Class 12

### Find the shortest distance between the given lines. Answer Given equations: r¯=(ı^+ȷ^)+λ(2ı^-ȷ^+k^) \overline{\mathrm{r}}=(\hat{\imath}+\hat{\jmath})+\lambda(2 \hat{\imath}-\hat{\jmath}+\hat{\mathrm{k}})...

### Find the shortest distance between the given lines. Answer Given equations: r¯=(ı^+ȷ^)+λ(2ı^-ȷ^+k^) \overline{\mathrm{r}}=(\hat{\imath}+\hat{\jmath})+\lambda(2 \hat{\imath}-\hat{\jmath}+\hat{\mathrm{k}})...

### If A(1, 2, 3), B(4, 5, 7), C(-4, 3, -6) and D(2, 9, 2) are four given points then find the angle between the lines AB and CD.

Answer Given - A = (1,2,3) B = (4,5,7) C = (-4,3,-6) D = (2,9,2) Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by ((a’-a),(b’-b),(c’-c))...

### Show that the lines x = – y = 2z and x + 2 = 2y – 1 = – z + 1 are perpendicular to each other.

To prove – The lines are perpendicular to each other Direction ratios of L1 = (2,-2,1) Direction ratios of L2 = (2,1,-2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be...

### prove that

To find – The value of λ Direction ratios of L1 = (-3,2λ,2) Direction ratios of L2 = (3λ,1,-5) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of the second, then...

To prove – The lines are perpendicular to each other Direction ratios of L1 = (2,-3,4) Direction ratios of L2 = (2,4,2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (-3,-2,0) Direction ratios of L2 = (1,-3,2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (1,0,-1) Direction ratios of L2 = (3,4,5) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (2,1,-3) Direction ratios of L2 = (3,2,-1) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (4,3,5) Direction ratios of L2 = (1,-1,1) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines

To find – Angle between the two pair of lines Direction ratios of L1 = (1,1,2) Direction ratios of L2 = (3,5,4) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (2,-2,1) Direction ratios of L2 = (1,2,-2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines

Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of the second, then the angle between these pair of lines is given by

### The position vectors of three points A, B and C are

It can thus be written as: A = (-4,2,-3) B = (1,3,-2) C = (-9,1,-4) To prove – A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line...

### Find the values of λ and μ so that the points A(-1, 4, -2), B(λ, μ 1) and C(0, 2, -1) are collinear.

Answer Given - A = (-1,4,-2) B = (λ,μ,1) C = (0,2,-1) To find – The value of λ and μ so that A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios...

### Find the values of λ and μ so that the points A(3, 2, -4), B(9, 8, -10) and C(λ, μ -6) are collinear.

Answer Given - A = (3,2,-4) B = (9,8,-10) C = (λ,μ,-6) To find – The value of λ and μ so that A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction...

### Find the value of λ for which the points A(2, 5, 1), B(1, 2, -1) and C(3, λ, 3) are collinear.

Answer Given - A = (2,5,1) B = (1,2,-1) C = (3,λ,3) To find – The value of λ so that A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the...

### Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.

Answer Given - A = (2,3,-4) B = (1,-2,3) C = (3,8,-11) To prove – A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by...

### Show that the points A(2, 1, 3), B(5, 0, 5) and C(-4, 3, -1) are collinear.

Answer Given - A = (2,1,3) B = (5,0,5) C = (-4,3,-1) To prove – A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by...

### Find the image of the point (2, -1, 5) in the line

Find the image of the point $(2,-1,5)$ in the line r→=(11i^-2j^-8k^)+λ(10i^-4j^-11k^) \overrightarrow{\mathrm{r}}=(11 \hat{i}-2 \hat{j}-8 \hat{\mathrm{k}})+\lambda(10 \hat{i}-4...

### Find the image of the point (5, 9, 3) in the line

Find the image of the point $(5,9,3)$ in the line $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}=2}{3}=\frac{\mathrm{z}-3}{4}$. Answer Given: Equation of line is...

### Find the image of the point (0, 2, 3) in the line

Find the image of the point $(0,2,3)$ in the line $\frac{\mathrm{x}+3}{5}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}+4}{3}$. Answer Given: Equation of line is...

### Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).

Find the coordinates of the foot of the perpendicular drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$ Answer Given: perpendicular drawn from point...

### Find the coordinates of the foot of the perpendicular drawn from the point A(1, 2, 1) to the line joining the points B(1, 4, 6) and C(5, 4, 4).

Find the coordinates of the foot of the perpendicular drawn from the point $A(1,2,1)$ to the line joining the points $B(1,4,6)$ and $C(5,4,4)$. Answer Given: perpendicular drawn from point...

### Find the vector equation of a line passing through the point having the position vector

Find the vector equation of a line passing through the point having the position vector $(\hat{i}+2 \hat{j}-3 \hat{k})$ and parallel to the line joining the points with position vectors...

### Find the vector equation of a line passing through the point A(3, -2, 1) and parallel to the line joining the points B(-2, 4, 2) and C(2, 3, 3). Also, find the Cartesian equations of the line.

Find the vector equation of a line passing through the point $A(3,-2,1)$ and parallel to the line joining the points $\mathrm{B}(-2,4,2)$ and $\mathrm{C}(2,3,3) .$ Also, find the Cartesian equations...

### Find the vector and Cartesian equations of the line joining the points whose position vectors are

Find the vector and Cartesian equations of the line joining the points whose position vectors are $(\hat{i}-2 \hat{j}+\hat{k})$ and $(\hat{i}+3 \hat{j}-2 \hat{k})$ Answer Given: line passes through...

### Find the vector and Cartesian equations of the line passing through the points A(2, -3, 0) and B(-2, 4, 3).

Find the vector and Cartesian equations of the line passing through the points $A(2,-3,0)$ and $B(-2,4,$, 3). Answer Given: line passes through the points $(2,-3,0)$ and $(-2,4,3)$ To find: equation...

### Find the vector and Cartesian equations of the line passing through the points A(3, 4, -6) and B(5, -2, 7).

Find the equations of the line passing through the point $(1,-2,3)$ and parallel to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-2}{-4}=\frac{\mathrm{Z}+7}{5}$. Also find the vector form of...

### Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line

Find the coordinates of the foot of the perpendicular drawn from the point $(1,2,3)$ to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}$. Also, find the length of the...

### Show that the lines and do not intersect each other.

Show that the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{3}=z$ and $\frac{\mathrm{x}+1}{5}=\frac{\mathrm{y}-2}{1}, Z=2$ do not intersect each other. Answer Given: The equations of the two...

### Show that the lines and intersect each other. Also, find the point of their intersection.

Show that the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{Z}-3}{4}$ and $\frac{\mathrm{x}-4}{5}=\frac{\mathrm{y}-1}{2}=\mathrm{Z}$ intersect each other. Also, find the point...

### Prove that the lines

Prove that the lines $\frac{\mathrm{x}-4}{1}=\frac{\mathrm{y}+3}{4}=\frac{\mathrm{z}+1}{7}$ and $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-3}=\frac{\mathrm{z}+10}{8}$ intersect each other and find...

### Find the Cartesian and vector equations of the line passing through the point (1, 2, -4) and

Find the Cartesian and vector equations of the line passing through the point $(1,2,-4)$ and perpendicular to each of the lines...

### Find the equations of the line passing through the point (-1, 3, -2) and perpendicular to each of the lines and

Find the equations of the line passing through the point $(-1,3,-2)$ and perpendicular to each of the lines $\frac{\mathrm{X}}{1}=\frac{\mathrm{y}}{2}=\frac{\mathrm{Z}}{3}$ and...

### Find the Cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line

Find the Cartesian and vector equations of a line which passes through the point $(1,2,3)$ and is parallel to the line $\frac{-\mathrm{X}-2}{1}=\frac{\mathrm{y}+3}{7}=\frac{2 Z-6}{3}$ Answer Given:...

### Find the equations of the line passing through the point (1, -2, 3) and parallel to the line Also find the vector form of this equation so obtained.

Find the equations of the line passing through the point $(1,-2,3)$ and parallel to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-2}{-4}=\frac{\mathrm{Z}+7}{5}$. Also find the vector form of...

### Find the Cartesian equations of the line which passes through the point (1, 3, -2) and is parallel to the line Also, find the vector form of the equations so obtained.

Find the Cartesian equations of the line which passes through the point $(1,3,-2)$ and is parallel to the line given by $\frac{\mathrm{x}+1}{3}=\frac{\mathrm{y}-4}{5}=\frac{\mathrm{z}+3}{-6}$. Also,...

### The Cartesian equations of a line are 3x + 1 = 6y – 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its vector equation.

The Cartesian equations of a line are $3 x+1=6 y-2=1-z$. Find the fixed point through which it passes, its direction ratios and also its vector equation. Answer Given: Cartesian equation of line are...

### The Cartesian equations of a line Find the vector equation of the line.

The Cartesian equations of a line are $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}+2}{-5}=\frac{\mathrm{Z}-6}{4}$. Find the vector equation of the line. Answer Given: Cartesian equation of line...

### A line is drawn in the direction and it passes through a point with position vector Find the equations of the line in the vector as well as Cartesian forms.

A line is drawn in the direction of $(\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and it passes through a point with position vector $(2 \hat{i}-\hat{j}-4 \hat{k}) .$ Find the equations...

Find the vector equation of the line passing through the point with position vector $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ and parallel to the vector $(\hat{i}+3... read more ### A line passes through the point (2, 1, -3) and is parallel to the vector Find the equations of the line in vector and Cartesian forms. A line passes through the point$(2,1,-3)$and is parallel to the vector$(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$. Find the equations of the line in vector and Cartesian forms.... read more ### A line passes through the point (3, 4, 5) and is parallel to the vector Find the equations of the line in the vector as well as Cartesian forms. Answer Given: line passes through point$(3,4,5)$and is parallel to$2 \hat{\imath}+2 \hat{\jmath}-3 \hat{k}$To find: equation of line in vector and Cartesian forms Formula Used: Equation of a... read more ### For the binary operation ×10 set S = {1, 3, 7, 9}, find the inverse of 3. Answer: 1 ×10 1 = remainder obtained by dividing 1 × 1 by 10 = 1 3 ×10 7 = remainder obtained by dividing 3 × 7 by 10 = 1 7 ×10 9 = remainder obtained by dividing 7 × 9 by 10 = 3 Composition table:... read more ### Construct the composition table for ×5 on set Z5 = {0, 1, 2, 3, 4} Answer: 1 ×5 1 = remainder obtained by dividing 1 × 1 by 5 = 1 3 ×5 4 = remainder obtained by dividing 3 × 4 by 5 = 2 4 ×5 4 = remainder obtained by dividing 4 × 4 by 5 = 1 Composition table: ×5 0 1... read more ### Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}. Answer: 1 ×6 1 = remainder obtained by dividing 1 × 1 by 6 = 1 3 ×6 4 = remainder obtained by dividing 3 × 4 by 6 = 0 4 ×6 5 = remainder obtained by dividing 4 × 5 by 6 = 2 Composition table: ×6 0 1... read more ### Construct the composition table for +5 on set S = {0, 1, 2, 3, 4} Answer: 1 +5 1 = remainder obtained by dividing 1 + 1 by 5 = 2 3 +5 1 = remainder obtained by dividing 3 + 1 by 5 = 2 4 +5 1 = remainder obtained by dividing 4 + 1 by 5 = 3 Composition Table: +5 0 1... read more ### Construct the composition table for ×4 on set S = {0, 1, 2, 3}. Answer: Given, ×4 on set S = {0, 1, 2, 3} 1 ×4 1 = remainder obtained by dividing 1 × 1 by 4 = 1 0 ×4 1 = remainder obtained by dividing 0 × 1 by 4 = 0 2 ×4 3 = remainder obtained by dividing 2 × 3... read more ### Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. (i) Show that * is both commutative and associative. (ii) Find the identity element in Z Answers: (i) Consider, a, b ∈ Z a * b = a + b – 4 = b + a – 4 = b * a a * b = b * a, ∀ a, b ∈ Z Then, * is commutative on Z. a * (b * c) = a * (b + c – 4) = a + b + c -4 – 4 = a + b + c – 8 (a * b)... read more ### Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by a * b = (3ab/5) for all a, b ∈ Q0. Show that * is commutative as well as associative. Also, find its identity element, if it exists. Answer: Consider, a, b ∈ Q0 a * b = (3ab/5) = (3ba/5) = b * a a * b = b * a, for all a, b ∈ Q0 a * (b * c) = a * (3bc/5) = [a (3 bc/5)] /5 = 3 abc/25 (a * b) * c = (3 ab/5) * c = [(3 ab/5)... read more ### Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, (i) Show that * is both commutative and associative on Q – {-1} (ii) Find the identity element in Q – {-1} Answers: (i) Consider, a, b ∈ Q – {-1} a * b = a + b + ab = b + a + ba = b * a a * b = b * a, ∀ a, b ∈ Q – {-1} a * (b * c) = a * (b + c + b c) = a + (b + c + b c) + a (b + c + b c) = a + b +... read more ### Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, Show that every element of Q – {-1} is invertible. Also, find inverse of an arbitrary element. Answer: Consider, a ∈ Q – {-1} and b ∈ Q – {-1} be the inverse of a. a * b = e = b * a a * b = e and b * a = e a + b + ab = 0 and b + a + ba = 0 b (1 + a) = – a Q – {-1} b = -a/1 + a Q – {-1}... read more ### Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation ‘O’ is defined on A as follows: (a, b) O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R. (i) Show that ‘O’ is commutative and associative on A (ii) Find the identity element in A Answers: (i) Consider, X = (a, b) Y = (c, d) ∈ A, ∀ a, c ∈ R0 b, d ∈ R X O Y = (ac, bc + d) Y O X = (ca, da + b) X O Y = Y O X, ∀ X, Y ∈ A O is not commutative on A. X = (a, b) Y = (c, d) a Z = (e,... read more ### Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation ‘O’ is defined on A as follows: (a, b) O (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R. Find the invertible element in A. Answer: Consider, F = (m, n) be the inverse in A ∀ m ∈ R0 and n ∈ R X O F = E F O X = E (am, bm + n) = (1, 0) and (ma, na + b) = (1, 0) Considering (am, bm + n) = (1, 0) am = 1 m = 1/a And bm + n =... read more ### Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z. Find the invertible element in Z. Answer: Consider, a ∈ Z and b ∈ Z be the inverse of a. a * b = e = b * a a * b = e and b * a = e a + b – 4 = 4 and b + a – 4 = 4 b = 8 – a ∈ Z Hence, 8 – a is the inverse of a ∈... read more ### Find the identity element in the set of all rational numbers except – 1 with respect to * defined by a * b = a + b + ab Answer: Consider, e be the identity element in I+ with respect to * such that a * e = a = e * a, ∀ a ∈ Q – {-1} a * e = a and e * a = a, ∀ a ∈ Q – {-1} a + e + ae = a and e + a + ea = a, ∀ a ∈ Q –... read more ### Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+. Answer: Consider, e be the identity element in I+ with respect to * a * e = a = e * a, ∀ a ∈ I+ a * e = a and e * a = a, ∀ a ∈ I+ a + e = a and e + a = a, ∀ a ∈ I+ e = 0, ∀ a ∈ I+ Hence, 0 is the... read more Let$I=\int _{\pi /6}^{\pi /2}\frac{\cos ecx\cot x}{1+\cos e{{c}^{2}}x}dx=\int _{\pi /6}^{\pi /2}\frac{\cos x}{1+{{\sin }^{2}}x}dx$Let$\sin x=t\Rightarrow \cos xdx=dt$Also, when... read more Let$I=\int _{1}^{2}\frac{1}{x{{(1+{{\log }_{e}}x)}^{2}}}dx$Let$1+{{\log }_{e}}x=t\Rightarrow \frac{1}{x}dx=dt$Also, when$x=1,t=1$and when$x=2t=1+{{\log }_{e}}2$Hence,$I=\int...

Let $I=\int _{0}^{{{\left( \frac{\pi }{2} \right)}^{\frac{1}{3}}}}{{x}^{2}}\sin \left( {{x}^{3}} \right)dx$ Let ${{x}^{3}}=t$ $\Rightarrow 3{{x}^{2}}=dt$ Also, when $x=0,t=0$and when $t=\frac{\pi... read more Let$I=\int _{\pi /4}^{\pi /2}\frac{\cos x}{{{\left( \cos \left( \frac{x}{2} \right)+\sin \left( \frac{x}{2} \right) \right)}^{3}}}dx\cos x={{\cos }^{2}}\left( \frac{x}{2} \right)-{{\sin...

Let $I=\int _{2}^{3}\frac{2-x}{\sqrt{5x-6-{{x}^{2}}}}dx$ Let $2-x=\frac{ad}{dx}\left( 5x-6-{{x}^{2}} \right)+b$ $=-2ax+5a+b$ Hence $-2a=-1$and $5a+b=2$ Solving these equations, We get and...

Let $I=\int _{0}^{\pi /2}\left( \sqrt{\tan x}+\sqrt{\cos tx} \right)dx=\int _{0}^{\pi /2}\frac{\sin x+\cos x}{\sqrt{\sin x\cos x}}dx$ Let $\sin x-\cos x=t$ $\Rightarrow \left( \cos x+\sin x... read more Let$I=\int _{0}^{1}\frac{1-{{x}^{2}}}{{{(1+{{x}^{2}})}^{2}}}dx$Let$x=\tan t\Rightarrow dx={{\sec }^{2}}tdt$Also, when$x=0,t=0$and when$x=1,t=\frac{\pi }{4}$Hence,$I'=\int _{0}^{\pi...

Let $I=\int _{0}^{1}{{x}^{3}}\sqrt{1+3{{x}^{4}}}dx$ Let $1+3{{x}^{4}}=t$ $\Rightarrow 12{{x}^{3}}dx=dt$ Also, when $x=0,t=1$and when $x=1,t=4$ $I=\frac{1}{12}\int _{1}^{4}\sqrt{t}dt$...

Let $I=\int _{0}^{9}\frac{1}{1+\sqrt{x}}$ Let $\sqrt{x}=u$ $\Rightarrow \frac{1}{2\sqrt{x}}=dx=du$ Or $dx=2udu$ $=\frac{1}{2u}dx$ Also, when $x=0,u=0$and $x=9,u=3$ Hence, $I=\int... read more Let$I=\int _{0}^{a}{{\sin }^{-1}}\sqrt{\frac{x}{a+x}}dx$Let$x=a{{\tan }^{2}}y\Rightarrow dx=2a\tan y{{\sec }^{2}}ydy$Also, when$x=0,y=0$And when$x=ay=\frac{\pi }{4}$Hence,$I=\int...

Let $I=\int _{0}^{1}{{\sin }^{-1}}\sqrt{x}dx$ Let $\sqrt{x}=1$ $\Rightarrow \frac{1}{2\sqrt{x}}=dx=dt$ Or $dx=2tdt$ When $x=0,t=0$ And when $x=1,t=1$ Hence, $I=2\int _{0}^{1}t{{\sin }^{-1}}tdt$...

Let $I=\int _{0}^{1}x{{(a{{\tan }^{-1}}x)}^{2}}dx$ Using integration by parts, we get $I=\frac{{{\left( {{\tan }^{-1}}x \right)}^{2}}{{x}^{2}}}{2}|_{0}^{1}-\int _{0}^{1}\frac{2{{\tan... read more Let$I=\int _{0}^{1}{{\left( {{\cos }^{-1}}x \right)}^{2}}dx$Let$x=\cos t\Rightarrow dx=-\sin tdt$Also, when$x=0,t=\frac{\pi }{2}$And when$x=1,t=0$Hence,$I=-\int _{\pi /2}^{0}{{t}^{2}}\sin...

Let $I=\int _{\pi /3}^{\pi /2}\frac{\sqrt{1+\cos x}}{(1-\cos x)\frac{5}{2}}dx$ Using $1+\cos x=2\cos {{x}^{2}}\left( \frac{x}{2} \right)$ $1-\cos x=2{{\sin }^{2}}\left( \frac{x}{2} \right)$ we get...

Let $I=\int _{0}^{\pi /2}\frac{\sin 2x}{{{\sin }^{4}}x+{{\cos }^{4}}x}dx$ Using $\sin 2x=2\sin x\cos x$, we get $=2\int _{0}^{\pi /2}\frac{\tan x{{\sec }^{2}}x}{({{\tan }^{4}}x+1)}dx$ $I=\int... read more Let$I=\int _{0}^{\pi /2}\frac{\sin x\cos x}{{{\cos }^{2}}x+3\cos x+2}dx$Let$\cos x=t\Rightarrow -\sin xdx=dt$Also, when$x=0,t=1$And when$t=0x=\frac{\pi }{2}$Hence,$I=-\int...

Let $I=\int _{0}^{\pi /4}\frac{{{\tan }^{3}}x}{1+\cos 2x}dx$ Using $1+\cos 2x=2{{\cos }^{2}}x$, we get $I=\frac{1}{2}\int _{0}^{\pi /4}{{\tan }^{3}}x{{\sec }^{2}}xdx$ Let $\tan x=t$ $\Rightarrow... read more Let$I=\int _{0}^{\pi }\frac{1}{3+\cos x+2\sin x}dx$Using$\sin x=\frac{2\tan \left( \frac{x}{2} \right)}{1+{{\tan }^{2}}\left( \frac{x}{2} \right)}\cos x=\frac{1-{{\tan }^{2}}\left( \frac{x}{2}...

$f(x)$is continuous in $\left[ a,b \right]$ $\int _{a}^{b}f(x)dx=\underset{n\to \infty }{\mathop{\lim }}\,h\sum _{r=0}^{n-1}f(a+rh)$, where $h=(b-a)/n$ Here $h=\left( b-a \right)/n$ $\int... read more$f\left( x \right)$is continuous in$\left[ 1,3 \right]\int _{a}^{b}f(x)dx=\underset{n\to \infty }{\mathop{\lim }}\,h\sum _{r=0}^{n-1}f(a+rh)$, where$h=(b-a)/n$Here$h=2/n\int _{1}^{3}\left(...

$f\left( x \right)$is continuous in $\left[ 0,2 \right]$ $\int _{a}^{b}f(x)dx=\underset{n\to \infty }{\mathop{\lim }}\,h\sum _{r=0}^{n-1}f(a+rh)$, where $h=(b-a)/n$ Here $h=2/n$ $\int _{0}^{2}\left(... read more Since it is modulus function so we need to break the function and then solve it$f\left( x \right)=\int _{0}^{1/3}\left( 1-3x \right)dx+\int _{1/3}^{1}(3x-1)dx$it is continuous in$\left[ 0,1...

$f\left( x \right)$is continuous in $\left[ 0,3 \right]$ $\int _{a}^{b}f(x)dx=\underset{n\to \infty }{\mathop{\lim }}\,h\sum _{r=0}^{n-1}f(a+rh)$, where $h=(b-a)/n$ Here $h=3/n$ $\int _{0}^{3}\left(... read more$f\left( x \right)$is continuous in$\left[ 0,2 \right]\int _{a}^{b}f(x)dx=\underset{n\to \infty }{\mathop{\lim }}\,h\sum _{r=0}^{n-1}f(a+rh)$, where$h=(b-a)/n$Here$h=2/n\int _{0}^{2}\left(...

### Form the differential equation of the family of curves where and are arbitrary constants.

Equation of the family of curves, $x=A \cos n t+B \sin n t$, where $A$ and $B$ are arbitrary constants. Differentiating the above equation with respect to $t$ on both sides, we have,...

### Mark the tick against the correct answer in the following: A. B. C. D. Solution: Option(C) To find: Value of $\left|\begin{array}{lll}b+c & a & b \\ c+a & c & a \\ a+b & b & c\end{array}\right|$ We have,...

### If the plane 2x – 3y – 6z = 13 makes an angle with the x-axis, then find the value of λ.

Answer: Direction ratios of the x-axis are 1,0,0 and direction ratios of normal to the plane are 2,−3,−6. The angle between the line and the plane,...

### Find the angle between the line and the plane 10x + 2y – 11z = 3.

Answer: Direction ratios of the given line are 2,3,6 Direction ratios of the normal to the given plane are 10,2,−11 The angle between the line and the plane: \begin{array}{l} \sin...

### Mark the tick against the correct answer in the following: A. B. C. 0 D. none of these

Solution: Option(B) To find: Value of $\left|\begin{array}{ccc}a & a+2 b & a+2 b+3 c \\ 3 a & 4 a+6 b & 5 a+7 b+9 c \\ 6 a & 9 a+12 b & 11 a+15 b+18 c\end{array}\right|$ We...

### Find the angle between the line and the plane Answer: The equation of line is: $\vec{r}=(i+2 \hat{j}-\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ Here, $\vec{b}=\hat{i}-j+\vec{k}$ The equation of plane is: r→·(2i^-j^+k^)=4 \vec{r}...

### Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Equation of the family of circles in the second quadrant and touching the coordinate axes can be represented by $(x-(-a))^{2}+(y-a)^{2}=a^{2}$, where $a$ is an arbitrary constants....

### Mark the tick against the correct answer in the following: A. 2 B. 6 C. 24 D. 120

Solution: Option(24) To find: Value of $\left|\begin{array}{lll}1 ! & 2 ! & 3 ! \\ 2 ! & 3 ! & 4 ! \\ 3 ! & 4 ! & 5 !\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following: A. 8 B. C. 16 D. 142

Solution: Option(A) To find: Value of $\left|\begin{array}{lll}1^{2} & 2^{2} & 3^{2} \\ 2^{2} & 3^{2} & 4^{2} \\ 3^{2} & 4^{2} & 5^{2}\end{array}\right|$ We have,...

### Mark the tick against the correct answer in the following: If is a complex cube root of unity then the value of is A. 2 B. 4 C. 0 D. Solution: Option(0) To find: Value of $\left|\begin{array}{ccc}1 & \omega & 1+\omega \\ 1+\omega & 1 & \omega \\ \omega & 1+\omega & 1\end{array}\right|$ Formula used: (i)...

### Mark the tick against the correct answer in the following: If is a complex root of unity then A. 1 B. C. 0 D. none of these

Solution: Option(C) To find: Value of $\left|\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right|$ We have,...

### Form the differential equation of the family of concentric circles x^{2} + y^{2}=a^{2}, where a>0 and a is a parameter.

X^2+y^2=a^2 Differentiating w.r.t x, 2x+2ydx/dy=0. Now as a is eliminated so this is the differential equation of the family of concentric circles. ⇒dx/dy=−y/x

### Form the differential equation of the family of straight lines where and are arbitrary constants.
The equation of a straight line is represented as, $Y=m x+c$ Differentiating the above equation with respect to $x$, $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{m}$ Differentiating the above equation...