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Factorize $\sqrt{5}{x}^2+8x+3\sqrt{5}=0$ into the form $\sqrt{5}{x}^2+px+qx+3\sqrt{5}=0$. What is the value of $p^2+q^2$ ?

If range of the numbers $x_1,x_2,x_3,x_4,x_5$ is $r$ where $x_i<x_{i+1}$. If each number is multiplied by $p$ and then $k$ is subtracted from each number then range of the new numbers is

Cofactor of 4 in the determinant $\left|\begin{array}{ccc}1& 2& -3\\ 4& 5& 0\\ 2& 0& 1\end{array}\right|$ is equal to ........................

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

If $f\left(x\right)=x^3+a{x}^2+bx+5{\sin }^{2}x$ is a strictly increasing function on the set of real numbers, then $a$ and $b$ must satisfy the relation

Find the equation of normal to the parabola $y^2=4axat(a{t}^2,2at)$ in terms of t, a.

The Expansion ${[x^2+{(x^6-1)}^{1/2}]}^5+{[x^2-{(x^6-1)}^{1/2}]}^5$ is a polynomial of degree.

1. 0.95

If the objective function $z=x+2y$ has to be optimized in subject to constraints : $x\ge 3,x+y\le 5,x+2y\ge 6,y\ge 0.$ Then which of the following(s) is correct

The number of irrational terms in the expansion of $(4^{1/5}+7^{1/10}{)}^{45}$ is

If $1+sinx+si{n}^{2}x+.........................\infty =4+2\sqrt{3},0<x<\pi$ and $x\ne \frac{1}{2}$, then x is equal to

In the given question, a part or the complete sentence is printed in bold. Below each sentence, some phrases are given which can substitute the bold part of the sentence. Find out the phrase which can correctly substitute that part of the sentence. If the sentence is correct as it is, mark the answer as 'No correction required' or 'No Improvement'.

You need not come unless you want to.

The range of the function

$f\left(x\right)={\log }_{\sqrt{5}}(3+\cos (\frac{3\pi }{4}+x)+\cos (\frac{\pi }{4}+x)+\cos (\frac{\pi }{4}-x)-\cos (\frac{3\pi }{4}-x))$

is

If common tangents of $x^2+y^2=r^2$ and $\frac{x^2}{16}+\frac{y^2}{9}=1$ form a square, then the length of the diagonal of the square is

Find the sum of the coefficients of two middle terms in the binomial expansion of $(1+x{)}^{2n-1}$

10^{2n
– 1} + 1 is divisible by 11.

If $2y={\left({\cot }^{-1}\left(\frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x}\right)\right)}^2,x\in (0,\frac{\pi }{2})$ then $\frac{dy}{dx}$ is equal to:

If $a,b,c$ are positive real numbers such that $a^{{\log }_{3}7}=27,b^{{\log }_{11}7}=49\text{and}{c}^{{\log }_{11}25}=\sqrt{11}$, then the middle digit in the value of $(a^{({\log }_{3}7{)}^2}+b^{({\log }_{7}11{)}^2}+c^{({\log }_{11}25{)}^2})$ equals to

Find the mean deviation about the mean for the data

Which of the following statements is/are correct statements?
1. Centroid of a triangle is the point of concurrency of medians
2. Incentre of a triangle is the point of concurrency of perpendicular bisectors of the sides of the triangle
3. Circumcentre of a triangle is the point of concurrency of internal bisectors of the angles of the triangle
4. Orthocentre of a triangle is the point of concurrency of altitudes of the triangle drawn from one vertex to opposite side
5. A triangle can have only one excentre

a/x-a + b/x-a=2c/x-c

Find the range of ${\sin }^{-1}\left(x\right)-{\cos }^{-1}\left(x\right)$

Simplified form of $(1-i{)}^{50}$ is

Points A,B,C and D are chosen on a semicircle and quadrilateral ABCD is drawn.

Then tan A + tan B + tan C + tan D is

Compute the indicated product.
$$\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]$$$$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$$

The acute angle between the pair of straight lines passing through $(-6,-8)$ and also through the points which divide the line $2x+y+10=0$ enclosed between coordinate axes in the ratio $1:2:2$ in the direction from the point of intersection with the $x-$axis to the point of intersection with $y-$axis is

If a>0, b>0, c>0 are in G.P., then $lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ are in

A transfer function has two zeros at infinity. Then the relation between the numerator degree (N) and the denominator degree (M) of the transfer function is

A father has 3 children with atleast one boy. The probability that he has 2 boys and one girl is: