Sample prmo-questions
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This document contains 10 sample questions for a PRMO exam with the answers provided. The questions cover a range of mathematical topics including: algebra, polynomials, number theory, geometry, and trigonometry. The answers are numerical values ranging from 00 to 84.
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![Sample Questions for PRMO 2017
1. Two positive integers a and b are such that a + b =
a
b
+
b
a
. What is the value of a2
+ b2
?
[Ans: 02]
2. The equations x2
− 4x + k = 0 and x2
+ kx − 4 = 0, where k is a real number, have exactly
one common root. What is the value of k? [Ans: 03]
3. Let P(x) be a non-zero polynomial with integer coefficients. If P(n) is divisible by n for each
positive integer n, what is the value of P(0)? [Ans: 00]
4. A natural number k is such that k2
< 2014 < (k + 1)2
. What is the largest prime factor of
k? [Ans: 11]
5. How many two-digit positive integers N have the property that the sum of N and the number
obtained reversing the order of the digits of N is a perfect square? [Ans: 08]
6. What is the greatest possible perimeter of a right-angled triangle with integer side lengths if
one of the sides has length 12? [Ans: 84]
7. In rectangle ABCD, AB = 8 and BC = 20. LetP be a point on AD such that ∠BPC = 90◦
.
If r1, r2, r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value
of r1 + r2 + r3? [Ans: 08]
8. Let n be the largest integer that is the product of exactly 3 distinct prime numbers, x, y and
10x + y, where x and y are digits. What is the sum of the digits of n? [Ans: 12]
9. A subset B of the set of first 100 positive integers has the property that no two elements of
B sum to 125. What is the maximum possible number of elements in B? [Ans: 62]
10. The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY
be a diameter of Ω which is also tangent to ω. Assume PY > PX. Let PY intersect ω at
Z. If Y Z = 2PZ ,what is the magnitude of ∠PY X in degrees? [Ans: 15]
1](https://image.slidesharecdn.com/sample-prmo-questions-170803062543/85/Sample-prmo-questions-1-320.jpg)
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This document contains two worksheets with mathematics problems involving concepts from chapters 9, 10, and 11 including: properties of arithmetic progressions; finding sums of series; equations of lines, circles, ellipses; and other geometric relationships. The Level-1 worksheet contains 8 problems involving finding terms of arithmetic progressions, summing series, finding equations of lines and circles, and calculating heights of arches. The Level-2 worksheet contains 8 more challenging problems involving finding sums of arithmetic progressions using given conditions, properties of geometric progressions, calculating postage costs in a mailing chain, relating lines and finding points of intersection, identifying concyclic points, and finding equations of concentric circles.
Class xii probability
This document provides examples of probability problems involving events such as drawing cards or balls from boxes/bags with varying compositions, rolling dice, coin tosses, and other scenarios. The problems cover concepts like probability distributions, means, variances, conditional probabilities, and solving for probabilities of individual or combined events.
Assignment of straight lines and conic section
1. The document contains 35 problems related to straight lines and conic sections like circles, ellipses, parabolas and hyperbolas. The problems involve finding equations of lines and conic sections given certain properties, finding properties like foci, vertices, axes, etc. from given equations.
2. Specific problems include finding slopes of lines, angle between lines, perpendicular lines, finding equations of lines passing through points or perpendicular/parallel to other lines.
3. For conic sections, problems include finding equations given foci, directrix, passing points, intersecting lines, centers and radii of circles, lengths of axes, foci, latus rectum and eccentricities of ellipses and hyperbol
Three dimensional geometry
This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. Specifically, it asks the reader to: 1) Find direction ratios and determine perpendicularity between lines; 2) Write equations of planes passing through given points; and 3) Calculate lengths and points of intersection between lines and planes.
Class xii assignment chapters (1 to 9)
1. Find the equation of tangent to the curve x = sin 3t, y = cos 2t at t = π/4.
2. Solve the differential equation: cos2x dy/dx + y = tan x.
3. Using integration, find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.
4. Evaluate several integrals using properties of integrals, derivatives, and determinants.
Mcq exemplar class 12
The document contains 30 multiple choice questions related to various mathematical concepts such as relations, functions, matrices, determinants, and trigonometric functions. The questions test knowledge of properties such as equivalence, injectivity, surjectivity, inverse functions, composition of functions, matrix operations, and evaluating determinants.
Mcq exemplar class 12
The document contains 30 multiple choice questions related to various mathematical concepts such as relations, functions, matrices, determinants, and trigonometric functions. The questions test knowledge of properties such as equivalence, injectivity, surjectivity, inverse functions, composition of functions, matrix operations, and evaluating determinants.
Application of integrals
This document provides examples of calculating areas of 2D regions bounded by curves, lines, and other geometric shapes using integration. There are 29 problems involving finding the area of regions bounded by parabolas, circles, ellipses, lines, and other curves through integration. Many problems provide the area as the solution or ask the reader to evaluate the integral to find the area of the region.
Assignment chapters 3 to 7
1. This document contains mathematics questions on matrices and determinants ranging from 1 to 6 marks. Questions involve finding determinants, inverses, solving systems of equations using matrices, properties of determinants, and continuity and differentiability.
2. The document is divided into sections on 1, 2, 4 and 6 mark questions related to matrices and determinants as well as questions on continuity and differentiability ranging from 2 to 4 marks.
3. The questions cover a wide range of matrix and calculus concepts and techniques including finding determinants, inverses, solving systems of equations, properties of determinants, continuity, differentiation, mean value theorem, Rolle's theorem and related calculus topics.
Surface chemistry notes
1. Adsorption is a surface phenomenon where adsorbate molecules accumulate on the surface of an adsorbent. It can occur due to unbalanced forces during crystallization, unpaired electrons, or free valencies in solids.
2. There are two types of adsorption - physical adsorption which occurs via weak van der Waals forces and chemisorption which involves chemical bonding. Physical adsorption is reversible and can form multilayers while chemisorption forms a unimolecular layer through stronger bonds.
3. Catalysis involves a substance that increases the rate of a chemical reaction without being consumed. Catalysts function by providing an alternative reaction pathway with lower activation energy through mechanisms
Assignment of solution
1. Henry's law relates the pressure of a gas in a solution to its solubility. It has applications.
2. Glucose solution will freeze at a specific temperature, calculated using information provided.
3. Raoult's law describes how volatile liquid solutions deviate from ideal behavior. Positive and negative deviations are explained with examples.
Class X1 assignment of chapters 1,3,4,5
This document provides 25 math problems across various topics for students to complete as part of their class assignment. The problems cover topics like sets, trigonometry, complex numbers, and polar forms. Students are asked to perform tasks like writing sets in set-builder form, finding intersections and unions of sets, solving trigonometric equations, working with complex numbers, and evaluating expressions in polar form.
Continuity and differentiability
This document provides 25 questions related to continuity, differentiability, and derivatives. Some key questions include:
- Finding the value of k for which a function f(x) is continuous at x=0
- Finding values of k or a for which a piecewise defined function is continuous at a point
- Taking derivatives of various trigonometric, logarithmic, inverse trigonometric and implicit functions
- Simplifying derivative expressions and proving identities about second derivatives
The questions cover a range of fundamental calculus topics including limits, continuity, differentiation rules, implicit differentiation and higher order derivatives.
Notes of Matrices and Determinants
This document covers matrices, determinants, and their properties. Some key points:
- A matrix is a rectangular arrangement of numbers. The order of a m×n matrix is defined by its m rows and n columns.
- A square matrix is symmetric if it is equal to its transpose, and skew-symmetric if it is equal to the negative of its transpose.
- The determinant of a square matrix provides a number associated with the matrix, denoted |A| or detA.
- Properties of determinants include that the value does not change under row/column interchange operations, and that multiplying a row/column by a constant k multiplies the determinant by k.
Class xii worksheet (chapters 2,5,6)
1) Show that the derivative of the function xy = ex-y is log(x)/(log(xe))^2
2) Calculate the rate of increase of the height of a cone formed by sand pouring from a pipe when the height is 4cm.
3) Find the equation of the tangent line to the curve y = √3x - 2 that is parallel to the line 4x - 2y + 5 = 0.
4) Prove the identity tan-1(1) + tan-1(2) + tan-1(3) = π.
Three dimensional geometry
This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. It covers topics such as direction cosines, perpendicular lines and planes, intersections of lines and planes, and vector and Cartesian representations of geometric entities in three dimensions.
Differential equation
This document lists 7 problems involving differential equations: 1) Find the order and degree of a given differential equation. 2) Find the differential equation of a family of lines passing through the origin. 3) Form the differential equation of a family of circles with centers on the y-axis and radius of 3 units. 4) Find the general solution of a given differential equation. 5) Solve a given differential equation involving radicals. 6) Show that a given differential equation is homogeneous and find its particular solution. 7) Solve a given differential equation involving radicals.
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