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If the roots of $x^2-2x-a^2+1=0$ lie between the roots of $x^2-2(a+1)x+a(a-1)=0$, then the range of $a$ is

Evaluate the definite integrals.
${\int }_2^3\frac{x}{x^2+1}dx.$

If $f\left(x\right)=\{\begin{array}{cc}x+3,& x<3\\ 3x^2+1,& x\ge 3\end{array}$

then the value of $\underset{2}{\overset{5}{\int }}f\left(x\right)dx$ is:

Euclidean norm (length) of the vector $[4-2-6{]}^T$ is

If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then which of the following statement(s) is/are correct ?

(where $[.]$ denotes $G.I.F.$)

The ratio of area of incircle and circumcircle of quadrilateral formed by lines $x=1,x=5,y=-1,y=3$ is

The middle term of the binomial expansion of $(a+b{)}^n$ is the $6^{th}$ term.Find the possible values of n,if n is even

The value of $\int se{c}^{3}xdx$ will be

Question 11

Which of the following is not the graph of a quadratic polynomial?

The distance of the point (2, 3, - 5) from the plane x + 2y - 2z = 9 is:

If the roots of the equation 6 $x^2$ -5x + k = 0 are in the ratio 2:3, then k

Prove that the lines x=py+q,z=ry+s and x=p'y+q',z=r'y+s' are perpendicular, if pp'+n'+1=0.

If $A,B$ are two square matrices of order $n$ and $A$ and $B$ commute $(K$ be a real number), then which of the following is correct

$(secAsecB+tanAtanB{)}^2-(secAtanB+tanAsecB{)}^2=1$

If the function $f\left(x\right)=\{\begin{array}{cc}\frac{\sqrt{2+\cos x}-1}{(\pi -x{)}^2},& x\ne \pi \\ k,& x=\pi \end{array}$

is continuous, then $k$ is equal to

The value of $(1-i{)}^n{(1-\frac{1}{i})}^n$ is equal to

The equation of circle of radius $\sqrt{17}$ unit, with centre on the positive side of $x$-axis and through the point $(0,1)$ is

Law of cosine is applicable if we know

If the equation ${\cos }^4\theta +{\sin }^4\theta +\lambda =0$ has real solutions for $\theta$, then $\lambda$ lies in the interval:

In a cricket board, Anil Jain is a cricket team selector. He can select a cricket team, 11 from 17 players in which only 5 players can bowl.

In how many ways exactly 4 bowlers must include out of 11 players?

State whether each of the following statements are true or false. If the statement is false, re-write the given statement correctly

(i) If P = {m, n} and Q = {n, m}. then $P\times Q=$ {(m, n), (n, m)}

(ii) If A and B are non-empty sets, then $A\times B$ is a non - empty set of ordered pairs (x, y) such that $xϵB$ and $yϵA.$

(iii) If A = {1, 2}, B = {3, 4}, then $A\times (B\cap \varphi )=\varphi$

(ay³+by^​3/2+c) is it a quadratic expression in y^{​​​​​​3/2}