Condition for Three Vectors to Form a Triangle

Q. Let $△xyz$ be a triangle and $\overrightarrow{a}=\overrightarrow{YZ},\overrightarrow{b}=\overrightarrow{ZX},\overrightarrow{c}=\overrightarrow{XY}$. If $|\overrightarrow{a}|=6,|\overrightarrow{c}|=14,\overrightarrow{a}\cdot \overrightarrow{b}=30,$ then the values of $\frac{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}{|}^2}{100}$ is

Q. In a triangle $ABC,$ if $2\overrightarrow{AC}=3\overrightarrow{CB}$ and $2\overrightarrow{OA}+3\overrightarrow{OB}=\lambda \cdot \overrightarrow{OC},$ then $\lambda$ is

Q. If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-collinear vectors and $ABC$ is a triangle whose side length satisfying $(5a-12b)\overrightarrow{x}+(26b-10c)\overrightarrow{y}+(12c-13a)(\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0},$ then which of the following is/are correct ?

1. $△ABC$ is an acute angle triangle
2. $△ABC$ is right angle triangle
3. $△ABC$ is an obtuse angle triangle
4. $\angle B={90}^{\circ }$

Q. Let $△xyz$ be a triangle and $\overrightarrow{a}=\overrightarrow{YZ},\overrightarrow{b}=\overrightarrow{ZX},\overrightarrow{c}=\overrightarrow{XY}$. If $|\overrightarrow{a}|=6,|\overrightarrow{c}|=14,\overrightarrow{a}\cdot \overrightarrow{b}=30,$ then the values of $\frac{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}{|}^2}{100}$ is

Q. If , then is it true that ? Justify your answer.

Q. In triangle ABC which of the following is not true: A. B. C. D.

Q.

3 vectors $\overline{a},\overline{b},\overline{c}$ can always form the sides of a triangle if $\overline{a}+\overline{b}+\overline{c}=0$

1. True

2. False

Q.

The given vectors can form sides of a triangle $\hat{i}-\hat{j}-\hat{k},-2\hat{i}+3\hat{j}-\hat{k},\hat{i}-2\hat{j}+2\hat{k}$

1. True

2. False

Q. If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-parallel vectors and $ABC$ is a triangle with side lengths $a,b$ and $c$ satisfying $(20a-15b)\overrightarrow{x}+(15b-12c)\overrightarrow{y}+(12c-20a)(\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0}$, then triangle $ABC$ is

1. an acute-angled triangle
2. an obtuse-angled triangle
3. a right-angled triangle
4. an isosceles triangle

Q. If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-parallel vectors and $ABC$ is a triangle with side lengths $a,b$ and $c$ satisfying $(20a-15b)\overrightarrow{x}+(15b-12c)\overrightarrow{y}+(12c-20a)(\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0}$, then triangle $ABC$ is

1. an acute-angled triangle
2. an obtuse-angled triangle
3. a right-angled triangle
4. an isosceles triangle

Q. Let $\overrightarrow{a}$ be a unit vector and $\overrightarrow{b}$ be a non-zero vector not parallel to $\overrightarrow{a}$. If two sides of a triangle are represented by the vectors $\sqrt{3}(\overrightarrow{a}\times \overrightarrow{b})$ and $\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{a}$, then the angles of the triangle are

1. ${30}^0,{90}^0,{60}^0$
2. ${45}^0,{45}^0,{90}^0$
3. ${60}^0,{60}^0,{60}^0$
4. $noneofthese$

Q. Determine the missing parts of triangle $ABC$ from the given information.
$a=7,b=24,c=26$

Q. The value of ${\int }_1^a\left[x\right]f^{\prime }\left(x\right)dx,a>1$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is

1. $af\left(\right[a\left]\right)-\left(f\right(1)+f(2)+\dots \dots +f(a\left)\right)$
2. $\left[a\right]f\left(\right[a\left]\right)-\left(f\right(1)+f(2)+\dots \dots +f(a\left)\right)$
3. $af\left(a\right)-\left(f\right(1)+f(2)+\dots \dots +f(a\left)\right)$
4. $\left[a\right]f\left(a\right)-\left(f\right(1)+f(2)+\dots \dots +f(a\left)\right)$

Q.

Fill $can/cannot$in the given blank.

We ________ construct $∆ABC$when $AC=2.5cm,$$AB=6cm$and $BC=4.2cm$

Q. Assertion :If $\left[x\right]$ stands for greatest integer function, then the value of ${\int }_3^{15}\frac{\left[x^2\right]dx}{[x^2-36x+324]+\left[x^2\right]}$ is $6$ Reason: ${\int }_a^b\frac{f\left(x\right)}{f\left(x\right)+f(a+b-x)}dx=\frac{b-a}{2}$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. $(-3x^2+5x-2)-2(x^2-2x-1)$
If the expression above is rewritten in the form $a{x}^2+bx+c$, where $a,b$ and $c$ are constants, what is the value of $b$?

1. $9$
2. $1$
3. $3$
4. $8$

Q. $\int (x^2+4x+3)dx$

Q. The value of ${\int }_0^2[x+[x+\left[x\right]\left]\right]dx$, (where $[.]$ denotes the greatest integer function), is equal to

1. $3$
2. $2$
3. none of these
4. $-3$

Q.

The given vectors can form sides of a triangle $\hat{i}-\hat{j}-\hat{k},-2\hat{i}+3\hat{j}-\hat{k},\hat{i}-2\hat{j}+2\hat{k}$

1. True

2. False

Q. If ABCDEF is a regular hexagon and $\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}=\lambda \overrightarrow{AD}$, then $\lambda$ =

1. 2
2. 3
3. 4
4. 6

Q. If ABCDEF is a regular hexagon and $\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}=\lambda \overrightarrow{AD}$, then $\lambda$ =

1. 2
2. 6
3. 3
4. 4

Q. Evaluate (using formulae) $\frac{0.3\times 0.3\times 0.3+0.1\times 0.1\times 0.1}{0.3\times 0.3-0.3\times 0.1+0.1\times 0.1}$

1. $0.5$
2. $0.4$
3. $0.3$
4. $0.2$

Q. If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-parallel vectors and $ABC$ is a triangle with side lengths $a,b$ and $c$ satisfying $(20a-15b)\overrightarrow{x}+(15b-12c)\overrightarrow{y}+(12c-20a)(\overrightarrow{x}\times \overrightarrow{y})=\overrightarrow{0}$, then triangle $ABC$ is

1. an acute-angled triangle
2. an obtuse-angled triangle
3. a right-angled triangle
4. an isosceles triangle

Q. $△$ABC with vertices A (-2, 0), B (2, 0) and C (0, 2) is similar to $△DEF$ with vertices D (-4, 0) E (4, 0) and F (0, 4).

Q.

3 vectors $\overline{a},\overline{b},\overline{c}$ can always form the sides of a triangle if $\overline{a}+\overline{b}+\overline{c}=0$

1. True

2. False