Grade VIII- Math Concept Maps

 

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Maths Concept Map 8 CBSE by Nilesh Gupta

Grade IX Math Concept Maps

 

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Maths Concept Map 9 CBSE by Nilesh Gupta

Grade X - Concept Maps

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Credit and thanks: Nilesh Gupta, Scribd

Maths Concept Map_X CBSE by Nilesh Gupta

Polynomials

Quick Revision Polynomials

1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

2. A quadratic polynomial in $x$ with real coefficients is of the form $a x^{2}+b x+c,$ where $a, b$, $c$ are real numbers with $a \neq 0$

3. The zeroes of a polynomial $p(x)$ are precisely the $x$ -coordinates of the points, where the graph of $y=p(x)$ intersects the $x$ -axis.

4. A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

5. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $a x^{2}+b x+c,$ then

$\alpha+\beta=-\frac{b}{a}, \quad \alpha \beta=\frac{c}{a}$

6. If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial $a x^{3}+b x^{2}+c x+d=0,$ then

$\alpha+\beta+\gamma=\frac{-b}{a} \alpha \beta+\beta \gamma+\gamma \alpha=\frac{c}{a}$

and $\alpha \beta \gamma=\frac{-d}{a}$

7. The division algorithm states that given any polynomial $p(x)$ and any non-zero polynomial $g(x)$, there are polynomials $q(x)$ and $r(x)$ such that

p(x)=g(x) q(x)+r(x)

where $\quad r(x)=0$ or degree $r(x)<$ degree $g(x)$

Real Number Theorems

Quick revision for theorems

Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b,

there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Theorem 1.2 (Fundamental Theorem of Arithmetic) : Every composite number

can be expressed ( factorised) as a product of primes, and this factorisation is

unique, apart from the order in which the prime factors occur.

Theorem 1.3: Let $p$ be a prime number. If $p$ divides $a^{2},$ then $p$ divides a, where $a$ is a positive integer.

Theorem 1.4: $\sqrt{2}$ is irrational.

Theorem 1.5: Let $x$ be a rational number whose decimal expansion terminates.

Then $x$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are coprime, and the

prime factorisation of $q$ is of the form $2^{\circ} 5^{m},$ where $n, m$ are non-negative integers.

Theorem 1.6: Let $x=\frac{p}{q}$ be a rational mamber, such that the prime factorisation of $q$ is of the form $2^{n} 5^{\text {" }},$ where $n, m$ are non-negative integers. Then $x$ has a

decimal expansion which terminates.

Theorem 1.7: Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of $q$ is not of the form $2^{n} 5$ ", where $n, m$ are non-negative integers. Then, $x$ has a decimal expansion which is non-terminating repeating (recurring).

So we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.

Grade 10 Revision Questions Set I

Fast track revision

 Q1 Find the largest four-digit number which when divided by 4,7 and 13 leaves a remainder 3 in each case.

Q2 Show that $\frac{2+3 \sqrt{2}}{7}$ is not a rational number, given that $\sqrt{2}$ is an irrational number.

Q3 Write the number of real roots of the equation $x^{2}+3|x|+2=0$

Q4 Solve for $\mathrm{x}$ $\frac{x+1}{x-1}+\frac{x-2}{x+2}=4-\frac{2 x+3}{x-2} ; x \neq 1,-2,2$

Q5 The longer side of a rectangular hall is $24 \mathrm{m}$ and the length of its diagonal is $26 \mathrm{m}$. Find the area of the hall.

Q6 Prove the trigonometric identity:

$\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta$

Q7 If $A=B=60^{\circ},$ verify that $\cos (A-B)=\cos A \cos B+\sin A \sin B$

Q8 From a point $\mathrm{P}$ on the ground the angle of elevation of the top of a $10 \mathrm{m}$ tall building is $30^{\circ} . \mathrm{A}$ flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from $\mathrm{P}$ is $45^{\circ} .$ Find the length of the flagstaff and distance of building from point $\mathrm{P}$. $\sqrt{3}=1.732]$

Q9 A pole casts a shadow of length $2 \sqrt{3} \mathrm{m}$ on the ground, when the Sun's elevation is $60^{\circ} .$ Find the height of the pole.

Q10 Find the values of $\mathrm{k}$ for which the given equation has real and equal roots:

$x^{2}-2 x(1+3 k)+7(3+2 k)=0$

Q11 If $\tan \theta=\frac{12}{13},$ evaluate $\frac{2 \sin \theta \cos \theta}{\cos ^{2} \theta-\sin ^{2} \theta}$

Q12 A solid metal cone with radius of base $12 \mathrm{cm}$ and height $24 \mathrm{cm}$ is melted to form solid spherical balls of diameter $6 \mathrm{cm}$ each. Find the number of balls formed.