Diagonals of a quadrilateral
Question: Diagonals of a quadrilateral ABCD bisect each other. If A= 35, determine B. Solution: Since, diagonals of a quadrilateral bisect each other, so it is a parallelogram. Therefore, the sum of interior angles between two parallel lines is 180 i.e., A+B = 180 = B = 180 A = 180- 35 [ A = 35, given] = B = 145...
Read More →In a circle of radius 10.5 cm, the minor arc is one-fifth of the major arc
Question: In a circle of radius 10.5 cm, the minor arc is one-fifth of the major arc. Find the area of the sector corresponding to the major arc. Solution: Let the length of the major arc be $x \mathrm{~cm}$ Radius of the circle = 10.5 cm $\therefore$ Length of the minor $\operatorname{arc}=\frac{x}{5} \mathrm{~cm}$ Circumference $=\left(x+\frac{x}{5}\right)=\frac{6 x}{5} \mathrm{~cm}$ Using the given data, we get: $\frac{6 x}{5}=2 \times \frac{22}{7} \times \frac{21}{2}$ $\Rightarrow \frac{6 x}...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:p2+ 6p+ 8 Solution: $p^{2}+6 p+8$ $=\mathrm{p}^{2}+6 \mathrm{p}+\left(\frac{6}{2}\right)^{2}-\left(\frac{6}{2}\right)^{2}+8 \quad$ [Adding and subtracting $\left(\frac{6}{2}\right)^{2}$, that is, $\left.3^{2}\right]$ $=\mathrm{p}^{2}+6 \mathrm{p}+3^{2}-3^{2}+8$ $=\mathrm{p}^{2}+2 \times \mathrm{p} \times 3+3^{2}-9+8$ $=\mathrm{p}^{2}+2 \times \mathrm{p} \times 3+3^{2}-1$ $=(\mathrm{p}+3)^...
Read More →Can all the angles of a quadrilateral
Question: Can all the angles of a quadrilateral be right angles? Give reason for your answer. Solution: Yes, all the angles of a quadrilateralcan be right angles. In this case, the quadrilateral becomes rectangle or square....
Read More →If x = –9 is a root of
Question: If $x=-9$ is a root of $\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|$ $=0$, then other two roots are___________ Solution: Given: $x=-9$ is a root of $\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|=0$ Let $\Delta=\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|$ $\Delta=\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|$ Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ $=\left|\begin{array}{ccc}x+2+7 3+x+6 7+2+x \\ 2 ...
Read More →Can all the angles of
Question: Can all the angles of a quadrilateral be acute angles? Give reason for your answer. Solution: No, all the angles of a quadrilateral cannot be acute angles. As, sum of the angles of a quadrilateral is 360. So, maximum of three acute angles will be possible....
Read More →If x = –9 is a root of ∣∣∣
Question: If $x=-9$ is a root of $\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|$ $=0$, then other two roots are___________ Solution: Given: $x=-9$ is a root of $\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|=0$ Let $\Delta=\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|$ $\Delta=\left|\begin{array}{lll}x 3 7 \\ 2 x 2 \\ 7 6 x\end{array}\right|$ Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$ $=\left|\begin{array}{ccc}x+2+7 3+x+6 7+2+x \\ 2 ...
Read More →In figure, ABCD and AEFG are
Question: In figure, ABCD and AEFG are two parallelograms. If C = 55, then determine F. Solution: We have, ABCD and AEFG are two parallelograms and C = 55. Since, ABCD is a parallelogram, then opposite angles of a parallelogram are equal. A = C = 55 ...(i) Also, AEFG is a parallelogram. A=F = 55 [from Eq. (i)]...
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:6x2y2, 9xy3, 3x3y2 Solution: The numerical coefficients of the given monomials are 6, 9 and 3. The greatest common factor of 6, 9 and 3 is 3. The common literals appearing in the three monomials are x and y. The smallest power of x in the three monomials is 1. The smallest power of y in the three monomials is 2. The monomial of common literals with the smallest powers is xy2. Hence, the greatest common factor is 3xy...
Read More →A chord of a circle of radius 30 cm makes an angle of 60° at the centre of the circle.
Question: A chord of a circle of radius 30 cm makes an angle of 60 at the centre of the circle. Find the areas of the minor major segments. Solution: Let the chord beAB. The ends of the chord are connected to the centre of the circleOto give the triangleOAB.OABis an isosceles triangle. The angle at the centre is 60 Area of the triangle $=\frac{1}{2}(30)^{2} \sin 60^{\circ}=450 \times \frac{\sqrt{3}}{2}=389.25 \mathrm{~cm}^{2}$ Area of the sector $O A C B O=\frac{60}{360} \times \pi \times 30 \ti...
Read More →In the figure, it is given that BDEF
Question: In the figure, it is given that BDEF and FDCE are parallelogram. Can you say that BD = CD? Why or why not? Solution: Yes, in the given figure, BDEF is a parallelogram.. BD || EF and BD = EF (i) Also, FDCE is a parallelogram. CD||EF and CD = EF(ii) From Eqs.(i)and(ii), BD = CD = EF...
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:4a2b3, 12a3b, 18a4b3 Solution: The numerical coefficients of the given monomials are 4, -12 and 18. The greatest common factor of 4, -12 and 18 is 2. The common literals appearing in the three monomials are a and b. The smallest power of a in the three monomials is 2. The smallest power of b in the three monomials is 1. The monomial of the common literals with the smallest powers is a2b. Hence, the greatest common fa...
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:9x2, 15x2y3, 6xy2and 21x2y2 Solution: The numerical coefficients of the given monomials are 9, 15, 6 and 21. The greatest common factor of 9, 15, 6 and 21 is 3. The common literal appearing in the three monomials is x. The smallest power of x in the four monomials is 1. The monomial of common literals with the smallest powers is x. Hence, the greatest common factor is3x....
Read More →In ΔABC, AB = 5 cm,
Question: In ΔABC, AB = 5 cm, BC = 8 cm and CA = 7 cm. If D and E are respectively the mid-points of AB and BC, determine the length of DE. Solution: In ΔABC, we have AB = 5cm, BC = 8 cm and CA = 7 cm. Since, D and E are the mid-points of AB and BC, respectively. By mid-point theorem, DE || AC and $D E=\frac{1}{2} A C=\frac{7}{2}=3.5 \mathrm{~cm}$...
Read More →Can all the four angles of
Question: Can all the four angles of a quadrilateral be obtuse angles? Give reason for your answer. Solution: No, all the four angles of a quadrilateral cannot be obtuse. As, the sum of the angles of a quadrilateral is 360, then may have maximum of three obtuse angles....
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:12ax2, 6a2x3and 2a3x5 Solution: The numerical coefficients of the given monomials are 12, 6 and 2. The greatest common factor of 12, 6 and 2 is 2. The common literals appearing in the three monomials are a and x. The smallest power of a in the three monomials is 1. The smallest power of x in the three monomials is 2. The monomial of common literals with the smallest powers is ax2. Hence, the greatest common factor i...
Read More →Diagonals of a rectangle are
Question: Diagonals of a rectangle are equal and perpendicular. Is this statement true? Give reason for your answer. Solution: No, diagonals of a rectangle are equal but need not be perpendicular....
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:42x2yzand 63x3y2z3 Solution: The numerical coefficients of the given monomials are 42 and 63. The greatest common factor of 42 and 63 is 21. The common literals appearing in the two monomials are x, y and z. The smallest power of x in the two monomials is 2. The smallest power of y in the two monomials is 1. The smallest power of z in the two monomials is 1. The monomial of the common literals with the smallest power...
Read More →If A is a matrix of order 3 × 3,
Question: If $A$ is a matrix of order $3 \times 3$, then the number of minors in $A$ is________ Solution: Ais a matrix of order 3 3⇒Ahas 9 elements⇒Ahas 9 minorsHence,the number of minors inAis9....
Read More →Find the areas of both the segments of a circle of radius 42 cm with central angle 120°.
Question: Find the areas of both the segments of a circle of radius 42 cm with central angle 120. Solution: Area of the minor sector $=\frac{120}{360} \times \pi \times 42 \times 42$ $=\frac{1}{3} \times \pi \times 42 \times 42$ $=\pi \times 14 \times 42$ $=1848 \mathrm{~cm}^{2}$ Area of the triangle $=\frac{1}{2} R^{2} \sin \theta$ Here,Ris the measure of the equal sides of the isosceles triangle and is the angle enclosed by the equal sides.Thus, we have: $\frac{1}{2} \times 42 \times 42 \times...
Read More →All the angles of a quadrilateral are equal.
Question: All the angles of a quadrilateral are equal. What special name is given to this quadrilateral? Solution: We know that, sum of all angles in a quadrilateral is 360. If ABCD is a quadrilateral, A+ B+ C + D = 360 (i) But it is given all angles are equal. A = B = C = D From Eq.(i) A + A + A + A = 360 = 4 A = 360 A = 90 So, all angles of a quadrilateral are 90. Hence, given quadrilateral is a rectangle....
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:7x, 21x2and 14xy2 Solution: The numerical coefficients of the given monomials are 7, 21 and 14. The greatest common factor of 7, 21 and 14 is 7. The common literal appearing in the three monomials is x. The smallest power of x in the three monomials is 1. The monomial of the common literals with the smallest powers is x. Hence, the greatest common factor is 7x....
Read More →In quadrilateral ABCD, ∠A + ∠D = 180°.
Question: In quadrilateral ABCD, A + D = 180. What special name can be given to this quadrilateral? Solution: It is a trapezium because sum of cointerior angles is 180....
Read More →Find the greatest common factor (GCF/HCF) of the following polynomial:
Question: Find the greatest common factor (GCF/HCF) of the following polynomial:6x3yand 18x2y3 Solution: The numerical coefficients of the given monomials are 6 and 18. The greatest common factor of 6 and 18 is 6. The common literals appearing in the two monomials are x and y. The smallest power of x in the two monomials is 2. The smallest power of y in the two monomials is 1. The monomial of the common literals with the smallest powers is x2y. Hence, the greatest common factor is 6x2y....
Read More →Can the angles 110°, 80°, 70° and 95°
Question: Can the angles 110, 80, 70 and 95 be the angles of a quadrilateral? Why or why not? Solution: No, we know that, sum of all angles of a quadrilateral is 360. Here, sum of the angles = 110+ 80 + 70 + 95 = 355 360 So, these angles cannot be the angles of a quadrilateral....
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