(B) strictly larger than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 }$ but strictly smaller than $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$
(C) less than or equal to $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 }$
(D) equal to $\binom { 10 } { 1 } ^ { 2 } \binom { 10 } { 2 } ^ { 2 } \binom { 10 } { 3 } ^ { 2 } \binom { 10 } { 4 } ^ { 2 } \binom { 10 } { 5 }$

Q35 4 marks Reciprocal Trig & Identities View

The value of $$\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$$ is
(A) 0
(B) $\pi / 6$
(C) $\pi / 4$
(D) $\pi / 2$

Q36 4 marks Differentiating Transcendental Functions Graphical identification of function or derivative 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 shown in the graphs.

Q37 4 marks Taylor series Limit evaluation using series expansion or exponential asymptotics View

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(A) 0
(B) 1
(C) $e$
(D) $\sqrt { e } / 2$

Q38 4 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

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) $\alpha > 2$

Q39 4 marks Circles Chord Length and Chord Properties View

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(A) $36 ^ { \circ }$
(B) $29 ^ { \circ }$
(C) $7 ^ { \circ }$
(D) $25 ^ { \circ }$

Q40 4 marks Combinations & Selection Selection with Group/Category Constraints View

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

Q41 4 marks Conic sections Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(A) 2
(B) $1 / 2$
(C) 4
(D) $1 / 4$

Q42 4 marks Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ represents, in the complex plane,
(A) a straight line
(B) a pair of intersecting straight lines
(C) a pair of distinct parallel straight lines
(D) a point

Q43 4 marks Number Theory Combinatorial Number Theory and Counting View

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

Q44 4 marks Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above

Q45 4 marks Standard trigonometric equations Inverse trigonometric equation View

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(A) 1
(B) 2
(C) 3
(D) 5

Q46 4 marks Sine and Cosine Rules Multi-step composite figure problem View

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(A) $75 ^ { \circ }$
(B) $85 ^ { \circ }$
(C) $95 ^ { \circ }$
(D) $110 ^ { \circ }$

Q47 4 marks Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto

Q48 4 marks Straight Lines & Coordinate Geometry Locus Determination View

Let $L$ be the point $(t, 2)$ and $M$ be a point on the $y$-axis such that $L M$ has slope $-t$. Then the locus of the midpoint of $L M$, as $t$ varies over all real values, is
(A) $y = 2 + 2 x ^ { 2 }$
(B) $y = 1 + x ^ { 2 }$
(C) $y = 2 - 2 x ^ { 2 }$
(D) $y = 1 - x ^ { 2 }$

Q49 4 marks Applied differentiation MCQ on derivative and graph interpretation View

Let $f : ( 0, 2 ) \cup ( 4, 6 ) \rightarrow \mathbb { R }$ be a differentiable function. Suppose also that $f ^ { \prime \prime } ( x ) = 1$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$. Which of the following is ALWAYS true?
(A) $f$ is increasing
(B) $f$ is one-to-one
(C) $f ( x ) = x$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$
(D) $f ( 5.5 ) - f ( 4.5 ) = f ( 1.5 ) - f ( 0.5 )$

Q50 4 marks Circles Circle-Related Locus Problems View

A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is
(A) a circle whose centre is the midpoint of $B C$
(B) a circle whose centre is on the line $B C$ but not the midpoint of $B C$
(C) a straight line
(D) none of the above

Q51 4 marks Circles Circle-Related Locus Problems View

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

Q52 4 marks Number Theory Modular Arithmetic Computation View

$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(A) 1
(B) 3
(C) 7
(D) 9

Q53 4 marks Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac { 3 } { 4 }$
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) $\frac { 2 } { 3 }$

Q54 4 marks Sequences and Series Estimation or Bounding of a Sum View

For any $n \geq 5$, the value of $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \cdots + \frac { 1 } { 2 ^ { n } - 1 }$ lies between
(A) 0 and $\frac { n } { 2 }$
(B) $\frac { n } { 2 }$ and $n$
(C) $n$ and $2 n$
(D) none of the above

Q55 4 marks Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View

Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is
(A) 4
(B) 5
(C) 7
(D) infinite

Q56 4 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $( 2, 3 )$
(D) is greater than 3

Q57 4 marks Areas Between Curves Select Correct Integral Expression View

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

Q58 4 marks Probability Definitions Combinatorial Counting (Non-Probability) View

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players A, B, C, D, E and F play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, B has 8 points and C has 4 points. It is also known that E won against F. In the next set of games D, E and F win their games against A, B and C respectively. If A, B and D move to the final round, the final scores of E and F are, respectively,
(A) 4 and 2
(B) 2 and 4
(C) 2 and 2
(D) 4 and 4

Q59 4 marks Combinations & Selection Selection with Adjacency or Spacing Constraints View

The number of ways in which one can select six distinct integers from the set $\{ 1, 2, 3, \cdots, 49 \}$, such that no two consecutive integers are selected, is
(A) $\binom { 49 } { 6 } - 5 \binom { 48 } { 5 }$
(B) $\binom { 43 } { 6 }$
(C) $\binom { 25 } { 6 }$
(D) $\binom { 44 } { 6 }$

Q60 4 marks Circles Circles Tangent to Each Other or to Axes View

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) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$

Q61 4 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View

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

Q62 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements View

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

Q63 4 marks Modulus function Differentiability of functions involving modulus View

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.
(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 $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ Then
(C) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(D) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$

Q64 4 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

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 $- \infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f ( x )$ is decreasing on $- \infty < x \leq 0$ and increasing on $0 \leq x < \infty$

Q65 4 marks Quadratic trigonometric equations View

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

Q66 4 marks Discriminant and conditions for roots Parameter 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) $0 \leq m < \frac { 1 } { 4 }$

Q67 4 marks Number Theory Properties of Integer Sequences and Digit Analysis View

The digit in the unit's place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(A) 3
(B) 0
(C) 1
(D) 7

Q68 4 marks Sequences and Series Limit Evaluation Involving Sequences View

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

Q69 4 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View

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 $n$

Q70 4 marks Straight Lines & Coordinate Geometry 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

Q71 4 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

For each positive integer $n$, define a function $f _ { n }$ on $[ 0, 1 ]$ as follows: $$f _ { n } ( x ) = \left\{ \begin{array} { c c c } 0 & \text { if } & x = 0 \\ \sin \frac { \pi } { 2 n } & \text { if } & 0 < x \leq \frac { 1 } { n } \\ \sin \frac { 2 \pi } { 2 n } & \text { if } & \frac { 1 } { n } < x \leq \frac { 2 } { n } \\ \sin \frac { 3 \pi } { 2 n } & \text { if } & \frac { 2 } { n } < x \leq \frac { 3 } { n } \\ \vdots & \vdots & \vdots \\ \sin \frac { n \pi } { 2 n } & \text { if } & \frac { n - 1 } { n } < x \leq 1 \end{array} \right.$$ Then, the value of $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { 1 } f _ { n } ( x ) d x$ is
(A) $\pi$
(B) 1
(C) $\frac { 1 } { \pi }$
(D) $\frac { 2 } { \pi }$

Q72 4 marks Number Theory Arithmetic Functions and Multiplicative Number Theory View

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

Q73 4 marks Groups Ring and Field Structure View

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

isi-entrance

Papers (32)

2016 UGB