Fill in the blanks:
Question: Fill in the blanks:(i) The probability of an impossible event is ....... .(ii) The probability of a sure event is ........ .(iii) For any eventE,P(E)+P(notE) = ........ .(iv) The probability of a possible but not a sure event lies between ......... and ........ .(v) The sum of probabilities of all the outcomes of an experiment is ....... . Solution: (i) 0(ii) 1(iii) 1(iv) 0, 1(v) 1...
Read More →Determine, algebraically, the vertices
Question: Determine, algebraically, the vertices of the triangle formed by the lines 3 x y = 3 2x 3y = 2 and x + 2y = 8 Solution: Given equation of lines are 3 x y = 3 ..(i) 2x 3y = 2 .(ii) and x + 2y = 8 .(iii) Let lines (i), (ii) and (iii) represent the sides of a ΔABC i.e., AB, BC and CA, respectively. On solving lines (i) and (ii), we will get the intersecting point B. On multiplying Eq. (i) by 3 in Eq. (i) and then subtracting, we get On putting the value of $x$ in Eq. (i), we get $3 \times...
Read More →Construct a quadrilateral ABCD in which AB = BC = 6 cm,
Question: Construct a quadrilateralABCDin whichAB=BC= 6 cm,AD=DC= 4.5 cm and B= 120. Solution: Steps of construction: Step I: Draw $\mathrm{AB}=6 \mathrm{~cm} .$ Step II : Construct $\angle \mathrm{ABC}=120^{\circ}$. Step III : With B as the centre and radius $6 \mathrm{~cm}$, cut off BC $=6 \mathrm{~cm} .$ Now, we can see that AC is about $10.3 \mathrm{~cm}$ which is greater than AD $+\mathrm{CD}=4.5+4.5=9 \mathrm{~cm} .$ We know that sum of the lengths of two sides of triangle is always greate...
Read More →Construct a quadrilateral ABCD in which AB = 2.8 cm,
Question: Construct a quadrilateralABCDin whichAB= 2.8 cm,BC= 3.1 cm,CD= 2.6 cm, andDA= 3.3 cm and A= 60. Solution: Steps of construction : Step I : Draw $\mathrm{AB}=2.8 \mathrm{~cm}$. Step II : Construct $\angle \mathrm{BAD}=60^{\circ}$. Step III : With A as the centre and radius $3.3 \mathrm{~cm}$, cut off $\mathrm{AD}=3.3 \mathrm{~cm}$. Step IV: With D as the centre and radius $2.6 \mathrm{~cm}$, draw an arc. Step V : With B as the centre and radius $3.1 \mathrm{~cm}$, draw an arc to interse...
Read More →Construct a quadrilateral ABCD in which AB = BC = 3 cm,
Question: Construct a quadrilateralABCDin whichAB=BC= 3 cm,AD=CD= 5 cm, and B= 120. Solution: Steps of construction: Step I: Draw $\mathrm{AB}=3 \mathrm{~cm} .$ Step II : Construct $\angle \mathrm{ABC}=120^{\circ}$. Step III : With B as the centre and radius $3 \mathrm{~cm}$, cut off BC $=3 \mathrm{~cm}$. Step IV : With C as the centre and radius $5 \mathrm{~cm}$, draw an arc. Step V : With A as the centre and radius $5 \mathrm{~cm}$, draw an arc to intersect the arc drawn in Step IV at D. Step ...
Read More →Construct a quadrilateral ABCD in which AB = 7.7 cm,
Question: Construct a quadrilateralABCDin whichAB= 7.7 cm,BC= 6.8 cm,CD= 5.1 cm,AD= 3.6 cm and C= 120. Solution: Steps of construction: Step I: Draw DC $=5.1 \mathrm{~cm} .$ Step II : Construct $\angle \mathrm{DCB}=120^{\circ}$. Step III : With C as the centre and radius $6.8 \mathrm{~cm}$, cut off BC $=6.8 \mathrm{~cm} .$ Step IV : With B as the centre and radius $7.7 \mathrm{~cm}$, draw an arc. Step V : With D as the centre and radius $3.6 \mathrm{~cm}$, draw an arc to intersect the arc drawn ...
Read More →Construct a quadrilateral ABCD, given that AB = 8 cm,
Question: Construct a quadrilateralABCD, given thatAB= 8 cm,BC= 8 cm,CD= 10 cm,AD= 10 cm and A= 45. Solution: Steps of Construction : Step I : Draw $\mathrm{AB}=8 \mathrm{~cm} .$ Step II : Construct $\angle \mathrm{BAD}=45^{\circ}$. Step III : With A as the centre and radius $10 \mathrm{~cm}$, cut off AD $=10 \mathrm{~cm} .$ Step IV : With D as the centre and radius $10 \mathrm{~cm}$, draw an arc. Step V : With B as the centre and radius $8 \mathrm{~cm}$, draw an arc to intersect the arc drawn i...
Read More →The cost of 4 pens and 4 pencils boxes is 1100.
Question: The cost of 4 pens and 4 pencils boxes is 1100. Three times the cost of a pen is ₹ 15 more than the cost of a pencil box. Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box. Solution: Let the cost of a pen be ₹ x and the cost of a pencil box be ₹ y. Then, by given condition, $4 x+4 y=100 \Rightarrow x+y=25$$\ldots($ (1) and $\quad 3 x=y+15$ $\Rightarrow \quad 3 x-y=15 \quad$...(ii) On adding Eqs. (i) and (ii), we get $4 x=40$ $\Rightarrow...
Read More →Find the values of
Question: Find $A^{-1}$, if $A=\left[\begin{array}{rrr}1 2 5 \\ 1 -1 -1 \\ 2 3 -1\end{array}\right]$. Hence solve the following system of linear equations: $x+2 y+5 z=10, x-y-z=-2,2 x+3 y-z=-11$ Solution: Here, $A=\left[\begin{array}{ccc}1 2 5 \\ 1 -1 -1 \\ 2 3 -1\end{array}\right]$ $|A|=\left|\begin{array}{ccc}1 2 5 \\ 1 -1 -1 \\ 2 3 -1\end{array}\right|$ $=1(1+3)-2(-1+2)+5(3+2)$ $=4-2+25$ $=27$ Let $\mathrm{C}_{i j}$ be the co factors of the elements $\mathrm{a}_{i j}$ in $\mathrm{A}\left[a_{i...
Read More →Construct a quadrilateral ABCD in which AB = 3.8 cm,
Question: Construct a quadrilateralABCDin whichAB= 3.8 cm,BC= 3.4 cm,CD= 4.5 cm,AD= 5 cm and B= 80. Solution: Steps of construction: Step I: Draw $\mathrm{AB}=3.8 \mathrm{~cm}$. Step II : Construct $\angle \mathrm{ABC}=80^{\circ}$. Step III : With B as the centre and radius $3.4 \mathrm{~cm}$, cut off BC $=3.4 \mathrm{~cm}$. Step IV : With C as the centre and radius $4.5 \mathrm{~cm}$, draw an arc. Step V : With A as the centre and radius $5.3 \mathrm{~cm}$, draw an arc to intersect the arc draw...
Read More →Draw the graphs of the equations x = 3,
Question: Draw the graphs of the equations x = 3, x = 5 and 2x y 4 = 0. Also find the area of the quadrilateral formed by the lines and the X-axis. Solution: Given equation of lines 2x y 4 = 0, x = 3 and x = 5 Table for line 2x y 4 = 0 Also find the area of the quadrilateral formed by the lines and the X-axis. Draw the points $P(0,-4)$ and $Q(2,0)$ and join these points and form a line $P Q$ also draw the lines $x=3$ and $x=5$. $\therefore$ Area of quadrilateral $A B C D=\frac{1}{2} \times$ dist...
Read More →Construct a quadrilateral ABCD in which BC = 4 cm,
Question: Construct a quadrilateralABCDin whichBC= 4 cm,CA= 5.6 cm,AD= 4.5 cm,CD= 5 cm andBD= 6.5 cm. Solution: Steps of construction: Step I: Draw $\mathrm{BC}=4 \mathrm{~cm}$. Step II: With B as the centre and radius $6.5 \mathrm{~cm}$, draw an arc. Step III : With $\mathrm{C}$ as the centre and radius $5 \mathrm{~cm}$, draw an arc to intersect the arc drawn in Step II at D. Step IV: With $\mathrm{C}$ as the centre and radius $5.6 \mathrm{~cm}$, draw an arc on the same side. Step V : With D as...
Read More →Construct a quadrilateral ABCD given AD = 5 cm,
Question: Construct a quadrilateralABCDgivenAD= 5 cm,AB= 5.5 cm,BC= 2.5 cm,AC= 7.1 cm andBD= 8 cm. Solution: Steps of construction: Step I : Draw $\mathrm{AB}=5.5 \mathrm{~cm} .$ Step II : With A as the centre and radius $7.1 \mathrm{~cm}$, draw an arc. Step III : With B as the centre and radius $2.5 \mathrm{~cm}$, draw an arc to intersect the arc drawn in Step II at C. Step IV : With B as the centre and radius $8 \mathrm{~cm}$, draw an arc. Step V : With A as the centre and radius $5 \mathrm{~c...
Read More →Determine graphically,
Question: Determine graphically, the vertices of the triangle formed by the lines y = x, 3y = x and x + y = 8 Solution: Given linear equations are $y=x$ ...(i) $3 y=x$ ....(ii) and $x+y=8$....(iii) For equation $y=x$ If $x=1$, then $y=1$ If $x=0$, then $y=0$ If $x=2$, then $y=2$ Table for line $y=x$, For equation $x=3 y$, If $x=0$, then $y=0$; if $x=3$, then $y=1$ and if $x=6$, then $y=2$ Table for line $x=3 y$, For equation. $x+y=8 \Rightarrow y=8-x$ If $x=0$, then $y=8$; if $x=8$, then $y=0$ a...
Read More →Construct a quadrilateral ABCD given AD = 3.5 cm, BC = 2.5 cm,
Question: Construct a quadrilateralABCDgivenAD= 3.5 cm,BC= 2.5 cm,CD= 4.1 cm,AC= 7.3 cm andBD= 3.2 cm. Solution: Steps of construction: Step I: Draw $\mathrm{CD}=4.1 \mathrm{~cm}$. Step II : With $\mathrm{C}$ as the centre and radius $7.3 \mathrm{~cm}$, draw an arc. Step III : With D as the centre and radius $3.5 \mathrm{~cm}$, draw an arc to intersect the arc drawn in Step II at A. Step IV : With D as the centre and radius $3.2 \mathrm{~cm}$, draw an arc on the other side of AC. Step V : With C...
Read More →Construct a quadrilateral ABCD, when AB = 3 cm,
Question: Construct a quadrilateralABCD, whenAB= 3 cm,CD= 3 cm,DA= 7.5 cm,AC= 8 cm andBD= 4 cm. Solution: If we consider a triangle ABD from the given data, then AB = 3 cm BD = 4 cm AD = 7.5 cm AB + BD = 3 + 4 = 7 cm However, we know that the sum of the lengths of two sides of a triangle is always greater than the third side. Therefore, construction is not possible from the given data....
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{rrr}2 -3 5 \\ 3 2 -4 \\ 1 1 -2\end{array}\right]$, find $A^{-1}$ and hence solve the system of linear equations $2 x-3 y+5 z=11,3 x+2 y-4 z=-5, x+y+2 z=-3$ Solution: Here, $A=\left[\begin{array}{ccc}2 -3 5 \\ 3 2 -4 \\ 1 1 -2\end{array}\right]$ $|A|=\left|\begin{array}{ccc}2 -3 5 \\ 3 2 -4 \\ 1 1 -2\end{array}\right|$ $=2(-4+4)+3(-6+4)+5(3-2)$ $=0-6+5$ $=-1$ Let $\mathrm{C}_{i j}$ be the co factors of the elements $\mathrm{a}_{i j}$ in $\mathrm{A}\left[a_{i j}...
Read More →Construct a quadrilateral ABCD in which BC = 7.5 cm,
Question: Construct a quadrilateralABCDin whichBC= 7.5 cm,AC=AD= 6 cm,CD= 5 cm andBD= 10 cm. Solution: Steps of construction: Step I: Draw AC $=6 \mathrm{~cm} .$ Step II : With A as the centre and radius $6 \mathrm{~cm}$, draw an arc. Step III : With C as the centre and radius $5 \mathrm{~cm}$, draw an arc to intersect the arc drawn in Step II at D. Step IV : With D as the centre and radius $10 \mathrm{~cm}$, draw an arc on the other side of the line segment AC. Step V : With C as the centre and...
Read More →Construct a quadrilateral ABCD in which AB = 3.8 cm,
Question: Construct a quadrilateralABCDin whichAB= 3.8 cm,BC= 3.0 cm,AD= 2.3 cm,AC= 4.5 cm andBD= 3.8 cm. Solution: Steps of construction : Step I : Draw $\mathrm{AC}=6 \mathrm{~cm} .$ Step II : With $\mathrm{A}$ as the centre and radius $3.8 \mathrm{~cm}$, draw an arc. Step III : With $\mathrm{C}$ as the centre and radius $3.0 \mathrm{~cm}$, draw an arc to intersect the arc drawn in Step II at B. Step IV: With B as the centre and radius $3.8 \mathrm{~cm}$, draw an arc on the other side of AC. S...
Read More →Graphically, solve the following pair
Question: Graphically, solve the following pair of equations 2x + y = 6 and 2x y + 2 = 0 Find the ratio of the areas of the two triangles formed by the lines representing these equations with the X-axis and the lines with the Y-axis. Solution: Given equations are $2 x+y=6$ and $2 x-y+2=0$ Table for equation $2 x+y=6$ Table for equation $2 x-y+2=0$, Let $A_{1}$ and $A_{2}$ represent the areas of $\triangle A C E$ and $\triangle B D E$, respectively. Now, $A_{1}=$ Area of $\triangle A C E=\frac{1}...
Read More →Construct, if possible, a quadrilateral ABCD in which AB = 6 cm, BC = 7 cm, CD = 3 cm,
Question: Construct, if possible, a quadrilateralABCDin whichAB= 6 cm,BC= 7 cm,CD= 3 cm,AD= 5.5 cm andAC= 11 cm. Give reasons for not being able to construct, if you cannot. (Not possible, because in triangleACD,AD+CDAC). Solution: Such a quadrilateral cannot be constructed because in a triangle, the sum of the length of its two sides must be greater than the that of the third side But here in triangle ACD, AD + CD = 5.5 + 3 = 8.5 cm and AC = 11 cm i.e., AD + CD AC, which is not possible. So, th...
Read More →Construct, if possible, a quadrilateral ABCD given AB = 6 cm,
Question: Construct, if possible, a quadrilateralABCDgivenAB= 6 cm,BC= 3.7 cm,CD= 5.7 cm,AD= 5.5 cm andBD= 6.1 cm. Give reasons for not being able to construct it, if you cannot. Solution: Steps of construction: Step I: Draw $\mathrm{AB}=6 \mathrm{~cm} .$ Step II : With $\mathrm{A}$ as the centre and radius $5.5 \mathrm{~cm}$, draw an arc. Step III : With B as the centre and radius $6.1 \mathrm{~cm}$, draw an arc to intersect the arc drawn in StepII at D. Step IV : With B as the centre and radiu...
Read More →A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep.
Question: A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km/hr, then in how much time will the tank be filled completely? Solution: We have, Internal radius of the pipe, $r=\frac{20}{2}=10 \mathrm{~cm}=0.1 \mathrm{~m}$, Radius of the cylindrical tank, $R=\frac{10}{2}=5 \mathrm{~m}$ and Height of the cylindrical tank, $H=2 \mathrm{~m}$ Also, the speed of the water...
Read More →Solve this
Question: If $A=\left[\begin{array}{rrr}1 -1 0 \\ 2 3 4 \\ 0 1 2\end{array}\right]$ and $B=\left[\begin{array}{rrr}2 2 -4 \\ -4 2 -4 \\ 2 -1 5\end{array}\right]$ are two square matrices, find $A B$ and hence solve the system of linear equations: $x-y=3,2 x+3 y+4 z=17, y+2 z=7$ Solution: Here, $A=\left[\begin{array}{ccc}1 -1 0 \\ 2 3 4 \\ 0 1 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 2 -4 \\ -4 2 -4 \\ 2 -1 5\end{array}\right]$ Now, $A B=\left[\begin{array}{ccc}1 -1 0 \\ 2 3 4 \\ 0 1 ...
Read More →Construct a kite ABCD in which AB = 4 cm,
Question: Construct a kiteABCDin whichAB= 4 cm,BC= 4.9cmand AC = 7.2 cm. Solution: Steps of construction: Step I : Draw AC $=7.2 \mathrm{~cm}$. Step II : With A as the centre and radius $4 \mathrm{~cm}$, draw arcs on both sides of the line segment AC. Step III : With C as the centre and radius $4.9 \mathrm{~cm}$, draw arcs on both sides of AC intersecting the previous arcs of step II at B and D. Step IV : Join BA, DA, BC and CD Thus, the quadrilateral ABCD so obtained is the required kite....
Read More →