In a cyclic quadrilateral ABCD, ∠A = (2x + 4)°, ∠B = (y + 3)°, ∠C = (2y + 10)°, ∠D = (4x − 5)°.
Question: In a cyclic quadrilateral $A B C D, \angle A=(2 x+4)^{\circ}, \angle B=(y+3)^{\circ}, \angle C=(2 y+10)^{\circ}, \angle D=(4 x-5)^{\circ}$. Find the four angles. Solution: We know that the sum of the opposite angles of cyclic quadrilateral is $180^{\circ}$ in the cyclic quadrilateral $A B C D$, angles $A$ and $C$ and angles $B$ and $D$ pairs of opposite angles Therefore $\angle A+\angle C=180^{\circ}$ and $\angle B+\angle D=180^{\circ}$ Taking $\angle A+\angle C=180^{\circ}$ By substit...
Read More →Evaluate each of the following
Question: Evaluate each of the following (a) $111^{3}-89^{3}$ (b) $46^{3}+34^{3}$ (c) $104^{3}+96^{3}$ (d) $93^{3}-107^{3}$ Solution: (a) Given, $111^{3}-89^{3}$ the above equation can be written as $(100+11)^{3}-(100-11)^{3}$ we know that, $(a+b)^{3}-(a-b)^{3}=2\left[b^{3}+3 a b^{2}\right]$ here, a = 100 b = 11 $(100+11)^{3}-(100-11)^{3}=2\left[11^{3}+3(100)^{2}(11)\right]$ = 2[1331 + 330000] = 2[331331] = 662662 The value of $111^{3}-89^{3}=662662$ (b) $46^{3}+34^{3}$ the above equation can be...
Read More →In a ∆ABC, ∠A = x°, ∠B = (3x − 2)°, ∠C = y°. Also, ∠C − ∠B = 9°
Question: In a $\triangle A B C, \angle A=x^{\circ}, \angle B=(3 x-2)^{\circ}, \angle C=y^{\circ}$. Also, $\angle C-\angle B=9^{\circ}$. Find the three angles. Solution: Let $\angle A=x^{\circ}, \angle B=(3 x-2)^{\circ}, \angle C=y^{\circ}$ and $\angle C-\angle B=9^{\circ}$ $\Rightarrow \angle C=9^{\circ}+\angle B$ $\Rightarrow \angle C=9^{\circ}+3 x^{\circ}-2^{\circ}$ $\Rightarrow \angle C=7^{\circ}+3 x^{\circ}$ Substitute $\angle C=y^{\circ}$ in above equation we get, $y^{\circ}=7^{\circ}+3 x^...
Read More →2 men and 7 boys can do a piece of work in 4 days.
Question: 2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long would it take one man and one boy to do it? Solution: A man can alone finish the work in $x$ days and one boy alone can finish it in $y$ days then One mans one days work $=\frac{1}{x}$ One boys one days work $=\frac{1}{y}$ 2 men one day work $=\frac{2}{x}$ 7boys one day work $=\frac{7}{y}$ Since 2 men and 7 boys can finish the work in 4 days $4\left(\frac{2}{x}+\frac{7}{y}\r...
Read More →There are two examination rooms A and B.
Question: There are two examination rooms A and B. If 10 candidates are sent from A to B, the number of students in each room is same. If 20 candidates are sent from B to A, the number of students in A is double the number of students in B. Find the number of students in each room. Solution: Let us take the A examination room will be x and the B examination room will be y If 10 candidates are sent from A to B, the number of students in each room is same. According to the above condition equation...
Read More →Evaluate each of the following
Question: Evaluate each of the following (a) $(103)^{3}$ (b) $(98)^{3}$ (c) $(9.9)^{3}$ (d) $(10.4)^{3}$ (e) $(598)^{3}$ (f) $(99)^{3}$ Solution: Given, (a) $(103)^{3}$ we know that $(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)$ $\Rightarrow(103)^{3}$ can be written as $(100+3)^{3}$ Here, a = 100 and b = 3 $(103)^{3}=(100+3)^{3}$ $=(100)^{3}+(3)^{3}+3(100)(3)(100+3)$ = 1000000 + 27 + (900*103) = 1000000 + 27 + 92700 = 1092727 The value of $(103)^{3}=1092727$ (b) $(98)^{3}$ we know that $(a-b)^{3}=a^{3}-b^{3...
Read More →A and B each has some money. If A gives Rs 30 to B,
Question: A and B each has some money. If A gives Rs 30 to B, then B will have twice the money left with A. But, if B gives Rs 10 to A, then A will have thrice as much as is left with B. How much money does each have? Solution: Let the money with A be Rs x and the money with B be Rs y. If A gives Rs 30 to B, Then B will have twice the money left with A, According to the condition we have, $y+30=2(x-30)$ $y+30=2 x-60$ $0=2 x-y-60-30$ $0=2 x-y-90 \cdots(i)$ If B gives Rs 10 to A, then A will have ...
Read More →The incomes of X and Y are in the ratio of 8 : 7 and their expenditures are in the ratio 19 : 16.
Question: The incomes of $X$ and $Y$ are in the ratio of $8: 7$ and their expenditures are in the ratio $19: 16$. If each saves Rs 1250, find their incomes. Solution: Let the income of $X$ be Rs $8 x$ and the income of $Y$ be Rs $7 x$.further let the expenditure of $X$ be $19 y$ and the expenditure of $Y$ be $16 y$ respectively then, Saving of $x=8 x-19 y$ Saving of $Y=7 x-16 y$ $8 x-19 y=1250$ $7 x-16 y=1250$ $8 x-19 y-1250=0 \cdots(i)$ $7 x-16 y-1250=0 \cdots(i i)$ Solving equation $(i)$ and $...
Read More →In a rectangle, if the length is increased by 3 meters and breadth is decreased by 4 meters,
Question: In a rectangle, if the length is increased by 3 meters and breadth is decreased by 4 meters, the area of the rectangle is reduced by 67 square meters. If length is reduced by 1 meter and breadth is increased by 4 meters, the area is increased by 89 Sq. meters. Find the dimensions of the rectangle. Solution: Let the length and breadth of the rectangle be $x$ and $y$ units respectively Then, area of rectangle $=x y$ square units If the length is increased by 3 meters and breath is reduce...
Read More →The area of a rectangle remains the same if the length is increased by 7 meters and the breadth is decreased by 3 meters.
Question: The area of a rectangle remains the same if the length is increased by 7 meters and the breadth is decreased by 3 meters. The area remains unaffected if the length is decreased by 7 meters and breadth in increased by 5 meters. Find the dimensions of the rectangle. Solution: Let the length and breadth of the rectangle be $x$ and $y$ units respectively Then, area of rectangle $=x y$ square units If length is increased by 7 meters and breadth is decreased by 3 meters when the area of a re...
Read More →If in a rectangle, the length is increased and breadth reduced each by 2 units,
Question: If in a rectangle, the length is increased and breadth reduced each by 2 units, the area is reduced by 28 square units. If, however the length is reduced by 1 unit and the breadth increased by 2 units, the area increases by 33 square units. Find the area of the rectangle. Solution: Let the length and breadth of the rectangle be $x$ and $y$ units respectively Then, area of rectangle $=x y$ square units If length is increased and breadth reduced each by 2 units, then the area is reduced ...
Read More →The polynomial which when divided by $-x^{2}+x-1$
Question: The polynomial which when divided by $-x^{2}+x-1$ gives a quotient $x-2$ and remainder 3 , is (a) $x^{3}-3 x^{2}+3 x-5$ (b) $-x^{3}-3 x^{2}-3 x-5$ (c) $-x^{3}+3 x^{2}-3 x+5$ (d) $x^{3}-3 x^{2}-3 x+5$ Solution: We know that $f(x)=g(x) q(x)+r(x)$ $=\left(-x^{2}+x-1\right)(x-2)+3$ $=-x^{3}+x^{2}-x+2 x^{2}-2 x+2+3$ $=-x^{3}+x^{2}+2 x^{2}-x-2 x+2+3$ $=-x^{3}+3 x^{2}-3 x+5$ Therefore, The polynomial which when divided by $-x^{2}+x-1$ gives a quotient $x-2$ and remainder 3 , is $-x^{3}+3 x^{2...
Read More →If x + 2 is a factor of x2 + ax + 2b and a + b = 4,
Question: If $x+2$ is a factor of $x^{2}+a x+2 b$ and $a+b=4$, then (a) $a=1, b=3$ (b) $a=3, b=1$ (c) $a=-1, b=5$ (d) $a=5, b=-1$ Solution: Given that $x+2$ is a factor of $x^{2}+a x+2 b$ and $a+b=4$ $f(x)=x^{2}+a x+2 b$ $f(-2)=(-2)^{2}+a(-2)+2 b$ $0=4-2 a+2 b$ $-4=-2 a+2 b$ By solving $-4=-2 a+2 b$ and $a+b=4$ by elimination method we get Multiply $a+b=4$ by 2 we get, $2 a+2 b=8$. So $4=4 b$ $\frac{4}{4}=b$ $1=b$ By substituting $b=1$ in $a+b=4$ we get $a+1=4$ $a=4-1$ $a=3$ Then $a=3, b=1$ Henc...
Read More →If 5–√ and −5–√ are two zeroes of the polynomial x3 + 3x2 − 5x − 15,
Question: If $\sqrt{5}$ and $-\sqrt{5}$ are two zeroes of the polynomial $x^{3}+3 x^{2}-5 x-15$, then its third zero is (a) 3(b) 3(c) 5(d) 5 Solution: Let $\alpha=\sqrt{5}$ and $\beta=-\sqrt{5}$ be the given zeros and $\gamma$ be the third zero of the polynomial $x^{3}+3 x^{2}-5 x-15$. Then, By using $\alpha+\beta+\gamma=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$ $\alpha+\beta+\gamma=\frac{-3}{1}$ $\alpha+\beta+\gamma=-3$ Substituting $\alpha=\sqrt{5}$ and $\beta=-\s...
Read More →If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3,
Question: If two zeroes of the polynomial $x^{3}+x^{2}-9 x-9$ are 3 and $-3$, then its third zero is (a) $-1$ (b) 1 (c) $-9$ (d) 9 Solution: Let $\alpha=3$ and $\beta=-3$ be the given zeros and $\gamma$ be the third zero of the polynomial $x^{3}+x^{2}-9 x-9$ then By using $\alpha+\beta+\gamma=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$ $\alpha+\beta+\gamma=\frac{-1}{1}$ $\alpha+\beta+\gamma=-1$ Substituting $\alpha=3$ and $\beta=-3$ in $\alpha+\beta+\gamma=-1$, we get...
Read More →A quadratic polynomial,
Question: A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is (a) $x^{2}-9$ (b) $x^{2}+9$ (c) $x^{2}+3$ (d) $x^{2}-3$ Solution: Since $\alpha$ and $\beta$ are the zeros of the quadratic polynomials such that $0=\alpha+\beta$ If one of zero is 3 then $\alpha+\beta=0$ $3+\beta=0$ $\beta=0-3$ $\beta=-3$ Substituting $\beta=-3$ in $\alpha+\beta=0$ we get $\alpha-3=0$ $\alpha=3$ Let S and P denote the sum and product of the zeros of the polynomial respectively then $S=\alpha+\b...
Read More →What should be subtracted to the polynomial x2 − 16x + 30,
Question: What should be subtracted to the polynomial $x^{2}-16 x+30$, so that 15 is the zero of the resulting polynomial? (a) 30 (b) 14 (c) 15 (d) 16 Solution: We know that, if $x=\alpha$, is zero of a polynomial then $x-\alpha$ is a factor of $f(x)$ Since 15 is zero of the polynomial $f(x)=x^{2}-16 x+30$, therefore $(x-15)$ is a factor of $f(x)$ Now, we divide $f(x)=x^{2}-16 x+30$ by $(x-15)$ we get Thus we should subtract the remainder 15 from $x^{2}-16 x+30$, Hence, the correct choice is (c)...
Read More →If x4 + (1/x4) = 119, Find the value of
Question: If $x^{4}+\left(1 / x^{4}\right)=119$, Find the value of $x^{3}-\left(1 / x^{3}\right)$ Solution: Given, $x^{4}+\left(1 / x^{4}\right)=119 \quad \ldots .1$ We know that $(x+y)^{2}=x^{2}+y^{2}+2 x y$ Substitute $x^{4}+\left(1 / x^{4}\right)=119$ in eq 1 $\left(x^{2}+\left(1 / x^{2}\right)\right)^{2}=x^{4}+\left(1 / x^{4}\right)+\left(2^{*} x^{2} * 1 / x^{2}\right)$ $=x^{4}+\left(1 / x^{4}\right)+2$ = 119 + 2 = 121 $\left(x^{2}+\left(1 / x^{2}\right)\right)^{2}=121$ $x^{2}+\left(\frac{1}...
Read More →What should be added to the polynomial x2 − 5x + 4,
Question: What should be added to the polynomial $x^{2}-5 x+4$, so that 3 is the zero of the resulting polynomial? (a) 1(b) 2(c) 4(d) 5 Solution: If $x=\alpha$, is a zero of a polynomial then $x-\alpha$ is a factor of $f(x)$ Since 3 is the zero of the polynomial $f(x)=x^{2}-5 x+4$, Therefore $x-3$ is a factor of $f(x)$ Now, we divide $f(x)=x^{2}-5 x+4$ by $(x-3)$ we get Therefore we should add 2 to the given polynomial Hence, the correct choice is$(b)$...
Read More →The product of the zeros of x3 + 4x2 + x − 6 is
Question: The product of the zeros of $x^{3}+4 x^{2}+x-6$ is (a) $-4$ (b) 4 (c) 6 (d) $-6$ Solution: Given $\alpha, \beta, \gamma$ be the zeros of the polynomial $f(x)=x^{3}+4 x^{2}+x-6$ Product of the zeros $=\frac{\text { Constant term }}{\text { Coefficient of } x^{3}}=\frac{-(-6)}{1}=6$ The value of Product of the zeros is 6. Hence, the correct choice is $(\mathrm{c})$...
Read More →If two zeros x3 + x2 − 5x − 5 are 5–√ and −5–√
Question: If two zeros $x^{3}+x^{2}-5 x-5$ are $\sqrt{5}$ and $-\sqrt{5}$, then its third zero is (a) 1 (b) $-1$ (c) 2 (d) $-2$ Solution: Let $\alpha=\sqrt{5}$ and $\beta=-\sqrt{5}$ be the given zeros and $\gamma$ be the third zero of $x^{3}+x^{2}-5 x-5=0$ then By using $\alpha+\beta+\gamma=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$ $\alpha+\beta+\gamma=\frac{+(+1)}{1}$ $\alpha+\beta+\gamma=-1$ By substituting $\alpha=\sqrt{5}$ and $\beta=-\sqrt{5}$ in $\alpha+\beta+...
Read More →If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero,
Question: If two of the zeros of the cubic polynomial $a x^{3}+b x^{2}+c x+d$ are each equal to zero, then the third zero is (a) $\frac{-d}{a}$ (b) $\frac{c}{a}$ (c) $\frac{-b}{a}$ (d) $\frac{b}{a}$ Solution: Let $\alpha=0, \beta=0$ and $\gamma$ be the zeros of the polynomial $f(x)=a x^{3}+b x^{2}+c x+d$ Therefore $\alpha+\beta+\gamma=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$ $=-\left(\frac{b}{a}\right)$ $\alpha+\beta+\gamma=-\frac{b}{a}$ $0+0+\gamma=-\frac{b}{a}$ $...
Read More →If 3x - 2y = 11 and xy = 12,
Question: If $3 x-2 y=11$ and $x y=12$, Find the value of $27 x^{3}-8 y^{3}$ Solution: Given, 3x - 2y = 11, xy = 12 We know that $(a-b)^{3}=a^{3}-b^{3}-3 a b(a+b)$ $(3 x-2 y)^{3}=11^{3}$ $\Rightarrow 27 x^{3}-8 y^{3}-(18 * 12 * 11)=1331$ $\Rightarrow 27 x^{3}-8 y^{3}-2376=1331$ $\Rightarrow 27 x^{3}-8 y^{3}=1331+2376$ $\Rightarrow 27 x^{3}-8 y^{3}=3707$ Hence, the value of $27 x^{3}-8 y^{3}=3707$...
Read More →If α, β are the zeros of the polynomial f(x) = ax2 + bx + c,
Question: If $\alpha, \beta$ are the zeros of the polynomial $f(x)=a x^{2}+b x+c$, then $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=$ (a) $\frac{b^{2}-2 a c}{a^{2}}$ (b) $\frac{b^{2}-2 a c}{c^{2}}$ (c) $\frac{b^{2}+2 a c}{a^{2}}$ (d) $\frac{b^{2}+2 a c}{c^{2}}$ Solution: We have to find the value of $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}$ Given $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $\mathrm{f}(\mathrm{x})=a x^{2}+b x+c$ $\alpha+\beta=\frac{-\text { Coefficient of } x}{\...
Read More →If 2x + 3y = 13 and xy = 6,
Question: If $2 x+3 y=13$ and $x y=6$, Find the value of $8 x^{3}+27 y^{3}$ Solution: Given, 2x + 3y = 13, xy = 6 We know that, $(2 x+3 y)^{3}=13^{2}$ $\Rightarrow 8 x^{3}+27 y^{3}+3(2 x)(3 y)(2 x+3 y)=2197$ $\Rightarrow 8 x^{3}+27 y^{3}+18 x y(2 x+3 y)=2197$ Substitute 2x + 3y = 13, xy = 6 $\Rightarrow 8 x^{3}+27 y^{3}+18(6)(13)=2197$ $\Rightarrow 8 x^{3}+27 y^{3}+1404=2197$ $\Rightarrow 8 x^{3}+27 y^{3}=2197-1404$ $\Rightarrow 8 x^{3}+27 y^{3}=793$ Hence, the value of $8 x^{3}+27 y^{3}=793$...
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