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Magnitude of the vector joining the points $P(a{t}_1^2,a{t}_2^2,a{t}_3^2)$ and $Q(2a{t}_1,2a{t}_2,2a{t}_3)$ is

The value of $\underset{0}{\overset{1}{\int }}\frac{\psi \left(x\right)}{\psi \left(x\right)+\psi \left(\right[x+1]-x)}dx$ is

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

Study the following information and answer the questions given below.

1. A, B, C, D, E, and F are six members of a family.

2. One is a student, one housewife, one doctor, one teacher, one lawyer and one engineer.

3. There are two married couples in the family.

4. B is a teacher and the mother of C.

5. D is the grandmother of C and is a housewife.

6. F is a lawyer and is the father of A.

7. C is the brother of A.

8. E is the father of F and is a doctor.

Q67. Which one is the student?

The value of the determinant $\left|\begin{array}{ccc}15!& 16!& 17!\\ 16!& 17!& 18!\\ 17!& 18!& 19!\end{array}\right|=$

The order of differential equation of all circles of given radius ‘a’ is …….

$IfF\left(x\right)=\left[\begin{array}{ccc}cosx& -sinx& 0\\ sinx& cosx& 0\\ 0& 0& 1\end{array}\right],thenF\left(x\right).F\left(y\right)=$

The area of the region $(x,y)$ : $xy\le 8,1\le y\le x^2$ is

If $f\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{\ln t}{1+t}dt,$ then

The maximum value of $(\sin \theta \cos \theta {)}^{42}$ is

The locus of orthocentre of the triangle formed by three tangents to the parabola $4x^2–8x–3y+10=0$ is

200 people were surveyed to see which of the four mentioned snacks below that people prefer. The data is as shown in the table.

$$\begin{array}{cc}Snack& Numberofpeople\\ Pizza& 42\\ Sandwich& 50\\ Burger& 74\\ Icecream& 34\end{array}$$

What is the probability that a person

If $1+2+2^2+2^3+\dots +2^{1999}$ is divided by $5$, then the remainder is

Given the terms $a_{10}=\frac{3}{512}$ and $a_{15}=\frac{3}{16384}$ of a geometric sequence, find the exact value of the term $a_{30}$ of the sequence.

Which of the following are the properties of transpose :-

If $\alpha ,\beta$ are zeroes of a polynomial $6x^2+x–2$, then the polynomial whose zeroes are $2\alpha +3\beta$ and $3\alpha +2\beta$ , is .

Determine order and degree (when defined) of differential equations.
y'''+2y''+y'=0.

The value of k so that the lines 2x – 3y + k = 0, 3x – 4y – 13 = 0 and 8x – 11y – 33 = 0 are concurrent, is

If $C_r={}^n{C}_{r}and(C_0+C_1)(C_1+C_2)....(C_{n-1}+C_n)=k\frac{(n+1{)}^n}{n!}$ , then the value of k is :

If $\int \frac{({\cos }^{3}x+{\cos }^{5}x)}{({\sin }^{2}x+{\sin }^{4}x)}dx=A\sin x+B\text{cosec}x+C{\tan }^{-1}\left(\sin x\right)+K,$ where $A,B,C$ are fixed constants and $K$ is constant of integration, then the value of $A+B-C$ is equal to

1. -13

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then $k$ is equal to

Determine whether an inclusive 'OR' or exclusive 'OR' is used. Also, give reason for your answer. 'Students can take Hindi or Sanskrit as their third language'.

The probability that atleast one of the events $A$ and $B$ occurs, is $0.6$. If $A$ and $B$ occur simultaneously with probability $0.2$. The value of $P\left(\overline{A}\right)+P\left(\overline{B}\right)$is

The product of perpendiculars drawn from the origin to the lines represented by the equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0,$ will be

If $f\left(x\right)=\cos x-1,$ then select the correct graph of $|f\left(x\right)|.$

If the point $P(4,2)$ undergoes the following transformations successively.

$i)$ Reflection about the line $y=x,$

$ii)$ Translation through a distance of $5$ units along the positive $x-$ axis,

$iii)$ Reflection about $y=0$.

Then the coordinates of final position of $P$ are