Operations on Rational Numbers using PEMDAS Rule

Q.

If $a=2\hat{i}+3\hat{j}-\hat{k},b=-\hat{i}+2j-4\hat{k}$and $c=\hat{i}+\hat{j}+\hat{k},$then what is the value of $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})$$?$

1. $47$

2. $74$

3. $-74$

4. None of these

Q.

A piece of wire $\frac{7}{8}m$ long broke into two pieces. One piece was $\frac{1}{4}m$ long. How long is the other piece $?$

Q.

Convert $15$ years $3$ months into year.

Q.

On the set $Q$of all rational numbers the operation $*$ which is both associative and commutative is given by $a*b$, is

1. $a+b+ab$

2. $a^2+b^2$

3. $ab+1$

4. $2a+3b$

Q.

On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are $20$ students in the class, then the total number of greeting cards exchanged by the students is

1. ${}^{20}C_2$

2. $2\times {}^{20}C_2$

3. $2\times {}^{20}P_2$

4. None of these

Q.

If $\overrightarrow{a}=\alpha \hat{i}+\beta \hat{j}+3k,\stackrel{\rightharpoonup }{b}=–\beta \hat{i}–\alpha \hat{j}–k$, $\overrightarrow{c}=\hat{i}-2\hat{j}$ such that $\overrightarrow{a}.\overrightarrow{b}=1$ and $\overrightarrow{b.}\overrightarrow{c}=-3$then $\frac{1}{3}\left(\right(\overrightarrow{a}\times \overrightarrow{b}).\overrightarrow{c})$ is equal to

Q.

From $12$books, the difference between the number of ways a selection of $5$ books when one specified book is always excluded and one specified book is always included, is equal to:

1. $64$

2. $118$

3. $132$

4. $330$

5. $462$

Q.

Find: $3÷2\frac{1}{3}$.

Q.

For non-negative integers $s$ and$r,$ let

$$\begin{gathered} \left(\begin{array}{c}\mathrm{s}\\ \mathrm{r}\end{array}\right)=\left\{\frac{\mathrm{s}!}{\mathrm{r}!\left(\mathrm{s}-\mathrm{r}\right)!}\begin{array}{l}\mathrm{if}\mathrm{r}\le \mathrm{s}\\ \mathrm{if}\mathrm{r}>\mathrm{s}\end{array}\right. \\ \mathrm{For}\mathrm{positive}\mathrm{integers}\mathrm{m}\mathrm{and}\mathrm{n},\mathrm{let} \\ \mathrm{g}\left(\mathrm{m},\mathrm{n}\right)=\sum _{\mathrm{p}=0}^{\mathrm{m}+\mathrm{n}}\frac{\mathrm{f}\left(\mathrm{m},\mathrm{n},\mathrm{p}\right)}{\left(\begin{array}{c}\mathrm{n}+\mathrm{p}\\ \mathrm{p}\end{array}\right)} \\ \mathrm{Where}\mathrm{for}\mathrm{any}\mathrm{nonnegative}\mathrm{integer}\mathrm{p}, \\ \mathrm{f}\left(\mathrm{m},\mathrm{n},\mathrm{p}\right)=\sum _{\mathrm{i}=0}^{\mathrm{p}}\left(\begin{array}{c}\mathrm{m}\\ \mathrm{i}\end{array}\right)\left(\begin{array}{c}\mathrm{n}+\mathrm{i}\\ \mathrm{p}\end{array}\right)\left(\begin{array}{c}\mathrm{p}+\mathrm{n}\\ \mathrm{p}-\mathrm{i}\end{array}\right). \end{gathered}$$

Then which of the following statements is/are TRUE?

1. $(m,n)=(n,m)forallpositiveintegersm,n$

2. $(m,n+1)=(m+1,n)forallpositiveintegersm,n$

3. $(2m,2n)=2(m,n)forallpositiveintegersm,n$

4. $(2m,2n)=({(m,n))}^{2}forallpositiveintegersm,n$

Q.

In a park, there is a trail of $\frac{2}{3}$$km$ length, There are $6$ parts of the trail and each part has the same length. How long is each part of the trail?

Q.

Mayank is going on a holiday trip. He wants to pack $3$ t-shirts from $5$ T-shirts he has. In how many ways can he make his choice

Q.

The product of two numbers is $435375$ .If one number is $225$ , find the another number.

Q.

Multiply and express as a mixed fraction: $7\times 2\frac{1}{4}$.

Q.

Convert the following mixed fraction into an improper fraction $1\frac{2}{7}$

Q.

How do you tell if a word problem is a permutation or combination $?$

Q.

Find the sum or difference of the following:

$\frac{5}{6}+\frac{4}{9}$

Q.

$8kg$ grapes cost $Rs196.80$. Find the cost of $1kg$ grapes$?$

Q.

Write the ratios in ascending order:

$2:3,3:7,5:6,11:14.$

Q.

$\frac{5}{12}of$₹144 is

1. ₹55

2. ₹60

3. ₹65

4. ₹70

Q.

$$\begin{gathered} \mathrm{Choose}\mathrm{an}\mathrm{appropriate}\mathrm{option}\mathrm{and}\mathrm{fill}\mathrm{in}\mathrm{the}\mathrm{blanks}: \\ 1250\mathrm{paise}=\mathrm{Rs}..................\left[1.25/12.5\right] \end{gathered}$$

Q.

Express improper fractions as mixed fraction :

$\frac{9}{4}$

Q.

Encircle the odd one of the following.

1. $(-1)\times (-1)$

2. $(-1)\times (-1)\times (-1)$

3. $(-1)\times (-1)\times (-1)\times (-1)$

4. $(-1)\times (-1)\times (-1)\times (-1)\times (-1)\times (-1)$

Q.

Convert the following into vulgar fractions: $0.\overline{2}$

Q.

A piece of rope $4.8m$ long is cut into $12$ equal small pieces. What is the length of each small piece?

Q.

Varun has $10$cards with numbers $1$to $10$. In how many ways can a total of $15$be made using only three cards at a time?

Q.

Convert the following into an improper fraction.

$11\frac{5}{8}$

Q.

Express the given as improper fractions :

$5\frac{6}{7}$

Q.

In an international seminar on Oncology, surgeons from different countries participated. $\frac{1}{8}$ of the participants were from the U.S, $\frac{1}{5}$ from UK, $\frac{1}{12}$ from other European countries and the remaining participants were Indian. If $71$ Indian surgeons participated, find the total number of surgeons who attended the seminar.

Q.

Convert the following improper fractions into mixed fractions:
(a) $\frac{25}{13}$ (b) $\frac{-46}{12}$

Q. $\frac{1}{3}$ part of an apple was given to each of the five students in a class . How many apples were there in the class if the teacher had $4$ apples$?$

1. $\frac{12}{3}$
2. $\frac{17}{3}$
3. $\frac{7}{3}$
4. $\frac{28}{3}$