Find the value
Question: Find the value of $2 \sec ^{-1} 2+\sin ^{-1}\left(\frac{1}{2}\right)$ Solution: $2 \sec ^{-1} 2+\sin ^{-1}\left(\frac{1}{2}\right)=2 \sec ^{-1}\left(\sec \frac{\pi}{3}\right)+\sin ^{-1}\left(\sin \frac{\pi}{6}\right)$ $=2 \times \frac{\pi}{3}+\frac{\pi}{6}$ $=\frac{5 \pi}{6}$...
Read More →Find the sum of odd integers from 1 to 2001.
Question: Find the sum of odd integers from 1 to 2001. Solution: The odd integers from 1 to 2001 are $1,3,5 \ldots \ldots 2001$. It is an AP with $a=1$ and $d=2$. $a_{n}=2001$ $\Rightarrow 1+(n-1) 2=2001$ $\Rightarrow 2 n-2=2000$ $\Rightarrow 2 n=2002$ $\Rightarrow n=1001$ Also, $S_{1001}=\frac{1001}{2}[2 \times 1+(1001-1) 2]$ $\Rightarrow S_{1001}=\frac{1001}{2}[2 \times 1+(1000) 2]$ $\Rightarrow S_{1001}=\frac{1001}{2} \times 2002=1002001$...
Read More →if
Question: If $\cos \left(\tan ^{-1} x+\cot ^{-1} \sqrt{3}\right)=0$, find the value of $x$. Solution: $\cos \left(\tan ^{-1} x+\cot ^{-1} \sqrt{3}\right)=0$ $\Rightarrow \cos \left(\tan ^{-1} x+\cot ^{-1} \sqrt{3}\right)=\cos \left(\frac{\pi}{2}\right)$ $\Rightarrow \tan ^{-1} x+\cot ^{-1} \sqrt{3}=\frac{\pi}{2}$ $\Rightarrow x=\sqrt{3} \quad\left[\because \tan ^{-1} y+\cot ^{-1} y=\frac{\pi}{2}\right]$...
Read More →In the given figure, if TP and TQ are tangents drawn from an external point
Question: In the given figure, ifTPandTQare tangents drawn from an external pointTto a circle with centreOsuch that TQP= 60, then OPQ= (a) 25(b) 30(c) 40(d) 60 Solution: Consider. We have, TP = TQ(Tangents from an external point will be equal) We know that angles opposite to equal sides will be equal. Therefore, $\angle T Q P=\angle T P Q$ It is given that, $\angle T Q P=60^{\circ}$ Therefore, $\angle T P Q=60^{\circ}$ We know that the radius will always be perpendicular to the tangent at the po...
Read More →Write the value
Question: Write the value of $\cos \left(\frac{\tan ^{-1} x+\cot ^{-1} x}{3}\right)$, when $x=-\frac{1}{\sqrt{3}}$ Solution: $\cos \left(\frac{\tan ^{-1} x+\cot ^{-1} x}{3}\right)=\cos \left(\frac{\pi}{6}\right) \quad\left[\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right]$ $=\frac{\sqrt{3}}{2}$...
Read More →If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116.
Question: If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term. Solution: The given A.P. is 25,.22,.19..... Here, $a=25, d=22-25=-3$ $S_{n}=116$ $\Rightarrow \frac{n}{2}[2 a+(n-1) d]=116$ $\Rightarrow n[2 \times 25+(n-1)(-3)]=232$ $\Rightarrow 50 n-3 n^{2}+3 n=232$ $\Rightarrow 3 n^{2}-53 n+232=0$ $\Rightarrow 3 n^{2}-29 n-24 n+232=0$ $\Rightarrow n(3 n-29)-8(3 n-29)=0$ $\Rightarrow(3 n-29)(n-8)=0$ $\Rightarrow n=\frac{29}{3}$ or 8 Since $n$ cannot be a fr...
Read More →In the adjoining figure, ABCD is a parallelogram whose diagonals intersect each other at O.
Question: In the adjoining figure,ABCDis a parallelogram whose diagonals intersect each other atO.Aline segmentEOFis drawn to meetABatEandDCatF. Prove thatOE=OF. Solution: In∆ODFand∆OBE, we have:OD = OB (Diagonals bisects each other)DOF= BOE (Vertically opposite angles)FDO= OBE (Alternate interior angles)i.e., ∆ODF∆OBEOF = OE (CPCT)Hence, proved....
Read More →Write the value
Question: Write the value of $\cot ^{-1}(-x)$ for all $x \in R$ in terms of $\cot ^{-1} x$ Solution: We know that $\cot ^{-1}(-x)=\pi-\cot ^{-1}(x)$ Therefore, the value of $\cot ^{-1}(-x)$ for all $x \in R$ in terms of $\cot ^{-1} x$ is $\pi-\cot ^{-1}(x)$....
Read More →In the adjoining figure, ABCD is a parallelogram.
Question: In the adjoining figure, $A B C D$ is a parallelogram. If $P$ and $Q$ are points on $A D$ and $B C$ respectively such that $A P=\frac{1}{3} A D$ and $C Q=\frac{1}{3} B C$, prove that $A Q C P$ is a parallelogram. Solution: We have:B= D [Opposite angles of parallelogramABCD]AD = BCandAB = DC [Opposite sides of parallelogramABCD]Also,AD|| BCandAB|| DC It is given that $A P=\frac{1}{3} A D$ and $C Q=\frac{1}{3} B C$. AP = CQ [∵AD = BC]In∆DPCand∆BQA, we have: AB = CD,B= DandDP = QB $\le...
Read More →Find the sum of all two digit numbers which when divided by 4,
Question: Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder. Solution: The two-digit numbers which when divided by 4 yield 1 as remainder are 13, 17....97. $\therefore a=13, d=4, a_{n}=97$ $\therefore a_{n}=a+(n-1) d$ $\Rightarrow 97=13+(n-1) 4$ $\Rightarrow 84=4 n-4$ $\Rightarrow 88=4 n$ $\Rightarrow 22=n \ldots(1)$ Also, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $S_{22}=\frac{22}{2}[2 \times 13+(22-1) \times 4] \quad($ From $(1))$ $\Rightarrow S_{22}=11[110]=1210$...
Read More →Write the value
Question: Write the value of $\tan ^{-1}\left(\frac{1}{x}\right)$ for $x0$ in terms of $\cot ^{-1} x$ Solution: $\tan ^{-1}\left(\frac{1}{x}\right)=\tan ^{-1}\left(-\frac{1}{x}\right) \quad$ for $x0$ $=-\tan ^{-1}\left(\frac{1}{x}\right)$ $=-\cot ^{-1} x$ $=-\left(\pi-\cot ^{-1} x\right)$ $=-\pi+\cot ^{-1} x$...
Read More →In the given figure, PQ and PR are tangents drawn from P to a circle with centre O. If ∠OPQ = 35°, then
Question: In the given figure,PQandPRare tangents drawn fromPto a circle with centreO. If OPQ= 35, then (a)a= 30,b= 60(b)a= 35,b= 55(c)a= 40,b= 50(d)a= 45,b= 45 Solution: Considerand. We have, POis the common side for both the triangles. OQ = OR(Radii of the same circle) PQ = PR(Tangents from an external point will be equal) Therefore, from SSS postulate of congruent triangles, we have, $\triangle P O Q \cong \triangle P O R$ Therefore, $\angle O P Q=\angle O P R$ That is, $\angle O P Q=\angle a...
Read More →The set of values
Question: The set of values of $\operatorname{cosec}^{-1}\left(\frac{\sqrt{3}}{2}\right)$ Solution: The value of $\operatorname{cosec}^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is undefined as it is outside the range i.e., $R-(-1,1)$....
Read More →Find the sum of n terms of the A.P.
Question: Find the sum ofnterms of the A.P. whosekth terms is 5k+ 1. Solution: We have: $a_{k}=5 k+1$ For $k=1, a_{1}=5 \times 1+1=6$ For $k=2, a_{2}=5 \times 2+1=11$ For $k=n, a_{n}=5 n+1$ $\therefore S_{n}=\frac{n}{2}\left[a+a_{n}\right]$ $\Rightarrow S_{n}=\frac{n}{2}[6+5 n+1]=\frac{n}{2}(5 n+7)$...
Read More →Write the principal value
Question: Write the principal value of $\sin ^{-1}\left\{\cos \left(\sin ^{-1} \frac{1}{2}\right)\right\}$ Solution: $\sin ^{-1}\left\{\cos \left(\sin ^{-1} \frac{1}{2}\right)\right\}=\sin ^{-1}\left\{\cos \left[\sin ^{-1}\left(\sin \frac{\pi}{3}\right)\right]\right\}$ $=\sin ^{-1}\left[\cos \left(\frac{\pi}{3}\right)\right]$ $=\sin ^{-1}\left[\frac{1}{2}\right]$ $=\sin ^{-1}\left[\sin \left(\frac{\pi}{3}\right)\right]$ $=\frac{\pi}{3}$...
Read More →In the given figure, if AB = 8 cm and PE = 3 cm, then AE =
Question: In the given figure, ifAB= 8 cm andPE= 3 cm, thenAE= (a) 11 cm(b) 7 cm(c) 5 cm(d) 3 cm Solution: We know that tangents drawn from the same external point will be equal in length. Therefore, AB = AC It is given that, AB= 8 cm Hence, AC= 8 cm (1) Similarly, PE = CE It is given that, PE= 3 Therefore, CE= 3 (2) Subtracting equations (1) and (2), we get, AC CE= 8 3 From the figure we can see that, AC CE = AE Therefore, AE= 8 3 AE= 5 cm Option (c) is the correct answer....
Read More →If the 5th and 12th terms of an A.P. are 30 and 65 respectively,
Question: If the 5thand 12thterms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms? Solution: We have; $a_{5}=30$ $\Rightarrow a+(5-1) d=30$ $\Rightarrow a+4 d=30 \quad \ldots(\mathrm{i})$ Also, $a_{12}=65$ $\Rightarrow a+(12-1) d=65$ $\Rightarrow a+11 d=65 \quad \ldots \ldots($ ii $)$ Solving (i) and (ii), we get: $7 d=35$ $\Rightarrow d=5$ Putting the value of $d$ in (i), we get: $a+4 \times 5=30$ $\Rightarrow a=10$ $\therefore S_{20}=\frac{20}{2}[2 \times 10+(20-1) \ti...
Read More →In a parallelogram ABCD, points M and N have been taken on opposite sides AB and CD respectively such that AM = CN.
Question: In a parallelogramABCD, pointsMandNhave been taken on opposite sidesABandCDrespectively such thatAM=CN. Show thatACandMNbisect each other. Solution: Given:In aparallelogramABCD,AM=CN.Toprove:ACandMNbisecteachother.Construction:JoinANandMC.Proof:Since,ABCDis a parallelogram. $\Rightarrow A B \| D C$ $\Rightarrow A M \| N C$ Also,AM = CN (Given)Thus,AMCNis a parallelogram.Since, diagonals of a parallelogram bisect each other.Hence,ACandMNbisect each other....
Read More →In the given figure, there are two concentric circles with centre O.
Question: In the given figure, there are two concentric circles with centre O.PRandPQSare tangents to the inner circle from point plying on the outer circle. IfPR= 7.5 cm, thenPSis equal to(a) 10 cm(b) 12 cm(c) 15 cm(d) 18 cm Solution: Let us first draw the radiiOS,OPandOQin the given figure, for our convenience. Consider. We have, PO = OS(Radii of the same circle) Therefore,is an isosceles triangle. We know that in an isosceles triangle, if a line is drawn perpendicular to the base of the trian...
Read More →Two concentric circles of radii 3 cm and 5 cm are given.
Question: Two concentric circles of radii 3 cm and 5 cm are given. Then length of chordBCwhich touches the inner circle atPis equal to(a) 4 cm(b) 6 cm(c) 8 cm(d) 10 cm Solution: Consider. We have, $\mathrm{OQ} \perp \mathrm{AB}$ Therefore, $\mathrm{QA}^{2}=\mathrm{OA}^{2}-\mathrm{OQ}^{2}$ $\mathrm{QA}^{2}=5^{2}-3^{2}$ $\mathrm{QA}^{2}=25-9$ $\mathrm{QA}^{2}=16$ $\mathrm{QA}=4$ Considering AB as the chord to the bigger circle, as OQ is perpendicular to AB, OQ bisects AB. AQ = QB = 4 cm. Now, as B...
Read More →In a rhombus ABCD show that diagonal AC bisects ∠A as well as
Question: In a rhombusABCDshow that diagonalACbisects Aas well as Cand diagonalBDbisects Bas well as D. Solution: Given:ArhombusABCD.Toprove:DiagonalACbisectsAaswellasCanddiagonalBDbisectsBaswellasD.Proof: In $\Delta A B C$, $A B=B C$ (Sides of rhombus are equal.) $\angle 4=\angle 2$ (Angles opposite to equal sides are equal.) ...(1) Now, $A D \| B C$ (Opposite sides of rhombus are parallel.) ACis transversal. So, $\angle 1=\angle 4$ (Alternate interior angles) ...(2) From (1) and (2), we get $\...
Read More →If 12th term of an A.P. is −13 and the sum of the first four terms is 24,
Question: If 12thterm of an A.P. is 13 and the sum of the first four terms is 24, what is the sum of first 10 terms? Solution: Letabe the first term anddbe the common difference. $a_{12}=-13$ $\Rightarrow a+(12-1) d=-13$ $\Rightarrow a+11 d=-13 \quad \ldots(i)$ Also, $S_{4}=24$ $\Rightarrow \frac{4}{2}[2 a+(4-1) d]=24$ $\Rightarrow 2(2 a+3 d)=24$ $\Rightarrow 2 a+3 d=12 \quad \ldots(i i)$ From (i) and (ii), we get: $19 d=-38$ $\Rightarrow d=-2$ Putting the value of $d$ in (i), we get: $a+11(-2)=...
Read More →Two circles of same radii r and centres O and O' touch each other at P as shown in Fig.
Question: Two circles of same radii r and centresOandO' touch each other atPas shown in Fig. 10.91. IfOO' is produced to meet the cireleC(O', r) atAandATis a tangent to the circleC(O,r) such thatO'QAT. ThenAO:AO' = (a) 3/2(b) 2(c) 3(d) 1/4 Solution: From the given figure we have, AO = r + r + r AO= 3r AO = r Therefore, $\frac{A O}{A O^{\prime}}=\frac{3 r}{r}$ $\frac{A O}{A O^{\prime}}=3$ Also as $O^{\prime} Q \| O T$ therefore $\frac{A T}{A Q}=\frac{A O}{A O^{\prime}}$ Therefore, option (c) is c...
Read More →Wnte the value of the expression
Question: Wnte the value of the expression $\tan \left(\frac{\sin ^{-1} x+\cos ^{-1} x}{2}\right)$, when $x=\frac{\sqrt{3}}{2}$ Solution: $\tan \left(\frac{\sin ^{-1} x+\cos ^{-1} x}{2}\right)=\tan \left(\frac{\pi}{4}\right)$ $\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right]$ $=1$...
Read More →Write the value
Question: Write the value of $\cos \left(\sin ^{-1} x+\cos ^{-1} x\right),|x| \leq 1$ Solution: We have $|x| \leq 1$ $\Rightarrow \pm x \leq 1$ $\Rightarrow x \leq 1$ or $-x \leq 1$ $\Rightarrow x \leq 1$ or $x \geq-1$ $\Rightarrow x \in[-1,1]$ Now, $\cos \left(\sin ^{-1} x+\cos ^{-1} x\right)=\cos \left(\frac{\pi}{2}\right)$ $\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right]$ $=0$...
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