Question 12 Marks
How much does $93 p^2-55 p+4$ exceed $13 p^3-5 p^2+17 p-90$?
AnswerRequired expression is
$93 p^2-55 p+4-\left(13 p^3-5 p^2+17 p-90\right)$
$=93 p^2-55 p+4-13 p^3+5 p^2-17 p+90$
On combining the like terms,
$=93 p^2+5 p^2-55 p-17 p+4+90-13 p^3$
$=98 p^2-72 p+94-13 p^3$
$=-13 p^3+98 p^2-72 p+94$
So,
$93 p^2-55 p+4 \text { is }-13 p^3+9 p^2-72 p+94 \text { exceed from } 13 p^3-5 p^2+17 p-90$
View full question & answer→Question 22 Marks
Critical Thinking Will the value of $11x$ for $x = -5$ be greater than $11$ or less than $11?$ Explain.
AnswerExpression given is:
$11x = 11 \times (-5)$
$= -55 [$put $x = -5]$
Clearly, $-55 < 11.$
Hence, the value is grater than $11.$
View full question & answer→Question 32 Marks
Add the following expression:
$a^2+3 a b-b c, b^2+3 b c-c a$ and $c^2+3 c a-a b$
AnswerWe have,
$\left(a^2+3 a b-b c\right)+\left(b^2+3 b c-c a\right)+\left(c^2+3 c a-a b\right)$
On combining the like terms.
$=a^2+3 a b-b c+b^2+3 b c-c a+c^2+3 c a-a b$
$=a^2+b^2+c^2+3 a b-a b-b c+3 b c-c a+3 c a$
$=a^2+b^2+c^2+2 a b+2 b c+2 c a$
View full question & answer→Question 42 Marks
Simplify the combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial:
$p^3 q^2 r+p q^2 r^3+3 p^2 q r^2-9 p^2 q r^2$
AnswerWe have
$p^3 q^2 r+p q^2 r^3+3 p^2 q r^2-9 p^2 q r^2$
By combining the like terms
$=p^3 q^2 r+p q^2 r^3+3 p^2 q r^2-9 p^2 q r^2$
$=p^3 q^2 r+p q^2 r^3-6 p^2 q r^2$
The expression contains three terms so, it is binomial.
View full question & answer→Question 52 Marks
If $A=3 x^2-4 x+1, B=5 x^2+3 x-8$ and $C=4 x^2-7 x+3$, then find:
$B+C-A$
AnswerGiven, $A=3 x^2-4 x+1, B=5 x^2+3 x-8$ and $C=4 x^2-7 x+3$
$B+C-A$
$\left.=5 x^2+3 x-8+4 x^2-7 x+3-\left(3 x^2-4 x+1\right)\right)$
On combining the like terms,
$=\left(5 x^2+4 x^2+3 x-7 x-8+3\right)-\left(3 x^2-4 x+1\right)$
$=\left(9 x^2-4 x-5\right)-\left(3 x^2-4 x+1\right)$
$=9 x^2-4 x-5-3 x^2+4 x-1$
$=9 x^2-3 x^2-4 x+4 x-5-1$
$=6 x^2-6$
View full question & answer→Question 62 Marks
How much is $21 a^3-17 a^2$ less than $89 a^3-64 a^2+6 a+16$ ?
AnswerRequired expression is
$89 a^3-64 a^2+6 a+16-\left(21 a^3-17 a^2\right)$
$=89 a^3-64 a^2+6 a+16-21 a^3+17 a^2$
On combining the like terms,
$=89 a^3-21 a^3-64 a^2+17 a^2+6 a+16$
$=68 a^3-47 a^2+6 a+16$
So,
$21 a^3-17 a^2 \text { is } 68 a^3-47 a^2+6 a+16 \text { less than } 89 a^3-64 a^2+6 a+16$.
View full question & answer→Question 72 Marks
Then find the value of:
Answer Given,
$=2 \times(2\times6+3)+\Big(\frac{3}{2}\times3 +7\Big)-(1-3)$
$=2\times(12+3)+\Big(\frac{9}{2}+7\Big)-(-2)$
$2\times 15 +\Big(\frac{23}{2}\Big)+2=30+2+\frac{23}{2}$
$=23+\frac{23}{2}=\frac{32\times2+23}{2}=\frac{64+23}{3}$
$=\frac{87}{2}$
View full question & answer→Question 82 Marks
Add the following expression: $uv - vw, vw - wu$ and $wu - uv$
AnswerWe have, $uv - vw + (vw - wu) + (wu - uv)$
$= uv -vw + vw - wu + wu - uv$
On combining the like terms.
$= uv - uv - vw + vw - wu + wu $
$= 0 + 0 + 0 = 0$
View full question & answer→Question 92 Marks
If $P = -(x - 2), Q = -2(y + 1) $ and $R = -x + 2y,$ find $a,$ when $P + Q + R = ax.$
AnswerGiven, $P = -(x-2),$
$Q = -2(y+ 1)$ and $R = -x + 2y $
Also given, $P+Q + R=ax$
On putting the values of $P,Q$ and $R$ on $LHS,$
we get: $-(x - 2) + [-2(y + 1)] + (-x + 2y) = ax$
$\Rightarrow -x+2 + (-2y - 2) -x + 2y = ax$
$\Rightarrow -x + 2 - 2y - 2 - x + 2y = ax$
On combining the like terms, $-x - x - 2y + 2y + 2 - 2 = ax$
$\Rightarrow -2x = ax$
By comparing $LHS$ and $RHS$ we get $a = -2$
View full question & answer→Question 102 Marks
Subtract:
$-2 a^2-2 b^2$ from $-a^2-b^2+2 a b$.
AnswerWe have,
$\left(-a^2-b^2+2 a b\right)-\left(-2 a^2-2 b^2\right)$
$=-a^2-b^2+2 a b+2 a^2+2 b^2$
On combining the like terms.
$=-a^2+2 a^2-b^2+2 b^2+2 a b$
$=a^2+b^2+2 a b$
View full question & answer→Question 112 Marks
Subtract:
$-a^2-a b$ from $b^2+a b$.
AnswerWe have,
$b^2+a b-\left(-a^2-a b\right)$
$=b^2+a b+a^2+a b$
Combining the like terms.
$=b^2+a^2+a b+a b$
$=a^2+b^2+2 a b$
View full question & answer→Question 122 Marks
Subtract: $ab - bc - ca$ from $- ab + bc + ca.$
AnswerWe have, $-ab + bc + ca - (ab - bc - ca)$
$= -ab + bc + ca - ab + ba + ca$
On combining the like terms.
$= -ab - ab - bc + bc + ca + ca$
$= -2ab + 2bc + 2ca$
View full question & answer→Question 132 Marks
Find the perimeter of the figure given below:
AnswerWe know that, perimeter is the sum of all sides.
Perimeter of the given figure
$= AB + BC + CD + DA$
$= (5x - y) + 2(x + y) + (5x - y) + 2(x + y)$
$= 5x - y + 2x + 2y + 5x - y + 2x + 2y$
On combining the like terms.
$= 5x + 2x + 5x + 2x - y + 2y - y + 2y$
$= 4x + 2y$
View full question & answer→Question 142 Marks
To what expression must $99 x^3-33 x^2-13 x-41$ be added to make the sum zero?
AnswerIn order to find the solution, we will subtract $99 x^3-33 x^2-13 x-41$ from 0 .
Required expression is
$0-\left(99 x^3-33 x^2-13 x-41\right)$
$=0-99 x^3+33 x^2+13 x+41$
$=-99 x^3+33 x^2+13 x+41$
So,
If we add $-99 x^3+33 x^2+13 x+41$ to $99 x^3-33 x^2-13 x-41$, then the sum is zero.
View full question & answer→Question 152 Marks
If $A=3 x^2-4 x+1, B=5 x^2+3 x-8$ and $C=4 x^2-7 x+3$, then find:
$(A+B)-C$
AnswerGiven, $A=3 x^2-4 x+1, B=5 x^2+3 x-8$ and $C=4 x^2-7 x+3$
$(A+B)-C$
$=\left(3 x^2-4 x+1+5 x^2+3 x-8\right)-\left(4 x^2-1 x+3\right)$
On combining the like terms,
$=\left(3 x^2+5 x^2-4 x+3 x+1-8\right)-\left(4 x^2-1 x+3\right)$
$=\left(8 x^2-x-7\right)-\left(4 x^2-7 x+3\right)$
$=8 x^2-x-7-4 x^2+7 x-3$
$=8 x^2-4 x^2-x+7 x-7-3$
$=4 x^2+6 x-10$
View full question & answer→Question 162 Marks
The sum of squares of first n natural numbers is given by $\frac{1}{6}\text{n}(\text{n} + 1)(2\text{n} + 1)$ of $\frac{1}{6}(2\text{n}^3+3\text{n}^2+\text{n)}$ Find the sum of squares of the first $10$ natural numbers.
AnswerGiven, the sum of squares of first $n$ natural numbers $\frac{1}{6}(\text{n} + 1)(2\text{n}+ 1)$
$\therefore\ $The sum of squares of first $10$ natural numbers
$ [\therefore\ $ put $n = 10]$
$\frac{1}{6}(10)(10 + 1)(2\times10 +1)=\frac{1}{6}\times10\times11\times 21 385$
View full question & answer→Question 172 Marks
Subtract:
$11-15 y 2$ from $y 3-15 y 2-y-11$.
AnswerWe have,
$y^3-15 x^2-y-11-\left(11-15 y^2\right)$
$=y^3-15 y^2-y-11-11+15 y^2$
On combining the like terms.
$=y^3-15 y^2+15 y^2-y-11-11$
$=y^3-y-22$
View full question & answer→Question 182 Marks
Add the following expression: $p^2-q+r, q^2-r+p$ and $r^2-p+q$
AnswerWe have,
$p^2-q+r+\left(q^2-r+p\right)+\left(r^2-p+q\right)$
$=p^2-q+r+q^2-r+p+r^2-p+q$
On combining the like terms.
$=p^2+q^2+r^2-q+q+r-r+p-p$
$=p^2+q^2+r^2+0+0+0=p^2+q^2+r^2$
View full question & answer→Question 192 Marks
At age of $2$ years, a cat or a dog is considered $24$ “human” years old. Each year, after age $2$ is equivalent to $4$ “human” years. Fill in the expression $[24 + \Box\ (\text{a} – 2)] $so that it represents the age of a cat or dog in human years. Also, you need to determine for what $‘a’$ stands for. Copy the chart and use your expression to complete it.
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Age
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$[24 + \Box\ (\text{a} – 2)] $
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Age (Human Years)
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$2$
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$3$
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$4$
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$5$
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$6$
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$7$
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AnswerThe expression is $[24 + \Box\ (\text{a} – 2)] $
Hare 'a' represents the present age of dog or cat.
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Age
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$[24 + \Box\ (\text{a} – 2)] $
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Age (Human Years)
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$2$
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$[24 + 4(2 - 2)]$
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$24$
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$3$
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$[24 + 4(3 - 2)]$
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$28$
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$4$
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$[24 + 4(4 - 2)]$
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$32$
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$5$
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$[24 + 4(5 - 2)]$
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$36$
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$6$
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$[24 + 4(6 - 2)]$
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$40$
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View full question & answer→Question 202 Marks
Shiv works in a mall and gets paid $Rs. 50$ per hour. Last week he worked for $7$ hours and this week he will work for $x$ hours. Write an algebraic expression for the money paid to him for both the weeks.
AnswerGiven, Money paid to shiv $= Rs. 50$ per h
Money paid last week $=Rs. 50 \times 7 = Rs. 350$
So, money paid this week $= Rs. 50 \times x = Rs. 50x$
Total money paid to shiv $= Rs. (350 + 50x) = Rs. 50(x + 7)$
View full question & answer→Question 212 Marks
Simplify the combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial: $2a + 2b + 2c - 2a - 2b - 2c - 2b + 2c + 2a$
Answer$2a + 2b + 2c - 2a - 2b - 2c - 2b + 2c + 2a$
By cimbining the like terms. $= 2a - 2a + 2a + 2b - 2b - 2b + 2c - 2c + 2c$
$= 2a - 2b + 2c$
The expression contains three terms so it is trinomial.
View full question & answer→Question 222 Marks
Find the numerical coefficient of the terms:
$10 x y z,-7 x y^2 z,-9 x y z, 2 x y^2 z, 2 x^2 y^2 z^2$
AnswerNumericla coefficient of, $10 x y z=10$
$-7 x y^2 z=-7$
$-9 x y z=-9$
$2 x y^2 z=2$
$2 x^2 y^2 z^2=2$
View full question & answer→Question 232 Marks
Rohan's mother gave him $Rs. 3 x y^2$ and his father gave him $Rs. 5\left(x y^2+2\right)$. Out of this total money he spent $Rs. (10 3 x y^2 ) $ on his birthday party. How much money is left with him?
AnswerGiven, amount given to Rohan by his mother $= Rs. 3 x y^2$ and amount given to Rohan by his father $= Rs.5\left(x y^2+2\right)$
Total amount Rohan have
$=\left[\left(3 x y^2\right)+\left(5 x y^2+10\right)\right]$
$=\text { Rs. }\left[3 x y^2+5 x y^2+10\right]$
$=\text { Rs. }\left(8 x y^2+10\right)$
Total amount spent by Rohan.
$=\text { Rs. }\left(10-3 x y^2\right)$
$\therefore$ After spending, Rohan have left money.
View full question & answer→Question 242 Marks
Simplify the combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial:
$x^4+3 x^3 y+3 x^2 y^2-3 x^3 y-3 x y^3+y^4-3 x^2 y^2$
AnswerWe have
$x^4+3 x^3 y+3 x^2 y^2-3 x^3 y-3 x y^3+y^4-3 x^2 y^2$
By combining the like terms
$=x^4+3 x^3 y-3 x^3 y+3 x^2 y^2-3 x^2 y^2-3 x^3 y+y^4$
$=x^4+0+0-3 x^3 y+y^4=x^4+y^4-3 x^3 y$
The expression contains three terms so, it is binomial.
View full question & answer→Question 252 Marks
If $A=3 x^2-4 x+1, B=5 x^2+3 x-8$ and $C=4 x^2-7 x+3$, then find:
$A+B+C$
AnswerGiven, $A=3 x^2-4 x+1, B=5 x^2+3 x-8$ and $C=4 x^2-7 x+3$
$B+C+A$
$=3 x^2-4 x+1+5 x^2+3 x-8+4 x^2-7 x+3$
On combining the like terms,
$=3 x^2+5 x^2+4 x^2-4 x+3 x-7 x+1-8+3$
$=12 x^2-8 x-4$
View full question & answer→Question 262 Marks
$= x + 6,$ then find the value of:
AnswerGiven,
$=2\times (12+6)-\frac{3}{2}\Big(\frac{3}{4}\times1-2\Big)$
$=36-\frac{3}{2}\Big(\frac{3-2\times4}{4}\Big)$
$=36-\frac{3}{2}\Big(\frac{3-8}{4}\Big)$
$36-\frac{3}{2}\Big(\frac{-5}{4}\Big)$
$=36+\frac{15}{8}$
$=\frac{36\times8+15}{8}$
$=\frac{288+15}{8}$
$=\frac{303}{8}$ View full question & answer→Question 272 Marks
A wire is $(7x - 3)$ metres long. A length of $(3x - 4)$ metres is cut for use. Now, answer the following questions: If this left out wire is used for making an equilateral triangle. What is the length of each side of the triangle so formed$?$
Answer$\therefore\ $Left wire. $= (4x +1)m$
$\therefore\ $Perimeter of equilateral triangle $=$ Length of wire left
$\Rightarrow3 \times \text{(side)} = 4\text{x} + 1$
$\Rightarrow\ \text{Side}=\frac{(4\text{x}+ 1)}{3}=\frac{1}{3}(4\text{x}+1)\text{m}$
View full question & answer→Question 282 Marks
$= x + 6,$ then find the value of:
AnswerGiven,
$=\frac{3}{4}\times10-2-4-6$ $=\frac{30}{4}-\frac{12}{1}$ $=\frac{30-48}{4}$ $=\frac{-18}{4}$ $=\frac{-9}{2}$ View full question & answer→Question 292 Marks
Add the following expression:
$p^2-7 p q-q^2$ and $-3 p^2-2 p q+7 q^2$
AnswerWe have,
$p^2-7 p q-q^2+\left(-3 p^2-2 p q+7 q^2\right)$
$=p^2-7 p q-q^2-3 p^2-2 p q+7 q^2$
On combining the like terms.
$=p^2-3 p^2-7 p q-2 p q-q^2+7 q^2=-2 q^2-9 p q+6 q^2$
View full question & answer→Question 302 Marks
Challenge Write an expression for the sum of $1$ and twice a number $n.$ If you let n be any odd number, will the result always be an odd number$?$
AnswerLet the number be $n.$ So, according to the statement, the expression can be written as $= 2n + 1.$ Yes, the result is always an odd number, because when a number becomes multiplied by $2,$ it becomes even and addition of $1$ in that even number makes it an odd number
View full question & answer→Question 312 Marks
Sonu and Raj have to collect different kinds of leaves for science project. They go to a park where Sonu collects $12$ leaves and Raj collects $x$ leaves. After some time Sonu loses $3$ leaves and Raj collects $2x$ leaves. Write an algebraic expression to find the total number of leaves collected by both of them.
AnswerAccording to the question, Sonu collected leaves $= 12 - 3 = 9$
Raj collected leaves $= x + 2x = 3$
Total leaves collected $=9 + 3x$
Hence, the required expression is $9 + 3x.$
View full question & answer→Question 322 Marks
Add the following expression: $ab + bc + ca$ and $- bc - ca - ab$
AnswerWe have, $ab + bc + ca + (-bc - ca - ab)$
$= ab + bc + ca - bc - ca - ab$
On combining the like terms.
$= ab - ab + bc - cb + ca -ca$
$= 0 + 0 + 0 = 0$
View full question & answer→Question 332 Marks
Add the following expression:
$x^3-x^2 y-x y^2-y^3$ and $x^3-2 x^2 y+3 x y^2+4 y$
AnswerWe have,
$x^3-x^2 y-x y^2-y^3+x^3-2 x^2 y+3 x y^2+4 y$
On combining the like terms.
$=x^3-x^3-x^2 y-2 x^2 y-x y^2+3 x y^2-y^3+4 y$
$=2 x^3-3 x^2 y+2 x y^2-y^3+4 y$
View full question & answer→Question 342 Marks
Add the following expression:
$p^2 q r+p q^2 r+p q r^2$ and $-3 p q^2 r-2 p q r^2$
AnswerWe have,
$p^2 q r+p q^2 r+p q r^2+\left(-3 p q^2 r-2 p q r^2\right)$
$=p^2 q r+p q^2 r+p q r^2-3 p q^2 r-2 p q r^2$
On combining the like terms.
$=p^2 q+p q^2 r-3 p q^2 r+p q r^2-2 p q r^2$
$=p^2 q r-2 p q^2 r-p q r^2$
View full question & answer→Question 352 Marks
The rate of planting the grass is $Rs. x$ per square metre. Find the cost of planting the grass on a triangular lawn whose base is $y$ metres and height is $z$ metres.
AnswerGiven, Base of triangular lawn is $y$ metres and height $z$ metres.
$\therefore\ $Area of triangular lawn
$=\frac{1}{2}\times\text{y}\times\text{z}=\frac{1}{2}\text{yz}\text{m}^2$
$\Big[\because\ $area of triangle $=\frac{1}{2}\times \text{height}\times\text{base}\Big]$
$\therefore\ $Cost of planting the grass on lawn
$=\frac{1}{2}\text{yz}\times\text{x}=\text{Rs.}\frac{1}{2}\text{xyz}$ View full question & answer→Question 362 Marks
A taxi service charges $Rs. 8$ per km and levies a fixed charge of $Rs. 50.$ Write an algebraic expression for the above situation, if the taxi is hired for $x\ km.$
AnswerAs per the given information, taxi service charges $Rs. 8$ per km and fixed charge of $50.$
If taxi is hired for $x\ km.$
Then, algebraic expression for the situation $= 8 \times x + 50 = 8x + 50$
Hence, the required expression is $8x + 50.$
View full question & answer→Question 372 Marks
Simplify the combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial:
$50 x^3-21 x+107+41 x^3-x+1-93+71 x-31 x^3$
Answer$50 x^3-21 x+107+41 x^3-x+1-93+71 x-31 x^3$
By cimbining the like terms.
$=50 x^3-21 x+107+41 x^3-x+1-93+71 x-31 x^3$
$=60 x^3+49 x+15$
The expression contains three terms so it is trinomial.
View full question & answer→Question 382 Marks
How much is $y^4-12 y^2+y+14$ greater than $17 y^3+34 y^2-51 y+68 ?$
Answer$y^4-12 y^2+y+14-\left(17 y^3+34 y^2-51 y+68\right)$
$=y^4-12 y^2+y+14-17 y^3-34 y^2+51 y-68$
On combining the like terms,
$=y^4-12 y^2-34 y^2+y+51 y+14-68-17 y^3$
$y^4-46 y^2+52 y$
$17 y^3-54=y 4-17 y^3-46 y^2+52 y-54$
So, $y^4-12 y^2+y+14$ is $y^4-17 y^3-46 y^2+52 y-54$ greater than $17 y^3+34 y^2-51 y+68$
View full question & answer→Question 392 Marks
Subtract:
$x^4+3 x^3 y^3+5 y^4$ from $2 x^4-x^3 y^3+7 y^4$.
AnswerWe have,
$2 x^4-x^3 y^3+7 y^4-\left(x^4+3 x^3 y+5 y^4\right)$
$2 x^4-x^3 y^3+7 y^4-x^4-4 x^3 y^3-5 y^3$
on combining the like terms.
$=2 x^4-x^4-x^3 y^3-3 x^3 y^3+7 y^4-5 y^4=x^4-4 x^3 y^3+2 y^4$
View full question & answer→Question 402 Marks
From the sum of $x^2-y^2-1, y^2-x^2-1$ and $1-x^2-y^2$ subtract $-\left(1+y^2\right)$.
AnswerSum of $x^2-y^2-1, y^2-x^2-1$ and $1-x^2-y^2$
$=x^2-y^2-1+y^2-x^2-1+1$
$-x^2-y^2$
On combining the like terms,
$=x^2-x^2-x^2-y^2+y^2-y^2-1-1+1$
$=-x^2-y^2-1$
Now, subtract $-\left(1+y^2\right)$ from $-x^2-y^2-1$
$=-x^2-y^2-1-\left[-\left(1+y^2\right)\right]$
$=-x^2-y^2-1+1+y^2$
$=-x^2-y^2+y^2-1+1$
$=-x^2$
View full question & answer→Question 412 Marks
Subtract:
$x^3 y^2+3 x^2 y^2-7 x y^3$ from $x^4+y^4+3 x^2 y^2-x y^3$.
AnswerWe have,
$x^4+y^4+3 x^2 y^2-x y^3-\left(x^3 y^3+3 x^2 y^2-7 x y^3\right)$
$=x^4+y^4+3 x^2 y^2-x y^3-x^3 y^3-3 x^2 y^2+7 x y^3$
On combining the like terms.
$=x^4+y^4+3 x^2 y^2-3 x^2 y^2-x y^3+7 x y^3-x^3 y^3$
$=x^4+y^4+6 x y^3-x^3 y^3$
View full question & answer→Question 422 Marks
The sum of first $n$ natural numbers is given by $\frac{1}{2}\text{n}^2+\frac{1}{2}\text{n}$ Find: The sum of first $5$ natural numbers.
AnswerGiven, Sum of first $n$ natural numbers
$=\frac{1}{2}\text{n}^2+\frac{1}{2}\text{n}$
$[$put $n= 5]$ Sum of first $5$
natural numbers $=\frac{1}{2}(5)^2+\frac{1}{2}(5) $
$=\frac{25}{2}+\frac{5}{2}=\frac{30}{2}=15$
View full question & answer→Question 432 Marks
Subtract:
$-7 p^2 q$ r from $-3 p^2 q r$.
AnswerWe have,
$-3 p^2 q r-\left(-7 p^2 q r\right)$
$=-3 p^2 q r+7 p^2 q r$
$=p^2 q r(-3+7)$
$=4 p^2 q r$
${[\therefore \text { with - ve sign }+ \text { ve sign will be change }]}$
View full question & answer→Question 442 Marks
The sum of first $n$ natural numbers is given by $\frac{1}{2}\text{n}^2+\frac{1}{2}\text{n}$ Find:
The sum of natural numbers from $11$ to $30.$
AnswerGiven,
Sum of first $n$ natural numbers $=\frac{1}{2}\text{n}^2+\frac{1}{2}\text{n} [$divide each term by $2]$
Sum of natural numbers from $11$ to $30$
$=$ sum of first $30$ natural numbers $-$ sum of first $10$ natural numbers
$=\Big[\frac{1}{2}(30)^2+\frac{1}{2}(30)\Big]-\Big[\frac{1}{2}(10)^2+\frac{1}{2}(10)\Big]$
$=\frac{900}{2}+\frac{30}{2}-\frac{100}{2}-\frac{10}{2}$
$= 450 + 15 - 50 - 5 = 410$
View full question & answer→Question 452 Marks
Each symbol given below represents an algebraic expression:
The symbols are then represented in the expression:
Find the expression which is represented by the above symbols.
AnswerGiven,
$=\left(2 x^2+3 y\right)+\left(5 x^2+3 x\right)-\left(8 x^2-3 x^2+2 x+3 y\right)$
$=2 x^2+3 y+5 x^2+3 x-8 y^2+3 x^2-2 x-3 y$
On combining the like terms.
$=2 x^2+5 x^2+3 x^2+3 y-3 y+3 x-2 x-8 y^2$
$=10 x^2-8 y^2+x$ View full question & answer→Question 462 Marks
The sum of first $n$ natural numbers is given by $\frac{1}{2}\text{n}^2+\frac{1}{2}\text{n}$ Find: The sum of first $11$ natural numbers.
AnswerGiven, Sum of first n natural numbers $=\frac{1}{2}\text{n}^2+\frac{1}{2}\text{n}$
$[$put $n = 11]$ Sum of first $11$
natural numbers $=\frac{1}{2}(11)^2+\frac{1}{2}(11) =\frac{1}{2}\times121+\frac{11}{2}$
$=\frac{121}{2}+\frac{11}{2}=\frac{132}{2}=66$
View full question & answer→Question 472 Marks
Add the following expression:
$x^3 y^2+x^2 y^3+3 y^4$ and $x^4+3 x^2 y^3+4 y^4$
AnswerWe have,
$\left(x^3 y^2+x^2 y^3+3 y^4\right)+\left(x^4+3 x^2 y^3+4 y^4\right)$
$=x^2 y^2+x^2 y^3+3 y^4+x^4+3 x^2 y^3+a y^4$
On combining the like terms.
$=x^3 y^2+x^2 y^3+3 x^2 y^3+x^4+3 y^4+4 y^4$
$=x^4+7 y^4+x^3 y^2+4 x^2 y^3$
View full question & answer→Question 482 Marks
Add the following expression:
$t-t^2-t^3-14,15 t^3+13+9 t-8 t^2, 12 t^2-19-24 t$ and $4 t-9 t^2+19 t^3$
AnswerWe have,
$\left(t-t^2-t^3-14\right)+\left(15 t^3+13+9 t-8 t^2\right)+\left(12 t^2-19-24 t\right)+\left(4 t-9 t^2+19 t^3\right)$
$=t-t^2-t^3-14+15 t^3+13+9 t-8 t^2+12 t^2-19-24 t+4 t-9 t^2+19 t^3$
On combining the like terms.
$=t+9 t-24 t+4 t-t^2-8 t^2+12 t^2-9 t^2-t^3+15 t^3+19 t^3-14+13-19$
$=-10 t-6 t^2+33 t^3-20$
$=33 t^3-6 t^2-10 t-20$
View full question & answer→Question 492 Marks
Simplify the combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial:
$3 x^2 y z^2-3 x y^2 z+x^2 y z^2+7 x y^2 z$
AnswerWe have
$3 x^2 y z^2-3 x y^2 z+x^2 y z^2+7 x y^2 z$
By combining the like terms
$=3 x^2 y z^2+x^2 y z^2-3 x y^2 z+7 x y^2 z$
$=4 x^2 y z^2+4 x y^2 z$
The expression contains two terms so, it is binomial.
View full question & answer→Question 502 Marks
Subtract:
$4.5 x^5-3.4 x^2+5.7 \text { from } 5 x^4-3.2 x^2-7.3 x$
AnswerWe have,
$5 x^4-3.2 x^2-7.3 x-\left(4.5 x^5-3.4 x^2+5.7\right)$
$=5 x^4-3.2 x^2-7.3 x-4.5 x^5+3.4 x^2-5.7$
On combining the like terms.
$=-4.5 x^5+5 x^4-3.2 x^2+3.4 x^2-7.3 x-5.7$
$=-4.5 x^5+5 x^4+0.2 x^2-7.3 x-5.7$
View full question & answer→