List of practice Questions

Let three fair coins be tossed. Let $A =$ {all heads or all tails}, $B =$ {atleast two heads), and $C =$ {atmost two tails). Which of the following events are independent ?

Let $t_r$ denotes the $r^{th}$ term of an $A.P$. Also, suppose that $t_{m} = \frac{1}{n}$ and $t_{n}= \frac{1}{m}, \left(m\ne n\right)$, for some positive integers $m$ and $n$, then which of the following is necessarily a root of the equation $(l+m- 2n)x^2 + (m + n- 2l)x +(n + l- 2m) = 0$?

Let the population of rabbits surviving at a time t be governed by the differential equation $dp(t)/dt = (1/2) p(t) - 200.$ If $ p(0) = 100,$ then p(t) equals

Let $S_n$ denote the sum of the cubes of the first $n$ natural numbers and $s_n$ denote the sum of the first $n$ natural numbers. Then $\sum\limits_{r=1}^{n} \frac{S_{r}}{s_{r}}$ equals

Let $S_n $ denote the sum of the first n terms of an A.P. If $S_{2n} = 3S_n$ then $S_{3n} : S_n$ is equal to

Let set $X= \{a, b, c\}$ and $Y = \phi$. The number of ordered pairs in $X \times Y$ are

Let $R$ be a relation from $N$ to $N$ defined by $R = \{(a, b) : a, b \in N$ and $a = b^2\}$. Which of the following is true?

Let $R$ be a relation on the set $N$ of natural numbers denoted by $nRm \Leftrightarrow n$ is a factor of m (i.e. $n \,| \,m$). Then, $R$ is

Let $S= \frac{4}{19} +\frac{44}{19^{2}} + \frac{444}{19^{3}} + .....\infty$. Then $S$ is equal to

Let $S$ be the sum, $P$ be the product and $R$ be the sum of the reciprocals of $3$ terms of a $G.P$. Then $P^{2}R^{3} : S^{3}$ is equal to

Let $Q^+$ be the set of all positive rational numbers. Let $\ast$ be an operation on $Q^+$ defined by $a \ast b = \frac{ab}{2} \forall \, a,b \in Q^+$. Then, the identity element in $Q^+$ for the operation $ \ast $ is:

Let R and S be two non-void relations on a set A. Which of the following statements is false ?

Let R be a reflexive relation on a finite set A having n-elements, and let there be m ordered pairs in R. Then

Let PS be the median of the triangle with vertices P(2, 2), Q(6, - 1) and R(7 , 3). The equation of the line passing through (1, - 1) and parallel to PS is

Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement $p \Rightarrow (q \vee r)$ is:

Let P be a set of squares, Q be set of parallelograms, R be a set of quadrilaterals and S be a set of rectangles. Consider the following : 1. P $\subset$ Q 2. R $\subset$ P 3. P $\subset$ S 4. S $\subset$ R Which of the above are correct?

Let p be the statement "x is an irrational number", q be the statement "y is a transcendental number", and r be the statement " x is a rational number iff y is a transcendental number". r is equivalent to either q or p r is equivalent to $\sim (p \leftrightarrow \sim q)$.

Let p: I am brave, q: I will climb the Mount Everest. The symbolic form of a statement, 'I am neither brave nor I will climb the mount Everest' is

Let $P_n(x) = 1 + 2x + 3x^2$ + ..... + $(n + 1)x^n$ be a polynomial such that $n$ is even. Then the number of real roots of $P_n(x) = 0$ is

Let I denote the $3 \times 3$ identity matrix and P be a matrix obtained by rearranging the columns of I. Then

Let $n(A - B) = 25 + x, n (B -A)= 2x$ and $n(A \cap B) = 2x$. If $n(A) = 2 (n(B)) $ then 'x' is

Let $n$ be a fixed positive integer. Let a relation $R$ be defined in $I$ (the set of all integers) as follows : $aRb$ iff $n|(a - b)$, that is, iff $a$ - $b$ is divisible by $n$. Then, the relation $R$ is

Let O (A) = m, O (B) = n. Then the number of relations from A to B is

Let $F(x) = x^3 + ax^2 + bx + 5 sin^2\, x$ be an increasing function in the set of real number $R$. Then a and b satisfy the condition.

Let f(x) = $\frac {x^4-5x^2+4} {|(x-1) (x-2)|}$ , x $\neq $ 1,2 = 6 ,x=1,12, x = 2, Then f (x) is continuous on the set