Try this beautiful problem from ISI BStat 2017 Subjective 2 based on right-angled triangle in a circle. Understand, solve, and learn.
Try this beautiful problem from ISI BStat 2017 Subjective 2 based on right-angled triangle in a circle. Understand, solve, and learn.
Problem: Let the sequence \( { a_n} _{n \ge 1 } \) be defined by \(a_n = \tan n \theta \) where \( \tan \theta = 2 \). Show that for all n \( a_n \) is a rational number which can be written with an odd denominator. Discussion: This is simple induction. The claim […]
Here, you will find all the questions of ISI Entrance Paper 2017 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1 : Let the sequence \( \{ a_n\} _{n \ge 1 } \) be defined by $$ a_n = \tan n \theta $$ […]
Here are the problems and their corresponding solutions from BStat Hons Objective Admission Test 2005. Try it yourself and then read the solutions.
Here, you will find all the questions of ISI Entrance Paper 2016 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems. Problem 1: In a sports tournament of $n$ players, each pair of players plays exactly one match against each other. There are no draws. […]
Try this problem from TOMATO Objective 288, useful for ISI BStat, BMath Entrance Exam based on finding big remainder in a small way. Problem: Tomato objective 288 The remainder R(x) obtained by dividing the polynomial [latex]x^{100}[/latex] by the polynomial [latex]x^2-3x+2[/latex] is (A) [latex]2^{100}-1[/latex] (B) [latex](2^{100}-1)x-(2^{99}-1)[/latex] (C) [latex]2^{100}x-3(2^{100})[/latex] (D) [latex](2^{100}-1)x+(2^{99}-1)[/latex] SOLUTION: (B) The the divisor is […]
Problem: If the roots of the equation ${(x-a)(x-b)}$+${(x-b)(x-c)}$+${(x-c)(x-a)}$=$0$, (where a,b,c are real numbers) are equal , then (A) $b^2-4ac=0$ (B) $a=b=c$ (C) a+b+c=0 (D) none of foregoing statements is correct Answer: $(B)$ ${(x-a)(x-b)}$+${(x-b)(x-c)}$+${(x-c)(x-a)}$=$0$ => $x^2-{(a+b)}x$+$ab+x^2-{(b+c)}x$+$bc+x^2-{(c+a)}x+ca$=$0$ => $3x^2-2{(a+b+c)}x$+$(ab+bc+ca)$=$0$ discriminant, of the equation is => $4{(a+b+c)^2}$-$4.3{(ab+bc+ca)}$=$0$ => $a^2+b^2+c^2+2(ab+bc+ca)$-$3(ab+bc+ca)$=$0$ => $a^2+b^2+c^2$-$(ab+bc+ca)$=$0$ => $a=b=c$ So, option (B) is correct.
Try this beautiful problem from TOMATO Objective no. 258 based on Real Roots of a Cubic Polynomial. Problem: Real Roots of a Cubic Polynomial Let a,b,c be distinct real numbers. Then the number of real solution of [latex](x-a)^3+(x-b)^3+(x-c)^3=0[/latex] is (A) 1 (B) 2 (C) 3 (D) depends on a,b,c Solution: Ans: (A) Let [latex]f(x)=(x-a)^3+(x-b)^3+(x-c)^3[/latex] [latex]=> f'(x)=3(x-a)^2+3(x-b)^2+3(x-c)^2=0[/latex] [latex]=> […]
Try this beautiful problem from TOMATO Objective no. 257 based on Roots of a Quintic Polynomial. Problem: Roots of a Quintic Polynomial The number of real roots of [latex] x^5+2x^3+x^2+2=0[/latex] is (A) 0 (B) 3 (C) 5 (D) 1 Solution: Answer: (D) [latex] x^5+2x^3+x^2+2=0[/latex] [latex] \implies x^3(x^2+2)+(x^2+2)=0[/latex] [latex] \implies (x^3+1)(x^2+2)=0[/latex] [latex] \implies (x+1)\bold{\underline{(x^2-x+1)(x^2+2)}}=0[/latex] The expression in underline doesn't have any […]
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Head Tail Problem.
Try this beautiful problem from Geometry:Radius of a circle.AMC-10A, 2003. You may use sequential hints to solve the problem
Try this beautiful problem from algebra, based on Sum of the digits from AMC-10A, 2007. You may use sequential hints to solve the problem
Try this beautiful problem from Geometry based on medians of triangle from PRMO 2018. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry based on Hexagon from AMC-10A, 2014. You may use sequential hints to solve the problem
Try this beautiful sum of Co-ordinates based on co-ordinate Geometry from AMC-10A, 2014. You may use sequential hints to solve the problem.
Try this beautiful problem from Singapore Mathematical Olympiad, SMO, 2010 - Problem 7 based on the combination of equations.
Try this beautiful problem from PRMO, 2019, problem-17, based on Largest Possible Value Problem. You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2019 based on the Diameter of a circle. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based on positive integers from PRMO 2019. You may use sequential hints to solve the problem.