SAT

Conducting Body: College Board
Primary Audience: High school students applying to college
Sections: Reading and Writing, Math
Question Format: Primarily multiple choice
Test Duration: About 2 hours 15 minutes
Scoring Scale: 400 to 1600
Frequency: Offered multiple times per year
Used For: College admissions and scholarship applications

The SAT is a standardized test used by many colleges and universities in the United States as part of their admissions process. It is designed to assess a high school student's readiness for college and provide schools with a common point of data to compare applicants from different backgrounds and high schools. Most students take the SAT during their junior or senior year of high school.

The exam is administered digitally and consists of two main sections: Reading and Writing, and Math. The questions are primarily multiple choice and test the knowledge and skills students are expected to have learned in high school. The test is adaptive, meaning a student's performance on the first part of each section determines the difficulty level of the questions in the second part.

In practice, the main difficulty of the SAT is managing the time constraints. Students must answer questions quickly and efficiently while maintaining accuracy. The questions often require careful reading and critical thinking, not just recall of facts. For the Math section, a broad understanding of algebra, geometry, and data analysis is necessary. The adaptive nature of the test can also be challenging, as students must perform well from the start to access the harder, higher scoring questions.

How a question gets worked here

From a mathematics paper rather than SAT, worked the way every question on this site is. SAT questions are being added.

JEE Advanced 2021

Let $\vec{u}$ , $\vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $\vec{u}\cdot\vec{w} = 1$ , $\vec{v}\cdot\vec{w} = 1$ , $\vec{w}\cdot\vec{w} = 4$ . If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{u}$ , $\vec{v}$ and $\vec{w}$ , is $\sqrt{2}$ , then the value of $|3\vec{u} + 5\vec{v}|$ is _______.

Vector Algebra Let

\vec{u}

\vec{v}

and

\vec{w}

be vectors in three-dimensional space, where

\vec{u}

and

\vec{v}

are unit vectors which are not perpendicular to each other and

\vec{u}\cdot\vec{w} = 1

\vec{v}\cdot\vec{w} = 1

\vec{w}\cdot\vec{w} = 4

. If the volume of the parallelepiped, whose adjacent sides are represented by the vectors

\vec{u}

\vec{v}

and

\vec{w}

, is

\sqrt{2}

, then the value of

|3\vec{u} + 5\vec{v}|

is _______.

Watch this solved on the board The full walkthrough, written out a line at a time with narration explaining each step as it appears. Pause anywhere and ask why a line is there. Free to watch

Or read the worked solution below. Jump to the steps ↓

$\text{Given:}\quad |\vec{u}| = 1,\quad |\vec{v}| = 1$

$\vec{u}\cdot\vec{v} \neq 0$

$\vec{u}\cdot\vec{w} = 1,\quad \vec{v}\cdot\vec{w} = 1,\quad \vec{w}\cdot\vec{w} = 4$

$\text{Volume} = \left|[\vec{u}\ \vec{v}\ \vec{w}]\right| = \sqrt{2}$

$\text{To find:}\quad \left|3\vec{u} + 5\vec{v}\right|$

$\text{Plan:}\quad (1)\ [\vec{u}\ \vec{v}\ \vec{w}]^2 = \det G = 2$ $(2)\ \text{find } p = \vec{u}\cdot\vec{v}$ $(3)\ \left|3\vec{u}+5\vec{v}\right|^2 = 34 + 30p$ $\text{Setup: the parallelepiped on } \vec{u},\ \vec{v},\ \vec{w}$ $\text{Volume} = \left|[\vec{u}\ \vec{v}\ \vec{w}]\right| = \sqrt{2}$ $[\vec{u}\ \vec{v}\ \vec{w}]^2 = \left(\sqrt{2}\right)^2 = 2$ $[\vec{u}\ \vec{v}\ \vec{w}]^2 = \det G$ $G = \begin{pmatrix} \vec{u}\cdot\vec{u} & \vec{u}\cdot\vec{v} & \vec{u}\cdot\vec{w} \\ \vec{v}\cdot\vec{u} & \vec{v}\cdot\vec{v} & \vec{v}\cdot\vec{w} \\ \vec{w}\cdot\vec{u} & \vec{w}\cdot\vec{v} & \vec{w}\cdot\vec{w} \end{pmatrix}$ $\text{Let } p = \vec{u}\cdot\vec{v}$ $G = \begin{pmatrix} 1 & p & 1 \\ p & 1 & 1 \\ 1 & 1 & 4 \end{pmatrix}$ $\det G = 1(4-1) - p(4p-1) + 1(p-1)$ $\det G = 3 - 4p^2 + p + p - 1$ $= 2 + 2p - 4p^2$ $\det G = [\vec{u}\ \vec{v}\ \vec{w}]^2 = 2$ $2 + 2p - 4p^2 = 2 \Rightarrow 2p - 4p^2 = 0$ $2p(1 - 2p) = 0 \Rightarrow p = 0 \text{ or } p = \dfrac{1}{2}$ $\vec{u}\cdot\vec{v} \neq 0 \Rightarrow p \neq 0$ $\therefore p = \dfrac{1}{2} \Rightarrow \cos\theta = \dfrac{1}{2} \Rightarrow \theta = 60^\circ$ $\left|3\vec{u}+5\vec{v}\right|^2 = (3\vec{u}+5\vec{v})\cdot(3\vec{u}+5\vec{v})$ $= 9|\vec{u}|^2 + 25|\vec{v}|^2 + 30\,\vec{u}\cdot\vec{v}$ $= 9(1) + 25(1) + 30\cdot\dfrac{1}{2}$ $= 34 + 15 = 49$ $\left|3\vec{u}+5\vec{v}\right| = \sqrt{49} = 7$

Get a question like this solved →