The function f:[0,2π]➝[-1,1] defined by f(x) = sin x is

Which one of the following is false?

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

If tan40⁰ = λ, then \(\frac { tan{ 140 }^{ 0 }-tan{ 130 }^{ 0 } }{ 1+tan{ 140 }^{ 0 }.tan{ 130 }^{ 0 } } \) =

If tan θ = \(\frac{-4}{3}\), then sin θ is _____________ 

The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is _________

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

The distance between the line 12x - 5y + 9 = 0 and the point (2, 1) is ______________

The image of the point (1, 2) with respect to the line y = x is ______________

If the square of the matrix \(\begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}\) is the unit matrix of order 2, then \(\alpha ,\beta \) and \(\gamma\) should satisfy the relation.

The value of\(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1+sin\theta & 1 \\ 1 & 1 & 1+cos\theta \end{matrix} \right| \) is _____________

The angle between two vectors \(\vec{a}\) and \(\vec{b}\) with magnitudes\(\sqrt{3}\) and 4 respectively and \(\vec{a}.\vec{b}=2\sqrt{3}\) is ________ .

The value of \(lim_{x\rightarrow k^-}x-\left\lfloor x \right\rfloor \) , where k is an integer is

If y= 6x -x³ and x increases at the ratio of 5 units per second, the rate of change of slope when x = 3 is ______ units/sec.

If f(x) = x² - 3x, then the points at which f(x) = f '(x) are

\(\int e^{\sqrt{x}} d x\) is

\(\int { \left| x \right| ^{ 3 } } \) dx is equal to ________+c.

If a and b are chosen randomly from the set {1,2,3,4} with replacement, then the probability of the real roots of the equation \(x^2+ax+b=0\) is

If P(B)=\(\frac { 3 }{ 5 } \), P(A/B)=\(\frac { 1 }{ 2 } \) and P(AUB)=\(\frac { 4 }{ 5 } \), then P(B/\(\bar { A } \)) =

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 symmetric

If the arcs of same lengths in two circles subtend central angles 30° and 80⁰, find the ratio of their radii.

Let p(n) be the statement "7 divides 2³ⁿ-1" What is p(n+1) =?

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

Find the locus of a point P that moves at a constant distant of 
(i) two units from the X-axis
(ii) three units from the Y-axis

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

Find the derivative of the tan (x + y) + tan (x - y) = x

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

The probability that student selected at random from a class will pass in Mathematics is \(\frac { 2 }{ 3 } \) and the probability that he passes in Mathematics and English is \(\frac { 1 }{ 3 } \). What is the probability that he will pass in English if it is known that he has passed in Mathematics?

Consider the functions:
(i) f(x) = |x|
(ii) f(x) = |x| − 1
(iii) f(x) = |x| + 1
 

Simplify: \(\sqrt { 98 } +\sqrt { 50 } -\sqrt { 18 } +\sqrt { 75 } -\sqrt { 27 } \)

Solve the equation sin 9\(\theta\) = sin \(\theta\).

An A.P. consists of 21 terms. The sum of the three terms in the middle is 129 and of the last three is 237. Find the series.

Find \(\sqrt [ 3 ]{ 65 } \).

Find the value of \(\begin{vmatrix} 1 & log_xy & log_x z \\ log_y x & 1 & log_yz \\ log_z x & log_zy & 1 \end{vmatrix}\)  if x, y, z \(\neq\) 1.

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 \(f^{ ' }\left( 1 \right) \)

Evaluate : \(\int{x^3+2\over x-1}dx\)

A firm manufactures PVC pipes in three plants viz, X, Y, and Z. The daily production volumes from the three firms X, Y and Z are respectively 2000 units, 3000 units, and 5000 units. It is known from the past experience that 3% of the output from plant X, 4% from plant Y and 2% from plant Z are defective. A pipe is selected at random from a day’s total production,
(i) find the probability that the selected pipe is a defective one.
(ii) if the selected pipe is a defective, then what is the probability that it was produced by plant Y?

If \(f:R-\{ -1,1\}\rightarrow R\) is defined by \(f(x)={x \over x^2-1},\) verify whether f is one-to-one or not.

Resolve into partial fractions \(\frac { { x }^{ 2 }-2x-9 }{ (x+1)({ x }^{ 2 }+x+6) } \)

Using Heron's formula, show that the equilateral triangle has the maximum area for any fixed perimeter. [Hint: In xyz \(\le\)  k, maximum occurs when x = y = z]

Using the Mathematical induction, show that for any integer
n\(\ge\) 2, 3n² > (n + 1)²

What will Rs. 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

Find all the equations of the straight lines in the family of the lines y = mx - 3, for which m and the x - coordinate of the point of intersection of the lines with x - y = 6 are integers.

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.

For what value of x, the matrix A = \(\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & x^3 \\ 2 & -3 & 0 \end{bmatrix}\) is skew-symmetric.

Find \(\lambda\), when the projection of \(\overrightarrow{a}=\lambda \hat{i}+\hat{j}+4\hat{k}\) on \(\overrightarrow{b}=2\hat{i}+6\hat{j}+3\hat{k}\) is 4 units.

Sketch the graph of a function f that satisfies the given values :
f(0) is undefined
\(lim_{x\rightarrow0}f(x)=4\)
f(2) = 6
\(lim_{x\rightarrow2}f(x)=3\)

Discuss the differentiability of \(f\left( x \right) =\begin{cases} x{ e }^{ -\left( \frac { 1 }{ \left| x \right| } +\frac { 1 }{ x } \right) } \\ 0,\quad x=0 \end{cases}, x\neq 0\) at x = 0

Integrate the following with respect to x : \({1\over xlog \ xlog(log \ x)}\)

Evaluate \(\int { \frac { { x }^{ 3 }dx }{ { x }^{ 4 }+{ 3x }^{ 2 }+2 } } \)

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.