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The negation of the conditional statement ‘If it rains, I shall go to school’ is___.

Find the eccentricity of ellipse whose minor axis is double the latus rectum

Evaluate $\int \frac{(4x+1)dx}{x^2+3x+2}$.

The mean and standard deviation (s.d.) of $10$ observations are $20$ and $2$ respectively. Each of these $10$ observations is multiplied by $p$ and then reduced by $q$, where $p\ne 0$ and $q\ne 0$. If the new mean and standard deviation become half of their original values, then $q$ is equal to:

If the third term in the binomial expansion of $(1+x{)}^m$ is -$\frac{1}{8}{x}^2$, then the rational value of m is

Which of the following is/are negative?

(angles in following options are mentioned in radians)

If one root of the equation $(a^2-5a+3)x^2+(3a-1)x+2=0$ be double the other, then $a=$

The perpendicular distance from the points $\hat{i}+\hat{j}+\hat{k}and5i+5j$ to the plane given by $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=10$ will be

If three unequal positive real numbers a,b,c are in G.P. and a-b,c-a, a-b are in H.P., then the values of a+b+c is independent of

The co-ordinates of the foot of perpendicular drawn from point P(1,0,3) to the line joining the points A(4,7,1) and B(3,5,3) is

Reduce $[\frac{1}{1-4i}-\frac{2}{1+i}]=\left[\frac{3-4i}{5+i}\right]$ to the standard form.

Let $Z$ be the set of integers. If
$A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

Find the coefficient of $x^n$ in the expansion of $(1+x{)}^{2n}$

Find the number of empty relation we can define on a non empty set A

A committee of 5 students is selected at random from a group consisting 10 boys and 5 girls. Given that there is at least one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.

If p and q are the lengths of perpendicular from the origin to the lines $xcos\theta -ysin\theta =kcos2\theta$ and $xsec\theta +ycosec\theta =k$, respectively.

Prove that $p^2+4q^2=k^2$

The sum of the infinite series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+\cdots$ is equal to

Determine the smallest positive value of x (in degrees) for which tan $(x+{100}^{\circ })=tan(x+{50}^{\circ })tanxtan(x-{50}^{\circ })$.

Prove that $\frac{cos7x-cos8x}{1+2cos5x}=cos2x-cos3x$

The number of four digit numbers that can be formed with the digits 1,2,3,4 and 5 in which atleast two digits are identical is

Which of the following is/are simple event?

The linear factor(s) of the equation $x^2+4xy+4y^2+3x+6y-4=0$ is/are

If $z=\frac{(\sqrt{3}+i{)}^3(3i+4{)}^2}{(8+6i{)}^2}$, then |$z$| is equal to

Let two fair six-faced dice A and B be thrown simulatneously. If $E_1$ is the event that die A shows up four, $E_2$ is the event that die B shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is/are true ?

If $x$ and $(20+y{)}^{\circ }$ are the supplementary angles and difference between them is ${60}^{\circ }$, then the value of $y$ is

Find the equation of director circle of a circle whose equation is $x^2+y^2-4x+6y+12=0$

If the length of latus rectum of a hyperbola $\frac{x^2}{k}-\frac{y^2}{25}=-1$ is $\frac{22}{5}$ units, then its $e$ (eccentricity) is

By the method of matrix inversion, solve the system.

$\left(\begin{array}{ccc}1& 1& 1\\ 2& 5& 7\\ 2& 1& -1\end{array}\right]\left(\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\\ x_3& y_3\end{array}\right]=\left(\begin{array}{cc}9& 2\\ 52& 15\\ 0& -1\end{array}\right]$

There are 60 students in a Mathematics class and 90 students in Physics class. Find the number of students which are either in Physics class or Mathematics class in the following cases.

(i) Two classes meet at the same hour.

(ii) Two classes meet at different hours and 30 students are enrolled in both the courses.

(iii) What value is shown here ?

If the ${10}^{th}$ term of an A.P. is $\frac{1}{20}$ and its ${20}^{th}$ term is $\frac{1}{10}$, then the sum of its first $200$ terms is

Find the coefficient of $x^8$ in the following expansion

$(1+x)(1+x^2)(1+x^3)(1+x^4)(1+x^5)...$

$....(1+x^{98})(1+x^{99})(1+x^{100})$.

If the harmonic mean between two positive numbers is to their geometric mean as 12:13. The numbers are in the ratio

The value(s) of the parameter $\alpha (\alpha \ge 2)$ for which the area of region bounded by pair of straight lines $y^2-3y+2=0$ and the curves $y=\left[\alpha \right]x^2$, $y=\frac{1}{2}\left[\alpha \right]x^2$ is greatest, where $[.]$ denotes the greatest integer function, is

Prove that:

$2{\text{sin}}^2\frac{pi}{6}+{\text{cosec}}^2\frac{7\pi }{6}{\text{cos}}^2\frac{\pi }{3}=\frac{3}{2}$

Mohit has the following transactions , prepare accounting equation

$\begin{array}{cc}& \\ & \left(Rs\right)\\ \text{(a)Business started with cash}& 1,75,000\\ \text{(b) Purchased goods from Rohit}& 50,000\\ \text{(c) Sales goods on credit to Manish (Costing Rs 17,500)}& 20,000\\ \text{(d) Purchased furniture for office use}& 10,000\\ \text{(e) Cash paid to Rohit in full settlement}& 48,500\\ \text{(f) Cash received from Manish}& 20,000\\ \text{(g) Rent paid}& 1,000\\ \text{(h) Cash withdraw for personal use}& 3,000\end{array}$

If $x^2-3x+2$ is a factor of $x^4-a{x}^2+b$ then the equation whose roots are $a,b$ is

Which of the following correctly represents the +I effects of the substituents?

If roots of the cubic equation $x^3$ + a $x^2$ + 146x + c = 0 are three consecutive positive integers. Find the sum of the roots.

Let $(1+x-2x^2{)}^{20}={\sum }_{r=0}^{40}{a}_r{x}^r.then{a}_1+a_3+a_5+....+a_{39}$ is equal to:

Two rods of lengths a and b slides along the x-axis and y-axis respectively in such a manner that their ends are concyclic. The locus of the centre of the circle passing through the end points is

The degree measure of an arc whose measure is $\frac{7\pi }{12}$ radian, is

$\text{Find the value of x if x if}|x+2|$= 3

$6^{th}$ term in expansion of $(2x^2-\frac{1}{3x^2}{)}^{10}$ is

The equation of the transverse and conjugate axis of the hyperbola $16x^2-y^2+64x+4y+44=0$ are

Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.



$\begin{array}{cc}ColumnI& ColumnII\left(Typeof△ABC\right)\\ \text{a.}\cot \frac{A}{2}=\frac{b+c}{a}& \text{p. always right angled}\\ \text{b.}a\tan A+b\tan B=(a+b)\tan \frac{A+B}{2}& \text{q. always isosceles}\\ \text{c.}a\cos A=b\cos B& \text{r. may be right angled}\\ \text{d.}\cos A=\frac{\sin B}{2\sin C}& \text{s. may be right angled isosceles}\end{array}$

The set of values of $x,$ satisfying the inequality $\left|\frac{1}{|x|-3}\right|>\frac{1}{2}$ is

Tangents are drawn from different points on the line $x-y+10=0$ to the parabola $y^2=4x$. If the chords of contact pass through a fixed point, then the coordinates of the fixed point is

If cos A =

$\frac{\sqrt{3}}{2}$, then tan 3A =

Tangents are drawn from the point $(-8,0)$ to the parabola $y^2=8x$ touch the parabola at $P$ and $Q$. If $F$ is the focus of the parabola, then the area of the triangle $PFQ$ (in sq. units) is equal to

Concentric circles of radius $1,2,3,\dots ,100$ units are drawn such that the interior of the smallest circle is coloured red and the angular regions are coloured alternatively green and red, so that no two adjacent regions are of the same colour. Then the area of green region (in sq. units) is