Write the coordinates of the vertices of
Question: Write the coordinates of the vertices of a rectangle whose length and breadth are 5 and 3 units respectively, one vertex at the origin, the longer side lies on the X-axis and one of the vertices lies in the third quadrant. Solution: Given, length of a rectangle = 5 units and breadth of a rectangle = 3 units One vertex is at origin i.e., (0, 0) and one of the other vertices lies in III quadrant. So, the length of the rectangle is 5 units in the negative direction of X-axis and then vert...
Read More →Points A(5, 3), B(-2, 3) and 0(5, – 4) are three vertices of a square ABCD.
Question: Points A(5, 3), B(-2, 3) and 0(5, 4) are three vertices of a square ABCD. Plot these points on a graph paper and hence, find the coordinates of the vertex C.Thinking Process (i)Firstly, plot the given points on a graph and join in order. (ii)Now, we extend a line from point D parallel to X-axis and extend an other line from point 8 parallel to Y-axis, which will meet at point C. (iii)Further, we measure the distance from point C to the coordinate axis. Solution: The graph obtained by p...
Read More →Express each of the following product as a monomials and verify the result in each case for x = 1:
Question: Express each of the following product as a monomials and verify the result in each case for x = 1:(x2)3 (2x) (4x) (5) Solution: We have to find the product of the expression in order to express it as a monomial. To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e.,$a^{m} \times a^{n}=a^{m+n}$ and $\left(a^{m}\right)^{n}=a^{m n}$ We have: $\left(x^{2}\right)^{3} \times(2 x) \times(-4 x) \times 5$ $=\left(x^{6}\right) \times(2 x...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:A kite is flying at a height of 30 m from the ground. The length of stringfrom the kite to the ground is 60 m. Assuming that there is no slackin the string, the angle of elevation of the kite at the ground is(a) 45 (b) 30 (c) 60 (d) 90 Solution: Let point A be the position of the kite and AC be its string.We have, $\mathrm{AB}=30 \mathrm{~m}$ and $\mathrm{AC}=60 \mathrm{~m}$ Let $\angle \mathrm{ACB}=\theta$ In $\triangle \mathrm{ABC}$...
Read More →Taking 0.5 cm as 1 unit, plot the following points on
Question: Taking 0.5 cm as 1 unit, plot the following points on the graph paper A( 1, 3), 6(-3, -1,), C( 1, -4), D(-2, 3), E(0, -8) and F(1, 0). Solution: Here, in point 4(1, 3) both x and y-coordinates are positive, so it lies in I quadrant. In point 8(-3, -1),both x and y-coordinates are negative, so it lies in III quadrant. In point C(1, -4), x-coordinate is positive and y-coordinate is negative, so it lies in IV quadrant. In point D(-2, 3), x-coordinate is negative and y-coordinate is positi...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The angle of depression of a car parked on the road from the top of a 150-m-high tower is 30. The distance of the car from the tower is (a) $50 \sqrt{3} \mathrm{~m}$ (b) $150 \sqrt{3} \mathrm{~m}$ (c) $150 \sqrt{2} \mathrm{~m}$ (d) $75 \mathrm{~m}$ Solution: Let AB be the tower and point C be the position of the car.We have, $\mathrm{AB}=150 \mathrm{~m}$ and $\angle \mathrm{ACB}=30^{\circ}$ In $\Delta \mathrm{ABC}$, $\tan 30^{\circ}=\...
Read More →Find the coordinates of the point
Question: Find the coordinates of the point (i)which lies on X and Y-axes both. (ii)whose ordinate is 4 and which lies on Y-axis. (iii)whose abscissa is 5 and which lies on X-axis. Solution: (i)The point which lies on X and Y-axes both is origin whose coordinates are (0, 0). (ii)The point whose ordinate is 4 and which lies on Y-axis, i.e., whose x-coordinate is zero, is (0,-4). (iii)The point whose abscissa is 5 and which lies on X-axis, i.e., whose y-coordinate is zero, is (5, 0)....
Read More →Express each of the following product as a monomials and verify the result in each case for x = 1:
Question: Express each of the following product as a monomials and verify the result in each case for x = 1:(5x4) (x2)3 (2x)2 Solution: We have to find the product of the expression in order to express it as a monomial. To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e.,$a^{m} \times a^{n}=a^{m+n}$ and $\left(a^{m}\right)^{n}=a^{m n}$ We have: $\left(5 x^{4}\right) \times\left(x^{2}\right)^{3} \times(2 x)^{2}$ $=\left(5 x^{4}\right) \t...
Read More →A point lies on positive direction of X-axis at
Question: A point lies on positive direction of X-axis at a distance of 7 units from the Y-axis. What are its coordinates? What will be the coordinates, if it lies on negative direction of Y-axis at a distance of 7 units from X-axis? Solution: Given, point lies on the positive direction of X-axis, so its y-coordinate will be zero and it is at a distance of 7 units from the X-axis, so its coordinates are (7, 0). If it lies on negative direction of X-axis, then its x-coordinate will be zero and it...
Read More →Plot the points (x, y) given by the following table.
Question: Plot the points (x, y) given by the following table. Use scale 1 cm= 0.25 unit. Solution: Let XOX and X OX be the coordinate axes. Plot the given points (1.25, -0.5), (0.25, 1), (1.5,1.5) and (-1.75, 0.25) on the graph paper....
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:From a point on the ground, 30 m away from the foot of a tower,the angle of elevation of the top of the tower is 30. The height of thetower is (a) $30 \mathrm{~m}$ (b) $10 \sqrt{3} \mathrm{~m}$ (c) $10 \mathrm{~m}$ (d) $30 \sqrt{3} \mathrm{~m}$ Solution: Let AB be the tower and point C be the point of observation on the ground.We have, $\mathrm{BC}=30 \mathrm{~m}$ and $\angle \mathrm{ACB}=30^{\circ}$ In $\Delta \mathrm{ABC}$, $\tan 30...
Read More →Which of the following points lies on Y-axis?
Question: Which of the following points lies on Y-axis? A(l, 1), B(1, 0), C(0, 1), D(0, 0), E(0, -1), F(-1, 0), G(0, 5), H(-7, 0) and I(3 ,3). Thinking Process The point lies on Y-axis means x-coordinate of point will be zero. Check this condition for every given point and find out the correct point. Solution: We know that, a point lies on the Y-axis, if its x-coordinate is zero. Here, x-coordinate of points C(0, 1), D(0, 0), E(0,-1) and G(0, 5) are zero. So, these points lie on Y-axis. Also, D(...
Read More →Express each of the following product as a monomials and verify the result in each case for x = 1:
Question: Express each of the following product as a monomials and verify the result in each case for x = 1: $\left(4 x^{2}\right) \times(-3 x) \times\left(\frac{4}{5} x^{3}\right)$ Solution: We have to find the product of the expression in order to express it as a monomial. 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: $\left(4 x^{2}\right) \times(-3 x) \times\left(\frac{4}{5} x^{3}\right)$ ...
Read More →In which quadrant or on which axis
Question: In which quadrant or on which axis each of the following points lie? (-3, 5), (4,-1), (2,0), (2, 2), (-3,-6) Solution: (i)In point (-3, 5), x-coordinate is negative and y-coordinate is positive, so it lies inII quadrant. (ii)In point (4,-1), x-coordinate is positive and y-coordinate is negative, so it lies in IV quadrant. (iii)In point (2,0), x-coordinate is positive and y-coordinate is zero, so it lies on X-axis. (iv)In point (2,2), x-coordinate and y-coordinate both are positive, so ...
Read More →In figure LM is a line parallel to the Y-axis at a distance of 3 units.
Question: In figure LM is a line parallel to the Y-axis at a distance of 3 units. (i)What are the coordinates of the points P, R and Q? (ii)Whatisthe difference between the abscissa of the points L and M? Solution: Given, LM is a line parallel to the Y-axis and its perpendicular distance from Y-axis is 3 units. (i)Coordinate of point P = (3, 2) [since, its perpendicular distance from X-axis is 2] Coordinate of point 0 = (3, -1) [since, its perpendicular distance from X-axis is 1 in negative dire...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:A ladder 15 m long makes an angle of 60 with the wall. Find the height of the point, where the ladder touches the wall. (a) $15 \sqrt{3} \mathrm{~m}$ (b) $\frac{15 \sqrt{3}}{2} \mathrm{~m}$ (c) $\frac{15}{2} \mathrm{~m}$ (d) $15 \mathrm{~m}$ Solution: Let AB be the wall and AC be the ladder.We have, $\mathrm{AC}=15 \mathrm{~m}$ and $\angle \mathrm{BAC}=60^{\circ}$ In $\triangle \mathrm{ABC}$ $\cos 60^{\circ}=\frac{\mathrm{AB}}{\mathrm...
Read More →Express each of the following product as a monomials and verify the result in each case for x = 1:
Question: Express each of the following product as a monomials and verify the result in each case for x = 1:(3x) (4x) (5x) Solution: We have to find the product of the expression in order to express it as a monomial. 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: $(3 x) \times(4 x) \times(-5 x)$ $=\{3 \times 4 \times(-5)\} \times(x \times x \times x)$ $=\{3 \times 4 \times(-5)\} \times\left(x^...
Read More →Solve this
Question: $x-4 y-z=11$ $2 x-5 y+2 z=39$ $-3 x+2 y+z=1$ Solution: Given: $x-4 y-z=11$ $2 x-5 y+2 z=39$ $-3 x+2 y+z=1$ $D=\left|\begin{array}{ccc}1 -4 -1 \\ 2 -5 2 \\ -3 2 1\end{array}\right|$ $=1(-5-4)-(-4)(2+6)+(-1)(4-15)$ $=1(-9)-(-4)(8)+(-1)(-11)=34$ $D_{1}=\left|\begin{array}{ccc}11 -4 -1 \\ 39 -5 2 \\ 1 2 1\end{array}\right|$ $=11(-5-4)-(-4)(39-2)+(-1)(78+5)$ $=11(-9)-(-4)(37)+(-1)(83)=-34$ $D_{2}=\left|\begin{array}{ccc}1 11 -1 \\ 2 39 2 \\ -3 1 1\end{array}\right|$ $=1(39-2)-11(2+6)+(-1)(2...
Read More →Without plotting the points indicate the quadrant in which they will lie,
Question: Without plotting the points indicate the quadrant in which they will lie, if (i)ordinate is 5 and abscissa is 3. (ii)abscissa is 5 and ordinate is 3. (iii)abscissa is 5 and ordinate is 3. (iv)ordinate is 5 and abscissa is 3. Thinking Process (i)Firstly, write the giver) coordinates in a point form and check the sign of each coordinate of a point. (ii)Signs of the coordinates of a point in first quadrant are (+, +) in the second quadrant (-, +), in the third quadrant and in the fourth q...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:A ladder makes an angle of 60 with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder is (a) $\frac{4}{\sqrt{3}} \mathrm{~m}$ (b) $4 \sqrt{3} \mathrm{~m}$ (c) $2 \sqrt{2} \mathrm{~m}$ (d) $4 \mathrm{~m}$ Solution: Let AB be the wall and AC be the ladder.We have, $\mathrm{BC}=2 \mathrm{~m}$ and $\angle \mathrm{ACB}=60^{\circ}$ In $\Delta \mathrm{ABC}$, $\cos 60^{\ci...
Read More →Find each of the following product:
Question: Find each of the following product:(2.3xy) (0.1x) (0.16) 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: $(2.3 x y) \times(0.1 x) \times(0.16)$ $=(2.3 \times 0.1 \times 0.16) \times(x \times x) \times y$ $=(2.3 \times 0.1 \times 0.16) \times\left(x^{1+1}\right) \times y$ $=0.0368 x^{2} y$ Thus, the answer is $0.0368 x^{2} y$....
Read More →Plot the following points and check
Question: Plot the following points and check whether they are collinear or not (i)(X 3), (-X -1), (-2, 3) (ii)(1,1), (2, 3), (-X 2) (iii) (0,0),(2,2),(5,5) Thinking Process (i)Firstly, plot all three points on a graph paper and join them. (ii)If it lives a straight line, then points are collinear otherwise non-collinear. Solution: (i)Plotting the points P (1, 3), Q (-1, -1) and R (-2, 3) on the graph paper and join these points, we get a straight line. Hence, these points are collinear. (ii)Plo...
Read More →Solve this
Question: $3 x+y+z=2$ $2 x-4 y+3 z=-1$ $4 x+y-3 z=-11$ Solution: Given: $3 x+y+z=2$ $2 x-4 y+3 z=-1$ $4 x+y-3 z=-11$ $D=\left|\begin{array}{ccc}3 1 1 \\ 2 -4 3 \\ 4 1 -3\end{array}\right|$ $=3(12-3)-2(-3-1)+4(3+4)$ $=27+8+28$ $=63$ $D_{1}=\left|\begin{array}{ccc}2 1 1 \\ -1 -4 3 \\ -11 1 -3\end{array}\right|$ $=2(12-3)+1(-3-1)-11(3+4)$ $=18-4-77$ $=-63$ $D_{2}=\left|\begin{array}{ccc}3 2 1 \\ 2 -1 3 \\ 4 -11 -3\end{array}\right|$ $=3(3+33)-2(-6+11)+4(6+1)$ $=108-10+28$ $=126$ $D_{3}=\left|\begin...
Read More →Find each of the following product:
Question: Find each of the following product: $\left(\frac{4}{3} p q^{2}\right) \times\left(-\frac{1}{4} p^{2} r\right) \times\left(16 p^{2} q^{2} r^{2}\right)$ 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: $\left(\frac{4}{3} p q^{2}\right) \times\left(-\frac{1}{4} p^{2} r\right) \times\left(16 p^{2} q^{2} r^{2}\right)$ $=\left\{\frac{4}{3} \times\left(-\frac{1}{4}\right) \times 16\...
Read More →Choose the correct answer of the following question:
Question: Choose the correct answer of the following question:The shadow of a 5-m-long stick is 2 m long. At the same time, the lengthof the shadow of a 12.5-m-high tree is(a) 3 m (b) 3.5m (c) 4.5 m (d) 5 m Solution: Let AB be a stick and BC be its shadow; and PQ be the tree and QR be its shadow.We have, $\mathrm{AB}=5 \mathrm{~m}, \mathrm{BC}=2 \mathrm{~m}, \mathrm{PQ}=12.5 \mathrm{~m}$ In $\Delta \mathrm{ABC}$, $\tan \theta=\frac{\mathrm{AB}}{\mathrm{BC}}$ $\Rightarrow \tan \theta=\frac{5}{2} ...
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