Plot the points (x, y) given by the following table.
Question: Plot the points (x, y) given by the following table. Solution: On plotting the given points on the graph, we get the points P(2,4), Q(4,2) R (-3, 0), S (-2, 5), T (3, 3)and O (0 0)...
Read More →Plot the following points and write the name
Question: Plot the following points and write the name of the figure obtained by joining, them in order P(-3, 2), Q(-7, -3), R(6, -3)andS(2, 2). Solution: Let X OX and Y OY be the coordinate axes and mark point on it. Here, point P(-3,2) lies in II quadrant, Q(-7,-3) lies in III quadrant, R(6, 3) lies in IV quadrant and S(2,2) lies in I quadrant. Plotting the points on the graph paper, the figure obtained is trapezium PQRS....
Read More →Find each of the following product:
Question: Find each of the following product: $(0.5 x) \times\left(\frac{1}{3} x y^{2} z^{4}\right) \times\left(24 x^{2} y z\right)$ Solution: To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$. We have: $(0.5 x) \times\left(\frac{1}{3} x y^{2} z^{4}\right) \times\left(24 x^{2} y z\right)$ $=\left(0.5 \times \frac{1}{3} \times 24\right) \times\left(x \times x \times x^{2}\right) \times\left(y^{2} \times y\r...
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Question: $x+2 y=1$ $3 x+y=4$ Solution: Given: $x+2 y=1$ $3 x+y=4$ $D=\left|\begin{array}{ll}1 2 \\ 3 1\end{array}\right|=-5$ $D_{1}=\left|\begin{array}{ll}1 2 \\ 4 1\end{array}\right|=-7$ $D_{2}=\left|\begin{array}{ll}1 1 \\ 3 4\end{array}\right|=1$ Now, $x=\frac{D_{1}}{D}=\frac{7}{5}$ $y=\frac{D_{2}}{D}=-\frac{1}{5}$ $\therefore x=\frac{7}{5}$ and $y=-\frac{1}{5}$...
Read More →Write the coordinates of each of the points P, Q, R, S, T and 0 from the figure
Question: Write the coordinates of each of the points P, Q, R, S, T and 0 from the figure Thinking Process (i)Firstly, draw the perpendicular lines from the point to the coordinates axes. (ii)Further, measure the distance from intersecting points to the origin along their sign. (iii)Finally, write the x unit distance and y unit distance in pair. Solution: Here, points P and S lie in I quadrant so their both coordinates will be positive. Now, perpendicular distance of P from both axes is 1, so co...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question: If a pole $12 \mathrm{~m}$ high casts a shadow $4 \sqrt{3} \mathrm{~m}$ long on the ground, then the sun's elevation is (a) 60 (b) 45 (c) 30 (d) 90 Solution: Let $A B$ be the pole, $B C$ be its shadow and $\theta$ be the sun's elevation. We have, $\mathrm{AB}=12 \mathrm{~m}$ and $\mathrm{BC}=4 \sqrt{3} \mathrm{~m}$ In $\Delta \mathrm{ABC}$, $\tan \theta=\frac{\mathrm{AB}}{\mathrm{BC}}$ $\Rightarrow \tan \theta=\frac{12}{4 \sqrt{3}}$ ...
Read More →Find each of the following product:
Question: Find each of the following product: $\left(\frac{4}{3} u^{2} v w\right) \times\left(-5 u v w^{2}\right) \times\left(\frac{1}{3} v^{2} w u\right)$ Solution: To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$. We have: $\left(\frac{4}{3} u^{2} v w\right) \times\left(-5 u v w^{2}\right) \times\left(\frac{1}{3} v^{2} w u\right)$ $=\left\{\frac{4}{3} \times(-5) \times \frac{1}{3}\right\} \times\left(u^...
Read More →Write whether the following statements are true or false?
Question: Write whether the following statements are true or false? Justify your answer. (i)Point (3, 0) lies in the first quadrant. (ii)Points (1 -1) and (-1, 1) lie in the same quadrant. (iii)The coordinates of a point whose ordinate is and abscissa is 1 are(- ,1). (iv)A point lies on Y-axis at a distance of 2 units from the X-axis. Its coordinates are (2, 0). (v)(-1,7) is a point in the second quadrant. Solution: (i)False, since the ordinate of the point (3, 0) is zero. So, the point lies on ...
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Question: $9 x+5 y=10$ $3 y-2 x=8$ Solution: Given : $9 x+5 y=10$ $3 y-2 x=8$ Rearranging the second equation, the two equations can be written as $9 x+5 y=10$ $-2 x+3 y=8$ Now, $D=\mid 9 \quad 5$ $-2 \quad 3 \mid=27+10=37$ $D_{1}=\mid 105$ $8 \quad 3 \mid=30-40=-10$ $D_{2}=\mid 9 \quad 10$ $-2 \quad 8 \mid=72+20=92$ Using Cramer's rule, we get $x=\frac{D_{1}}{D}=\frac{-10}{37}$ $y=\frac{D_{2}}{D}=\frac{92}{37}$ $\therefore x=\frac{-10}{37}$ and $y=\frac{92}{37}$...
Read More →The perpendicular distance of the point
Question: The perpendicular distance of the point P(3, 4) from the Y-axis is (a)3 (b)4 (c)5 (d)7 Solution: (a)We know that, abscissa or the x-coordinate of a point is its perpendicular distance from the Y-axis. So, perpendicular distance of the point P(3, 4)from Y-axis= Abscissa = 3...
Read More →If the length of the shadow of a tower is
Question: If the length of the shadow of a tower is $\sqrt{3}$ times its height then the angle of elevation of the sun is (a) 45(b) 30(c) 60(d) 90 Solution: (b) 30 Let $A B$ be the pole and $B C$ be its shadow. Let $A B=h$ and $B C=x$ such that $x=\sqrt{3} h$ (given) and $\theta$ be the angle of elevation. From $\triangle A B C$, we have: $\frac{A B}{B C}=\tan \theta$ $\Rightarrow \frac{h}{x}=\frac{h}{\sqrt{3} h}=\tan \theta$ $\Rightarrow \tan \theta=\frac{1}{\sqrt{3}}$ $\Rightarrow \theta=30^{\...
Read More →Find each of the following product:
Question: Find each of the following product: $\left(\frac{-24}{25} x^{3} z\right) \times\left(-\frac{15}{16} x z^{2} y\right)$ Solution: To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$. We have: $\left(-\frac{24}{25} x^{3} z\right) \times\left(-\frac{15}{16} x z^{2} y\right)$ $=\left\{\left(-\frac{24}{25}\right) \times\left(-\frac{15}{16}\right)\right\} \times\left(x^{3} \times x\right) \times\left(z \t...
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Question: $5 x+7 y=-2$ $4 x+6 y=-3$ Solution: Given: $5 x+7 y=-2$ $4 x+6 y=-3$ Using Cramer's Rule, we get $D=\mid 5 \quad 7$ $4 \quad 6 \mid=30-28=2$ $D_{1}=\mid \begin{array}{ll}-2 7\end{array}$ $-3 \quad 6 \mid=-12+21=9$ $D_{2}=\mid 5-2$ $4-3 \mid=-15+8=-7$ Now, $x=\frac{D_{1}}{D}=\frac{9}{2}$ $y=\frac{D_{2}}{D}=\frac{-7}{2}$ $\therefore x=\frac{9}{2}$ and $y=\frac{-7}{2}$...
Read More →The point which lies on Y-axis at a distance
Question: The point which lies on Y-axis at a distance of 5 units in the negative direction of Y-axis is (a)(0,5) (b)(5,0) (c)(0,-5) (d)(-5,0) Solution: (C)Given the point lies on X-axis this shows that its ^-coordinate is zero. Also, it is at a distance of 5 units in negative direction of X-axis, so its y-coordinate is negative.Hence, the required point is (0, 5)....
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question: If the height of a vertical pole is $\sqrt{3}$ times the length of its shadow on the ground, then the angle of elevation of the sun at that time is (a) 30 (b) 45 (c) 60 (d) 75 Solution: Here, $A O$ be the pole; $B O$ be its shadow and $\theta$ be the angle of elevation of the sun. Let $\mathrm{BO}=x$ Then, $\mathrm{AO}=x \sqrt{3}$ In $\Delta \mathrm{AOB}$, $\tan \theta=\frac{\mathrm{AO}}{\mathrm{BO}}$ $\Rightarrow \tan \theta=\frac{x...
Read More →Which of the points P(0, 3), Q(l, 0), R(0, – 1),
Question: Which of the points P(0, 3), Q(l, 0), R(0, 1), S(-5, 0) and T(1, 2) do not lie on the X-axis? (a)P and R only (b)Q and S only (c)P,R and T (d)Q,S and T Solution: (c)We know that, if a point is of the form (x, 0)i.e., its y-coordinate is zero, then it will lie on X-axis otherwise not. Here, y-coordinates of points P(0, 3), R (0, -1) and T (1,2) are not zero, so these points do not lie on the X-axis....
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Question: $2 x+3 y=10$ $x+6 y=4$ Solution: Given : $2 x+3 y=10$ $x+6 y=4$ Using Cramer's Rule, we get $\mathrm{D}=\mid 2 \quad 3$ $1 \quad 6 \mid=12-3=9$ $\mathrm{D}_{1}=\mid 10 \quad 3$ $4 \quad 6 \mid=60-12=48$ $\mathrm{D}_{2}=\mid 2 \quad 10$ $1 \quad 4 \mid=8-10=-2$ Now, $\mathrm{x}=\frac{\mathrm{D}_{1}}{\mathrm{D}}=\frac{48}{9}=\frac{16}{3}$ $\mathrm{y}=\frac{\mathrm{D}_{2}}{\mathrm{D}}=\frac{-2}{9}$ $\therefore x=\frac{16}{3}$ and $\mathrm{y}=\frac{-2}{9}$...
Read More →Find each of the following product:
Question: Find each of the following product: $\left(\frac{7}{9} a b^{2}\right) \times\left(\frac{15}{7} a c^{2} b\right) \times\left(-\frac{3}{5} a^{2} c\right)$ Solution: To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$. We have: $\left(\frac{7}{9} a b^{2}\right) \times\left(\frac{15}{7} a c^{2} b\right) \times\left(-\frac{3}{5} a^{2} c\right)$ $=\left\{\frac{7}{9} \times \frac{15}{7} \times\left(-\frac...
Read More →The point whose ordinate is 4 and which lies on K-axis is
Question: The point whose ordinate is 4 and which lies on K-axis is (a)(4,0) (b)(0,4) (c)(1,4) (d)(4,2) Solution: (b)Given ordinate of the point is 4 arid the point lies on Y-axis, so its abscissa is zero. Hence, the required point is (0, 4)....
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:If the height of a vertical pole is equal to the length of its shadow on theground, the angle of elevation of the sun is(a) 0 (b) 30 (c) 45 (d) 60 Solution: LetAB represents the vertical poleand BC represents the shadow on the ground andrepresents angle ofelevation the sun. In $\triangle \mathrm{ABC}$, $\tan \theta=\frac{\mathrm{AB}}{\mathrm{BC}}$ $\Rightarrow \tan \theta=\frac{x}{x} \quad(\mathrm{As}$, the height of the pole, $\mathr...
Read More →In following figure, the point identified by the coordinates (-5, 3) is
Question: In following figure, the point identified by the coordinates (-5, 3) is (a) $T$ (b) $R$ (c) $L$ (d) $S$ Solution: (c)In point (-5, 3), x-coordinate is negative and y-coordinate is positive, so it will lie in II quadrant. Now, we see that perpendicular distance of L from V-axis is 5 and from X-axis is 3. So, the required point is L....
Read More →Find each of the following product:
Question: Find each of the following product: $\left(-\frac{2}{7} a^{4}\right) \times\left(-\frac{3}{4} a^{2} b\right) \times\left(-\frac{14}{5} b^{2}\right)$ Solution: To multiply algebraic expressions, we use commutative and associative laws along with the law of indices, i.e., $a^{m} \times a^{n}=a^{m+n}$. We have: $\left(-\frac{2}{7} a^{4}\right) \times\left(-\frac{3}{4} a^{2} b\right) \times\left(-\frac{14}{5} b^{2}\right)$ $=\left\{\left(-\frac{2}{7}\right) \times\left(-\frac{3}{4}\right) ...
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Question: $3 x+a y=4$ $2 x+a y=2, a \neq 0$ Solution: Given: $3 x+a y=4$ $2 x+a y=2$ Using Cramer's rule, we get $\mathrm{D}=\mid 3 \mathrm{a}$ $2 \mathrm{a} \mid=3 \mathrm{a}-2 \mathrm{a}=\mathrm{a}$ $\mathrm{D}_{1}=\mid 4 \mathrm{a}$ $2 \mathrm{a} \mid=4 \mathrm{a}-2 \mathrm{a}=2 \mathrm{a}$ $\mathrm{D}_{2}=\mid 3 \mathrm{4}$ $2 \quad 2 \mid=6-8=-2$ Now, $\mathrm{x}=\frac{\mathrm{D}_{1}}{\mathrm{D}}=\frac{2 \mathrm{a}}{\mathrm{a}}=2$ $\mathrm{y}=\frac{\mathrm{D}_{2}}{\mathrm{D}}=\frac{-2}{\mat...
Read More →The angles of depression of the top and bottom of a 8 m tall building from the top of a tower are 30° and 45°
Question: The angles of depression of the top and bottom of a 8 m tall building from the top of a tower are 30 and 45 respectively. Find the height of the tower and the distance between the tower and the building. Solution: Let AB be the building and CD be the tower.Draw AE CD. Suppose the height of the tower behm.Here, AB = 8 mDAE = ADX = 30 (Alternate angles)DBC = BDX = 45 (Alternate angles)CE = AB = 8 m DE = CD CE = (h 8) mDistance between the building and tower = BCIn right ∆BCD, $\tan 45^{\...
Read More →In following figure, coordinates of P are
Question: In following figure, coordinates of P are (a) $(-4,2)$ (b) $(-2,4)$ (c) $(4,-2)$ (d) $(2,-4)$ Solution: (b)Here, given point P lies in II quadrant, so its abscissa will be negative and ordinate wilt be positive. Also, its perpendicular distance from X-axis is 4, so y-coordinate of P is 4 and its perpendicular distance from Y-axis is 2, so x-coordinate is -2. Hence, coordinates of P are (-2, 4)....
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