eSaral - JEE | NEET | Class 8-10 Exam Prep App

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Question: If $\cos ^{-1} \frac{x}{2}+\cos ^{-1} \frac{y}{3}=\alpha$, then prove that $9 x^{2}-12 x y \cos \alpha+4 y^{2}=36 \sin ^{2} \alpha$ Solution: We know $\cos ^{-1} x+\cos ^{-1} y=\cos ^{-1}\left[x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right]$ Now, $\cos ^{-1} \frac{x}{2}+\cos ^{-1} \frac{y}{3}=\alpha$ $\Rightarrow \cos ^{-1}\left[\frac{x}{2} \frac{y}{3}-\sqrt{1-\frac{x^{2}}{4}} \sqrt{1-\frac{y^{2}}{3}}\right]=\alpha$ $\Rightarrow \frac{x}{2} \frac{y}{3}-\sqrt{1-\frac{x^{2}}{4}} \sqrt{1-\frac{y...

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Question: If $\cos ^{-1} \frac{x}{2}+\cos ^{-1} \frac{y}{3}=\alpha$, then prove that $9 x^{2}-12 x y \cos \alpha+4 y^{2}=36 \sin ^{2} \alpha$ Solution: We know $\cos ^{-1} x+\cos ^{-1} y=\cos ^{-1}\left[x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right]$ Now, $\cos ^{-1} \frac{x}{2}+\cos ^{-1} \frac{y}{3}=\alpha$ $\Rightarrow \cos ^{-1}\left[\frac{x}{2} \frac{y}{3}-\sqrt{1-\frac{x^{2}}{4}} \sqrt{1-\frac{y^{2}}{3}}\right]=\alpha$ $\Rightarrow \frac{x}{2} \frac{y}{3}-\sqrt{1-\frac{x^{2}}{4}} \sqrt{1-\frac{y...

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29,

Question: Ifnarithmetic means are inserted between 1 and 31 such that the ratio of the first mean andnth mean is 3 : 29, then the value ofnis (a) 10 (b) 12 (c) 13 (d) 14 Solution: (b) 12 The given series is 1, . . . . . . . . . . . , 31 There are $\mathrm{n}$ A.M.s between 1 and 31: $1, A_{1}, A_{2}, A_{3}, \ldots . A_{n}, 31$ Common difference, $d=\frac{31-1}{n+1}=\frac{30}{n+1}$ Here, we have: $\frac{A_{1}}{A_{n}}=\frac{3}{29}$ $\Rightarrow \frac{1+d}{1+n d}=\frac{3}{29}$ $\Rightarrow \frac{1+...

From the top of a light house, the angles of depression of two ships on

Question: From the top of a light house, the angles of depression of two ships on the opposite sides of it are observed to be and . If the height of the light house be h metres and the line joining the ships passes through the foot of the light house, show that the distance between the ship is $\frac{h(\tan \alpha+\tan \beta)}{\tan \alpha+\tan \beta}$ metres. Solution: Letbe the height of light house. And an angle of depression of the top of light house from two ships areandrespectively. Let,. A...

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least,

Question: If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are (a) 5, 10, 15, 20 (b) 4, 10, 16, 22 (c) 3, 7, 11, 15 (d) none of these Solution: (a) 5, 10, 15, 20 Let the four numbers in A.P. be as follows: $a-2 d, a-d, a, a+d$ Their sum = 50 (Given) $\Rightarrow(a-2 d)+(a-d)+a+(a+d)=50$ $\Rightarrow 2 a-d=25$ Also, $(a+d)=4(a-2 d)$ $\Rightarrow a+d=4 a-8 d$ $\Rightarrow 3 d=a$ ....(2) From equations(1)and(2),we get: d = 5,a= 15 ...

From the top of a building AB, 60 m high, the angles of depression of the top and bottom of

Question: From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30 and 60 respectively. Find(i) the horizontal distance between AB and CD(ii) the height of the lamp post.(iii) the difference between the heights of the building and the lamp post. Solution: Let $A B$ be the building of height 60 and $C D$ be the lamp post of height $h$, an angle of depression of the top and bottom of vertical lamp post are $30^{\circ}...

Which of the following figures lie on the same base and between the same parallels.

Question: Which of the following figures lie on the same base and between the same parallels. In such a case, write the comon base and the two parallels. Solution: (i) No, it doesnt lie on the same base and between the same parallels.(ii) No, it doesntlie on the same base and between the same parallels.(iii)Yes, it lies on the same base and between the same parallels. The same base is AB and the parallels are AB and DE.(iv)No, it doesntlie on the same base and between the same parallels.(v)Yes, ...

with common difference d, then the sum of the series sin

Question: Ifa1,a2,a3, ....anare in A.P. with common differenced, then the sum of the series sind[seca1seca2+ seca2seca3+ .... + secan 1secan], is (a) seca1 secan (b) coseca1 cosecan (c) cota1 cotan (d) tanan tana1 Solution: (d) $\tan a_{n}-\tan a_{1}$ We have: $\sin d\left(\sec a_{1} \sec a_{2}+\sec a_{2} \sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right)$ $=\frac{\sin d}{\cos a_{1} \cos a_{2}}+\frac{\sin d}{\cos a_{2} \cos a_{3}}+\ldots+\frac{\sin d}{\cos a_{n-1} \cos a_{n}}$ $=\frac{\sin \left(a...

Solve the following:

Question: Solve the following: (i) $\sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{3}$ (ii) $\cos ^{-1} x+\sin ^{-1} \frac{x}{2}=\frac{\pi}{6}$ Solution: (i) We know $\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right]$ $\therefore \sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{3}$ $\Rightarrow \sin ^{-1} x+\sin ^{-1} 2 x=\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ $\Rightarrow \sin ^{-1} x-\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)=-\sin ^{-1} 2 x$ $\Rightarrow \sin ^{-1}\left[x \s...

Solve the following:

Question: Solve the following: (i) $\sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{3}$ (ii) $\cos ^{-1} x+\sin ^{-1} \frac{x}{2}=\frac{\pi}{6}$ Solution: (i) We know $\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right]$ $\therefore \sin ^{-1} x+\sin ^{-1} 2 x=\frac{\pi}{3}$ $\Rightarrow \sin ^{-1} x+\sin ^{-1} 2 x=\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ $\Rightarrow \sin ^{-1} x-\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)=-\sin ^{-1} 2 x$ $\Rightarrow \sin ^{-1}\left[x \s...

In the arithmetic progression whose common difference is non-zero,

Question: In the arithmetic progression whose common difference is non-zero, the sum of first 3nterms is equal to the sum of nextnterms. Then the ratio of the sum of the first 2 n terms to the next 2nterms is (a) 1/5 (b) 2/3 (c) 3/4 (d) none of these Solution: (a) 1/5 $S_{3 n}=S_{4 n}-S_{3 n}$ $\Rightarrow 2 S_{3 n}=S_{4 n}$ $\Rightarrow 2 \times \frac{3 n}{2}\{2 a+(3 n-1) d\}=\frac{4 n}{2}\{2 a+(4 n-1) d\}$ $\Rightarrow 3\{2 a+(3 n-1) d\}=2\{2 a+(4 n-1) d\}$ $\Rightarrow 6 a+9 n d-3 d=4 a+8 n d...

Two boats approach a light house in mid-sea from opposite directions.

Question: Two boats approach a light house in mid-sea from opposite directions. The angles of elevation of the top of the light house from two boats are 30 and 45 respectively. If the distance between two boats is 100 m, find the height of the light house. Solution: Letbe the height of light house. Angle of elevation of the top of light house from two boats are 30 and 45. Let,and it is given thatm. So. And, Here we have to find height of light house. The corresponding figure is as follows So we ...

with common difference d, then the sum of the series sin d

Question: Ifa1,a2,a3, ....anare in A.P. with common differenced, then the sum of the series sind[coseca1coseca2+ coseca1coseca3+ .... + cosecan 1cosecan] is (a) sec a1 secan (b) coseca1 cosecan (c) cota1 cotan (d) tana1 tanan Solution: (c) $\cot a_{1}-\cot a_{n}$ We have: $\sin d\left(\operatorname{cosec} a_{1} \operatorname{cosec} a_{2}+\operatorname{cosec} a_{2} \operatorname{cosec} a_{3}+\ldots+\operatorname{cosec} a_{n-1} \operatorname{cosec} a_{n}\right)$ $=\frac{\sin d}{\sin a_{1} \sin a_{...

Match the following columns:

Question: Match the following columns: (a) ......,(b) ......,(c) ......,(d) ......, Solution: (a) - (r), (b) - (s), (c) - (p), (d) - (q)Explanation: (a) $P Q=\frac{1}{2}(A B+C D)=\frac{1}{2}(17)=8.5 \mathrm{~cm}$ (b) $O R=\frac{1}{2}(P R)=\frac{1}{2}(13)=6.5 \mathrm{~cm}$...

The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation

Question: The angle of elevation of the top of a hill at the foot of a tower is 60 and the angle of elevation of the top of the tower from the foot of the hill is 30. If the tower is 50 m high, what is the height of the hill? Solution: Letbe the height of hill. Andbe the tower of heightm. Angle of elevation of the top of hill from the foot of tower is 60 and angle of elevation of top of tower from foot of hill is 30. Letand, Here we have to find height of hill. The corresponding figure is as fol...

If the sum of n terms of an A.P.

Question: If the sum ofnterms of an A.P. is 2n2+ 5n,then its nth term is (a) 4n 3 (b) 3n 4 (c) 4n+ 3 (d) 3n+ 4 Solution: (c) 4n + 3 $S_{n}=2 n^{2}+5 n$ $S_{1}=2 \cdot 1^{2}+5 \cdot 1=7$ $\therefore a_{1}=7$ $S_{n}=2 \cdot 2^{2}+5 \cdot 2=18$ $\therefore a_{1}+a_{2}=18$ $\Rightarrow a_{2}=11$ Common difference, $d=11-7=4$ $a_{n}=a+(n-1) d$ $=7+(n-1) 4$ $=4 n+3$...

Match the following columns:

Question: Match the following columns: (a) .....,(b) .....,(c) .....,(d) ....., Solution: (a) - (q), (b) - (r), (c) - (s), (d) - (p)...

A boy is standing on the ground and flying a kite with 100 m of string at an elevation of 30°.

Question: A boy is standing on the ground and flying a kite with 100 m of string at an elevation of 30. Another boy is sanding on the roof of a 10 m high building and is flying his kite at an elevation of 45. Both the boys are on opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet. Solution: Letbe the string of stringm. letbe the ground and a boy flying kite ofm string at an elevation of.And another boy flying kite of 10 m high...

Assertion: The diagonals of a || gm bisect each other.

Question: Assertion:The diagonals of a || gm bisect each other.Reason:If the diagonals of a || gm are equal and intersect at right angles, then the parallelogram is a square.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (b)Both Assertion and Reason are true but Reason i...

If the sum of n terms of an A.P.,

Question: If the sum ofnterms of an A.P., is 3n2+ 5nthen which of its terms is 164? (a) 26th (b) 27th (c) 28th (d) none of these. Solution: (b) 27th $S_{n}=3 n^{2}+5 n$ $S_{1}=3(1)^{2}+5(1)=8$ $\therefore a_{1}=8$ $S_{2}=3(2)^{2}+5(2)=22$ $\therefore a_{1}+a_{2}=22$ $\Rightarrow a_{2}=14$ Common difference, $d=14-8=6$ Also, $a_{n}=164$ $\Rightarrow a+(n-1) d=164$ $\Rightarrow 8+(n-1) 6=164$ $\Rightarrow n=27$...

Assertion: Every parallelogram is a rectangle.

Question: Assertion:Every parallelogram is a rectangle.Reason:The angle bisectors of a parallelogram form a rectangle.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (d)Assertion is false and Reason is true.Explanation:We can easily prove reason (R). So, reason (R) is tru...

A carpenter makes stools for electricians with a square top of side 0.5 m and at a height of 1.5 m

Question: A carpenter makes stools for electricians with a square top of side 0.5 m and at a height of 1.5 m above the ground. Also, each leg is inclined at an angle of 60 to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances. Solution: Let the length of stool,m, heightm and its leg inclined at an angle ofto the ground. Let length of leg $A E=h \mathrm{~m}$. We have to find length of leg, lengths of two steps equal in length. $\ln _{\triangle A...

Assertion: In a rhombus ABCD, the diagonal AC bisects ∠A as well as ∠C.

Question: Assertion:In a rhombusABCD, the diagonalACbisectsAas well as C.Reason:The diagonals of a rhombus bisect each other at right angles.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assertion is false and Reason is true. Solution: (b)Both Assertion and Reason are true but Reason is not a correct explanation of Ass...

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36,

Question: The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be (a) 5 (b) 6 (c) 7 (d) 8 Solution: (b) 6 $a=1, a_{n}=11, S_{n}=36$ $\because a_{n}=11$ $\Rightarrow a+(n-1) d=11$ $\Rightarrow 1+(n-1) d=11$ $\Rightarrow(n-1) d=10$ ....(1) Also, $S_{n}=36$ $\Rightarrow \frac{n}{2}\{2 a+(n-1) d\}=36$ $\Rightarrow n\{2+10\}=72$ (Using (1)) $\Rightarrow n=6$...

Assertion: ABCD is a quadrilateral in which P, Q, R and S are the mid-points of AB, BC, CD and DA respectively.

Question: Assertion:ABCDis a quadrilateral in whichP,Q,RandSare the mid-points ofAB,BC,CDandDArespectively. Then,PQRSis a parallelogram.Reason:The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.(c) Assertion is true and Reason is false.(d) Assert...

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