The ages of two friends Ani and Biju differ by 3 years.
Question: The ages of two friends Ani and Biju differ by 3 years. Ani's father Dharma is twice as old as Ani and Biju as twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju. Solution: Let the present ages of Ani, Biju, Dharamand Cathy bex, y, zandtyears respectively. The ages of Ani and Biju differ by 3 years. Thus, we have $x-y=\pm 3$ $\Rightarrow x=y \pm 3$ Dharam is twice as old as Ani. Thus, we have $z=2 x$ Biju is twice as old as ...
Read More →If f(x) = cos
Question: Iff(x) = cos [2]x+ cos [2]x, where [x] denotes the greatest integer less than or equal tox, then write the value off(). Solution: f(x) = cos [2]x+ cos [2]x Thus, $f(\pi)=\cos \left[\pi^{2}\right] \pi+\cos \left[-\pi^{2}\right] \pi$ $\Rightarrow f(\pi)=\cos [9.8] \pi+\cos [-9.8] \pi$ $\Rightarrow f(\pi)=\cos 10 \pi+\cos 9 \pi$ $\Rightarrow f(\pi)=1+(-1)=0$...
Read More →If f(x) = cos
Question: Iff(x) = cos [2]x+ cos [2]x, where [x] denotes the greatest integer less than or equal tox, then write the value off(). Solution: f(x) = cos [2]x+ cos [2]x Thus, $f(\pi)=\cos \left[\pi^{2}\right] \pi+\cos \left[-\pi^{2}\right] \pi$ $\Rightarrow f(\pi)=\cos [9.8] \pi+\cos [-9.8] \pi$ $\Rightarrow f(\pi)=\cos 10 \pi+\cos 9 \pi$ $\Rightarrow f(\pi)=1+(-1)=0$...
Read More →ABCD is a rectangle with ∠ABD = 40°. Determine ∠DBC
Question: ABCD is a rectangle with ABD = 40. Determine DBC Solution: We have, ABC = 90 ⇒ ABD + DBC = 90 [∵ ABD = 40] ⇒ 400 + DBC = 90 DBC = 50...
Read More →Write the range of the function f(x) = sin [x],
Question: Write the range of the function $1(x)=\sin [x]$, where $\frac{-\pi}{4} \leq x \leq \frac{\pi}{4}$. Solution: Given: $f(x)=\sin [x]$, where $\frac{-\pi}{4} \leq x \leq \frac{\pi}{4}$. $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ $\Rightarrow-7.85 \leq x \leq 0.785$ $\therefore x \in[-7.85,7.85]$ Or $x=\{0,1\}$ Thus, range of $f(x)=\sin [x]$ is $\{\sin 0, \sin 1\}=\{0, \sin 1\}$...
Read More →If f is a real function satisfying
Question: If $t$ is a real function satisfying $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}$ for all $x \in \mathrm{R}-\{0\}$, then write the expression for $f(x)$. Solution: Given: $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}$ $=x^{2}+\frac{1}{x^{2}}+2-2$ $=\left(x+\frac{1}{x}\right)^{2}-2$ Thus, $f\left(x+\frac{1}{x}\right)=\left(x+\frac{1}{x}\right)^{2}-2$ Hence, f(x) =x2-2 , where |x| 2....
Read More →ABCD is a square. AC and BD intersect at O. State the measure of ∠AOB.
Question: ABCD is a square. AC and BD intersect at O. State the measure of AOB. Solution: Since, diagonals of a square bisect each other at right angle. AOB = 90...
Read More →Five years ago, Nuri was thrice as old as Sonu.
Question: Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu? Solution: Let the present age of Nuri bexyears and the present age of Sonu beyyears. After 10 years, Nuris age will be(x+ 10) years and the age of Sonu 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 Nuri was(x 5)years and the age of Sonu was(y 5)years. ...
Read More →Show that
Question: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{7} x d x$ Solution: Let $I=\int_{\frac{\pi}{2}}^{\pi} \sin ^{7} x d x$ ...(1) As $\sin ^{7}(-x)=(\sin (-x))^{7}=(-\sin x)^{7}=-\sin ^{7} x$, therefore, $\sin ^{2} x$ is an odd function. It is known that, if $f(x)$ is an odd function, then $\int_{-a}^{a} f(x) d x=0$ $\therefore I=\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{7} x d x=0$...
Read More →Write the range of the real function f(x)
Question: Write the range of the real functionf(x) = |x|. Solution: Given: f(x) = |x|,xR We know that $|x|= \begin{cases}x, x \geq 0 \\ -x x0\end{cases}$ It can be observed that the range off(x) = |x| is all real numbers except negative real numbers. The range offis [0, ) ....
Read More →In a parallelogram ABCD, if ∠B = 135°, determine the measures of its other angles.
Question: In a parallelogram ABCD, if B = 135, determine the measures of its other angles. Solution: GivenB = 135 ABCD is a parallelogram A = C, B = D and A + B = 180 ⇒ A + 135 = 180 ⇒ A = 45 ⇒ A = C = 45 and B = C = 135...
Read More →Two years ago, a father was five times as old as his son.
Question: Two years ago, a father was five times as old as his son. Two year later, his age will be 8 more than three times the age of the son. Find the present ages of father and son. Solution: Let the present age of father bexyears and the present age of his son beyyears. After 2 years, father's age will be $(x+2)$ years and the age of son will be $(y+2)$ years. Thus using the given information, we have $x+2=3(y+2)+8$ $\Rightarrow x+2=3 y+6+8$ $\Rightarrow x-3 y-12=0$ Before 2 years, the age o...
Read More →The number of elements of an identity function defined on a set containing four elements is
Question: The number of elements of an identity function defined on a set containing four elements is __________ . Solution: The number of elements of an identity function defined on a set of containing. Four elements is four only Since identity function maps same element to same element. i.exx yy zz and w w There are four such elements....
Read More →In a parallelogram ABCD, determine the sum of angles ∠C and ∠D.
Question: In a parallelogram ABCD, determine the sum of angles C and D. Solution: C and Dare consecutive interior angles on the same side of the transversal CD. C + D = 180...
Read More →Show that
Question: $\int_{0}^{\pi} \frac{x d x}{1+\sin x}$ Solution: Let $I=\int_{0}^{\pi} \frac{x d x}{1+\sin x}$ ...(1) $\Rightarrow I=\int_{0}^{\pi} \frac{(\pi-x)}{1+\sin (\pi-x)} d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$ $\Rightarrow I=\int_{0}^{\pi} \frac{(\pi-x)}{1+\sin x} d x$ ...(2) Adding (1) and (2), we obtain $2 I=\int_{0}^{\pi} \frac{\pi}{1+\sin x} d x$ $\Rightarrow 2 I=\pi \int_{0}^{\pi} \frac{(1-\sin x)}{(1+\sin x)(1-\sin x)} d x$ $\Rightarrow 2 I=\pi \int_{0}^{\pi} ...
Read More →If f(2x + 3) =
Question: Iff(2x+ 3) = 4x2+ 12x+15, then the value off(3x+ 2) is __________ . Solution: Givenf(2x+ 3) = 4x2+ 12x+ 15 = (2x)2+ 2(2x) (3) + (3)2+ 6 f(2x+ 3) = (2x+ 3)2+ 6 i.ef(y) =y2+ 6 wherey= 2x+ 3 f(3x+ 2) = (3x+ 2)2+ 6 = 9x2+ 4 + 12x+ 6 Hencef(3x+ 2) = 9x2+ 12x+ 10...
Read More →Which of the following statements are true (T) and which are false (F)?
Question: Which of the following statements are true (T) and which are false (F)? (i) In a parallelogram, the diagonals are equal. (ii) In a parallelogram, the diagonals bisect each other. (iii) In a parallelogram, the diagonals intersect each other at right angles. (iv) In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram. (v) If all the angles of a quadrilateral are equal, it is a parallelogram. (vi) If three sides of a quadrilateral are equal, it is a parallelogra...
Read More →Father's age is three times the sum of age of his two children.
Question: Father's age is three times the sum of age of his two children. After 5 years his age will be twice the sum of ages of two children. Find the age of father. Solution: Let the present age of father bexyears and the present ages of his two childrens beyandzyears. The present age of father is three times the sum of the ages of the two childrens. Thus, we have $x=3(y+z)$ $\Rightarrow y+z=\frac{x}{3}$ After 5 years, father's age will be $(x+5)$ years and the children's age will be $(y+5)$ a...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\frac{|x|-2}{|x|-3}$ is _________ . Solution: $f(x)=\frac{|x|-2}{|x|-3}$ heref(x) is not define if |x| 3 = 0 i.e |x| = 3 i.ex= 3 or 3 Domain off(x) isR {3, 3}...
Read More →Show that
Question: $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x$ Solution: Let $I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x$ As $\sin ^{2}(-x)=(\sin (-x))^{2}=(-\sin x)^{2}=\sin ^{2} x$, therefore, $\sin ^{2} x$ is an even function. It is known that if $f(x)$ is an even function, then $\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x$ $I=2 \int_{0}^{\frac{\pi}{2}} \sin ^{2} x d x$ $=2 \int_{0}^{\frac{\pi}{2}} \frac{1-\cos 2 x}{2} d x$ $=\int_{0}^{\frac{\pi}{2}}(1-\cos 2 x) d x$ $=\lef...
Read More →The domain of the function f(x)
Question: The domain of the function $f(x)=\sum_{n=1}^{10} \frac{1}{|2 x-n|}$ is ________ . Solution: $f(x)=\sum_{n=1}^{10} \frac{1}{|2 x-n|}$ f(x) is not defined if 2xn= 0 i. e $x=\frac{n}{2} ; n=1$ to 10 $\therefore$ Domain of $f(x)$ is $\mathbb{R} \sim\left\{\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3, \frac{7}{2}, 4, \frac{9}{2}, 5\right\}$...
Read More →In figure, ABCD is a parallelogram and E is the mid-point of side BC.
Question: In figure, ABCD is a parallelogram and E is the mid-point of side BC. If DE and AB when produced meet at F, prove that AF = 2AB. Solution: InΔBEF and ΔCED BEF = CED [Verified opposite angle] BE = CE [Since, E is the mid-point of BC] EBF = ECD [Since, Alternate interior angles are equal] ΔBEF ΔCED [ASA congruence] BF = CD [CPCT] AF = AB + AF AF = AB + AB AF = 2AB. Hence proved....
Read More →A father is three times as old as his son.
Question: A father is three times as old as his son. In 12 years time, he will be twice as old as his son. Find the present ages of father and the son. Solution: Let the present age of father bexyears and the present age of his son beyyears. The present age of father is three times the age of the son. Thus, we have $x=3 y$ $\Rightarrow x-3 y=0$ After 12 years, father's age will be $(x+12)$ years and son's age will be $(y+12)$ years. Thus using the given information, we have $x+12=2(y+12)$ $\Righ...
Read More →The range of the function f(x)
Question: The range of the function $f(x)=\sqrt{1-x^{2}}$ is _________ . Solution: $f(x)=\sqrt{1-x^{2}}$ Sincef(x) 0 ( Being a square root) Range off(x) = [0, )...
Read More →In figure, ABCD is a parallelogram in which ∠DAB = 75° and ∠DBC = 60°.
Question: In figure, ABCD is a parallelogram in which DAB = 75 and DBC = 60. Compute CDB, and ADB. Solution: To findCDB and ADB CBD = ABD = 60 [Alternate interior angle. AD∥ BC and BD is the transversal] InBDC CBD + C + CDB = 180 [Angle sum property] ⇒ 60 + 75 + CDB = 180 ⇒ CDB = 180 (60 + 75) ⇒ CDB = 45 Hence, CDB = 45, ADB = 60...
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