Find the product of the following binomials:
Question: Find the product of the following binomials: (i) (2x+y)(2x+y) (ii) (a+ 2b)(a 2b) (iii) (a2+bc)(a2bc) (iv) $\left(\frac{4 x}{5}-\frac{3 y}{4}\right)\left(\frac{4 x}{5}+\frac{3 y}{4}\right)$ (v) $\left(2 x+\frac{3}{y}\right)\left(2 x-\frac{3}{y}\right)$ (vi) (2a3+b3)(2a3b3) (vii) $\left(x^{4}+\frac{2}{x^{2}}\right)\left(x^{4}-\frac{2}{x^{2}}\right)$ (viii) $\left(x^{3}+\frac{1}{x^{3}}\right)\left(x^{3}-\frac{1}{x^{3}}\right)$ Solution: (i) We will use the identity $(a+b)^{2}=a^{2}+2 a b+...
Read More →In the figure, POQ is a line.
Question: In the figure, POQ is a line. The value of $x$ is (a)20 (b)25 (c)30 (d) 35 Thinking Process When two or more rays are initiated from a same point of a line, then the sum of all angles made between the rays and line at the same point is 180. Solution: (a) Since, $P O Q$ is a line segment. $\therefore \quad \angle P O Q=180^{\circ}$ $\Rightarrow \quad \angle P O A+\angle A O B+\angle B O Q=180^{\circ}$ $\Rightarrow \quad 40^{\circ}+4 x+3 x=180^{\circ}$ [putting $\angle P O A=40^{\circ}, ...
Read More →The area of a rectangular plot is 462 m2 and its length is 28 m.
Question: The area of a rectangular plot is 462 m2and its length is 28 m. Find the perimeter of the plot. Solution: Area of the rectangular plot = 462 m2Length (l) = 28 m Area of a rectangle $=$ Length $(l) \times$ Breadth $(b)$ $\Rightarrow 462=28 \times b$ $\Rightarrow b=16.5 \mathrm{~m}$ Perimeter of the plot $=2(l+b)$ $=2(28+16.5)$ $=2 \times 44.5$ $=89 \mathrm{~m}$...
Read More →If one of the angles of a triangle is 130°,
Question: If one of the angles of a triangle is 130, then the angle between the bisectors of the other two angles can be (a) 50 (b) 65 (c) 145 (d) 155 Solution: (d) Let angles of a triangle be A, B and C. In $\triangle A B C$, $\angle A+\angle B+\angle C=180^{\circ}$ [sum of all interior angles of a triangle is $180^{\circ}$ ] $\Rightarrow \quad \frac{1}{2} \angle A+\frac{1}{2} \angle B+\frac{1}{2} \angle C=\frac{180^{\circ}}{2}=90^{\circ}$ [dividing both sides by 2] $\Rightarrow \quad \frac{1}{...
Read More →One side of a rectangle is 12 cm long and its diagonal measures 37 cm.
Question: One side of a rectangle is 12 cm long and its diagonal measures 37 cm. Find the other side and the area of the rectangle. Solution: One side of the rectangle = 12 cmDiagonal of the rectangle = 37 cmThe diagonal of a rectangle forms the hypotenuse of a right-angled triangle. The other two sides of the triangle are the length and the breadth of the rectangle.Now, using Pythagoras' theorem, we have: (one side) $^{2}+(\text { other side })^{2}=$ (hypotenuse) $^{2}$ $\Rightarrow(12)^{2}+(\t...
Read More →If the angles of a triangle are in the ratio 5:3:7,
Question: If the angles of a triangle are in the ratio 5:3:7, then the triangle is (a) an acute angled triangle (b) an obtuse angled triangle (c) a right angled triangle (d) an isosceles triangle Solution: (a) Given, the ratio of angles of a triangle is 5 : 3 : 7. Let angles of a triangle be A,B and C. Then, A = 5x, B = 3x and C = 7x In ΔABC, A + B + C = 180 [since, sum of all angles of a triangle is 180] 5x + 3x + 7x = 180 = 15x = 180 x = 180/15= 12 A = 5x = 5 x 12 = 60 B = 3x= 3 x 12= 36 and C...
Read More →The length of a rectangular park is twice its breadth, and its perimeter measures 840 m.
Question: The length of a rectangular park is twice its breadth, and its perimeter measures 840 m. Find the area of the park. Solution: Let the breadth of the rectangular park be $b$. $\therefore$ Length of the rectangular park $=l=2 b$ Perimeter $=840 \mathrm{~m}$ $\Rightarrow 840=2(l+b)$ $\Rightarrow 840=2(2 b+b)$ $\Rightarrow 840=2(3 b)$ $\Rightarrow 840=6 b$ $\Rightarrow b=140 \mathrm{~m}$ Thus, we have: $l=2 b$ $=2 \times 140$ $=280 \mathrm{~m}$ Area $=l \times b$ $=280 \times 140$ $=39200 ...
Read More →Write the following squares of binomials as trinomials:
Question: Write the following squares of binomials as trinomials: (i) (x+ 2)2 (ii) (8a+ 3b)2 (iii) (2m+ 1)2 (iv) $\left(9 a+\frac{1}{6}\right)^{2}$ (v) $\left(x+\frac{x^{2}}{2}\right)^{2}$ (vi) $\left(\frac{x}{4}-\frac{y}{3}\right)$ (vii) $\left(3 x-\frac{1}{3 x}\right)^{2}$ (viii) $\left(\frac{x}{y}-\frac{y}{x}\right)^{2}$ (ix) $\left(\frac{3 a}{2}-\frac{5 b}{4}\right)^{2}$ (x) $\left(a^{2} b-b c^{2}\right)^{2}$ (xi) $\left(\frac{2 a}{3 b}+\frac{2 b}{3 a}\right)^{2}$ (xii) $\left(x^{2}-a y\righ...
Read More →An exterior angle of a triangle is 105°
Question: An exterior angle of a triangle is 105 and its two interior opposite angles are equal. Each of these equal angles is (a) 37 (b) 52 (c)72 (d) 75 Solution: Let one of interior angle be x. Sum of two opposite interior angles = Exterior angle x + x = 105 2x = 105 x = 105/2 x=52 Hence, each angle of a triangle is 52 ....
Read More →Solve this
Question: $3 x+y=5$ $-6 x-2 y=9$ Solution: Given: $3 x+y=5$ $-6 x-2 y=9$ $D=\left|\begin{array}{cr}3 1 \\ -6 -2\end{array}\right|=-6+6=0$ $D_{1=}\left|\begin{array}{rr}5 1 \\ 9 -2\end{array}\right|=-10-9=-19$ $D_{2}=\left|\begin{array}{cc}3 5 \\ -6 9\end{array}\right|=27+30=57$ Here, $D_{1}$ and $D_{2}$ are non-zero, but $D$ is zero. Thus, the system of linear equations is inconsistent....
Read More →If one angle of a triangle is equal to
Question: If one angle of a triangle is equal to the sum of the other two angles, then the triangle is (a) an isosceles triangle (b) an obtuse triangle (c) an equilateral triangle (d) a right triangle Solution: (d) Let the angles of a AABC be A, B and C. Given, A = B+C (i) InMBC, A+ B+ C-180 [sum of all angles of atriangle is 180](ii) From Eqs. (i) and (ii), A+A = 180 = 2 A = 180 = 180 /2 A = 90 Hence, the triangle is a right triangle....
Read More →In figure, if AB || CD || EE, PQ || RS,
Question: In figure, if AB || CD || EE, PQ || RS, RQD = 25 and CQP = 60, then QRS is equal to (a) $85^{\circ}$ (b) $135^{\circ}$ (c) $145^{\circ}$ (d) $110^{\circ}$ Solution: (c) Given, PQ || RS PQC = BRS = 60 [alternate exterior angles and PQC = 60 (given)] and DQR = QRA = 25 [alternate interior angles] [DQR = 25, given] QRS = QRA + ARS = QRA + (180 BRS) [linear pair axiom] = 25 + 180 60= 205 60= 145...
Read More →Solve this
Question: $2 x-y=5$ $4 x-2 y=7$ Solution: Given: $2 x-y=5$ $4 x-2 y=7$ $D=\left|\begin{array}{ll}2 -1 \\ 4 -2\end{array}\right|=-4+4=0$ $D_{1}=\left|\begin{array}{ll}5 -1 \\ 7 -2\end{array}\right|=-10+7=-3$ $D_{2=}\left|\begin{array}{ll}2 5 \\ 4 7\end{array}\right|=14-20=-6$ Here, $D_{1}$ and $D_{2}$ are non-zero, but $D$ is zero. Thus, the given system of linear equations is inconsistent....
Read More →The perimeter of a rectangular plot of land is 80 m and its breadth is 16 m.
Question: The perimeter of a rectangular plot of land is 80 m and its breadth is 16 m. Find the length and area of the plot. Solution: As, perimeter $=80 \mathrm{~m}$ $\Rightarrow 2($ length $+$ breadth $)=80$ $\Rightarrow 2($ length $+16)=80$ $\Rightarrow 2 \times$ length $+32=80$ $\Rightarrow 2 \times$ length $=80-32$ $\Rightarrow$ length $=\frac{48}{2}$ $\therefore$ length $=24 \mathrm{~m}$ Now, the area of the plot $=$ length $\times$ breadth $=24 \times 16$ $=384 \mathrm{~m}^{2}$ So, the le...
Read More →Solve this
Question: $2 x-3 z+w=1$ $x-y+2 w=1$ $-3 y+z+w=1$ $x+y+z=1$ Solution: $D=\left|\begin{array}{cccc}2 0 -3 1 \\ 1 -1 0 2 \\ 0 -3 1 1 \\ 1 1 1 0\end{array}\right|$ $2\left|\begin{array}{ccc}-1 0 2 \\ -3 1 1 \\ 1 1 0\end{array}\right|-0-3\left|\begin{array}{ccc}1 -1 2 \\ 0 -3 1 \\ 1 1 0\end{array}\right|-1\left|\begin{array}{ccc}1 -1 0 \\ 0 -3 1 \\ 1 1 1\end{array}\right|$ $=2[-1(0-1)-0(0-1)+2(-3-1)]-3[1(0-1)+1(0-1)+2(0+3)]-1[1(-3-1)+1(0-1)+0(0+3)]$ $=-21$ $D_{1}=\left|\begin{array}{cccc}1 0 -3 1 \\ ...
Read More →Simplify:
Question: Simplify:(x3 2x2+ 3x 4) (x1) (2x 3)(x2x+ 1) Solution: To simplify,we will proceed as follows: (x3 2x2+ 3x 4) (x1) (2x 3)(x2x+ 1) $=\left[\left(x^{3}-2 x^{2}+3 x-4\right)(x-1)\right]-\left[(2 x-3)\left(x^{2}-x+1\right)\right]$ $=\left[x\left(x^{3}-2 x^{2}+3 x-4\right)-1\left(x^{3}-2 x^{2}+3 x-4\right)\right]-\left[2 x\left(x^{2}-x+1\right)-3\left(x^{2}-x+1\right)\right]$ (Distributive law) $=\left[x\left(x^{3}-2 x^{2}+3 x-4\right)-1\left(x^{3}-2 x^{2}+3 x-4\right)\right]-\left[2 x\left(...
Read More →In the given figure, ∆ABC is an equilateral triangle the length of whose side is equal to 10 cm,
Question: In the given figure, $\triangle A B C$ is an equilateral triangle the length of whose side is equal to $10 \mathrm{~cm}$, and $\triangle D B C$ is right-angled at $D$ and $B D=8 \mathrm{~cm}$. Find the area of the shaded region. Take $\sqrt{3}=1.732$. Solution: Given:Side of equilateral triangleABC= 10 cmBD= 8 cm Area of equilateral $\Delta A B C=\frac{\sqrt{3}}{4} \mathrm{a}^{2}$ (where $a=10 \mathrm{~cm}$ ) Area of equilateral $\triangle A B C=\frac{\sqrt{3}}{4} \times 10^{2}$ $=25 \...
Read More →Simplify:
Question: Simplify:(x2 3x+ 2)(5x 2) (3x2+ 4x 5)(2x 1) Solution: To simplify, we will proceed as follows: (x2 3x+ 2)(5x 2) (3x2+ 4x 5)(2x 1) $=\left[\left(x^{2}-3 x+2\right)(5 x-2)\right]-\left[\left(3 x^{2}+4 x-5\right)(2 x-1)\right]$ $=\left[5 x\left(x^{2}-3 x+2\right)-2\left(x^{2}-3 x+2\right)\right]-\left[2 x\left(3 x^{2}+4 x-5\right)-1 \times\left(3 x^{2}+4 x-5\right)\right]$ (Distributive law) $=\left[5 x^{3}-15 x^{2}+10 x-\left(2 x^{2}-6 x+4\right)\right]-\left[6 x^{3}+8 x^{2}-10 x-3 x^{2}...
Read More →solve the problem
Question: $x+y+z+w=2$ $x-2 y+2 z+2 w=-6$ $2 x+y-2 z+2 w=-5$ $3 x-y+3 z-3 w=-3$ Solution: $D=\left|\begin{array}{cccc}1 1 1 1 \\ 1 -2 2 2 \\ 2 1 -2 2 \\ 3 -1 3 -3\end{array}\right|$ $1\left|\begin{array}{ccc}-2 2 2 \\ 1 -2 2 \\ -1 3 -3\end{array}\right|-1\left|\begin{array}{ccc}1 2 2 \\ 2 -2 2 \\ 3 3 -3\end{array}\right|+1\left|\begin{array}{ccc}1 -2 2 \\ 2 1 2 \\ 3 -1 -3\end{array}\right|-1\left|\begin{array}{ccc}1 -2 2 \\ 2 1 -2 \\ 3 -1 3\end{array}\right|$ $=1[-2(6-6)-2(-3+2)+2(3-2)]-1[1(6-6)-...
Read More →Find the area and perimeter of an isosceles right-angled triangle,
Question: Find the area and perimeter of an isosceles right-angled triangle, each of whose equal sides measures $10 \mathrm{~cm}$. [Given: $\sqrt{2}=1.41$ ] Solution: Let:Length of each of the equal sides of the isosceles right-angled triangle =a= 10 cmAnd,Base = Height =a Area of isosceles right $-$ angled triangle $=\frac{1}{2} \times$ Base $\times$ Height $=\frac{1}{2} \times 10 \times 10$ $=50 \mathrm{~cm}^{2}$ The hypotenuse of an isosceles right-angled triangle can be obtained using Pythag...
Read More →Simplify: (3x + 2y)(4x + 3y) − (2x − y)(7x − 3y)
Question: Simplify:(3x+ 2y)(4x+ 3y) (2xy)(7x 3y) Solution: To simplify, we will proceed as follows: (3x + 2y)(4x + 3y) (2x y)(7x 3y) $=[(3 x+2 y)(4 x+3 y)]-[(2 x-y)(7 x-3 y)]$ $=[3 x(4 x+3 y)+2 y(4 x+3 y)]-[2 x(7 x-3 y)-y(7 x-3 y)]$ (Distributive law) $=12 x^{2}+9 x y+8 x y+6 y^{2}-\left[14 x^{2}-6 x y-7 x y+3 y^{2}\right]$ $=12 x^{2}+9 x y+8 x y+6 y^{2}-14 x^{2}+6 x y+7 x y-3 y^{2}$ (Rearranging) $=12 x^{2}-14 x^{2}+9 x y+8 x y+6 x y+7 x y+6 y^{2}-3 y^{2}$ (Combining like terms) $=-2 x^{2}+30 x...
Read More →Read the following axioms
Question: Read the following axioms (i)Things which are equal to the same thing are equal to one another. (ii)If equals are added to equals, the wholes are equal. (iii)Things which are double of the same things are equal to one another. Check whether the given system of axioms is consistent or inconsistent. Thinking Process To check the given system is consistent or inconsistent, we have to find that whether we can deduce a statement from these axioms which contradicts any axiom or not. Solution...
Read More →Read the following two statements which are taken as axioms:
Question: Read the following two statements which are taken as axioms: (i)If two lines intersect each other, then the vertically opposite angles are not equal. (ii)If a ray stands on a line, then the sum of two adjacent angles, so formed is equal to 180. Is this system of axioms consistent ? Justify your answer. Solution: We know that, if two lines intersect each other, then the vertically opposite angles are equal. It is a theorem, So given Statement I is false and not an axiom. Also, we know t...
Read More →Read the following statements which are taken as axioms
Question: Read the following statements which are taken as axioms (i)If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal. (ii)If a transversal intersect two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent ? Justify your answer. Solution: A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axiom. We know that, if a transvers...
Read More →Simplify: (5x − 3)(x + 2) − (2x + 5)(4x − 3)
Question: Simplify:(5x 3)(x+ 2) (2x+ 5)(4x 3) Solution: To simplify, we will proceed as follows: (5x 3)(x+ 2) (2x+ 5)(4x 3) $=[(5 x-3)(x+2)]-[(2 x+5)(4 x-3)]$ $=[5 x(x+2)-3(x+2)]-[2 x(4 x-3)+5(4 x-3)]$(Distributive law) $=5 x^{2}+10 x-3 x-6-8 x^{2}+6 x-20 x+15$ $=5 x^{2}-8 x^{2}+10 x-3 x+6 x-20 x-6+15$ (Rearranging) $=5 x^{2}-8 x^{2}+10 x-3 x+6 x-20 x-6+15$ (Combining like terms) $=-3 x^{2}-7 x+9$ Hence, the answer is $-3 x^{2}-7 x+9$....
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