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The points (5, – 4, 2), (4, –3, 1), (7, – 6, 4) and (8, –7, 5) are the vertices of [RPET 2002]

The value of ${lim}_{x\to \infty }x[ta{n}^{-1}\left(\frac{x+1}{x+2}\right)-ta{n}^{-1}\left(\frac{x}{x+2}\right)]is$

Number 1, 2, 3, . . . . 100 are written down on each of the cards A, B, and C. One number is selected at random from each of the cards. Then find the probability that the numbers so selected can be the measures (in cm) of three sides of right-angled triangles, no two of which are similar.

If $A=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]$ and $B=\left[b_{ij}\right]$ is a matrix such that $A^n-A^2+I=B$ (where $n\in N$). If $2(b_{21}+b_{22})=7$ then, $n$ is

How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?

Find the 4th term form the end of the G.P. $\frac{2}{27},\frac{2}{9},\frac{2}{3},.....,162$

If $x-\frac{1}{x}=\sqrt{5},$ then the value of $x+\frac{1}{x}$ is
$(x>0)$

On a set {a,b,c}, we define an equivalence relation 'R'. If this relation is {(a,a),(b,b),(c,c),(a,b),(b,a)}. How many equivalence classes will be formed from this relation?

If the median of $21$ observations is $40$ and if the observations greater than the median are increased by $6$ then the median of new data will be

The equation(s) of standard ellipse which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$, is/are

The logical statement $\sim (\sim p\to q)\vee (p\wedge \sim q)$ is equivalent to

Neetu goes 35 km towards east from a point P to point Q. From Q, she moves 45 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from P?

The sides AB, BC, CD and DA of a quadrilateral are x + 2y = 3, x = 1, x – 3y = 4, 5x + y + 12 = 0 respectively. The angle between diagonals AC and BD is

An ellipse, with foci at $(0,2)$ and $(0,–2)$ and minor axis of length $4$, passes through which of the following points ?

If ${(1+x+x^2)}^{25}={\sum }_{r=0}^{50}{a}_r{x}^r,$ then ${\sum }_{r=0}^{16}{a}_{3r}$ equals

The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})$ is :

Find :

(i) 10th term of the A.P. 1, 4, 7, 10, .....

(ii) 18th term of the A.P. $\sqrt{2},3\sqrt{2},5\sqrt{2},\dots \dots$

(iii) nth term of the A.P. 13, 8, 3, -2, .....

If $cos4x=1+ksi{n}^{2}xco{s}^{2}x$, then write the value of k.

If $\frac{\pi }{2}<\theta <\pi$, then $\sqrt{\frac{1-sin\theta }{1+sin\theta }}+\sqrt{\frac{1+sin\theta }{1-sin\theta }}$ is equal to

Find the points of local maxima or minima for the function
f(x) =sin²x on [0,π]

Let $f\left(x\right)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime }\left(x\right)=f^{\prime }(2-x)$ for all $x\in (0,2),$ $f\left(0\right)=1$ and $f\left(2\right)=e^2.$ Then the value of $\underset{0}{\overset{2}{\int }}f\left(x\right)dx$ is

The proposition $\sim (p\iff q)$ is equivalent to

In a group of $115$ people, each has at least one among passport and voter ID. If $65$ had passport and $30$ had both, how many had only voter ID but not passport?

The general solution(s) of the equation $4{\cos }^{2}x+6{\sin }^{2}x=5$ is/are

$sin{47}^{\circ }+sin{61}^{\circ }-sin{11}^{\circ }-sin{25}^{\circ }$ is equal to

$Ifxcos\theta =ycos(\theta +\frac{2\pi }{3})=zcos(\theta +\frac{4\pi }{3}),thenthevalueof\frac{1}{x}+\frac{1}{y}+\frac{1}{z}isequalto$

Equation of pair of tangents drawn from $(4,3)$ to the circle $x^2+y^2=4$ is

Poor Dolly’s T.V. has only $4$ channels, all of them quite boring, hence it is not surprising that she desires to switch (change) channel after every one minute. Find the number of ways in which she can change the channels so that she is back to her original channel for the first time after $4$ minutes.

From the vertex of the parabola $y^2=4ax$ pair of perpendicular chords are drawn. If this chords are adjacent sides of a rectangle, then the locus of the vertex of the rectangle diagonally opposite to the vertex of parabola is

A line meets x-axis and y-axis at A and B respectively and O is the origin.

$\begin{array}{cccccc}& \text{Column I}& & \text{Column 2}& & \text{Column 3}\\ & & & \text{Equation of AB}& & \text{Area of}\Delta OAB\\ \text{(I)}& \text{Centroid}\Delta OAB\text{is (1, 2)}& \text{(i)}& 2x+y=2& \text{(P)}& \text{6 sq. units}\\ \text{(II)}& \text{Circumcenter of}\Delta OAB\text{is (1, 2)}& \text{(ii)}& 3x+4y=12& \text{(Q)}& \text{9 sq. units}\\ \text{(III)}& \text{Distance of the orthocentre of}\Delta OAB& \text{(iii)}& 2x+y=6& \text{(R)}& \text{1 sq. units}\\ & \text{From A and B is 1 and 2 respectively}& & & & \\ \text{(IV)}& \text{Incenter of}\Delta OAB\text{is (1, 1)}& \text{(iv)}& 2x+y=4& \text{(S)}& \text{4 sq. units}\end{array}$

Which of the following is correct combination?

The sum of the series $1\times n+2(n-1)+3(n-2)+\dots \dots +(n-1)\times 2+n\times 1$ is

Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points A (0, -4) and B (8, 0).

Find the values of the other five trigonometric functions in each of the following:

$\left(i\right)cos\theta =\frac{12}{5},\theta$ in quadrant III

$\left(ii\right)cos\theta =-\frac{1}{2},\theta$ in quadrant II

$\left(iii\right)tan\theta =\frac{3}{4},\theta$ in quadrant III

$\left(iv\right)sin\theta =\frac{3}{5},\theta$ in quadrant I

The range of the function $f\left(x\right)=\sqrt{9-x^2}$ is

A relation R on A = {1,2,3,4} is defined as x R y if x divides y. Then R is

A is a square matrix such that $A^2=A$, then dot (A) is equal to

Three vertices of a paralleiogram taken in order are (-1, - 6), (2, -5) and (7, 2). The fourth vertex is

Expand the values of x, if

(i)$\left|\begin{array}{cc}2& 4\\ 5& 1\end{array}\right|=\left|\begin{array}{cc}2x& 4\\ 6& x\end{array}\right|$

(ii) $\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right|$

Let $\overrightarrow{A}=a\hat{i}+b\hat{j}+c\hat{k}$ be a unit vector and $\overrightarrow{B}$ is another vector in $R^3$ such that $|\overrightarrow{A}\times \overrightarrow{B}|=1,\overrightarrow{C}=\frac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})$ and $(\overrightarrow{A}\times \overrightarrow{B}).\overrightarrow{C}=$ 1, then which of the following statement(s) is/are correct?

Two vectors have magnitudes 3 unit and 4 unit respectively. What should be the angle between them if the magnitude of the resultant is (a) 1 unit, (b) 5 unit and (c) 7 unit.