If a > 0 and discriminant
Question: If $a0$ and discriminant of $a x^{2}+2 b x+c$ is negative, then $\Delta=\left|\begin{array}{ccc}a b a x+b \\ b c b x+c \\ a x+b b x+c 0\end{array}\right|$ is (a) positive (b) $\left(a c-b^{2}\right)\left(a x^{2}+2 b x+c\right)$ (c) negative (d) 0 Solution: (c) negative Discriminant D of $a x^{2}+2 b x+c=(2 b)^{2}-4 a c0 \quad$ [Given] $\Rightarrow 4 \mathrm{~b}^{2}-4 \mathrm{ac}0$ $\Rightarrow \mathrm{b}^{2}-\mathrm{ac}0$, where $\mathrm{a}0 \quad \ldots(1)$ $\Delta=\mid \begin{array}{...
Read More →If a > 0 and discriminant
Question: If $a0$ and discriminant of $a x^{2}+2 b x+c$ is negative, then $\Delta=\left|\begin{array}{ccc}a b a x+b \\ b c b x+c \\ a x+b b x+c 0\end{array}\right|$ is (a) positive (b) $\left(a c-b^{2}\right)\left(a x^{2}+2 b x+c\right)$ (c) negative (d) 0 Solution: (c) negative Discriminant D of $a x^{2}+2 b x+c=(2 b)^{2}-4 a c0 \quad$ [Given] $\Rightarrow 4 \mathrm{~b}^{2}-4 \mathrm{ac}0$ $\Rightarrow \mathrm{b}^{2}-\mathrm{ac}0$, where $\mathrm{a}0 \quad \ldots(1)$ $\Delta=\mid \begin{array}{...
Read More →The area of a square field is 6050 m2.
Question: The area of a square field is 6050 m2. The length of its diagonal is(a) 135 m(b) 120 m(c) 112 m(d) 110 m Solution: (d) 110 mLet the diagonal of the square field bed m. In case of a square field, $d^{2}=2 a^{2}$, where $a$ is the side of the square field. Now, Area of a square field $=a^{2}$ $d^{2}=2 a^{2}$ $\Rightarrow d^{2}=2 \times$ Area of the square field $\Rightarrow d=\sqrt{2 \times \text { Area of the square field }}$ $\therefore d=\sqrt{2 \times 6050}=\sqrt{12100}=110$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:ab a b+ 1 Solution: $a b-a-b+1=(a b-b)+(1-a)$ [Regrouping the expressions] $=b(a-1)+(1-a)$ $=b(a-1)-(a-1)$ $[\because(1-a)=-(a-1)]$ $=(a-1)(b-1)$ [Taking out the common factor $(a-1)$ ]...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:x2 11xyx+ 11y Solution: $x^{2}-11 x y-x+11 y=\left(x^{2}-x\right)+(11 y-11 x y) \quad[$ Regrouping the expressions $]$ $=x(x-1)+11 y(1-x)$ $=x(x-1)-11 y(x-1)$ $[\because(1-x)=-(x-1)]$ $=(x-11 y)(x-1)$ [Taking out the common factor $(x-1)$ ]...
Read More →The length of the diagonal of a square is
Question: The length of the diagonal of a square is $10 \sqrt{2} \mathrm{~cm}$. Its area is (a) 200 cm2(b) 100 cm2(c) 150 cm2 (d) $100 \sqrt{2} \mathrm{~cm}^{2}$ Solution: (b) 100 cm2A diagonal of a square forms the hypotenuse of a right-angled triangle with base and height equal to sidea. Diagonal $^{2}=a^{2}+a^{2}$ $\Rightarrow$ Diagonal $^{2}=2 a^{2}$ $\Rightarrow a=\frac{1}{\sqrt{2}}$ Diagonal $=\frac{1}{\sqrt{2}} \times 10 \sqrt{2}$ $=10 \mathrm{~cm}$ $\therefore$ Area of the square $=a^{2}...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a(a + b c) bc Solution: $a(a+b-c)-b c=a^{2}+a b-a c-b c$ $=\left(a^{2}-a c\right)+(a b-b c)$ [Regrouping the expressions] $=a(a-c)+b(a-c)$ $=(a+b)(a-c)$ $[$ Taking $(a-c)$ as the common factor $]$...
Read More →In given figure $l | m$ and $M$ is the mid-point of a
Question: In given figure $l \| m$ and $M$ is the mid-point of a line segment $A B$. Show that $M$ is also the mid-point of any line segment $C D$, having its end points on $I$ and $m$, respectively. Solution: Given In the figure, $l \| m$ and $M$ is the mid-point of a line segment $A B$ i.e., $A M=B M$. To show $M C=M D .$ Proof $\because$ $l \| m$ [given] $\angle B A C=\angle A B D$ [alternate interior angles] $\angle A M C=\angle B M D$ [vertically opposite angles] In $\triangle A M C$ and $\...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a(a 2bc) + 2bc Solution: $a(a-2 b-c)+2 b c=a^{2}-2 a b-a c+2 b c$ $=\left(a^{2}-a c\right)+(2 b c-2 a b)$ [Regrouping the terms] $=a(a-c)+2 b(c-a)$ $=a(a-c)-2 b(a-c)$ $[\because(c-a)=-(a-c)]$ $=(a-2 b)(a-c)$ $[$ Taking $(a-c)$ as the common factor $]$...
Read More →A rectangular ground 80 m × 50 m has a path 1 m wide outside around it.
Question: A rectangular ground 80 m 50 m has a path 1 m wide outside around it. The area of the path is(a) 264 m2(b) 284 m2(c) 400 m2(d) 464 m2 Solution: (a) 264 m2 Length of the ground including the path $=80+2=82 \mathrm{~m}$ Breadth of the ground including the path $=50+2=52 \mathrm{~m}$ Total area (including the path) $=$ Length $\times$ Breadth $=82 \times 52=4264 \mathrm{~m}^{2}$ Area of the field $=80 \times 50=4000 \mathrm{~m}^{2}$ Area of the path $=4264-4000=264 \mathrm{~m}^{2}$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:a2x2+ (ax2+ 1)x+a Solution: $a^{2} x^{2}+\left(a x^{2}+1\right) x+a=a^{2} x^{2}+a x^{3}+x+a$ $=\left(a x^{3}+a^{2} x^{2}\right)+(x+a)$ [Regrouping the expressions] $=a x^{2}(x+a)+(x+a)$ $=\left(a x^{2}+1\right)(x+a)$ $[$ Taking $(x+a)$ as the common factor $]$...
Read More →On increasing the length of a rectangle by 20% and decreasing its breadth by 20%, what is the change in its area?
Question: On increasing the length of a rectangle by 20% and decreasing its breadth by 20%, what is the change in its area?(a) 20% increase(b) 20% decrease(c) No change(d) 4% decrease Solution: (d) 4% decrease Let: Length $=x$ breadth $=y$ Area $=x y$ Now, New length $=x+20 \% x=x+\frac{1}{5} x=\frac{6}{5} x$ New breadth $=y-20 \% y=y-\frac{1}{5} y=\frac{4}{5} y$ New area $=\frac{6}{5} x \times \frac{4}{5} y=\frac{24}{25} x y$ Difference in the areas $=x y-\frac{24}{25} x y=\frac{1}{25} x y$ Dif...
Read More →D is any point on side AC of a ΔABC
Question: D is any point on side AC of a ΔABC with AB = AC. Show that CD BD. Solution: Given in $\triangle A B C, D$ is any point on side $A C$ such that $A B=A C$. To show $C DB D$ or $B DC D$. Proof In $\triangle A B C$, $A C=A B$ [qiven] $\Rightarrow$ $\angle A B C=\angle A C B$ ...(i) [angles opposite to equal sides are equal] In $\triangle A B C$ and $\triangle D B C$, $\angle A B C\angle D B C$ [since, $\angle D B C$ is a internal angle of $\angle B$ ] $\Rightarrow$ $\angle A C B\angle D B...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:ab(x2+ 1) +x(a2+b2) Solution: $a b\left(x^{2}+1\right)+x\left(a^{2}+b^{2}\right)=a b x^{2}+a b+a^{2} x+b^{2} x$ $=\left(a b x^{2}+a^{2} x\right)+\left(b^{2} x+a b\right)$ [Regrouping the expressions] $=a x(b x+a)+b(b x+a)$ $=(a x+b)(b x+a)$ $[$ Taking $(b x+a)$ as the common factor $]$...
Read More →On increasing the length of a rectangle by 20% and decreasing its breadth by 20%, what is the change in its area?
Question: On increasing the length of a rectangle by 20% and decreasing its breadth by 20%, what is the change in its area?(a) 20% increase(b) 20% decrease(c) No change(d) 4% decrease Solution: (d) 4% decrease Let: Length $=x$ breadth $=y$ Area $=x y$ Now, New length $=x+20 \% x=x+\frac{1}{5} x=\frac{6}{5} x$ New breadth $=y-20 \% y=y-\frac{1}{5} y=\frac{4}{5} y$ New area $=\frac{6}{5} x \times \frac{4}{5} y=\frac{24}{25} x y$ Difference in the areas $=x y-\frac{24}{25} x y=\frac{1}{25} x y$ Dif...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:16(a b)3 24 (a b)2 Solution: $16(a-b)^{3}-24(a-b)^{2}$ $=8(a-b)^{2}[2(a-b)-3]$ $\left\{\right.$ Taking $\left[8(a-b)^{2}\right]$ as the common factor $\}$ $=8(a-b)^{2}(2 a-2 b-3)$...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:(ax + by)2+ (bx ay)2 Solution: $(a x+b y)^{2}+(b x-a y)^{2}=a^{2} x^{2}+2 a b x y+b^{2} y^{2}+b^{2} x^{2}-2 a b x y+a^{2} y^{2}$ $=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}$ $=\left(a^{2} x^{2}+a^{2} y^{2}\right)+\left(b^{2} x^{2}+b^{2} y^{2}\right)$ [Regrouping the expressions] $=a^{2}\left(x^{2}+y^{2}\right)+b^{2}\left(x^{2}+y^{2}\right)$ $=\left(a^{2}+b^{2}\right)\left(x^{2}+y^{2}\right)$ [Taking $\left(x^{2}+y^{2}\right)$ as the comm...
Read More →The length of a rectangle is thrice its breadth and the length of its diagonal is
Question: The length of a rectangle is thrice its breadth and the length of its diagonal is $8 \sqrt{10} \mathrm{~cm}$. The perimeter of the rectangle is (a) $15 \sqrt{10} \mathrm{~cm}$ (b) $16 \sqrt{10} \mathrm{~cm}$ (c) $24 \sqrt{10} \mathrm{~cm}$ (d) 64 cm Solution: (d) 64 cmLet the breadth of the rectangle bexcm. Length of the rectangle = 3xcmWe know: Diagonal $=\sqrt{(\text { Length })^{2}+(\text { Breadth })^{2}}$ $\Rightarrow 8 \sqrt{10}=\sqrt{x^{2}+(3 x)^{2}}$ $\Rightarrow 8 \sqrt{10}=\s...
Read More →The length of a rectangle is thrice its breadth and the length of its diagonal is
Question: The length of a rectangle is thrice its breadth and the length of its diagonal is $8 \sqrt{10} \mathrm{~cm}$. The perimeter of the rectangle is (a) $15 \sqrt{10} \mathrm{~cm}$ (b) $16 \sqrt{10} \mathrm{~cm}$ (c) $24 \sqrt{10} \mathrm{~cm}$ (d) 64 cm Solution: (d) 64 cmLet the breadth of the rectangle bexcm. Length of the rectangle = 3xcmWe know: Diagonal $=\sqrt{(\text { Length })^{2}+(\text { Breadth })^{2}}$ $\Rightarrow 8 \sqrt{10}=\sqrt{x^{2}+(3 x)^{2}}$ $\Rightarrow 8 \sqrt{10}=\s...
Read More →Factorize each of the following expression:
Question: Factorize each of the following expression:abx2+ (ay b)x y Solution: $a b x^{2}+(a y-b) x-y=a b x^{2}+a x y-b x-y$ $=\left(a b x^{2}-b x\right)+(a x y-y)$ [Regrouping the expressions] $=b x(a x-1)+y(a x-1)$ $=(b x+y)(a x-1)$ $[$ Taking $(a x-1)$ as the common factor $]$...
Read More →Solve the following equations
Question: If $A_{r}=\left|\begin{array}{ccc}1 r 2^{r} \\ 2 n n^{2} \\ n \frac{n(n+1)}{2} 2^{n+1}\end{array}\right|$, then the value of $\sum_{r=1}^{n} A_{r}$ is (a) $n$ (b) $2 n$ (c) $-2 n$ (d) $n^{2}$ Solution: $A_{r}=\mid 1 \quad r \quad 2^{r}$ $\begin{array}{lll}2 n n^{2}\end{array}$ $n \quad \frac{n(n+1)}{2} \quad 2^{n+1} \mid$ $\Rightarrow \sum_{r=1}^{n} A_{r}=\mid \sum_{r=1}^{n} 1 \quad \sum_{r=1}^{n} r \quad \sum_{r=1}^{n} 2^{r} \sum_{r=1}^{n} 2 \quad n \quad n \quad n^{2} n \quad \frac{n...
Read More →S is any point on side QR of a ΔPQR. Show that PQ + QR + RP > 2 PS.
Question: S is any point on side QR of a ΔPQR. Show that PQ + QR + RP 2 PS. Thinking Process Use the inequality of a triangle i.e., sum of two sides of a triangle is greater than the third side. Further, show the required result. Solution: Given In $\triangle P Q R, S$ is any point on side $Q R$. To show $P Q+Q R+R P2 P S$ Proof $\ln \triangle P Q S$, ... (i) [sum of two sides of a triangle is greater than the third side] Similarly, in $\triangle P R S$, $S R+R PP S$.....(ii) [sum of two sides o...
Read More →Factorize each of the following expressions:
Question: Factorize each of the following expressions:x3 2x2y+ 3xy2 6y3 Solution: $x^{3}-2 \mathrm{x}^{2} \mathrm{y}+3 \mathrm{xy}^{2}-6 \mathrm{y}^{3}$ $=\left(x^{3}-2 x^{2} y\right)+\left(3 x y^{2}-6 y^{3}\right)$ [Grouping the expressions] $=x^{2}(x-2 y)+3 y^{2}(x-2 y)$ $=\left(x^{2}+3 y^{2}\right)(x-2 y)$ $[$ Taking $(x-2 y)$ as the common factor $]$...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The cost of carpeting a room 15 m long with a carpet 75 cm wide, at₹70per metre, is₹8400. The width of the room is(a) 9 m (b) 8 m (c) 6 m (d) 12 m Solution: We have, Width of the carpet $=75 \mathrm{~cm}=0.75 \mathrm{~m}$ and length of the room $=15 \mathrm{~m}$ Length of the carpet $=\frac{\text { Cost of carpeting }}{\text { Rate of carpeting }}=\frac{8400}{70}=120 \mathrm{~m}$ Now, area of the carpet required for carpeting $=120 \t...
Read More →Factorize each of the following expressions:
Question: Factorize each of the following expressions:x2 2ax 2ab+bx Solution: $x^{2}-2 a x-2 a b+b x$ $=\left(x^{2}-2 a x\right)+(b x-2 a b)$ [Regrouping the expressions] $=x(x-2 a)+b(x-2 a)$ $=(x+b)(x-2 a)$ $[$ Taking $(x-2 a)$ as the common factor $]$ $=(x-2 a)(x+b)$...
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