If \(\overrightarrow{A} \times \overrightarrow{B} = \overrightarrow{C} + \overrightarrow{D}\), then which of the following statement is correct?

1.	\(\overrightarrow B\) must be perpendicular to \(\overrightarrow C\)
2.	\(\overrightarrow A\) must be perpendicular to \(\overrightarrow C\)
3.	Component of \(\overrightarrow C\) along \(\overrightarrow A\) = Component of \(\overrightarrow D\) along \(\overrightarrow A\)
4.	Component of \(\overrightarrow C\) along \(\overrightarrow A\)  = - (Component of \(\overrightarrow D\) along \(\overrightarrow A\))

What is the maximum value of \(5\sin\theta-12\cos\theta \text{?}\)
1. \(12\)

2. \(17\)

3. \(7\)

4. \(13\)

A block of weight \(W\) is supported by two strings inclined at \(60^{\circ}\) and \(30^{\circ}\) to the vertical. The tensions in the strings are \(T_1\) and \(T_2\) as shown. If these tensions are to be determined in terms of \(W\) using the triangle law of forces, which of these triangles should you draw? (block is in equilibrium):

1.		2.
3.		4.

The magnitude of the resultant of two vectors of magnitude \(3\) units and \(4\) units is \(1\) unit. What is the value of their dot product?

1. \(-12\) units

2. \(-7\) units

3. \(-1\) unit

4. \(0\)

The value of $\int \frac{1}{x+1}d\mathrm{x\; is:}$

1. ln (\(x\) + 1) + C

2. ${\left(x+1\right)}^{-2}+C$

3. $\frac{1}{{\left(x-1\right)}^2}+C$

4. ln (\(x\) – 1) + C

If \(\overrightarrow {A} = 2\hat{i} + \hat{j} - \hat{k},\) \(\overrightarrow {B} = \hat{i} + 2\hat{j} + 3\hat{k},\) and \(\overrightarrow {C} = 6 \hat{i} - 2\hat{j} - 6\hat{k},\) then the angle between \(\left(\overrightarrow {A} + \overrightarrow{B}\right)\) and \(\overrightarrow{C}\) will be:
1. \(30^{\circ}\)
2. \(45^{\circ}\)
3. \(60^{\circ}\)
4. \(90^{\circ}\)

The maximum or the minimum value of the function \(y= 25x^{2}-10x +5\) is:
1. \(y_{\text{min}}= 4\)
2. \(y_{\text{max}}= 8\)
3. \(y_{\text{min}}= 8\)
4. \(y_{\text{max}}= 4\)$\mathrm{}$

The unit vector perpendicular to vectors \(\overrightarrow a= \left(3 \hat{i}+\hat{j}\right)  \) and \(\overrightarrow B = \left(2\hat i - \hat j -5\hat k\right)\) is:
1. \(\pm \frac{\left(\right. \hat{i} - 3 \hat{j} + \hat{k} \left.\right)}{\sqrt{11}}\)
2. \(\pm \frac{\left(3 \hat{i} + \hat{j}\right)}{\sqrt{11}}\)
3. \(\pm \frac{\left(\right. 2 \hat{i} - \hat{j} - 5 \hat{k} \left.\right)}{\sqrt{30}}\)
4. None of these

If \(y = t^3+1\) and \(x = t^2+3,\) what is the value of \(\dfrac{dy}{dx}?\)
1. \(\dfrac{t^2}{3}\)
2. \(\dfrac{t}{2}\)
3. \(\dfrac{3t}{2}\)
4. \(t^2\)

The velocity of a body moving along the \(x\)-axis varies with \(x\) as \(v   =   \left(x^{3} -x^{2}\right)\) m/s. Find the acceleration of the body at \(x= 2~\text{m}\), if the acceleration is defined as \(a = v\frac{dv}{dx}\).
1. \(132~\text{m/s}^2\)
2. \(32~\text{m/s}^2\)
3. \(8~\text{m/s}^2\)
4. \(4~\text{m/s}^2\)