(1 + cot A + tan A) (sin A – cos A) = sin A tan A – cot A cos A
Question: (1 + cotA+ tanA) (sinA cosA) = sinAtanA cotAcosA Solution: $(1+\cot A+\tan A)(\sin A-\cos A)$ $=\sin A+\cot A \sin A+\tan A \sin A-\cos A-\cot A \cos A-\tan A \cos A$ $=\sin A+\frac{\cos A}{\sin A} \times \sin A+\tan A \sin A-\cos A-\cot A \cos A-\frac{\sin A}{\cos A} \times \cos A$ $=\sin A+\cos A+\tan A \sin A-\cos A-\cot A \cos A-\sin A$ $=\sin A \tan A-\cot A \cos A$...
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Question: $(\tan \theta+\sec \theta-1)(\tan \theta+\sec \theta+1)=\frac{2 \sin \theta}{(1-\sin \theta)}$ Solution: $(\tan \theta+\sec \theta-1)(\tan \theta+\sec \theta+1)$ $=(\tan \theta+\sec \theta)^{2}-1 \quad\left[(a-b)(a+b)=a^{2}-b^{2}\right]$ $=\tan ^{2} \theta+\sec ^{2} \theta+2 \tan \theta \sec \theta-1$ $=2 \tan ^{2} \theta+2 \tan \theta \sec \theta \quad\left(1+\tan ^{2} \theta=\sec ^{2} \theta\right)$ $=2 \tan \theta(\tan \theta+\sec \theta)$ $=2 \times \frac{\sin \theta}{\cos \theta} ...
Read More →sin2θ tanθ + cos2θ cot θ + 2sin θ cos θ = tan θ + cot θ
Question: sin2 tan + cos2 cot + 2sin cos = tan + cot Solution: $\sin ^{2} \theta \tan \theta+\cos ^{2} \theta \cot \theta+2 \sin \theta \cos \theta$ $=\frac{\sin ^{3} \theta}{\cos \theta}+\frac{\cos ^{3} \theta}{\sin \theta}+2 \sin \theta \cos \theta$ $=\frac{\sin ^{4} \theta+\cos ^{4} \theta+2 \sin ^{2} \theta \cos ^{2} \theta}{\sin \theta \cos \theta}$ $=\frac{\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}}{\sin \theta \cos \theta}$ $=\frac{1}{\sin \theta \cos \theta} \quad\left(\sin ^{2} ...
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Question: $\frac{\operatorname{cosec} A-\sin A}{\operatorname{cosec} A+\sin A}=\frac{\sec ^{2} A-\tan ^{2} A}{\sec ^{2} A+\tan ^{2} A}$ Solution: $\frac{\operatorname{cosec} A-\sin A}{\operatorname{cosec} A+\sin A}$ $=\frac{\frac{1}{\sin A}-\sin A}{\frac{1}{\sin A}+\sin A}$ $=\frac{\frac{1-\sin ^{2} A}{\sin A}}{\frac{1+\sin ^{2} A}{\sin A}}$ $=\frac{1-\sin ^{2} A}{1+\sin ^{2} A}$ $=\frac{\frac{1-\sin ^{2} A}{\cos ^{2} A}}{\frac{1+\sin ^{2} A}{\cos ^{2} A}}$ (Dividing numerator and denominator by...
Read More →If I is the identity matrix and A is a square matrix
Question: If $I$ is the identity matrix and $A$ is a square matrix such that $A^{2}=A$, then what is the value of $(I+A)^{2}=3 A$ ? Solution: Given: A is a square matrix, such that $A^{2}=A$. Here, $(I+A)^{2}-3 A=(I+A)(I+A)-3 A$ $\Rightarrow(I+A)^{2}-3 A=I \times I+I \times A+A \times I+A \times A-3 A \quad$ (using distributive property) $\Rightarrow(I+A)^{2}-3 A=I+A+A+A^{2}-3 A \quad($ using $I \times I=I$ and $I A=A I=A)$ $\Rightarrow(I+A)^{2}-3 A=I+2 A+A-3 A \quad\left(\because A^{2}=A\right)...
Read More →Find the smallest number which must be added to 2300
Question: Find the smallest number which must be added to 2300 so that it becomes a perfect square. Solution: To find the square root of 2300, we use the long division method: 23000 is 4 (704 700) less than 482. Hence, 4 must be added to 2300 to get a perfect square....
Read More →Find the greatest number of three digits which is a perfect square.
Question: Find the greatest number of three digits which is a perfect square. Solution: The greatest number with three digits is 999. To find the greatest perfect square with three digits, we must find the smallest number that must be subtracted from 999 in order to get a perfect square. For that, we have to find the square root by the long division method as shown below:So, 38 must be subtracted from 999 to get a perfect square. 999 38 = 961 961 = 312 Hence, the greatest perfect square with thr...
Read More →Prove that
Question: $\frac{\sin \theta}{(\cot \theta+\operatorname{cosec} \theta)}-\frac{\sin \theta}{(\cot \theta-\operatorname{cosec} \theta)}=2$ Solution: $\mathrm{LHS}=\frac{\sin \theta}{(\cot \theta+\operatorname{cosec} \theta)}-\frac{\sin \theta}{(\cot \theta-\operatorname{cosec} \theta)}$ $=\sin \theta\left\{\frac{(\cot \theta-\operatorname{cosec} \theta)-(\cot \theta+\operatorname{cosec} \theta)}{(\cot \theta+\operatorname{cosec} \theta)(\cot \theta-\operatorname{cosec} \theta)}\right\}$ $=\sin \t...
Read More →The cost of levelling and turfing a square lawn at Rs 2.50 per
Question: The cost of levelling and turfing a square lawn at Rs 2.50 per m2is Rs 13322.50. Find the cost of fencing it at Rs 5 per metre. Solution: First, we have to find the area of the square lawn, which the total cost divided by the cost of levelling and turfing per square metre: Area of a square $=\frac{13322.5}{2.5}=5329 \mathrm{~m}^{2}$ The length of one side of the square is equal to the square root of the area. We will use the long division method to find it as shown below: $\therefore$ ...
Read More →Two objects of masses m1 and m2 having
Question: Two objects of masses $m_{1}$ and $m_{2}$ having the same size are dropped simultaneously from heights h $_{1}$ and $h_{2}-$ respectively. Find out the ratio of time they would take in reaching the ground. Will this ratio remain the same if 1. one of the object is hollow and the other one is solid and 2. both of them are hollow, size remaining the same in each case. Give reason. Solution: $t=\sqrt{\frac{2 h}{g}} \therefore \frac{t_{1}}{t_{2}}=\sqrt{\frac{h_{1}}{h_{2}}} .$ Ratio will be...
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Question: If $\left[\begin{array}{ll}1 2 \\ 3 4\end{array}\right]\left[\begin{array}{ll}3 1 \\ 2 5\end{array}\right]=\left[\begin{array}{ll}7 11 \\ k 23\end{array}\right]$, then write the value of $k$ Solution: Given: $\left[\begin{array}{ll}1 2 \\ 3 4\end{array}\right]\left[\begin{array}{ll}3 1 \\ 2 5\end{array}\right]=\left[\begin{array}{ll}7 11 \\ k 23\end{array}\right]$ $\Rightarrow\left[\begin{array}{ll}3+4 1+10 \\ 9+8 3+20\end{array}\right]=\left[\begin{array}{ll}7 11 \\ k 23\end{array}\ri...
Read More →sec4θ – tan4θ = 1 + 2 tan2θ
Question: sec4 tan4 = 1 + 2 tan2 Solution: $\sec ^{4} \theta-\tan ^{4} \theta$ $=\left(\sec ^{2} \theta\right)^{2}-\left(\tan ^{2} \theta\right)^{2}$ $=\left(\sec ^{2} \theta-\tan ^{2} \theta\right)\left(\sec ^{2} \theta+\tan ^{2} \theta\right) \quad\left[a^{2}-b^{2}=(a-b)(a+b)\right]$ $=1 \times\left(1+\tan ^{2} \theta+\tan ^{2} \theta\right) \quad\left(1+\tan ^{2} \theta=\sec ^{2} \theta\right)$ $=1+2 \tan ^{2} \theta$...
Read More →How does the force of attraction between
Question: How does the force of attraction between two bodies depend upon their masses and distance between them? A student thought that two bricks tied together would fall faster than a single one under the action of gravity. Do you agree with this hypothesis or not? Comment. Solution: The force of attraction between two bodies of masses $m_{1}$ and $m_{2}$ and separated by a distance $r$ is given by $\mathrm{F}=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}$, where $\mathrm{G}$ is constant. This force i...
Read More →The area of a square field is 60025 m
Question: The area of a square field is 60025 m2. A man cycles along its boundary at 18 km/hr. In how much time will he return at the starting point? Solution: Area of the square field = 60025 m2 The length of the square field would be the square root of 60025. Using the long division method:Hence, the length of the square field is 245 m. The square has four sides, so the number of boundaries of the field is 4. The distancescovered by the man = 245 m4 = 980 m = 0.98 km If the velocityvis 18 km/h...
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Question: If $A=\left[\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]$ is identity matrix, then write the value of $\alpha$. Solution: Here, $A=\left[\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]=I$ $\Rightarrow\left[\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$ The corresponding elements of equal matrices are equal. $\there...
Read More →How does the weight of an object vary
Question: How does the weight of an object vary with respect to mass and radius of the earth. In a hypothetical case, if the diameter of the earth becomes half of its present value and its mass becomes four times of its present value, then how would the weight of any object on the surface of the earth be affected? (CBSE 2012) Solution: Weight of an object, $\mathrm{W}=m g=\frac{\mathrm{GM} m}{\mathrm{R}^{2}}\left(\because g=\frac{\mathrm{GM}}{\mathrm{R}^{2}}\right)$ So, weight of an object is di...
Read More →A General arranges his soldiers in rows to form a perfect square.
Question: A General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the number of soldiers in each row. Solution: 60 soldiers are left out. $\therefore$ Remainaing soldiers $=8160-60=8100$ The number of soldiers in each row to form a perfect square would be the square root of 8100. We have to find the square root of 8100 by the long division method as shown below: Hence, the number of soldi...
Read More →If A is a matrix of order 3 × 4 and B is a matrix
Question: IfAis a matrix of order 3 4 andBis a matrix of order 4 3, find the order of the matrix ofAB. Solution: IfAis a matrix of order 3 4 andBis a matrix of order 4 3, then the order of matrixABis given by the number of rows inAand number of columns inB,respectively.Thus, the order of matrixABis33....
Read More →The earth is acted upon by gravitation of sun,
Question: The earth is acted upon by gravitation of sun, even though it does not fall into the sun. Why? (CBSE 2012) Solution: The earth revolves around the sun. The centripetal force is needed by the earth to revolve around the sun. This centripetal force is provided by the gravitational force between the sun and the earth. The earth keeps on moving around the sun as long as gravitational force between the earth and the sun acts on it....
Read More →If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.
Question: IfAis a matrix of order 3 4 andBis a matrix of order 4 3, find the order of the matrix ofAB. Solution: IfAis a matrix of order 3 4 andBis a matrix of order 4 3, then the order of matrixABis given by the number of rows inAand number of columns inB,respectively.Thus, the order of matrixABis33....
Read More →Calculate the average density of earth in
Question: Calculate the average density of earth in terms of $g, G, m$ ? Solution: Average density $=\frac{\text { Mass of earth }}{\text { Volume of earth }}=\frac{\mathrm{M}_{e}}{\frac{4}{3} \pi \mathrm{R}^{3}}$ Now $\quad g=\frac{\mathrm{GM}_{e}}{\mathrm{R}^{2}}$ or $\mathrm{M}_{e}=g \mathrm{R}^{2} / \mathrm{G} \quad \therefore$ Average density of earth $=\frac{g \mathrm{R}^{2} / \mathrm{G}}{\frac{4}{3} \pi \mathrm{R}^{3}}=\frac{3 g}{4 \pi \mathrm{GR}}$...
Read More →Find the greatest number of 4 digits which is a perfect square.
Question: Find the greatest number of 4 digits which is a perfect square. Solution: The greatest number with four digits is 9999. To find the greatest perfect square with four digits, we must find the smallest number that must be subtracted from 9999 in order to make a perfect square. For that, we have to find the square root of 9999 by the long division method as shown below:We must subtract 198 from 9999 to make a perfect square: 9999 198 = 9801 Hence, the greatest perfect square with four dig...
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Question: If $\left[\begin{array}{ll}a+b 2 \\ 5 b\end{array}\right]=\left[\begin{array}{ll}6 5 \\ 2 2\end{array}\right]$, then find $a$ Solution: The corresponding elements of two equal matrices are equal. $\Rightarrow\left[\begin{array}{cc}a+b 2 \\ 5 b\end{array}\right]=\left[\begin{array}{ll}6 5 \\ 2 2\end{array}\right]$ $\Rightarrow a+b=6$ ....(1) $\therefore b=2$ Putting the value of $b$ in eq. (1) $a+2=6$ $\Rightarrow a=6-2$ $\therefore a=4$...
Read More →The weight of any person on the moon is about
Question: The weight of any person on the moon is about $1 / 6$ times that on the earth. He can lift a mass of $15 \mathrm{~kg}$ on the earth. Whatwill be the maximum mass, which can be lifted by the same force applied by the person on the moon? (CBSE Sample Paper) Solution: Weight on moon $=\frac{1}{6} \times$ Weight on earth ie., $m g_{m}=\frac{1}{6} m g_{e}$ or $g_{m}=\frac{g_{e}}{6}$ Force applied by person to lift $15 \mathrm{~kg}$ mass on earth, $\mathrm{F}=m g_{e}=\left(15 \mathrm{~g}_{e}...
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Question: If $\left[\begin{array}{ll}a+b 2 \\ 5 b\end{array}\right]=\left[\begin{array}{ll}6 5 \\ 2 2\end{array}\right]$, then find $a$ Solution: The corresponding elements of two equal matrices are equal. $\Rightarrow\left[\begin{array}{cc}a+b 2 \\ 5 b\end{array}\right]=\left[\begin{array}{ll}6 5 \\ 2 2\end{array}\right]$ $\Rightarrow a+b=6$ ....(1) $\therefore b=2$ Putting the value of $b$ in eq. (1) $a+2=6$ $\Rightarrow a=6-2$ $\therefore a=4$...
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