\[ \mathop {\lim }\limits_{x \to 2} \frac{x+3 x^2+5 x^3+7 x^4-166}{x-2}= \]
If \(\mathrm{f}(x)=\frac{10^x+7^x-14^x-5^x}{1-\cos x}, x \neq 0\) is continuous at \(x=0\), then the value of \(\mathrm{f}(0)\) is
If A and B are non-singular matrices of order 2 such that \(\quad(A B)^{-1}=\frac{1}{6}\left[\begin{array}{cc}-7 & -3 \\ 2 & 3\end{array}\right] \quad\) and \(A^{-1}=\frac{1}{3}\left[\begin{array}{cc}4 & 3 \\ -1 & 0\end{array}\right]\) then \(B^{-1}=\)
Let \(\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}\) and \(\bar{c}=\hat{j}-\hat{k}\) be three vectors such that \(\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}\) and \(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0\). If the length of projection vector of the vector \(\overline{\mathrm{b}}\) on the vector \(\overline{\mathrm{a}} \times \overline{\mathrm{c}}\) is \(l\), then the value of \(3 l^2\) is
In \(\triangle \mathrm{ABC}\), with usual notations, if \(\cos \frac{B}{2}=\sqrt{\frac{c+a}{2 a}}\), then \(a^2=\)
The joint equation of two lines passing through \((-2,3)\) and parallel to the bisectors of the angle between the co-ordinate axes is
The distance of the point \((1,2)\) from the line \(x+y=0\) measured parallel to the line \(3 x-y=2\) is
Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC . If \(A \equiv(2,4)\), then the length of the median through A is
Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}, \overline{\mathrm{d}}\) are vectors such that \(\overline{\mathrm{a}} \times \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}\) and \(\overline{\mathrm{c}} \times \overline{\mathrm{d}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}\) and if \(\left|\begin{array}{ll}\overline{\mathrm{a}} \cdot \overline{\mathrm{c}} & \overline{\mathrm{b}} \cdot \overline{\mathrm{c}} \\ \overline{\mathrm{a}} \cdot \overline{\mathrm{d}} & \overline{\mathrm{b}} \cdot \overline{\mathrm{d}}\end{array}\right|=0\), then \(\lambda=\)
A fair coin is tossed 100 times. The chance of getting a head even number of times is
The assets of a person reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹ 10 lakhs and due to loss they reduce to ₹ 10000 after 3 years, then the number of years required for the person to be bankrupt will be
\[ 3 \tan ^6 10^{\circ}-27 \tan ^4 10^{\circ}+33 \tan ^2 10^{\circ}= \]
The lines \(\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda(\overline{\mathrm{b}} \times \overline{\mathrm{c}})\) and \(\overline{\mathrm{r}}=\overline{\mathrm{c}}+\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})\) will intersect if
The minimum distance and maximum distance of the point \(\mathrm{P}(2,-7)\) from the circle \(x^2+y^2-14 x-10 y-151=0\) are respectively _______units
If the line \(2 x+y=\mathrm{k}\) passes though the point which divides the line segment joining the points \((1,1)\) and \((2,4)\) internally in the ratio \(3: 2\) then, \((k+1):(k-1)=\)
If the statement pattern \((p \wedge q) \rightarrow(r \vee \sim s)\) is false, then the truth values of \(p, q, r\) and \(s\) are respectively
The Y-intercept of the common tangent to the parabolas \(y^2=32 x\) and \(x^2=108 y\) is
The line of intersection of the planes \(\bar{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1 \quad\) and \(\quad \bar{r} \cdot(\hat{i}+4 \hat{j}-2 \hat{k})=2 \quad\) is parallel to the vector
If \(f(x)= \begin{cases}\frac{8^x-4^x-2^x+1}{x^2}, & \text { if } x>0 \\ e^x \sin x+x+\lambda \log 4, & \text { if } x \leqslant 0\end{cases}\)
is continuous at \(x=0\) then the value of \(1000 \mathrm{e}^\lambda=\)
The L.P.P. , minimize \(z=30 x+20 y, x+y \leq 8\), \(x+2 y \geq 4,6 x+4 y \geq 12, x \geqslant 0, y \geqslant 0\) has
If the distance of the point \(\mathrm{P}(1,-2,1)\) from the plane \(x+2 y-2 z=\alpha\), where \(\alpha>0\) is 5 units, then the foot of the perpendicular from P to the plane is
The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two 'I's do not come together is
The complex numbers \(\sin x+i \cos 2 x\) and \(\cos x\) - \(\mathrm{i} \sin 2 x,(\mathrm{i}=\sqrt{-1})\) are conjugate to each other for,
The logical statement
\[ [\sim(\sim p \vee q) \vee(p \wedge r) \wedge(\sim q \wedge r)] \]
is equivalent to
The equation of the plane containing the line \(\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-2}\) and the point \((0,5,0)\) is