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If Q is the foot of the perpendicular of the point P(3, 6) on the line x - 2y + 4 = 0, then the equation of PQ is

Express $x^{\frac{3}{4}}({\log }_{3}x{)}^2+{\log }_{3}x-\frac{5}{4}=\sqrt{3}$​ in terms of t wheret = ${\log }_3$ x.

Find the distance between $P(x_1,y_1)$ and $Q(x_2,y_2)$ when: (i) PQ is parallel to the y-axis (ii) PQ is parallel to the x-axis.

The chord AB of the parabola $y^2=4ax$ cuts the axis of the parabola at C. If $A=(a{t}_1^2,2a{t}_1),B=(a{t}_2^2,2a{t}_2)$ and AC : AB = 1:3 then

Given are the observations of runs scored by batsmen between India - Sri Lanka combined. Find the mean deviation about mean for the scores of batsmen

$\begin{array}{ccccccc}Score& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60\\ numberofBatsmen& 12& 18& 27& 20& 17& 6\end{array}$

A five digit number divisible by 3 has to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is

The domain of the function $f\left(x\right)={\log }_{1/2}(x-\frac{1}{2})+{\log }_2\sqrt{4x^2-4x+5}$ is

If sum of the coefficient of the first, the second and the third term of the expansion of ${(x^2+\frac{1}{x})}^m$ is $46$, then the coefficient of the term that does not contain $x$ is

If $\cos \theta =-\frac{7}{25}$ and $\pi <\theta <\frac{3\pi }{2}$, then $\tan \theta$ is equal to

The value of $\underset{x\to a}{lim}\left(\right[\frac{100x}{sinx}]+[\frac{99sinx}{x}\left]\right)$,where [.] denotes the greatest integer function, is

If $\sqrt{2(x+24)}-\sqrt{x-7}\ge \sqrt{x+7},$ then $x\in$

Prove that:

sin x + sin 3x +sin 5x + sin 7x = 4 cos x cos 2x sin 4x

The domain of the function $f\left(x\right)=\frac{7}{\left[|x|\right]-5}$ is

(where $[.]$ denotes the greatest integer function)

An ordinary cube has 4 blank faces, one face marked 2 and another marked 3. Then the probability of obtaining 12 in 5 throws is:

The equation of the plane passing through the points (1,-1,2) and (2, -2 2) and which is perpendicular to the plane 6x -2y +2z =9 is

Find the equation to the chord of contact of tangents drawn from a point p(4, 3) to the hyperbola

$\frac{x^2}{16}-\frac{y^2}{9}=1$

If $a_1,a_2,\dots \dots a_n$ are positive real numbers whose product is a fixed number c, then the minimum value of $a_1+a_2+\dots \dots +a_{n-1}+2a_n$ is

If the first term of a G.P. is 5 and the common ratio is -5, then 3125 is the ____ term.

If $(1+x+2x^2{)}^{20}=a_0+a_1+a_2{x}^2+...+a_{40}{x}^{40},then{a}_1+a_3+a_5+...+a_{37}equals$ :

If $a_n=\sqrt{7+\sqrt{7+\sqrt{7}+....}}$ has n radical signs, then by the method of mathematical induction, which is true?

1 - 2$si{n}^2(\frac{\pi }{4}+\theta )$ =

An n-digit number is a positive number with exactly n digits. Nine hundred

distinct n-digit numbers are to be formed using only the three digits 2, 5 and

7. The smallest value of n for which this is possible is

In a triangle the sum of two sides is x and the product of the same sides is y. If $x^2-c^2=y$, where c is the third side of the triangle, then the ratio of the in radius to the circum – radius of the triangle is

Calculate mean and standard deviation :

$\begin{array}{cccccc}\text{Class Interval}& 0-10& 10-20& 20-30& 30-40& 40-50\\ \text{Frequency}& 8& 13& 16& 8& 5\end{array}$

The area (in sq. units) bounded by $y=\cos x,y=1+x$ and $x-$axis is

Let $y=y\left(x\right)$ be a solution of the differential equation, $\sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0,|x|<1$.

If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to:

Arjun and Karna had an archery competition where they had to shoot a target board with 7 concurrent circles dividing the board into 8 concurrent circular strips. Score for each outcome decreases by unity with centermost circle having score of 8. Dronacharya had difficulty in deciding who the winner was after the scores were recorded. Calculate the mean deviation about the score of innermost circle so as to decide who the winner is, assuming the winner is less deviated from the bull's eye.

$\begin{array}{ccccccccc}Points& 8& 7& 6& 5& 4& 3& 2& 1\\ Arjun\left(f_i\right)& 5& 7& 14& 23& 11& 10& 4& 3\\ Karna\left(f_i\right)& 5& 8& 11& 24& 14& 8& 5& 3\end{array}$

Point D, E are taken on the side BC of the triangle ABC, such that BD = DE = EC. If $\angle BAD=x,$ $\angle DAE=y,$ $\angle EAC=z,$ then the value of $\frac{sin(x+y)sin(y+z)}{sinxsinz}$

Solution of the equation $co{s}^{2}x\frac{dy}{dx}-\left(tan2x\right)y=co{s}^{4}x,|x|<\frac{\pi }{4},$ where $\left(\frac{\pi }{6}\right)=\frac{3\sqrt{3}}{8}$ is

In the sum of first n terms of an A.P. is $c{n}^2$. then the sum of squares of these n terms is

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ where $y_1,y_2<0,$ be the end points of the latus rectum of the ellipse $x^2+4y^2=4.$ Then equation(s) of the parabola with latus rectum $PQ$ is/are

If $\left|\frac{(x^2-x-6)(x-5)}{(x^2+1)(x-4)}\right|=-\frac{(x^2-x-6)(x-5)}{(x^2+1)(x-4)}$, then $x$ lies in

Which of the following is not a geometric progression?

For the sequence $1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,\dots ,$ the ${1025}^{\text{th}}$ term is

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for $n\ge 3.$ And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal. Further consider a matrix ${\cup }_{3\times 3}$ with its column as ${\cup }_1,{\cup }_2,{\cup }_3$ such that $A^{50}{\cup }_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],A^{50}{\cup }_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],A^{50}{\cup }_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ Then,



The value of $|\cup |$ equals

For every integer $n,$ let $a_n$ and $b_n$ be real numbers. Let function $f:\to$ be given by



$f\left(x\right)=\{\begin{array}{cc}{a}_n+\sin \pi x,& \text{for}x\in [2n,2n+1]\\ b_n+\cos \pi x,& \text{for}x\in (2n-1,2n)\end{array}$



for all integers $n$. If $f$ is continuous, then which of the following hold(s) for all n?

Values of ${}^{\prime }{m}^{\prime }$ such that the roots of the equation $2x^2-x-1=0$ lie inside the roots of the equation $x^2+(2m-m^2)x-2m^3=0,$ is

There was a survey conducted in a city about number of people reading newspaper A, B and C. There are 42% of people read newspaper A; 51% of people read newspaper B and 68% of people read paper C. 30% of people read both newspaper A and B. 28% reads B and C and 36% read C and A. 8% do not read any newspaper. Find the percentage of people who read all the three newspapers.

The number of solution(s) of ${\sec }^2\theta {\text{cosec}}^2\theta +2{\text{cosec}}^2\theta =8$ for $\theta \in [0,2\pi ]$ is