List of practice Questions

Let f(x) = x2 and g(x) = 2X , for all real x. Then the value of f( f(g(x)) + g(f(x)) ) at x = 1 is

The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

If 9x-1/2-22x-2=4x-32x-3, then x is

If log (2a × 3b × 5C) is the arithmetic mean of log (22 × 33 × 5), log (26 × 3 × 57 ), and log (2 × 32 × 54 ), then a equals

Let a1 , a2 , a3 , a4 , a5 be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with 2a3.
If the sum of the numbers in the new sequence is 450, then a5 is

How many different pairs (a, b) of positive integers are there such that a ≤ b and 1/a+1/b=1/9=?

In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens?

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(l) is

How many four digit numbers, which are divisible by 6, can be formed using the digits 0, 2, 3, 4, 6, such that no digit is used more than once and 0 does not occur in the left-most position?

Let f(x) = 2x-5and g(x) = 7-2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

An infinite geometric progression a1 , a2 , a3 ,... has the property that an = 3(an+ l + an+2 +….) for every n ≥ 1. If the sum a1 + a2 + a3 +….... = 32, then a5 is

If a1=1/2×5, a2=1/5×8, a3=1/8×11,...,then a1, +a2, +a3, +....a100 is

If \(9^{2x – 1} – 81^{x-1} = 1944\), then x is

If tan (A + B) = m and tan (A - B) = n then value of tan 2A is

For how many integers n, will the inequality \((n – 5) (n – 10) – 3(n – 2) ≤ 0\) be satisfied?

If a, b, c, and d are integers such that \(a + b + c + d = 30\), then the minimum possible value of \((a - b)^2 + (a - c)^2 + (a - d)^2\) is

If \(f_1 (x) = x^2 + 11x + n\) and \(f_2 (x) = x,\) then the largest positive integer n for which the equation \(f_1 (x) = f_2 (x)\) has two distinct real roots, is

If \(f(x)=\frac{5x+2 }{3x-5}\) and \(g(x)=x^2-2x-1\),then the value of \(g(f(f(3)))\) is 

If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference is

In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

Let \(a_1 , a_2 ,……..a_{3n}\) be an arithmetic progression with \(a_1 = 3\) and \(a_2 = 7.\) If \(a_1 + a_2 + ….+a_{3n} = 1830\), then what is the smallest positive integer m such that m \((a_1 + a_2 + …. + a_n ) > 1830?\)

Let AB, CD, EF, GH, and JK be five diameters of a circle with center at O. In how many ways can three points be chosen out of A,B, C, D, E, F, G, H, J, K, and O so as to form a triangle?

The number of solutions \((x, y, z)\) to the equation \(x\ –\  y \ –\ z = 25\), where x, y, and z are positive integers such that \(x ≤ 40, y ≤ 12\), and \(z ≤ 12\) is

The shortest distance of the point \((\frac{1}{2},1)\) from the curve \(y = |x -1| + |x + 1|\) is