Mathematics questions
18011 published questions with worked solutions.
- Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0)=p, p 0, and f(1)= 13. If the equations f(x)=0 and f...
- Let R and let the equation E be |x|^2-2|x|+| -3|=0. Then the largest element in the set S= x+ :x is an integer solution of E is
- Let m and n be the numbers of real roots of the quadratic equations x^2-12x+[x]+31=0 and x^2-5|x+2|-4=0 respectively, where [x]...
- The number of elements in the set n Z: |n^2-10n+19 |<6 is
- Let S= z C- i,2i : z^2+8iz-15 z^2-3iz-2 R . If - 13 11 i S, R- 0 , then 242 ^2 is equal to
- The least positive value of a for which the equation, (the equation is not legible in the source), has real roots is
- If x=-1,-4 are roots of the equation x=-1 and C-3:x - ,-2) for each positive integer -x^2+5x+16=0, then the value of x^2-5x-16=0...
- The sum of all integral values of k (k 0) for which the equation 2 x-1 - 1 x-2 = 2k in x has no real roots, is
- If for some p,q,r R, not all have same sign, one of the roots of the equation (p^2+q^2 )x^2-2q(p+r)x+q^2+r^2=0 is also a root of...
- Let a,b (a>b) be the roots of the quadratic equation x^2-x-4=0. If P_n= ^n- ^n, n N, then P_ 15 P_ 16 -P_ 14 P_ 16 -P_ 15 ^2+P_...
- The sum of the cubes of all the roots of the equation x^4-3x^3-2x^2+3x+1=0 is
- Let =- ^2 and ^3=1 be two real numbers such that ^ 2009 + ^ 2009 =(- )^ 2009 +(- ^2)^ 2009 and =- [ ( ^3 )^ 869 ^2+ ( ^3 )^ 1337...
- Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x))=8x^2-2x and g(f(x))=4x^2+6x+1, then the...
- Let α, β be the roots of the equation x² − 3x + r = 0 and 2, 2β be the roots of the equation x² + 3x + r = 0. If the roots of the...
- Let the circles C₁ : |z| = r and C₂ : |z − 3 − 4i| = 5, z ∈ ℂ, be such that C₂ lies within C₁. If z₁ moves on C₁, z₂ moves on C₂...
- If the system of equations x + 5y + 6z = 4, 2x + 3y + 4z = 7, x + 6y + az = b has infinitely many solutions, then the point (a,...
- Let a₁, a₂, a₃, … be an A.P. and g₁ = a₁, g₂, g₃, … be an increasing G.P. If a₁ = a₂ + g₂ = 1 and a₃ + g₃ = 4, then a₁₀ + g₅ is...
- The sum 1^3 1 + 1^3+2^3 1+3 + 1^3+2^3+3^3 1+3+5 + up to 8 terms, is:
- If for 3 ≤ r ≤ 30, 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = mr, then m equals:
- The mean and variance of n observations are 8 and 16, respectively. If the sum of the first (n − 1) observations is 48 and the...