If sin α + sin β=a and cos α−cos β=b
Question: If $\sin \alpha+\sin \beta=a$ and $\cos \alpha-\cos \beta=b$ then $\tan \frac{\alpha-\beta}{2}=$ (a) $-\frac{a}{b}$ (b) $-\frac{b}{a}$ (c) $\sqrt{a^{2}+b^{2}}$ (d) none of these Solution: (b) $-\frac{b}{a}$ Given: $\sin \alpha+\sin \beta=a$ $\Rightarrow 2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=a \quad \ldots(1)$ Also, $\cos \alpha+\cos \beta=b$ $\Rightarrow-2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}=b \quad \ldots(2)$ On dividing $(1)$ by $(2)$, we get $...
Read More →Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Question: Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes: TWO HEADS: 95 times ONE HEADS: 290 times NO HEADS: 115 times Find the probability of occurrence of each of these events Solution: Probability (E) $=\frac{\text { Number of trialsin which events happen }}{\text { Total no of trials }}$ P(Getting two heads)= 95/500 = 0.19 P(Getting one tail)= 290/500 = 0.58 P(Getting no head) = 115/500 = 0.23...
Read More →A coin is tossed 1000 times with the following sequence:
Question: A coin is tossed 1000 times with the following sequence: Head: 455, Tail = 545. Compute the probability of each event Solution: It is given that the coin is tossed 1000 times. The number of trials is 1000 Let us denote the event of getting head and of getting tails be E and F respectively. Then Number of trials in which the E happens = 455 So, Probability of $\mathrm{E}=\frac{\text { Number of even theads }}{\text { Total no of trials }}$ i.e. $P(E)=\frac{455}{1000}=0.455$ Similarity, ...
Read More →If tan α=
Question: If $\tan \alpha=\frac{1-\cos \beta}{\sin \beta}$, then (a) $\tan 3 \alpha=\tan 2 \beta$ (b) $\tan 2 \alpha=\tan \beta$ (c) $\tan 2 \beta=\tan \alpha$ (d) none of these Solution: (b) $\tan 2 \alpha=\tan \beta$ $\tan \alpha=\frac{1-\cos \beta}{\sin \beta}$ $=\frac{2 \sin ^{2} \frac{\beta}{2}}{2 \sin \frac{\beta}{2} \cos \frac{\beta}{2}}$ $=\frac{\sin \frac{\beta}{2}}{\cos \frac{\beta}{2}}$ $\Rightarrow \tan \alpha=\tan \frac{\beta}{2}$ $\Rightarrow \alpha=\frac{\beta}{2}$ $\Rightarrow 2 ...
Read More →If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines,
Question: If $I_{1}, m_{1}, n_{1}$ and $I_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_{1} n_{2}$ $m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1}$. Solution: It is given thatl1,m1,n1andl2,m2,n2are the direction cosines of two mutually perpendicular lines. Therefore, $l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=0$ ...(1) $l_{1}^{2}+m_{1}^{2}+n_{1}^{2}=1$ ...(2) $l_{2}^{2}+...
Read More →If 2 tan α=3 tan β,
Question: If $2 \tan \alpha=3 \tan \beta$, then $\tan (\alpha-\beta)=$ (a) $\frac{\sin 2 \beta}{5-\cos 2 \beta}$ (b) $\frac{\cos 2 \beta}{5-\cos 2 \beta}$ (c) $\frac{\sin 2 \beta}{5+\cos 2 \beta}$ (d) none of these Solution: (a) $\frac{\sin 2 \beta}{5-\cos 2 \beta}$ Given: $2 \tan \alpha=3 \tan \beta$ Now, $\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}$ $=\frac{\frac{3}{2} \tan \beta-\tan \beta}{1+\left(\frac{3}{2} \tan \beta\right) \tan \beta}$ $=\frac{3 \tan \beta...
Read More →If cos x=
Question: If $\cos x=\frac{1}{2}\left(a+\frac{1}{a}\right)$, and $\cos 3 x=\lambda\left(a^{3}+\frac{1}{a^{3}}\right)$, then $\lambda=$ (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) 1 (d) none of these Solution: (b) $\frac{1}{2}$ Given: $\cos x=\frac{1}{2}\left(a+\frac{1}{a}\right)$ $\cos 3 x=\lambda\left(a^{3}+\frac{1}{a^{3}}\right)$ Now, $\cos ^{3} x=\frac{1}{8}\left[a^{3}+\frac{1}{a^{3}}+3 a \frac{1}{a}\left(a+\frac{1}{a}\right)\right]$ $\Rightarrow \cos ^{3} x=\frac{1}{8}\left(a^{3}+\frac{1}{a^{3}}...
Read More →Show that the line joining the origin to the point
Question: Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, 1), (4, 3, 1). Solution: Let OA be the line joining the origin, O (0, 0, 0), and the point, A (2, 1, 1). Also, let BC be the line joining the points, B (3, 5, 1) and C (4, 3, 1). The direction ratios of OA are 2, 1, and 1 and of BC are (4 3) = 1, (3 5) = 2, and (1 + 1) = 0 OA is perpendicular to BC, if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$ $\therefore a_{1} a_{2}+...
Read More →In the following cases, find the distance of each of the given points from the corresponding given plane.
Question: In the following cases, find the distance of each of the given points from the corresponding given plane. PointPlane (a) (0, 0, 0) (b) (3, 2, 1) (c) (2, 3, 5) (d) (6, 0, 0) Solution: It is known that the distance between a point,p(x1,y1,z1), and a plane,Ax+By+Cz=D, is given by, $d=\left|\frac{A x_{1}+B y_{1}+C z_{1}-D}{\sqrt{A^{2}+B^{2}+C^{2}}}\right|$ ...(1) (a)The given point is (0, 0, 0) and the plane is $\therefore d=\left|\frac{3 \times 0-4 \times 0+12 \times 0-3}{\sqrt{(3)^{2}+(-...
Read More →Evaluate each of the following
Question: Evaluate each of the following $\sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ}$ Solution: We have, $\sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}-2 \cos ^{2} 90^{\circ}+\frac{1}{24} \cos ^{2} 0^{\circ}$.....(1) Now, $\sin 30^{\circ}=\frac{1}{2}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}, \tan 30^{\circ}=\frac{1}{\sqrt{3}}, \sin 90^{\cir...
Read More →In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
Question: In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ (b) $2 x+y+3 z-2=0$ and $x-2 y+5=0$ (c) $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$ (d) $2 x-y+3 z-1=0$ and $2 x-y+3 z+3=0$ (e) $4 x+8 y+z-8=0$ and $y+z-4=0$ Solution: The direction ratios of normal to the plane, $L_{1}: a_{1} x+b_{1} y+c_{1} z=0$, are $a_{1}, b_{1}, c_{1}$ and $L_{2}: a_{1} x+b_{2}...
Read More →The demand of different shirt sizes, as obtained by a survey, is given below:
Question: The demand of different shirt sizes, as obtained by a survey, is given below: Find the modal shirt sizes, as observed from the survey. Solution: Since, maximum frequency 39 corresponds to the value 39then mode = 39...
Read More →Find the mode of the following data in each case:
Question: Find the mode of the following data in each case: (i) 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18 (ii) 7, 9, 12, 13, 7, 12, 15, 7, 12, 7, 25, 18, 7 Solution: (i) Arranging the numbers in ascending order: 14, 14, 14, 14, 17, 18, 18, 18, 22, 23, 25, 28 Here the observation 14 is having the highest frequency i. e, 4 in given data, so mode = 14. (ii) 7, 9, 12, 13, 7, 12, 15, 7, 12, 7, 25, 18, 7 Since maximum frequency 5 corresponds to the value 7 then mode = 7...
Read More →If in a Δ ABC,
Question: If in a $\Delta A B C, \tan A+\tan B+\tan C=0$, then $\cot A \cot B \cot C=$ (a) 6 (b) 1 (c) $\frac{1}{6}$ (d) none of these Solution: (d) none of these $A B C$ is a triangle. $\therefore A+B+C=\pi$ $\Rightarrow A+B=\pi-C$ $\Rightarrow \tan (\mathrm{A}+\mathrm{B})=\tan (\pi-\mathrm{C})$ $\Rightarrow \frac{\tan A+\tan B}{1-\tan A \tan B}=-\tan C$ $\Rightarrow \tan A+\tan B=-\tan C+\tan A \tan B \tan C$ $\Rightarrow \tan A+\tan B+\tan C=\tan A \tan B \tan C$ $\Rightarrow 0=\tan A \tan B ...
Read More →Find the mode for the following series:
Question: Find the mode for the following series: 7. 5, 7. 3, 7. 2, 7. 2, 7. 4, 7. 7, 7. 7, 7. 5, 7. 3, 7. 2, 7. 6, 7.2 Solution: Since maximum frequency 4 corresponds to the value 7. 2 then mode = 7.2...
Read More →The value of 2 tan
Question: The value of $2 \tan \frac{\pi}{10}+3 \sec \frac{\pi}{10}-4 \cos \frac{\pi}{10}$ is (a) 0 (b) $\sqrt{5}$ (c) 1 (d) none of these Solution: (a) 0 We have, $2 \tan \frac{\pi}{10}+3 \sec \frac{\pi}{10}-4 \cos \frac{\pi}{10}$ $=2 \tan 18^{\circ}+3 \sec 18^{\circ}-4 \cos 18^{\circ}$ $=2 \times \frac{\frac{\sqrt{5}-1}{4}}{\frac{\sqrt{10+2 \sqrt{5}}}{4}}+3 \times \frac{1}{\frac{\sqrt{10+2 \sqrt{5}}}{4}}-4 \times \frac{\sqrt{10+2 \sqrt{5}}}{4}$ $=2 \times \frac{\sqrt{5}-1}{\sqrt{10+2 \sqrt{5}}...
Read More →Evaluate each of the following
Question: Evaluate each of the following $2 \sin ^{2} 30^{\circ}-3 \cos ^{2} 45^{\circ}+\tan ^{2} 60^{\circ}$ Solution: We have to find the following expression $2 \sin ^{2} 30^{\circ}-3 \cos ^{2} 45^{\circ}+\tan ^{2} 60^{\circ}$ Now, $\sin 30^{\circ}=\frac{1}{2}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}, \tan 60^{\circ}=\sqrt{3}$ So by substituting above values in equation (1) We get, $2 \sin ^{2} 30^{\circ}-3 \cos ^{2} 45^{\circ}+\tan ^{2} 60^{\circ}$ $=2 \times\left(\frac{1}{2}\right)^{2}-3 \times\...
Read More →Find out the mode from the following data:
Question: Find out the mode from the following data: 125, 175, 225, 125, 225, 175, 325, 125, 375, 225, 125 Solution: Since maximum frequency 4 corresponds to the value 125 then mode = 125...
Read More →For all real values of x,
Question: For all real values of $x, \cot x-2 \cot 2 x$ is equal to (a) $\tan 2 x$ (b) $\tan x$ (c) $-\cot 3 x$ (d) none of these Solution: (b) $\tan x$ We have, $\cot x-2 \cot 2 x=\cot x-2 \frac{\cot ^{2} x-1}{2 \cot x}$ $=\frac{\cot ^{2} x-\cot ^{2} x+1}{\cot x}$ $=\frac{1}{\cot x}$ $=\tan x$...
Read More →Find the angle between the planes whose vector equations are
Question: Find the angle between the planes whose vector equations are $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=5$ and $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$. Solution: The equations of the given planes are $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=5$ and $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$ It is known that if $\vec{n}_{1}$ and $\vec{n}_{2}$ are normal to the planes, $\vec{r} \cdot \vec{n}_{1}=d_{1}$ and $\vec{r} \cdot \vec{n}_{2}=d_{2}$, then the angle between th...
Read More →Find out the mode of the following marks obtained by 15 students in a class:
Question: Find out the mode of the following marks obtained by 15 students in a class: Marks: 4, 6, 5, 7, 9, 8, 10, 4, 7, 6, 5, 9, 8, 7, 7. Solution: Since maximum frequency corresponds to the value 7 then mode = 7 marks....
Read More →If cos 2x+2 cos x=1 then,
Question: If $\cos 2 x+2 \cos x=1$ then, $\left(2-\cos ^{2} x\right) \sin ^{2} x$ is equal to (a) 1 (b) $-1$ (c) $-\sqrt{5}$ (d) $\sqrt{5}$ Solution: (a) 1 We have, $\cos 2 x+2 \cos x=1$ $\Rightarrow 2 \cos ^{2} x-1+2 \cos x=1$ $\Rightarrow \cos ^{2} x+\cos x-1=0$ $\Rightarrow \cos x=\frac{-1 \pm \sqrt{1^{2}+4}}{2}$ $\Rightarrow \cos x=\frac{-1 \pm \sqrt{5}}{2}$ $\Rightarrow \cos x=\frac{-1+\sqrt{5}}{2}$ Now, $\left(2-\cos ^{2} x\right) \sin ^{2} x=\left[2-\left(\frac{-1+\sqrt{5}}{2}\right)^{2}\...
Read More →Evaluate each of the following
Question: Evaluate each of the following $\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45^{\circ}$ Solution: We have to find the following expression $\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45 \ldots \ldots(1)$ Now, $\tan 30^{\circ}=\frac{1}{\sqrt{3}}, \tan 60^{\circ}=\sqrt{3}, \tan 45^{\circ}=1$ So by substituting above values in equation (1) We get, $\tan ^{2} 30^{\circ}+\tan ^{2} 60^{\circ}+\tan ^{2} 45$ $=\left(\frac{1}{\sqrt{3}}\right)^{2}+(\sqrt{3})^{2}+(1)^{2}$ $=\frac...
Read More →Find the equation of the plane through the line of intersection of the planes
Question: Find the equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$ which is perpendicular to the plane $x-y+z=0$ Solution: The equation of the plane through the intersection of the planes, $x+y+z=1$ and $2 x+3 y+4 z=5$, is $(x+y+z-1)+\lambda(2 x+3 y+4 z-5)=0$ $\Rightarrow(2 \lambda+1) x+(3 \lambda+1) y+(4 \lambda+1) z-(5 \lambda+1)=0$ ...(1) The direction ratios, $a_{1}, b_{1}, c_{1}$, of this plane are $(2 \lambda+1),(3 \lambda+1)$, and $(4 \la...
Read More →The following observation s have been arranged in ascending order.
Question: The following observation s have been arranged in ascending order. If the median of the data is 63, find the value of x: 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95. Solution: Total number of observation in the given data is 10 (even number). So median of this data will be mean of $10 / 2 \mathrm{i}$. $e$, $5^{\text {th }}$ observation and $10 / 2+1$ i. e, 6 the observation. So, Median of data $=\frac{5^{\text {th }} \text { observation }+6^{\text {th }} \text { observation }}{2}$ $\Right...
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