If two sets A and B have 17 elements in common, then the number of elements common to the set A \(\times\)B and B \(\times\)A is

The domain of the function \(f(x)=\sqrt{ x - 5 }+ \sqrt{6 - x}\) is 

If a and b are the real roots of the equation x²- kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

\(\left( 1+cos\frac { \pi }{ 8 } \right) \left( 1+cos\frac { 3\pi }{ 8 } \right) \left( 1+cos\frac { 5\pi }{ 8 } \right) \left( 1+cos\frac { 7\pi }{ 8 } \right) \) =

In any ΔABC, a(b cosC - c Cos B) = __________

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

The number of parallelogram formed if 5 parallel lines intersect with 4 other paralle llines is _________

\(\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+...\) is ______________

The length of \(\bot\) from the origin to the line \(\frac{x}{3}-\frac{y}{4}=1,\) is 

The equation x²+ kxy + y²- 5x - 7y + 6 = 0 represents a pair of straight lines then k = ______________

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

The negative of a matrix is obtained by multiplying it by ___________ .

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

If \(lim_{x \rightarrow 0}{sin \ px\over tan \ 3x}=4\) , then the value of p is

The points of discontinuity of the function \(\frac { { x }^{ 2 }+6x+8\quad }{ { x }^{ 2 }-5x+6\quad } is\)

If g(x) = (x² + 2x + 3) f(x) and f(0) = 5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\), then g'(0) is

\(\int \sqrt{\frac{1-x}{1+x}} d x\) is

\(\int { \frac { sin\sqrt { x } }{ x } } \) dx = ________ +c.

If X and Y be two events such that P(X/Y) = \({1\over2},P(Y/X)={1\over3}\) and \(P(X\cap Y)={1\over6}\), then P(X\(\cup\)Y) is

Two dice are thrown. It is known that the sum of the numbers on the dice was less than 6, the probability of getting a sum 3 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

A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 30⁰. If after 100 km, the target has an angle of depression of 60⁰, how far is the target from the fighter jet at that instant?

A family of 4 brothers and 3 sisters is to be arranged in a row for a photograph, in how many ways can they be seated if
(i) all the sisters sit together
(ii) all the sisters are not together.

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

Find the equation of the line through the point of intersection of the line 5x - 6y = 1 and 3x + 2y + 5 = 0 and cutting off equal intercepts on the coordinate axis.

Find the position vector of a point R which divides the line joining points P and Q whose position vectors are \(\hat{i}+2\hat{j}-\hat{k}\) and \(-\hat{i}+\hat{j}+\hat{k}\) respectively in the ratio 2:1 externally.

\(If\lim _{ x\rightarrow 2 }{ \frac { { x }^{ n }-{ 2 }^{ n } }{ x-2 } } =80\quad and\quad n\in N,\quad find\quad n.\)

Find the derivatives of the following functions using first principle. f(x) = - x² + 2

Evaluate : \(\int e^{xlog2}e^x dx\)

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

In the set Z of integers, define mRn if m - n is divisible by 7. Prove that R is an equivalence relation.

Find x if \({{1}\over{2}}\) log₁₀ \((11+4\sqrt{7})\) = log₁₀ (2 + x).

Find the general solution of sin\(\theta =-{\sqrt{3}\over 2}\).

Evaluate the following:
\(\sqrt [ 3 ]{ 1003 } \) correct to 4 places of decimals

The Pamban Sea Bridge is a railway bridge of length about 2065m constructed on the PalkStrait, which connects the Island town of Rameswaram to Mandapam, the main land of India. The Bridge is restricted to a uniform speed of only 12.5 m/s. If a train of length 560 m starts at the entry point of the bridge from Mandapam, then
(i) find an equation of the motion of the train.
(ii) when does the engine touch island.
(iii) when does the last coach cross the entry point of the bridge.
(iv) what is the time taken by a train to cross the bridge.

Determine the values of b so that the following matrices are singular:\(\begin{bmatrix}b-1 &2 &3 \\3 & 1 & 2 \\ 1 & -2 &4 \end{bmatrix}\)

Calculate \(\lim _{ x\rightarrow0}{|x| } \).

Differentiate \(\log _{ 7 }{ (\log _{ 7 }{ x } ) } \)

Integrate the following functions with respect to x : sin 3x

The probability that a car being filled with petrol will also need an oil change is 0.30; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and filter need changing is 0.15.
(i) If the oil had to be changed, what is the probability that a new oil filter is needed?
(ii) If a new oil filter is needed, what is the probability that the oil has to be changed?

Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

Determine the region in the Plane determined by the inequalities x+y≥4,2x-y>0

Solve \(\sqrt{3}tan^2\theta+(\sqrt{3}-1)tan\theta-1=0\)

By the principle of mathematical induction, prove that for n > 1,
\(1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = {n(2n-1)(2n+1)\over 3}\)

Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 4²) + (1 + 4 + 4² + 4³) + ...

A spring was hung from a hook in the ceiling. A number of different weights were attached to the spring to make it stretch, and the total length of the spring was measured each time shown in the following table.

(i) Draw a graph showing the results.
(ii) Find the equation relating the length of the spring to the weight on it.
(iii) What is the actual length of the spring.
(iv) If the spring has to stretch to 9 cm long, how much weight should be added?
(v) How long will the spring be when 6 kilograms of weight on it?

A point moves so that square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x -12y = 3. The equation of its locus is .................

If A and B are symmetric matrices of same order, prove that
AB - BA is a skew-symmetric matrix.

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)

According to Einstein’s theory of relativity, the mass m of a body moving with velocity v is m = \({m_O\over \sqrt{1-{v^2\over c^2}}},\)  where m₀ is the initial mass and c is the speed of light. What happens to m as \(v\rightarrow{c}^-\). Why is a left hand limit necessary?

If \(f\left( 2 \right) =4\ and f^{ ' }\left( 2 \right) =1, \) then find \(\lim _{ x\rightarrow 2 }{ \frac { xf\left( 2 \right) -(2)f\left( x \right) }{ x } }\)

Integrate the following with respect to x : \(\left(1-x^2\right)^{-\frac{1}{2}}\)

Evaluate \(\int { \frac { { e }^{ x }dx }{ { e }^{ 2x }+6{ e }^{ x }+5 } } \)

The probability of simultaneous occurrence of atleast one of two events A and B is p. if the probability that exactly one A, B occurs is q then prove that P(\(\bar { A } \)) + P(\(\bar { B } \)) = 2-2 p+q.