/ f ( x ) = x 3 - 5 x 2 - 8 x 3 is

negative and monotonically increasing for x ∈ - ∞ , 5 - 57 2 and positive and monotonically increasing for x ∈ 5 + 57 2 , + ∞

positive and monotonically increasing for x ∈ - ∞ , 5 - 57 2 and x ∈ 5 + 57 2 , + ∞

negative and monotonically decreasing for x ∈ - ∞ , 5 - 57 2 and x ∈ 5 + 57 2 , + ∞

positive and monotonically increasing for x ∈ - ∞ , 5 - 57 2 and negative and monotonically decreasing for x ∈ 5 + 57 2 , + ∞

If 2 x + 4 = A ( 2 x + 3 ) + B , find A and B .

Given that 2 x + 4 = A ( 2 x + 3 ) + B ⇒ 2 x + 4 = 2 A x + 3 A + B Equating the coefficients on both sides, we get 2 A = 2 , 3 A + B = 4 ⇒ A = 1 , B = 4 - 3 A = 4 - 3 1 = 1 Therefore, the value of A = 1 , B = 1 .

Let f x be a non-constant polynomial with real coefficients such that f 1 2 = 100 & f x ≤ 100 for all real x . Which of the following statements is NOT necessarily true?

The coefficient of the highest degree term in f x is negative

f x has at least two real roots

At least one of the coefficients of f x is bigger than 50

Let n ≥ 2 be an integer and X = 1 , … , n . Define Σ = σ : σ is a bijection of X and σ k - k ≤ 1 for all k ∈ X and for any σ ∈ Σ , Y σ = m ∈ X : σ m = m . Then

n is even implies number of elements of Y σ is even for every σ ∈ Σ

MEDIUM

Earn 100

Graph the exponential $y = 2^{x}$ . To graph the function, plot the points on the graph with $x$ -values $2, 3, 4$ , and $- 1 .$

Important Questions on Real Numbers

HARD

Mathematics>Arithmetic>Real Numbers>Exponents

f (x) = \frac{x^{3} - 5 x^{2} - 8 x}{3}

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

The value of function $f (x) = x^{3} - 3 x^{2} + 1$ at $x = 0$ will be

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

2 x + 4 = A (2 x + 3) + B

, find

A and B

HARD

Mathematics>Arithmetic>Real Numbers>Exponents

Let

f (x)

be a non-constant polynomial with real coefficients such that

f (\frac{1}{2}) = 100 & f (x) \leq 100

for all real

x .

Which of the following statements is NOT necessarily true?

MEDIUM

Mathematics>Arithmetic>Real Numbers>Exponents

The minimum value of

f (x) = \max \{x, 1 + x, 2 - x\}

MEDIUM

Mathematics>Arithmetic>Real Numbers>Exponents

The value of

\cos (\cos^{- 1} \frac{5}{13} + \sin^{- 1} \frac{3}{5} + \tan^{- 1} \frac{63}{16})

is equal to

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

Write the function whose graph is shown below.

Question Image

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

For

x

[0, π / 2),

\cot^{- 1} (\frac{\sqrt{3} \cos (x) - \sin (x)}{\cos (x) + \sqrt{3} \sin (x)}) = \frac{π}{3}

then

x =

HARD

Mathematics>Arithmetic>Real Numbers>Exponents

f (x)

and

g (x)

are two polynomials such that the polynomial

P (x) = f (x^{3}) + x g (x^{3})

is divisible by

x^{2} + x + 1

, then

P (1)

is equal to ___ .

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

Let

f : [- 5, 5] \to R

be a continuous function such that

f (x)

assumes only irrational values. If

f (3) = \sqrt{3}

, then

f (\sqrt{3} + 1)

is equal to

MEDIUM

Mathematics>Arithmetic>Real Numbers>Exponents

The value of

\sin^{- 1} \frac{1}{3} + \sin^{- 1} \frac{2 \sqrt{2}}{3} + \sin^{- 1} (\sin \frac{3 π}{5})

is equal to

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

Let

S

be the set of all functions

f : [0, 1] \to R

which are continuous on

[0, 1]

and differentiable on

(0, 1)

. Then for every

f

S

, there exists

c \in (0, 1)

depending on

f

, such that.

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

The maximum possible value of

3 \cos θ + 4 \sin θ + 5

is equal to

HARD

Mathematics>Arithmetic>Real Numbers>Exponents

Let

n \geq 2

be an integer and

X = \{1, \dots, n\}

. Define

Σ = \{σ : σ is a bijection of X and |σ (k) - k| \leq 1 for all k \in X\}

and for any

σ \in Σ, Y_{σ} = \{m \in X : σ (m) = m\}

. Then

MEDIUM

Mathematics>Arithmetic>Real Numbers>Exponents

The total number of solutions

θ \in [- 3 π, 3 π]

of the equation

\cos^{2} θ - \sin θ + 1 = 0

is equal to

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

The value of

\sin^{- 1} (\sin (\frac{4 π}{5})) + \cos^{- 1} (\cos (\frac{13 π}{5}))

is equal to

EASY

Mathematics>Arithmetic>Real Numbers>Exponents

The value of

2 \sec^{- 1} (2) + {cosec}^{- 1} (- 2)

is equal to

MEDIUM

Mathematics>Arithmetic>Real Numbers>Exponents

f (10 - x) = 3 x^{2} + 4 x - 5

and

f (x) = p x^{2} + q x + r

then

p + q + r =

MEDIUM

Mathematics>Arithmetic>Real Numbers>Exponents

Let

f

and

g

be differentiable functions on

R

such that

f o g

is the identity function. If for some

a, b \in R, g^{'} (a) = 5

and

g (a) = b,

then

f^{'} (b)

is equal to:

HARD

Mathematics>Arithmetic>Real Numbers>Exponents

Let $R$ be the set of real numbers and $f : R \to R$ be defined by $f (x) = \frac{\{x\}}{1 + {[x]}^{2}},$ where $[x]$ is the greatest integer less than or equal to $x,$ and $\{x\} = x - [x] .$ Which of the following statement are true?

I. The range of $f$ is closed interval
II. $f$ is continuous on $R$
III. $f$ is one-one on $R$

Graph the exponential y=2x. To graph the function, plot the points on the graph with x-values 2, 3, 4, and -1.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Graph the exponential $y = 2^{x}$ . To graph the function, plot the points on the graph with $x$ -values $2, 3, 4$ , and $- 1 .$