The domain of the function f(x) = x + [x]
Question: The domain of the functionf(x) =x+ [x] is __________ . Solution: f(x) =x+ [x] Since Domain of [x] =R and Domain ofx=R Domain ofx+ [x] =RR=R...
Read More →In Figure, ABCD is a parallelogram in which ∠A = 60°.
Question: In Figure, ABCD is a parallelogram in which A = 60. If the bisectors of A, and B meet at P, prove that AD = DP, PC = BC and DC = 2AD. Solution: AP bisectsA Then,DAP = PAB = 30 Adjacent angles are supplementary Then,A + B = 180 B + 600 = 180 B = 180 60 B = 120 BP bisectsB Then,PBA = PBC = 30 PAB = APD = 30[Alternate interior angles] Therefore, AD = DP [Sides opposite to equal angles are in equal length] Similarly PBA = BPC = 60[Alternate interior angles] Therefore, PC = BC DC = DP + PC ...
Read More →The domain of the function f(x) = x + [x]
Question: TThe domain of the functionf(x) =x+ [x] is __________ .__________ . Solution: f(x) =x+ [x] Since Domain of [x] =R and Domain ofx=R Domain ofx+ [x] =RR=R...
Read More →The present age of a father is three years more than three times the age of the son.
Question: The present age of a father is three years more than three times the age of the son. Three years hence father's age will be 10 years more than twice the age of the son. Determine their present ages. Solution: Let the present age of father bexyears and the present age of his son beyyears. The present age of father is three years more than three times the age of the son. Thus, we have $x=3 y+3$ $\Rightarrow x-3 y-3=0$ After 3 years, father's age will be $(x+3)$ years and son's age will b...
Read More →The present age of a father is three years more than three times the age of the son.
Question: The present age of a father is three years more than three times the age of the son. Three years hence father's age will be 10 years more than twice the age of the son. Determine their present ages. Solution: Let the present age of father bexyears and the present age of his son beyyears. The present age of father is three years more than three times the age of the son. Thus, we have $x=3 y+3$ $\Rightarrow x-3 y-3=0$ After 3 years, father's age will be $(x+3)$ years and son's age will b...
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Question: $\int_{0}^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x) d x$ Solution: Let $I=\int_{0}^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x) d x$ $\Rightarrow I=\int_{0}^{\frac{\pi}{2}}\{2 \log \sin x-\log (2 \sin x \cos x)\} d x$ $\Rightarrow I=\int_{0}^{\frac{\pi}{2}}\{2 \log \sin x-\log \sin x-\log \cos x-\log 2\} d x$ $\Rightarrow I=\int_{0}^{\frac{\pi}{2}}\{\log \sin x-\log \cos x-\log 2\} d x$ ...(1) It is known that, $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$ $\Rightarro...
Read More →The range of the function f(x)
Question: The range of the function $f(x)=\frac{|x-4|}{x-4}$ is ______ . Solution: $f(x)=\frac{|x-4|}{x-4}$ $\frac{-\frac{(x-4)}{x-4} \frac{(x-4)}{x-4}}{4}$ $= \begin{cases}\frac{x-4}{x-4} ; x \geq 4 \\ -\frac{(x-4)}{x-4} ; x+4\end{cases}$ $f(x)= \begin{cases}1 ; x \geq 4 \\ -1 ; x4\end{cases}$ Range off(x) is {1, 1}...
Read More →The domain of the function f (x)
Question: The domain of the function $f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]+2}}$ is __________ . Solution: $f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]+2}}$ Since [x]2 3[x] + 2 = 0 if [x]2 2[x] [x] + 2 = 0 [x] ([x] 2) 1 ([x] 2) = 0 i.e [x] = 2 or [x] = 1 f(x) is defined only if [x]2 3[x] + 2 0 i.e ([x] 2) ([x]1) 0 [x] 2 or [x] 1 i.e [x] 3 or [x] 0 i.ex[3, ) orx(, 1) i.e Domain off(x) is [3, ) (, 1) i.e (, 1) [3, )...
Read More →The domain and range of the function f(x)
Question: The domain and range of the function $f(x)=\frac{2-x}{x-2}$ are _______ and ___________respectively. Solution: $f(x)=\frac{2-x}{x-2}$ f(x) is defined forx 2 0 i.ex 2 Domain off(x) isR{2} Since $f(x)=-\frac{(x-2)}{x-2}$ f(x) = 1 Range off(x) is {1}...
Read More →Ten years ago, a father was twelve times as old as his son and ten years hence,
Question: Ten years ago, a father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be then. Find their present ages. Solution: Let the present age of father bexyears and the present age of his son beyyears. After 10 years, father's age will be $(x+10)$ years and son's age will be $(y+10)$ years. Thus using the given information, we have $x+10=2(y+10)$ $\Rightarrow x+10=2 y+20$ $\Rightarrow x-2 y-10=0$ Before 10 years, the age of father was $(x-10)$ ...
Read More →ABCD is a parallelogram in which ∠A = 700. Compute ∠B, ∠C and ∠D.
Question: ABCD is a parallelogram in which A = 700. Compute B, C and D. Solution: In a parallelogram ABCD A = 70 A + B = 180 [Since, adjacent angles are supplementary] 70 + B = 180 [∵A = 70] B = 1800 70 B = 110 In a parallelogram opposite sides are equal. A = C = 70 B = D = 110...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\sqrt{9-x}+\frac{1}{\sqrt{x^{2}-16}}$ is equal to Solution: $f(x)=\sqrt{9-x}+\frac{1}{\sqrt{x^{2}-16}}$ heref(x) is defined for 9 x 0 i.e 9x i.ex 9 andx2 16 0 i.ex2 16 i.ex 4 orx 4 f(x) is defined for common region is (, 4) (4, 9] Domain off(x) is(, 4) (4, 9]...
Read More →In a parallelogram ABCD, ∠D = 135°.
Question: In a parallelogram ABCD, D = 135. Determine the measures of A and B. Solution: In a parallelogram ABCD Adjacent angles are supplementary So,D + C = 180 C = 180 135 C = 45 In a parallelogram opposite sides are equal. A = C = 45 B = D = 135...
Read More →Six years hence a man's age will be three times the age
Question: Six years hence a man's age will be three times the age of his son and three years ago he was nine times as old as his son. Find their present ages. Solution: Let the present age of the man bexyears and the present age of his son beyyears. After 6 years, the man's age will be $(x+6)$ years and son's age will be $(y+6)$ years. Thus using the given information, we have $x+6=3(y+6)$ $\Rightarrow x+6=3 y+18$ $\Rightarrow x-3 y-12=0$ Before 3 years, the age of the man was $(x-3)$ years and ...
Read More →If f(x) =
Question: Iff(x) = [x]2 5 [x] + 6, then the set of values ofxsatisfyingf(x) = 0 is __________ . Solution: f(x) = [x]2 5 [x] + 6 Givenf(x) = 0 i.e [x]25 [x] + 6 = 0 Let [x] =y i.ey2 5y+ 6 = 0 i.ey2 3y 2y+ 6 = 0 y(y 3) 2 (y 3) = 0 i.ey= 2 ory= 3 i.e [x] = 2 or [x] = 3 here, [x] = 2 ifx [2, 3) and [x] = 3 ifx[3, 4) value ofxfor whichf(x) = 0 are [2, 3) [3, 4) i.ex [2, 4)...
Read More →The perimeter of a parallelogram is 22 cm.
Question: The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm what is the measure of the shorter side? Solution: Let the shorter side be 'x'. Therefore, perimeter = x + 6.5 + 6.5 + x [Sum of all sides] 22 = 2(x + 6.5) 11 = x + 6.5 ⟹ x = 11 - 6.5 = 4.5 cm Therefore, shorter side = 4.5 cm...
Read More →If f(x)=
Question: If $f(x)=\frac{x-1}{x+1}$, then $f(x) f\left(-\frac{1}{x}\right)$ is equal to _________ . Solution: If $f(x)=\frac{x-1}{x+1}$ $f\left(\frac{-1}{x}\right)=\frac{\frac{-1}{x}-1}{\frac{-1}{x}+1}$ $=\frac{\frac{(-1-x)}{x}}{\frac{(-1+x)}{x}}$ $=-\frac{(1+x)}{x-1}$ $f\left(\frac{-1}{x}\right)=\frac{1+x}{1-x}$ $\therefore f(x) f\left(\frac{-1}{x}\right)=\frac{x-1}{x+1} \times \frac{1+x}{1-x}$ $\therefore f(x) f\left(\frac{-1}{x}\right)=-1$...
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Question: $\int_{0}^{2} x \sqrt{2-x} d x$ Solution: Let $I=\int_{0}^{2} x \sqrt{2-x} d x$ $I=\int_{0}^{2}(2-x) \sqrt{x} d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$ $=\int_{0}^{2}\left\{2 x^{\frac{1}{2}}-x^{\frac{3}{2}}\right\} d x$ $=\left[2\left(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right)-\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\right]_{0}^{2}$ $=\left[\frac{4}{3} x^{\frac{3}{2}}-\frac{2}{5} x^{\frac{5}{2}}\right]_{0}^{2}$ $=\frac{4}{3}(2)^{\frac{3}{2}}-\frac{2}{5}(2)^{\frac{5}{...
Read More →Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
Question: Find the measure of all the angles of a parallelogram, if one angle is 24 less than twice the smallest angle. Solution: x + 2x - 24 = 180 ⟹ 3x - 24 = 180 ⟹ 3x = 108+ 24 ⟹ 3x = 204 ⟹ x =204/3 = 68 ⟹ x = 68 ⟹ 2x - 24= 2*68- 24= 112 Hence, four angles are 68, 112, 68, 112....
Read More →A is elder to B by 2 years. A's father F is twice as old as A and B is twice as old as his sister S.
Question: A is elder to B by 2 years. A's father F is twice as old as A and B is twice as old as his sister S. If the ages of the father and sister differ by 40 years, find the age of A. Solution: Let the present ages of $A, B, F$ and $S$ be $x, y, z$ and $t$ years respectively. $A$ is elder to $B$ by 2 years. Thus, we have $x=y+2$ $F$ is twice as old as $A$. Thus, we have $z=2 x$ $B$ is twice as old as $S$. Thus, we have $y=2 t$ The ages of $F$ and $S$ is differing by 40 years. Thus, we have $z...
Read More →If an angle of a parallelogram is two-third of its adjacent angle,
Question: If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram. Solution: Let the measure of the angle be x. Therefore, the measure of the angle adjacent is2x/3 We know that the adjacent angle of a parallelogram is supplementary. Hence,x + 2x/3 = 180 2x + 3x = 540 ⟹ 5x = 540 ⟹ x = 108 Adjacent angles are supplementary ⟹ x + 108= 180 ⟹ x = 180- 108= 72 ⟹ x = 72 Hence, four angles are 180, 72, 180, 72...
Read More →If f(x)=
Question: If $f(x)=\frac{x-1}{x+1}$, then $f\left(\frac{1}{x}\right)+f(x)$ is equal to ______ . Solution: If $f(x)=\frac{x-1}{x+1}$ $f\left(\frac{1}{x}\right)=\frac{\frac{1}{x}-1}{\frac{1}{x}+1}$ $=\frac{\frac{(1-x)}{x}}{\frac{(1+x)}{x}}$ $f\left(\frac{1}{x}\right)=\frac{1-x}{1+x}$ $\therefore f\left(\frac{1}{x}\right)+f(x)=\frac{1-x}{1+x}+\frac{x-1}{x+1}$ $=\frac{1-x+x-1}{x+1}$ $\therefore f\left(\frac{1}{x}\right)+f(x)=0$...
Read More →Two opposite angles of a parallelogram are (3x - 2)° and (50 - x)°.
Question: Two opposite angles of a parallelogram are (3x - 2)and (50 - x). Find the measure of each angle of the parallelogram. Solution: We know that, Opposite sides of a parallelogram are equal. (3x - 2)= (50 - x) ⟹ 3x + x = 50 + 2 ⟹ 4x = 52 ⟹ x = 13 Therefore, (3x - 2)= (3*13 - 2) = 37 (50 - x)= (50 - 13) = 37 Adjacent angles of a parallelogram are supplementary. x + 37 = 180 x = 180 37 = 143 Hence, four angles are: 37, 143, 37, 143....
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Question: $\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$ Solution: Let $I=\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$ ...(1) $\therefore I=\int_{0}^{\frac{\pi}{4}} \log \left[1+\tan \left(\frac{\pi}{4}-x\right)\right] d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$ $\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log \left\{1+\frac{\tan \frac{\pi}{4}-\tan x}{1+\tan \frac{\pi}{4} \tan x}\right\} d x$ $\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log \left\{1+\frac{1-\tan x}{1+\tan }\right\...
Read More →Ten years later, A will be twice as old as B and five years ago,
Question: Ten years later, A will be twice as old as B and five years ago, A was three times as old as B. What are the present ages of A and B? Solution: Let the present age ofAbexyears and the present age ofBbeyyears. After 10 years, $A$ 's age will be $(x+10)$ years and $B$ 's age will be $(y+10)$ years. Thus using the given information, we have $x+10=2(y+10)$ $\Rightarrow x+10=2 y+20$ $\Rightarrow x-2 y-10=0$ Before 5 years, the age of $A$ was $(x-5)$ years and the age of $B$ was $(y-5)$ year...
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