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Papers (30)
2025
pgmath 18
2024
pgmath 4 ugmath 20
2023
pgmath 12 ugmath 16
2022
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2021
pgmath 11 ugmath 15
2020
pgmath 18 ugmath 16
2019
pgmath 20 ugmath 15
2018
pgmath 4 ugmath 6
2017
ugmath 15
2016
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2015
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2014
ugmath 9
2013
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2012
pgmath 24 ugmath 14
2011
pgmath 15 ugmath 12
2010
pgmath 4 ugmath 19
2011 ugmath

12 maths questions

QA1 3 marks Permutations & Arrangements Word Permutations with Repeated Letters View
The word MATHEMATICS consists of 11 letters. The number of distinct ways to rearrange these letters is
(A) $11 !$
(B) $\frac { 11 ! } { 3 }$
(C) $\frac { 11 ! } { 6 }$
(D) $\frac { 11 ! } { 8 }$
QA2 3 marks Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
In a rectangle ABCD , the length BC is twice the width AB . Pick a point P on side BC such that the lengths of AP and BC are equal. The measure of angle CPD is
(A) $75 ^ { \circ }$
(B) $60 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above
QA3 3 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of $\theta$ with $0 \leq \theta < 2 \pi$ such that $4 \sin ( 3 \theta + 2 ) = 1$ is
(A) 2
(B) 3
(C) 6
(D) none of the above
QA5 3 marks Differentiating Transcendental Functions Determine parameters from function or curve conditions View
A function $f$ is defined by $f ( x ) = e ^ { x }$ if $x < 1$ and $f ( x ) = \log _ { e } ( x ) + a x ^ { 2 } + b x$ if $x \geq 1$. Here $a$ and $b$ are unknown real numbers. Can $f$ be differentiable at $x = 1$ ?
(A) $f$ is not differentiable at $x = 1$ for any $a$ and $b$.
(B) There exist unique numbers $a$ and $b$ for which $f$ is differentiable at $x = 1$.
(C) $f$ is differentiable at $x = 1$ whenever $a + b = e$.
(D) $f$ is differentiable at $x = 1$ regardless of the values of $a$ and $b$.
QA6 3 marks Discriminant and conditions for roots Nature of roots given coefficient constraints View
The equation $x ^ { 2 } + b x + c = 0$ has nonzero real coefficients satisfying $b ^ { 2 } > 4 c$. Moreover, exactly one of $b$ and $c$ is irrational. Consider the solutions $p$ and $q$ of this equation.
(A) Both $p$ and $q$ must be rational.
(B) Both $p$ and $q$ must be irrational.
(C) One of $p$ and $q$ is rational and the other irrational.
(D) We cannot conclude anything about rationality of $p$ and $q$ unless we know $b$ and $c$.
QA7 3 marks Factor & Remainder Theorem Divisibility and Factor Determination View
When does the polynomial $1 + x + \cdots + x ^ { n }$ have $x - a$ as a factor? Here $n$ is a positive integer greater than 1000 and $a$ is a real number.
(A) if and only if $a = - 1$
(B) if and only if $a = - 1$ and $n$ is odd
(C) if and only if $a = - 1$ and $n$ is even
(D) We cannot decide unless $n$ is known.
QB1 7 marks Permutations & Arrangements Handshake / Product Counting View
In a business meeting, each person shakes hands with each other person, with the exception of Mr. L. Since Mr. L arrives after some people have left, he shakes hands only with those present. If the total number of handshakes is exactly 100 , how many people left the meeting before Mr. L arrived? (Nobody shakes hands with the same person more than once.)
QB2 7 marks Binomial Theorem (positive integer n) Find the Largest Term or Coefficient in a Binomial Expansion View
Show that the power of $x$ with the largest coefficient in the polynomial $\left( 1 + \frac { 2 x } { 3 } \right) ^ { 20 }$ is 8 , i.e., if we write the given polynomial as $\sum _ { i } a _ { i } x ^ { i }$ then the largest coefficient $a _ { i }$ is $a _ { 8 }$.
QB4 7 marks Permutations & Arrangements Forming Numbers with Digit Constraints View
Let S be the set of all 5-digit numbers that contain the digits $1,3,5,7$ and 9 exactly once (in usual base 10 representation). Show that the sum of all elements of S is divisible by 11111. Find this sum.
QB5 7 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
It is given that the complex number $i - 3$ is a root of the polynomial $3 x ^ { 4 } + 10 x ^ { 3 } + A x ^ { 2 } + B x - 30$, where $A$ and $B$ are unknown real numbers. Find the other roots.
QB7 7 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
To find the volume of a cave, we fit $\mathrm { X } , \mathrm { Y }$ and Z axes such that the base of the cave is in the XY-plane and the vertical direction is parallel to the Z-axis. The base is the region in the XY-plane bounded by the parabola $y ^ { 2 } = 1 - x$ and the Y-axis. Each cross-section of the cave perpendicular to the X-axis is a square.
(a) Show how to write a definite integral that will calculate the volume of this cave.
(b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals?
QB8 7 marks Stationary points and optimisation Count or characterize roots using extremum values View
$f ( x ) = x ^ { 3 } + x ^ { 2 } + c x + d$, where $c$ and $d$ are real numbers. Prove that if $c > \frac { 1 } { 3 }$, then $f$ has exactly one real root.