If \(\lambda \hat{i}+2\lambda \hat{j}+2\lambda \hat{k}\) is a unit vector, then the value of \(\lambda\) is

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

\(lim_{x\rightarrow o}{8^x-4^x-2^x+1^x\over x^2}=\)

\(lim_{n \rightarrow \infty}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

The function \(f(x)= \begin{cases}\frac{x^{2}-1}{x^{3}+1} & x \neq-1 \\ P & x=-1\end{cases}\)is not defined for x = −1. The value of f(−1) so that the function extended by this value is continuous is

Let \(\vec a\) and \(\vec b\) be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are \(​​​​\frac{\vec{a}+2 \vec{b}}{3} \text { and } \frac{\vec{b}+2 \vec{a}}{3} \text {. }\)

Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\) - 3\(\hat{k}\) + 4\(\hat{j}\)

Find  \(\overrightarrow{a}\).\(\overrightarrow{b}\)when \(\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}\) and \(\overrightarrow{b}=3\hat{i}-4\hat{j}-2\hat{k}\)

Let \(f(x)= \begin{cases}x+1, & x>0 \\ x-1, & x<0\end{cases}\)
Verify the existence of limit as x\(\rightarrow\)0.

Evaluate the following limits :
\(lim_{x\rightarrow1}{x^m-1\over x^n-1}\) ,m and n are integers.

If \(\overrightarrow{PO}\) +\(\overrightarrow{OQ}\) = \(\overrightarrow{QO}\) +\(\overrightarrow{OR}\), prove that the points P, Q, R are collinear.

Show that the vectors 2\(\hat{i}\) − \(\hat{j}\) + \(\hat{k}\), 3 \(\hat{i}\) − 4\(\hat{j}\) - 4\(\hat{k}\),  \(\hat{i}\) − 3\(\hat{j}\) - 5 \(\hat{k}\) form a right angled triangle.

If \(\vec{a}=2 \hat{i}+3 \hat{j}-4 \hat{k}, \vec{b}=3 \hat{i}-4 \hat{j}-5 \hat{k} \text {, and } \vec{c}=-3 \hat{i}+2 \hat{j}+3 \hat{k} \text {, }\) find the magnitude and direction cosines of \((i)\ \vec{a}+\vec{b}+\vec{c}\ (ii)\ 3 \vec{a}-2 \vec{b}+5 \vec{c}.\)

Calculate \(\lim _{ x\rightarrow0}{|x| } \).

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
\(lim_{x\rightarrow1}(x^2+2)\)

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
\(lim_{x\rightarrow{0}}sec \ x\)

Evaluate the following limits :
\(lim_{x-1}{3\sqrt{7+x^3}-\sqrt{3+x^2}\over x-1}\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{2^x-3^x\over x}\)

At the given point x_o discover whether the given function is continuous or discontinuous citing the reasons for your answer :\(x_{0}=3, f(x)= \begin{cases}\frac{x^{2}-9}{x-3}, & \text { if } x \neq 3 \\ 5, & \text { if } x=3\end{cases}\)

Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of . \(f(x)= \begin{cases}(x-1)^{3}, & \text { if } x<0 \\ (x+1)^{3}, & \text { if } x \geq 0\end{cases}\)

Let A, B and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that \(\overrightarrow{AD}\) + \(\overrightarrow{BE}\) +\(\overrightarrow{CF}\) = \(\overrightarrow{0}\).

Calculate \(lim_{x\rightarrow0}{1\over (x^2+x^3)}\)

Evaluate : \(lim_{x \rightarrow {\pi\over 4}}{4\sqrt{2}-(cos \ x+sin \ x)^5\over 1-sin 2x}\)

State how continuity is destroyed at x = x _o for each of the following graphs.

State how continuity is destroyed at x= x _o for each of the following graphs.