If \(a \in R\) and the equation \( - 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0\) (where \([x]\) denotes the greatest integer \(\leq\,x\))has no integral solution ,then all possible values of \(a\) lie in the interval

Let \(f\) and \(g\) be twice differentiable functions on \(R\) such that \(f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x\) \(f^{\prime}(1)=4 g^{\prime}(1)-3=9\) \(f(2)=3 g(2)=12\) Then which of the following is NOT true ?

If the coefficients of \(x^2\) and \(x^3\) are both zero, in the expansion of the expression \((1 + ax + bx^2) (1 -3x)^{t5}\) in powers of \(x\), then the ordered pair \((a, b)\) is equal to

Suppose \({\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right)\) , where \(\left| x \right| < \frac{1}{{\sqrt 3 }}\), then one of the value of \( y \) is 

Let \(f(x)=a x^{2}+b x+c\) be such that \(f(1)=3, f(-2)\) \(=\lambda\) and \(f (3)=4\). If \(f (0)+ f (1)+ f (-2)+ f (3)=14\), then \(\lambda\) is equal to\(...\)

In an increasing geometric progression ol positive terms, the sum of the second and sixth terms is \(\frac{70}{3}\) and the product of the third and fifth terms is \(49\). Then the sum of the \(4^{\text {th }}, 6^{\text {th }}\) and \(8^{\text {th }}\) terms is :-

If \(\frac{{dy}}{{dx}} + \frac{3}{{{{\cos }^2}\,x}}\,y = \frac{1}{{{{\cos }^2}\,x}},\) \(x \in \left( {\frac{{ - \pi }}{3},\frac{\pi }{3}} \right)\) and \(y\left( {\frac{\pi }{4}} \right) = \frac{4}{3}\), then \(y\left( { - \frac{\pi }{4}} \right)\) equals

If \(x,y,z\)  are in \(A.P.\) and  \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\) and \({\tan ^{ - 1}}z\) are also in other \(A.P.\) then  . . . 

Let \(y=y(x)\) be the solution of the differential equation \(x^3 d y+(x y-1) d x=0, x>0\), \(y\left(\frac{1}{2}\right)=3-e\). Then \(y(1)\) is equal to

Let \(P_{1}, P_{2}, \ldots \ldots, P_{15}\) be \(15\) points on a circle. The number of distinct triangles formed by points \(P_{i}, P_{j}, P_{k}\) such that \(i+j+k \neq 15\), is :

Let sets \(A\) and \(B\) have \(5\) elements each. Let the mean of the elements in sets \(A\) and \(B\) be \(5\) and \(8\) respectively and the variance of the elements in sets \(A\) and \(B\) be \(12\) and \(20\) respectively \(A\) new set \(C\) of \(10\) elements is formed by subtracting \(3\) from each element of \(A\) and adding 2 to each element of B. Then the sum of the mean and variance of the elements of \(C\) is \(.......\).

Let \(A=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha \gt 0\), such that \(\operatorname{det}(A)=0\) and \(\alpha+\beta=1\). If I denotes \(2 \times 2\) identity matrix, then the matrix \((1+\mathrm{A})^8\) is:

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is