Functions Questions (992)

Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let \(z = 4x + 6y\) be the objective function. The minimum value of z occurs at

Let \(f: (-1, 1) \to B\), be a function defined by \(f(x) = \tan^{-1}\dfrac{2x}{1-x^2}\), then \(f\) is both one-one and onto when \(B\) is the interval

If $f:\{1,2,3,4\}\to\{1,2,3,4\}$ is a function such that $|f(\alpha)-\alpha|\leq 1$ for $\alpha\in\{1,2,3,4\}$, then total number of such functions is

Given \( f(x) = a^x \) (\( a > 0 \)) and \( f(x) = f_1(x) + f_2(x) \), where \( f_1(x) \) is an even function and \( f_2(x) \) is an odd function. Then \( f_1(x+y) + f_1(x-y) \) equals

The function $f:\mathbb{R}-\{-1,1\}\to\mathbb{R}$ defined by $f(x)=\dfrac{x}{1-x^2}$ is

In a linear programming problem, the objective function is \(z = 4x + 3y\). The corner points of the feasible region are (0, 8), (2, 5), (4, 3), and (9, 0). Find the minimum value of \(z\).

\((\sim(p \vee q)) \vee (\sim p \wedge q\) is logically equivalent to

Since \(P\) is true, \(Q\) is false and \(R\) is true, the true statement among the following is:

Consider the following two binary relations on the set \(A = \{a, b, c\}\):\(R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}\) and\(R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}\).Then:

The function $f(x) = \dfrac{\sqrt{x^2 + kx + 1}}{x^2 - k}$ is continuous for all real $x$. Find the range of $k$.

Which of the following is NOT a tautology?

The range of the function \(f(x) = 7 - {}^{x}P_{x-3}\) is

Given \( f(x) = \dfrac{x^2}{1-x^2} \). The set A should be chosen so that f is a function from A to \([0, \infty)\). Which of the following is the correct set A?

Let \(P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\}\) and \(Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta\}\) be two sets. Then

Let a function \(f : (0, \infty) \to (0, \infty)\) be defined by \(f(x) = \left|1 - \dfrac{1}{x}\right|\). Then f is:

Here, n(A) = 5, n(B) = 7. What is the minimum number of elements in A ∪ B?

The numbers of subsets that can be formed from the set \(A = \{4, 5, 6\}\) are

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

Let $f(x) = a\sin x + b\sqrt[3]{x+4}$. If $f\bigl(\log_{10}(\log_3 10)\bigr) = 5$ and $f\bigl(\log_{10}(\log_3 3)\bigr) = 3$, then $f\bigl(\log_{10}(\log_3 3)\bigr)$ is

Sets A and B have 5 and 7 elements respectively. What can be the minimum number of elements in \(A \cup B\) is __________.

The negation of the statement "If I become a teacher, then I will open a school" is

In a group of 140 students, 70 opted Mathematics, 46 opted Physics and 28 opted Chemistry. 23 opted both Mathematics and Physics, 9 opted both Physics and Chemistry, 14 opted both Mathematics and Chemistry, and 4 opted all three subjects. The number of students who did not opt for any of the three courses is:

Let S = {x ∈ ℝ : x ≥ 0 and \(2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0\)}. Then, S

Let $R=\{(a,b): a/b$ is a prime number$\}$ on first twenty natural numbers. Consider: (I) $R$ is reflexive and symmetric but not transitive; (II) Range of $R^{-1}$ has 20 elements; (III) Domain of $R^{-1}$ has 10 elements. Which statements are true?

Given \(P(n) = x^2 - n + 41\) is prime. Which of the following options is correct regarding \(P(3)\) and \(P(5)\)?

The feasible region is bounded. The maximum value of z = 4x + 3y is to be found. The corner points are (0, 0), (25, 0), (16, 16), and (0, 24). Find the maximum value of z.

The contrapositive of the following statement,"If the side of a square doubles, then its area increases four times", is

Given that \( |x - 1| \leq 5 \) and \( |x| \geq 2 \), the solution set is:

If $f(x+y)=f(x)\cdot f(y)$ for all $x,y$ and $f(1)=2$, then $\sum_{k=1}^n f(k)=$

$f(x)$ is differentiable satisfying $f^2(x)+f^2(y)+2(xy-1)=f^2(x+y)$ $\forall x,y\in\mathbb{R}$. Also $f(x)>0$, $f(\sqrt{2})=2$. Then $f(\sqrt{7})=$

If $x = \log_b a = \log_a b = \dfrac{1}{2}\log_b c$ and $\log_c c = n(x)^{n+1}$, then the value of $n$ is

Suppose \(f(x) = (x-1)^2\) for \(x \geq -1\). If \(g(x)\) is the function whose graph is reflection of the graph of \(f(x)\) with respect to the line \(y = x\), then \(g(x)\) equals

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} and A = {1, 2, 3, 4}. The relation R is

Let F1 be the set of parallelograms, F2 the set of rectangles, F3 be the set of rhombuses, F4 be the set of squares and F5 be the set of trapeziums in a plane. Then F1 may be equal to

The contrapositive of \((p \vee q) \rightarrow r\) is

Negation of 'Paris is in France and London is in England' is

The inverse of the proposition \((p \wedge \sim q) \rightarrow r\) is

The Boolean Expression \((p \wedge \sim q) \vee q \vee (\sim p \wedge q)\) is equivalent to

Let $f(x)$ be a real function. Number of correct statements among: I) If $f(x)$ is odd on $\mathbb{R}$ and increasing for $x>0$ then it is decreasing for $x<0$. II) If $f(x)$ is odd and $\lim_{x\to 0}f(x)$ exists then $\lim_{x\to 0}f(x)=0$. III) If $f(x)$ is even on $\mathbb{R}$ and known to be increasing for $x>0$, then it is decreasing for $x<0$. IV) If $f(x)$ is even and $f(0)$ exists then $\lim_{x\to 0^+}f(x)=\lim_{x\to 0^-}f(x)$.

The domain of $f(x)$ is $(0,1)$, therefore, the domain of $y=f(e^x)+f(\ln|x|)$ is

Let $f(x)=x^2+x$ be written as $f(x)=g(x)+h(x)$ where $g(x)$ is an odd function and $h(x)$ is an even function. Then $g(xy)+h\!\left(\dfrac{x}{y}\right)$ equals

If the domain of the function $f(x)=\dfrac{\sqrt{x^2-16}}{x^2-4}+\log_{10}(x^2+3x-10)$ is $(-\infty,p)\cup[q,\infty)$, then $p^2+q=$