List of practice Questions

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

Let $f \left(x\right)=g\left(x\right).\frac{e^{1/x}-e^{-1/x}}{e^{1/x}+e^{-1/x}},$ where g is a continuous function then $\displaystyle \lim_{x \to 0} f (x)$ does not exist if

Let $f \left( x\right) = \alpha\left( x\right)\beta\left( x\right) \gamma \left( x\right)$ for all real x, where $\alpha\left(x\right), \beta\left(x\right)$ and $\gamma \left( x\right)$ are differentiable functions of x. If $f ' \left(2\right) = 18 f \left(2\right),\alpha' \left(2\right) = 3\alpha\left(2\right), \beta' \left(2\right) = -4\beta\left(2\right)$ and $\gamma'\left(2\right) = k\gamma \left(2\right)$ , then the value of k is

Let $f(x) = \cos x \sin 2x$, then

Let $f (a) = g(a) = k$ and their nth derivatives $f^n (a), g^n (a)$ exist and are not equal for some n. Further if $\displaystyle\lim_{x \to a} \frac{f\left(a\right)g\left(x\right) -f\left(a\right) -g\left(a\right)f\left(x\right)+f\left(a\right)}{g\left(x\right)-f\left(x\right)} = 4 $ then the value of k is

Let f and g be functions from the interval $[0, \infty)$ to the interval $[0, \infty)$,f being an increasing and g being a decreasing function. If f{g(0)} = 0 then

Let $B = 2 \, \sin^2 \, x - \cos \, 2x$, then

Let $ { \frac{C}{5} = \frac{F - 32}{9}}$. If C lies between 10 and 20, then :

Let $F_1$ be the set of parallelograms, $F_2$ the set of rectangles, $F_3$ the set of rhombuses, $F_4$ the set of squares and $F_5$ the set of trapeziums in the plane. Then $F_1$ may be equal to

Let a, b, c are three non-coplanar vectors such that $r_{1}=a-b+c, r_{2}=b+c-a, r_{3}=c+a+b,$ $r=2a-3b+4c. If \, r=\lambda_{1}r_{1}+\lambda_{2}r_{2}+\lambda_{3}r_{3}, $then

Let $A=(a_{ij})_{m\times n}$ be a matrix such that $a_{ij}=1$ for all I, j. Then