Find the roots of the quadratic equations
Question: Find the roots of the quadratic equations by using the quadratic formula irt each of the following Solution: (i) Given equation is $2 x^{2}-3 x-5=0$. On comparing with $a x^{2}+b x+c=0$, we get $a=2, b=-3$ and $c=-5$ By quadratic formula, $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ $=\frac{-(-3) \pm \sqrt{(-3)^{2}-4(2)(-5)}}{2(2)}=\frac{3 \pm \sqrt{9+40}}{4}$ $=\frac{3 \pm \sqrt{49}}{4}=\frac{3 \pm 7}{4}=\frac{10}{4}, \frac{-4}{4}=\frac{5}{2},-1$ So, $\frac{5}{2}$ and $-1$ are the roots ...
Read More →Which of the following cannot be the probability of an event?
Question: Which of the following cannot be the probability of an event? (a) 1.5 (b) $\frac{3}{5}$ (c) 25%(d) 0.3 Solution: The probability of an event cannot be greater than 1.Thus, the probability of an event cannot be 1.5.Hence, the correct answer is option (a)....
Read More →The area of a trapezium is 1586 cm
Question: The area of a trapezium is 1586 cm2and the distance between the parallel sides is 26 cm. If one of the parallel sides is 38 cm, find the other. Solution: Given: Area of the trapezium $=1586 \mathrm{~cm}^{2}$ Distance between the parallel sides $=26 \mathrm{~cm}$ And, length of one parallel side $=38 \mathrm{~cm}$ Let us suppose the length of the other side to be $x \mathrm{~cm} .$ Now, area of the trapezium $=\frac{1}{2} \times($ Sum of the parallel sides $) \times($ Distance between t...
Read More →Mohan wants to buy a trapezium shaped field.
Question: Mohan wants to buy a trapezium shaped field. Its side along the river is parallel and twice the side along the road. If the area of this field is 10500 m2and the perpendicular distance between the two parallel sides is 100 m, find the length of the side along the river. Solution: Given: Area of the trapezium shaped field $=10500 \mathrm{~m}^{2}$ It is also given that the length of the side along the river is double the length of the side along the road. Let us suppose the length of the...
Read More →What is the probability of a sure event?
Question: What is the probability of a sure event? (a) 0 (b) $\frac{1}{2}$ (c) 1(d) none of these Solution: (c) 1The probability of a sure event is 1....
Read More →A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'.
Question: A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21 . Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60 . While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70 . Using matrix method, find cost of each variety of pen. Solution: As there are 3 varieties of pen $A, B$ and $C$ Meenu purchased 1 pen of each variety which costs her Rs ...
Read More →The area of a trapezium is 384 cm
Question: The area of a trapezium is 384 cm2. Its parallel sides are in the ratio 3 : 5 and the perpendicular distance between them is 12 cm. Find the length of each one of the parallel sides. Solution: Given: Area of the trapezium $=384 \mathrm{~cm}^{2}$ The parallel sides are in the ratio $3: 5$ and the perpendicular height between them is $12 \mathrm{~cm}$. Suppose that the sides are in $\mathrm{x}$ multiples of each other. Then, length of the shorter side $=3 \mathrm{x}$ Length of the longer...
Read More →What is the probability of an impossible event?
Question: What is the probability of an impossible event? (a) $\frac{1}{2}$ (b) 0(c) 1(d) none of these Solution: (b) 0The probability of an impossible event is 0....
Read More →If the probability of occurence of an event is p then the probability of non-happening of this event is
Question: If the probability of occurence of an event ispthen the probability of non-happening of this event is [CBSE 2013C](a) (p 1)(b) (1p)(c)p (d) $\left(1-\frac{1}{p}\right)$ Solution: P(occurence of an event) =pP(non-occurence of this event) =1pHence, the correct answer is option (b)....
Read More →The area of a trapezium is 91 cm
Question: The area of a trapezium is 91 cm2and its height is 7 cm. If one of the parallel sides is longer than the other by 8 cm, find the two parallel sides. Solution: Given: Area of the trapezium $=91 \mathrm{~cm}^{2}$ Height $=7 \mathrm{~cm}$ Let the length of the smaller side be $x .$ Then, the length of longer side will be 8 more than smaller side, i.e. $8+x .$ Area of trapezium $=\frac{1}{2} \times($ Sum of the parallel sides $) \times($ Height $)$ $\Rightarrow 91=\frac{1}{2} \times[(8+x)+...
Read More →If b = 0, c < 0, is it true that the roots
Question: If b = 0, c 0, is it true that the roots of x2+ bx + c = 0 are numerically equal and opposite in sign? Justify your answer. Solution: Given that, b = 0andc 0and quadratic equation, $x^{2}+b x+c=0$$\ldots(i)$ Put $b=0$ in Eq. (i), we get $x^{2}+0+c=0$ $\Rightarrow$ $x^{2}=-c$ $\left[\begin{array}{l}\text { here } c0 \\ \therefore-c0\end{array}\right]$ $\therefore$ $x=\pm \sqrt{-C}$ So, the roots of x2+ bx+c = O are numerically equal and opposite in sign....
Read More →The cross-section of a canal is a trapezium in shape.
Question: The cross-section of a canal is a trapezium in shape. If the canal is 10 m wide at the top 6 m wide at the bottom and the area of cross-section is 72 m2determine its depth. Solution: Let the depth of canal be $d$. Given: Lengths of the parallel sides of the trapezium shape canal are $10 \mathrm{~m}$ and $6 \mathrm{~m}$. And, the area of the cross section of the canal is $72 \mathrm{~m}^{2}$. Area of trapezium $=\frac{1}{2} \times$ (Sum of the parallel sides) $\times$ (Perpendicular dis...
Read More →Is 0.2 a root of the equation
Question: Is 0.2 a root of the equation x2 0.4 = 0? Justify your answer. Solution: No, since 0.2 does not satisfy the quadratic equation i.e., ( 0.2)2 0.4 = 0.04 0.4 0....
Read More →Does there exist a quadratic equation
Question: Does there exist a quadratic equation whose coefficient are all distinct irrationals but both the roots are rationals? why? Solution: Yes, consider the quadratic equation with all distinct irrationals coefficients i.e., 3x2 7 3x + 123 = 0. The roots of this quadratic equation are 3 and 4, which are rationals....
Read More →A total amount of ₹7000 is deposited in three different saving bank accounts
Question: A total amount of₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and812812% respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices. Solution: Let the amount deposited in each of the three accounts be₹x,₹xand₹yrespectively. Since, the total amount deposited is ₹7,000. $\t...
Read More →Top surface of a table is trapezium in shape.
Question: Top surface of a table is trapezium in shape. Find its area if its parallel sides are 1 m and 1.2 m and perpendicular distance between them is 0.8 m. Solution: The given figure is: Lengths of the parallel sides are $1.2 \mathrm{~m}$ and $1 \mathrm{~m}$ and the perpendicular distance between them is $0.8 \mathrm{~m}$. $\therefore$ Area of the trapezium shaped surface $=\frac{1}{2} \times($ Sum of the parallel sides $) \times($ Perpendicular distance $)$ $=\frac{1}{2} \times(1.2+1) \time...
Read More →Does there exist a quadratic equation
Question: Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer. Solution: Yes, consider the quadratic equation $2 x^{2}+x-4=$ Owith rational coefficient. The roots ofthe given quadratic equation are $\frac{-1+\sqrt{33}}{4}$ and $\frac{-1-\sqrt{33}}{4}$ are irrational....
Read More →A total amount of ₹7000 is deposited in three different saving bank accounts
Question: A total amount of₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and812812% respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices. Solution: Let the amount deposited in each of the three accounts be₹x,₹xand₹yrespectively. Since, the total amount deposited is ₹7,000. $\t...
Read More →A quadratic equation with integral
Question: A quadratic equation with integral coefficient has integral roots. Justify your answer. Solution: No, consider the quadratic equation $2 x^{2}+x-6=0$ with integral coefficient. The roots of the given quadratic equation are $-2$ and $\frac{3}{2}$ which are not integers....
Read More →Find the area of Fig. 20.35 as the sum of the areas of two trapezium and a rectangle.
Question: Find the area of Fig. 20.35 as the sum of the areas of two trapezium and a rectangle. Solution: The given figure is: In the given figure, we have a rectangle of length $50 \mathrm{~cm}$ and width $10 \mathrm{~cm}$, and two similar trapeziums with parallel sides as $30 \mathrm{~cm}$ and $10 \mathrm{~cm}$ at both ends. Suppose $\mathrm{x}$ is the perpendicular distance between the parallel sides in both the trapeziums. We have: Total length of the given figure $=$ Length of the rectangle...
Read More →Write whether the following statements are true or false.
Question: Write whether the following statements are true or false. Justify your answers. (i) Every quadratic equation has exactly one root. (ii) Every quadratic equation has atleast one real root. (iii) Every quadratic equation has atleast two roots. (iv) Every quadratic equation has atmost two roots. (v) If the coefficient of x2and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots. (vi)If the coefficient of x2and the constant term have th...
Read More →State whether the following quadratic
Question: State whether the following quadratic equations have two distinct real roots. Justify your answer. (i) $x^{2}-3 x+4=0$ (ii) $2 x^{2}+x-1=0$ (iii) $2 x^{2}-6 x+\frac{9}{2}=0$ (iv) $3 x^{2}-4 x+1=0$ (v) $(x+4)^{2}-8 x=0$ (vi) $(x-\sqrt{2})^{2}-\sqrt{2}(x+1)=0$ (vii) $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+\frac{1}{\sqrt{2}}=0$ (viii) $x(1-x)-2=0$ (ix) $(x-1)(x+2)+2=0$ (x) $(x+1)(x-2)+x=0$ Solution: (i) Given equation is $x^{2}-3 x+4=0$. On comparing with $a x^{2}+b x+c=0$, we get $a=1, b=-3...
Read More →The area of a trapezium is 960 cm
Question: The area of a trapezium is 960 cm2. If the parallel sides are 34 cm and 46 cm, find the distance between them. Solution: Given; Area of the trapezium $=960 \mathrm{~cm}^{2}$ And the length of the parallel sides are $34 \mathrm{~cm}$ and $46 \mathrm{~cm}$. Area of trapezium $=\frac{1}{2} \times($ Sum of the parallel sides $) \times($ Perpendicular distance between the parallel sides $)$ $\Rightarrow 960=\frac{1}{2} \times(34+46) \times($ Height $)$ $\Rightarrow 960=40 \times($ Height $)...
Read More →Find the area of a trapezium whose parallel sides of lengths 10 cm and 15 cm are at a distance of 6 cm from each other.
Question: Find the area of a trapezium whose parallel sides of lengths 10 cm and 15 cm are at a distance of 6 cm from each other. Calculate this area as (i) the sum of the areas of two triangles and one rectangle. (ii) the difference of the area of a rectangle and the sum of the areas of two triangles. Solution: Given: Length of the parallel sides of a trapezium are $10 \mathrm{~cm}$ and $15 \mathrm{~cm}$. The distance between them is $6 \mathrm{~cm}$. Let us extend the smaller side and then dra...
Read More →If P(E) denotes the probability of an event E then
Question: If P(E) denotes the probability of an event E then (a) P(E) 0(b) P(E) 1(c) 0 P(E) 1(d) 1 P(E) 1 Solution: Since, number of elements in the set of favourable cases is less than or equal to the number of elements in the set of whole number of cases,their ratio always end up being 1 or less than 1.Also,their ratio can never be negative.Thus,probability of an event always lies between 0 and 1.i.e.0 P(E) 1Hence, the correct answer is option (c)....
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