The shaded region in the adjoining diagram represents.

If n((A \(\times\) B) ∩(A \(\times\) C)) = 8 and n(B ∩ C) = 2, then n(A) is

The value of log₃ 11.log₁₁ 13.log₁₃ 15.log₁₅ 27.log₂₇ 81 is

The maximum value of 4sin²x + 3 cos²x + \(sin\frac { x }{ 2 } +cos\frac { x }{ 2 } \) is

The maximum value of 3 sin θ+4 cos θ is _______________

If ⁽ⁿ⁺⁵⁾P_(n+1)=\((\frac { 11(n-1) }{ 2 } )\).⁽ⁿ⁺³⁾P_n, then the value of n are

The number of ways to average the letters of the word CHEESE are _________

The sum up to n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\).....is

Slope of x-axis or a line parallel to x-axis is ______________

The lines x cos \(\alpha\) + y sin \(\alpha\) = p and xcos\(\beta\) + y sin\(\beta\) = q will be perpendicular if ______________

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂, y₂), (x₃, y₃ ) are

A matrix which is not a square matrix is called a_________matrix.

If \(|\overrightarrow { a } |=10,|\overrightarrow { b } |=2,\) and \(|\overrightarrow { a } .\overrightarrow { b } |=12\) then the value of \(|\overrightarrow { a } \times \overrightarrow { b } |\) is ___________ .

\(lim_{x \rightarrow \infty}{\sqrt{x^2-1}\over 2x+1}=\)

For what values of x is the rate of increase of x³ - 2x² + 3x + 8 is twice the rate of increase of x?

The derivative of f(x) = x |x| at x = −3 is

\(\int e^{-4 x} \cos x d x\) is

\(\int { \frac { 4\left( sin^{ -1 }x \right) ^{ 3 } }{ \sqrt { 1-{ x }^{ 2 } } } } \) dx = ________+c.

If A and B are two events such that A ⊂ B and P(B)\(\neq o\), then which of the following is correct?

Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd is

On the set of natural number let R be the relation defined by aRb if a + b \(\le\) 6. Write down the relation by listing all the pairs. Check whether it is transitive

Find the value of \(cot(\frac{-15\pi}{4})\).

How many three digit odd numbers can be formed by using the digits 4, 5, 6, 7, 8, 9 if
(i) repetition of digit is not allowed,
(ii) repetition is allowed.

Find the sum of first n terms of the series 1² + 3² + 5²+...

If the equation 12x² - 10xy + 2y² + 14x - 5y + k = 0 represents a pair of straight lines, find k, find separate equation and also angle between them.

Let \(\vec a\) and \(\vec b\) be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are \(​​​​\frac{\vec{a}+2 \vec{b}}{3} \text { and } \frac{\vec{b}+2 \vec{a}}{3} \text {. }\)

Discuss the continuity at x = 0 for \(f\left( x \right) =\begin{cases} \frac { 1-\cos { x } }{ { x }^{ 2 } } ,\quad x\neq 0 \\ \frac { 1 }{ 2 } ,\quad \quad \quad x=0 \end{cases}\)

Differentiate the following: y = sin(e^x)

Evaluate : \(\int \sqrt{1+\cos 2 x} d x\)

A die is tossed thrice. find the probability of getting an odd number atleast once?

The owner of a small restaurant can prepare a particular meal at a cost of Rupee 100. He estimate that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200 - x. Express his day revenue total cost and profit on this meal as a function of x.

Resolve into partial fractions: \({{x^3+1}\over{x(x+1)^2}}\)

Solve sin x - 3 sin 2x+ sin 3x = cos x - 3 cos 2x + cos 3x.

Show that the sequence where log a,\(log\frac { { a }^{ 2 } }{ b^{ 1 } } log\frac { { a }^{ 2 } }{ { b }^{ 2 } } \)  ..is an A.P

Find the equations of lines parallel to 3x - 4y- 5 = 0 at a unit distance from it. 

If A =\(\begin{bmatrix} 1 &0 &2 \\0 & 2 & 1 \\2 &0 &3 \end{bmatrix}\) and A³ - 6A² + 7A + KI = O, find the value of k.

Evaluate the following limits :\(lim_{x\rightarrow 0}{1-cosx\over x^2}\)

Show  that\(f\left( x \right) ={ x }^{ 2 }\) is differentiable at x = 1 and find \(f^{ ' }\left( 1 \right) \)

Integrate the following with respect to x : \({8\over \sqrt{1-(4x)^2}}+{27\over \sqrt{1-9x^2}}-{15\over 1+25x^2}\)

Let the matrix M = \(\left[ \begin{matrix} x & y \\ z & 1 \end{matrix} \right] \), If x,y and z are chosen at random from the set {1, 2,3, } and repetition is allowed (i.e., x = y = z ), what is the probability that the given matrix M is a singular matrix?

Let A = {a, b, c}, What is the equivalence relation of smallest cardinality on A? What is the equivalence relation of largest cardinality on A?

Prove that \(\sqrt { 5 } \) is an irrational number

If \(\frac { cos^{ 4 }\alpha }{ { cos }^{ 2 }\beta } +\frac { { sin }^{ 4 }\alpha }{ { sin }^{ 2 }\beta } =1\) prove that \({ sin }^{ 4 }\alpha +{ sin }^{ 4 }\beta =2{ sin }^{ 2 }\alpha { sin }^{ 2 }\beta\)

How many three-digit numbers are there with 3 in the unit place?
(i) with repetition
(ii) without repetition.

In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day, the virus which caused the disease spread in Geometric Progression. The amount of infectious virus particle gets doubled each day, being 5 particles on the first day. Find the day when the infectious virus particles just grow over 1,50,000 units?

Find the distance between the parallel lines
12x + 5y = 7 and 12x + 5y + 7 = 0.

Show that the locus of the mid-point of the segment intercepted between the axes of the variable line x cos \(\alpha\) + y sin \(\alpha\) = p is \(\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}\) where p is a constant.

Determine the roots of the equation \(\begin{vmatrix} 1 &4 &2 0 \\ 1 & -2 & 5 \\ 1 &2x &5x^2 \end{vmatrix}=0\)

Let \(\overrightarrow { a } ,\overrightarrow { b } \) and \(\overrightarrow { c } \) be unit vectors such that \(\overrightarrow { a } \) is perpendicular to both \(\overrightarrow { b } \) and \(\overrightarrow { c } \) and further the angle between \(\overrightarrow { b } \) and \(\overrightarrow { c } \) is \(\frac { \pi }{ 6 } \). Then \(\overrightarrow { a } =\pm 2\left( \overrightarrow { b } \times \overrightarrow { c } \right) \)

The function \(f(x)={x^2-1\over x^3-1}\) is not defined at x = 1. What value must we give f(1) inorder to make f(x) continuous at x = 1?

Differentiate x^x with respect to x log x

Integrate the following with respect to x : \(\frac{1}{3} \cos \left(\frac{x}{3}-4\right)+\frac{7}{7 x+9}+e^{\frac{x}{5}+3}\)

Evaluate \(\int { \frac { { x }^{ 2 }{ tan }^{ -1 }\left( { x }^{ 3 } \right) }{ 1+{ x }^{ 6 } } } \)dx

Two integers are selected at random from integers 1 to 11. If the sum is even, find the probability that both the numbers are odd.