### Optional Mathematics Chapter wise Model question Class 10

# Algebra

**For Q.No. 1(a) **

- Define an algebraic function.
- Write down the period of function defined as y = tan x.
- If f = {(2, 1), (4, 1), (5, 1) } Does $f^{-1}$ exist?
- If (x+2) is a factor of f(x) = $x^3$+8, find f(-2).
- If the 10th term of an A.P. is 100 and the common difference is 10, find the first term.
- If A.M. between 3x+7 and 5x+8 is $\frac{23}{2}$, find the value of x.

**For Q.No. 1(b)**

- What is the minimum value of the cosine function y = cos x?
- Define a composite function of two functions.
- If f(x) = (x-2)×($x^2$+4x+2)+3, write the degree of f(x).
- Find arithmetic mean between the following two terms. -60 and -20
- Check if the following sequences are in G.P. or not. 3, 9, 27, 81, ……..
- Define common ratio in G.P.

**For Q.No. 6(a) **

- If f = (1, 5), (2, 1), (3, 3), (5, 2)} and g = {(1, 3), (2, 1), (3, 2), (5, 5)}, find gof and fog.
- If f(x+2) = 4x+5, find f(x) and (fof) (x)
- If k(x) = $\frac{5x+7}{x-2}$, x≠2, find k-1(x) and k-1(4).
- If 4$x^3$-10$x^2$+25x-20 = (4x-3). Q(x)+R(x). Find the value of Q(x) and R(x).
- Find the quotient and remainder in each of the following cases when f(x) divided by d(x) (use synthetic division method): f(x) = 4$x^3$-7$x^2$+6x-2, d(x) = 3x+2
- If 2$x^3$+4$x^2$+px+7 has a factor of (x+2), find the value of p.

**For Question No. 6(b) **

- Which term of the series is 5+10+15+............+200?
- Find the arithmetic means as indicated, If 6,x,y, and 18 are in A.P., find the value of x and y.
- Find the number of terms in an A.P. which has its first term 16; common difference 4; and the sum, 120.
- How many terms a G.P. must be taken in the series 64+96+144+216+...........so that the sum may be 2059?
- Find the sum of the first 50 natural numbers and 30 even natural numbers.
- Find the common ratio in the following G.P. $\sqrt{2}+\frac{1}{\sqrt{2}}+\frac{1}{2\sqrt{2}}$

**For Question No. 6(c) **

- If f(2x+5) = 10x+7, find $f^{-1}$(x)
- Which quadrant does an inequality y ≥ 0 represent as its half plane?
- Find the quotient and remainder in each of the following cases when f(x) divided by d(x) (use synthetic division method): f(x) = 4$x^3$-7$x^2$+6x-2, d(x) = 3x+2
- Find the three numbers in an A.P. whose sum is 12 and the sum of their squares is 50.
- What will be the point of intersection of the curve f(x) = $x^2$-5x and f(x) = -5?
- Write the boundary line equation of 4x+3y ≤ 12.

**For Question No. 11**

- If f(x) = $\frac{7x+q}{2}$ and g(x) = x+8, (fog)(4) = 24, find the value of q.
- Solve the following polynomial equations: $x^3-4x^2$+x+6 = 0
- The 6th and the 9th terms of an A.P. 23 and 35. Which term is 67?
- The 5th mean between two numbers 7 and 71 is 27. Find the number of A.M’s.
- The first and the second terms of a G.P. are 32 and 8 respectively. Find:

- The first term
- Common ratio
- 10th term

6. The sum of the first 2 terms of a G.P. is 8 and that of the first 4 terms is 80. Find the sum of the first 6 terms.

**For question No. 12 **

- If g(x) = 5x+3 and (gof)(x) = 2x+5 and f(x) is a linear function, find the value of f(x).
- If f(x) 4x-17 and g(x) = $\frac{2x+8}{5}$ and ($fof^{-1}$)(x) = $g^{-1}$(x), find the value of x.
- When the polynomial 2$x^3$+9$x^2$-7x+k, is exactly divisible by 2x+3, find the value of k by using synthetic division.
- Find the values of a and b in each of the following cases: (x+3) and (2x-7) are factors of a$x^2$-bx-21.
- If x+6, 3x, and 2x+9 are in A.P., find the value of x and the next three terms the progression.
- There are three geometric means between a and b. If the first mean and third mean are $\frac{1}{4}$ and $\frac{1}{64}$, find the values of a and b.

**For Question No. 22**

- The fifth and twelve terms of an A.P. are respectively 17 and 45. Find the sum of the first 15 terms of the progression?
- Graph the following system of inequalities and find the vertices of convex polygons if they exist. X+2y≤ 12 and x≥ 6, y ≥0
- 2. In the given diagram, β³ABC is the feasible region. AB is parallel to the Y-axis and the coordinates of B are (5, 10), one of constraints is x+y ≥5, find the remaining constraints and minimize C = 3x+5y.

4. Draw the graphs for the following quadratic functions. Y = $x^2+2x-3$

5. Find the equation of the parabola under the given condition: Vertex at the origin and passing through the point (1, -4)

6. Find the equations of parabola under the following stated conditions. Passing through the points (0, -3), (-1, -1), and (3, 3)

7. Solve the following equation graphically: $x^2-2x-15$ = 0

8. Solve the following equation graphically: Y = $6x^2$-2x-15, y = 4x-3

**Limit and Continuity**

**For Question No. 2(a)**

- What are natural numbers used for?
- Define continuity of a function?
- Write down the point of discontinuity of a function given as h(x) = $\frac{1}{x+5}$.
- What are the starting and ending numbers in each of the number lines?
- Write conditions for continuity of function f(x) at a point x = a, using notations.
- Find the initial and the terminating points of the curve.

**For Question No. 13**

1. Examine the continuity or discontinuity of the following functions at the points mentioned:

f(x) = $\begin{cases}\frac{x^2-7x}{x-7} & when x\# 7 \\3 & when x = 7 \end{cases} at x = 7$

2. A function is defined as follows:

f(x) = $\begin{cases}3+2x & for -3/2\leq<0 \\3-2x & for 0\leq x< 3/2 \\ -3-2x & for x\geq 3/2 \end{cases}$

3. Find the value of n, if f(x) is continuous at x = 2

f(x) = $\begin{cases}\frac{x^2-4}{x-2} & for x \# 2 \\n & for x = 0\end{cases}$

4. Draw the graph of p(x) = $\frac{2}{x-5}$ and find, at which point the function is discontinuous?

**Matrices **

**For Question No. 2(b)**

- Evaluate the determinant of the following matrices: $\begin{bmatrix}2 & 3 \\4 & -5 \end{bmatrix}$
- Factorise $\begin{bmatrix}2x+3y\\4x-7y \end{bmatrix}$
- If D = 4, $D_x$ = $\frac{1}{2}$, $D_y$ = $\frac{1}{4}$ find the values of x and y.
- What is the determinant of a square matrix D = [-5]?
- If a square matrix is given M = $\begin{bmatrix}a & b \\c & d \end{bmatrix}$ then what does $\begin{bmatrix}d & b \\c & a \end{bmatrix}$ denote?
- Find the value of $D_y$ using Cramer’s rule for the equations 2x+3y = 4 and x+2y = 6

**For Question No. 7(a)**

- Which of the following matrices have their inverse? B =$\begin{bmatrix}4 & 10 \\2& 5 \end{bmatrix}$
- Find the inverse matrices of the following matrices if possible: E = $\begin{bmatrix}cos \theta & -sin theta \\ sin \theta & cos \theta \end{bmatrix}$
- Define: i. Singular Matrix ii. Inverse of a matrix
- Find the value of x, when = 0
- For what value of m, the equations 2x-4y = 5 and x+y = m+3 have $D_x$ = 12
- Find the R whose inverse $R^{-1}$=

**For Question No. 7(b)**

1. Solve the following system of equations by using Cramer’s rule:

- 2x+3y = 17
- $\frac{x}{3}-\frac{y}{4}$ = 2

2. If A = $\begin{bmatrix}1 & 2 \\4 & 5 \end{bmatrix}$ , B = $\begin{bmatrix}2 & 3 \\3 & 4 \end{bmatrix}$ Find the determinants of AB.

3. For what value of m, matrix $\begin{bmatrix}m+3 & m \\2& 4 \end{bmatrix}$ has no inverse? Find it.

4. If D = then write the corresponding equations to these values.

5. Find the inverse of a matrix K =

6. Find the value of y from the equations x +3y = 7 and 2x-y = 3 calculating the value of D and $D_y$.

**For Question No. 14**

1. If $\begin{bmatrix}-1 & 2 \\2p & -7 \end{bmatrix}$ and$\begin{bmatrix}q & 2 \\4 & 1 \end{bmatrix}$ are inverse to each other, find the values of o and

2. If A =$\begin{bmatrix}5& 3 \\3 & 2 \end{bmatrix}$ then, show following:

- $(A^{-1})^-1$= A
- $(A^{-1})^T$= $(A^T)^{-1}$
- $A^{-1}$A = A$A^{-1}$ = I

3. Solve the following equation by matrix method:

- $\frac{x}{2}+\frac{y}{2}$ = 7
- $\frac{x}{6}+\frac{y}{4}$= 1

4. If M = $\begin{bmatrix}2 & 4 \\5 & 2 \end{bmatrix}$, N = $\begin{bmatrix}2 & 1 \\3 & 4 \end{bmatrix}$, Find the determinant:

- of 2M +3N
- 3M-2N
- M2-N2

5. One fourth of the sum of two numbers is 7 and five times of the difference of them is 20. Find the numbers matrix method.

6. A cinema hall was full of audiences with a capacity of 120 seats a day. The hall had provision of a balcony and special each had to pay Rs. 40 and Rs. 20, respectively, for a ticket. If the ticket of Rs. 3600 was sold altogether on the day. Find the number of audiences in each i.e. balcony and special using Cramer’s rule.

**Coordinate Geometry **

**For Q.No. 3(a) **

- Write the acute between two lines in each of the following pair of straight lines: 2x+4y = 7 and x+2y = 5
- Find the separate equations of straight lines represented by the following equations: 9$x^2$-25$y^2$ = 0
- Write down the equation of the circle with centre (h, k) and radius r.
- Write the equation of a circle whose ends of a diameter are ($x_1, y_1$) and ($x_2, y_2$).
- Define a homogeneous equation.
- Write the general equation of second degree in x and y.

**For Q. No. 3(b)**

- Define circle with respect to conic section.
- State the condition under which the pair of straight lines a$x^2$+2hxy+$y^2$ = 0 will be coincident.
- Write the name of the locus represented by $x^2+y^2$ =$r^2$.
- Write down the formula to find the angle between two lines y = $m_1$x+$c_1$ and y = $m_2$x+$c_2$.
- Find the separate equations of straight lines represented by the following equations: $x^2-y^2$ +x-y= 0
- Write the acute between two lines in each of the following pair of straight lines: 3x+5y = 8 and 2x+y = 6

**For Q. No. 8(a) **

- Find the acute angle between two lines in each of the following pair of straight lines: 2x+4y= 7 and x+2y = 5
- The pair of straight lines 2x+5y+2 = 0 and kx+y+6 = 0 are parallel.
- Find the value of π when the following equations represents a pair of line perpendicular to each other: (3π+4)$x^2$-48xy-π$^2$y$^2$ = 0
- Find the equations of circles under the following condition. Centre at (4, -1) and through (-2, -3)
- Find the centre and the radius of a circle having equation $x^2+y^2$+2gx+2fy+c = 0.
- Show that the lines 4x-8y+7 = 0 and x-2y+8 = 0 are parallel to each other.

**For Q. No. 8(b)**

- Show the lines x+y = 2 and 2x+2y = 3 are parallel to each other.
- find the equation the straight line which: Passes through the point (4, 3) and parallel to the line 2x+3y+12 = 0
- Find the two separate equation of two lines represented by the following equations: $x^2$-2xy cot2πͺ-$y^2$ = 0
- Find the equations of circles under the following condition. Touching the coordinate axes at (a, 0) and (a, 0).
- Determine the lines represented by $x^2$-2xy+$y^2$-3x+3y-10 = 0
- Draw the conic of parabola.

**For Q. No. 15**

- Find the separate equations of two lines represented by $x^2$-5xy+4$y^2$ = 0. Find the angle between the lines and the points of intersection of the lines.
- Show that the pair of lines, 4$x^2$-9$y^2$ = 0 and 9$x^2$-4$y^2$ = 0 are perpendicular to each other.
- The equation of two diameters of a circle passing through (4, 7) are 2x-3y = 12 and 4x+2y = 10. Determine the equation of the circle.
- Find the equation of a circle that touches the positive X-axis at a distance of 6 units from the origin and cuts off an intercept of 4 units on the Y-axis positively.
- The equation of diagonal PS of a square POST is 3x-5y = 6 and the coordinates of vertex O is O(3, -1). Find the equation of diagonal OT.
- Find the equation of the altitude AD of ABC with vertices A(2, 3), B(-4, 1) and C(2, 0) drawn from the vertex A.

**For Q. No. 23 **

- Find the centre and radius of circles. Passing through the points P(-4, -2), Q(2, 6) and R(2, -2)
- Show that the two circles $x^2+y^2$ = 36 and $x^2+y^2$-12x-16y+84 = 0 touch externally.
- From point A(5, 6). Perpendicular AB is drawn to the line PQ: 2x+3y = 12. Find the equation of AB.
- FInd the equation of the sides of an equilateral triangle whose vertex is (1, 2) and base is y = 0.
- Find the equation of a circle with radius 4 units, whose centre lies on the line 12x+3y = 48 and which touches the line 3x+2y+23 = 0.
- Find the angle between a pair of straight lines represented by a$x^2$+2hxy+b$y^2$ = 0 also derives the condition of the lines being perpendicular and coincident.

**Trigonometry**

**For Q.No. 4(a) **

- Define multiple angles with an example.
- Express cos π in terms of cos $\frac{\theta}{2}$.
- Express each of the following as product form: sin x- sin y
- Define conditional trigonometric identities with an example.
- If Sec $\alpha$ = cosec $\alpha$ , what is the acute value of A?

**For Q. No. 4(b) **

- Find the value of π, $0^o$ ≤ π ≤$90^o$: sec π = cosec π
- Find the value of x in the following diagrams:

3. Express cos 3M-cos2N in the form of a product of sine or cosine.

4. What is meant by angle of elevation?

5. Express cos 10B in term of sin 5B

**For Q.No. 9(a) **

- If sin π = $\frac{\sqrt{3}}{2}$, find the value of sin 3π and cos 3π.
- If cos 2π = $\frac{-1}{2}$ ($p^2$ +$\frac{1}{p^2}$
- Prove that: $\frac{1-cos 2π}{1+cos2 π}$= $tan^2$π

**For Question No. 9(b)**

- Prove that: $\frac{1+sin 2π-cos 2π}{1+sin 2π+cos2π}$ = tanπ
- Prove that: Sin x= 3sin ($\frac{x}{3}$) - 4$sin^2(\frac{x}{3})$
- Prove that: cot( $\frac{π}{2}$+$\frac{\pi^c}{4}$) -tan($\frac{π}{2}$+$\frac{\pi^c}{4})$ =$\frac{2cos \theta }{1+sin \theta}$

**For question No. 9(c) **

- Prove that: $\frac{cos 7A+cos 3A-cos5A-cosA}{sin 7A-sin3A-sin 5A+sinA}$ = cot2A
- If A+B+C = $\pi^c$, prove that: cotA.cotB+cotB.cotC+cotC.cotA-1 = 0
- Express cos $105^o$. Cos15o into sum or difference of sine or cosine. Hence, find its value.

**For Q.No. (16)**

- If 2tanπͺ = 3 tan π, then prove that, tan(πͺ+π) = $\frac{5sin2 π}{5cos 2π-1}$
- Prove that: $\frac{2 sinπ -sin 2π}{2sinπ+sin2π}$ = $tan^2$ $\frac{π}{2}$
- If A+B+C = $\pi^c$, prove that:sinA-sinB-sinC = -4cos $\frac{A}{2}$.sin$\frac{B}{2}$.sin$\frac{C}{2}$
- Solve: 0o≤π≤$360^0$ : 4$cos^2$π+4sinπ = 5
- A pole is divided by a point in the ratio 1:9 from bottom to top. If the two parts of the pole subtend equal angles at 20m away from the foot of the pole, find the height of the pole.

**For Q.No. 17**

- Prove that: $\frac{cos7A+cos3A-cos5A-cosA}{sin7A-sin3A-sin5A+sinA}$ = cot2A
- If A+B+C = $\pi^c$, prove that: $cos^2$A-$cos^2$B+$cos^2$C = -1+2cosA.sinB.sinC
- Solve for π: $0^o$≤π≤$180^o$: Sinπ+sin2π+sin3π = 0
- From a helicopter flying vertically above a straight road, the angles of depressions of two consecutive kilometer stones on the same side are found to be $45^o$ and $60^o$. Find the height of the helicopter.
- The shadow of a pole on the ground level increases by x m when the sun's altitude changes from $60^o$ to $45^o$. If the height of the pole is 50 ft, find the value of x.

**For Q.No. 18 **

- Prove that: (cosB-cosA)$^2$ +(sinA-sinB)$^2$= 4sin$^2$($\frac{A-B}{2}$)
- If A+B+C = $\pi^c$, prove that: $\frac{sin 2A+sin2B+sin2C}{4 cos\frac{A}{2}. cos\frac{B}{2}. cos \frac{C}{2}}$ = 8sin $\frac{A}{2}$ sin $\frac{B}{2}$ sin $\frac{C}{2}$
- Solve: $0^0$≤x≤$360^o$ : $\sqrt{3}$cosx = $\sqrt{3}$-sinx
- From the top and bottom of a tower, the angle of depression of the top of the house and angle of elevation of the house are found to be $60^o$ and $30^0$ respectively. If the height of the building is 20m. Find the height of the tower.
- A tower of a radio station is divided by a point in the ratio 9:1, Form the top if both the parts subtend equal angles at a point on the ground level 600m away from its bottom, find the height of the tower.

**Vectors **

**For Q.No. 5(a) **

- Define scalar product of two vectors.
- Find the dot product of |$\vec{a}$| and |$\vec{b}$| in following cases: |$\vec{a}$| = 2, |$\vec{b}$| = 5, angle between them = $90^o$
- If p = (3, 4) and q = (-1, 6), What is the dot product of p and q?

**For Q.No. 10(a) **

- Find the angles between given pair of vectors: |$\vec{a}$| = 3, |$\vec{b}$| = 5, $\vec{a}$.$\vec{b}$ = 7.5
- In PQR if $\vec{P}$Q = 5 $\vec{i}$-9$\vec{j}$, $\vec{Q}$R = 4$\vec{i}$+14$\vec{j}$ prove that $\triangle$PQR is a right angled triangle.
- The position vectors of A and B are 2 $\vec{a}$- 3$\vec{b}$ and 5$\vec{a}$-4$\vec{b}$ respectively. Find the position vector of the point which divides AB in ratio 3:2 internally.

**For Q.No. 10(b) **

- In ABC, the coordinates of two ends of a median AD are A(2, 5) and D(7, -1) then find the position vector of centroid G of ABC.
- Find the angle between two vectors P and q when p = 6 $\vec{i}$+8$\vec{j}$ and q = 4$\vec{i}$ +3$\vec{j}$
- If the position vector of the mid point of the line segment AB is (3$\vec{i}$-2$\vec{j}$), where the position vector of B is (5$\vec{i}$+2$\vec{j}$), find the position vector of A.

**For Q. No. 24. **

- In the figure O is the centre of the semi-circle. Prove that <PRQ is right angle.

2. In the adjoining figure, PQRS is a parallelogram. G is the point of intersection of the diagonals. If O is any point prove that

$\vec{O}$G = $\frac{1}{4}$ ($\vec{O}$A+$\vec{O}$B + $\vec{O}$C +$\vec{O}$D)

- Prove by vector method that diagonals KN and ID of a rectangle KIND are equal to each other.
- If a triangle XYZ, <XYZ = $90^o$and A is the midpoint of side XZ, prove by vector method that: XA = YA = ZA.

**Transformations **

**For Q.No. 5. (b) **

- Define combined transformation.
- Find the image of P(5, 3) under enlargement E[(1, 1), 2].
- Write down the transformation which is associated with the reflection on Y-axis.
- In a circle with centre ‘O’and radius ‘r’, ‘P’is the inversion point of P. If OP = 2cm and OP’= 18cm, find the value of r.

**For Question No. 19 **

- Let $T_1$ = $\begin{bmatrix}3\\2 \end{bmatrix}$ and $T_2$ =$\begin{bmatrix}1\\2 \end{bmatrix}$ be two translations, find the image of the following points under the combined translations $T_1$ o $T_2$ and $T_2$o$T_1$: A(-6, 7)
- If $T_1$ =$\begin{bmatrix}1\\2 \end{bmatrix}$ and $T_2$ = $\begin{bmatrix}-2\\4 \end{bmatrix}$and $T_1$ o $T_2$ (x, y) = (8, 8) find the values of x and y.
- Find the image of S(7, 5) under combined transformation $R_2$ o $R_1$, where,

- R1 = Rotation of 180o about (2, 3)
- R2 = rotation of +90o about origin.

4. Find the inverse of the point (2, -5) with respect to the circle $x^2+y^2$ = 1

5. A triangle with vertices P(2, 1), Q(5, 3), and R(6, 7) is transformed by the matrix $\begin{bmatrix}0 & -1 \\-1 & 0 \end{bmatrix}$ find the image of the quadrilateral PQRS.

**For Question No. 25**

- Let M(1, 1), N(-3, -6) and (6, -1) be the vertices of MNP. The vertices are transformed by a single transformation obtained by the combination of the rotation [(0, 0) , + $90^o$] and in the same direction of a rotation [(0, 0), +$180^o$]. Find the coordinates of the images of these points.
- On a graph paper draw ABC having the vertices A(5, 4), B(2, 2) and C(5, 2). Find the image of ABC by stating coordinates and graphing them after successive reflections on x-axis following by reflection on the line y = x.
- A triangle having vertices A(2,5), B(-1, 5) and C(4, 1) is rotated through +90o about the origin. The image obtained is reflected on the line x = 0. Find the vertices of image triangles. Show all the triangles in the same paper and also write the single transformation to represent these two transformations.
- Find the image of ABC with the vertices A(2, 1), B(3, 5) and C(5, 4) under enlargement $E_1$ [(0, 0), 3/2] followed by another enlargement $E_2$ [(0, 0), -2]. Draw ABC and its images on the same graph paper.
- M(3, 4) , N(1, 1), and P(4, 1) are the vertices of MNP. find the image of MNP. Under the enlargement with centre (1, -1) and scale factor -2 followed by the rotation about the origin through negative quarter turn. Also show the image on the same graph paper.
- PQR with vertices P(5, 1), Q(12, 4) and R(4, 5) maps onto the P’Q’R’ with the vertices P’(-5, -1) Q’(-12, -4) and R’(-4, -5). Which is the single transformation for this mapping? Also find the 2×2 matrix that represents the transformation.

**Statistics **

**For Q.No. 10(c) **

- Define quartile deviation.
- Calculate the coefficient of quartile deviation from the following: $Q_1$ = 20, $Q_3$ = 40
- In a continuous series, mean = 20, N = 50 and ∑f|m$\vec{x}$| = 208, find the mean deviation and its coefficient.
- If the standard deviation of a data is 0.35, find its variance.

**For Question No. 20**

1. Calculate the quartile deviation and its coefficient from the following data:

Volume of water (l) |
5-10 |
10-15 |
15-20 |
20-25 |
25-30 |
30-40 |
---|---|---|---|---|---|---|

No. of families |
4 |
12 |
16 |
6 |
2 |
1 |

2. Calculate quartile deviation and its coefficient from the following data:

Income (Rs) |
Below 50 |
50-70 |
70-90 |
90-110 |
110-130 |
130-150 |
150 above |

workers |
5 |
10 |
20 |
25 |
18 |
12 |
10 |

3. Calculate the quartile deviation and its coefficient from the given data:

Class interval |
20-29 |
30-39 |
40-49 |
50-59 |
60-69 |
70-79 |

frequency |
100 |
80 |
75 |
95 |
70 |
40 |

4. Calculate the mean, deviation from (i) mean (ii) median for the following data:

Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |

Number of students |
9 |
6 |
4 |
12 |
9 |

5. Calculate mean deviation from median and its coefficient for the following grouped data:

Mid point |
2 |
6 |
10 |
14 |
18 |
22 |
26 |

frequency |
7 |
13 |
10 |
5 |
8 |
4 |
3 |

6. Calculate the standard deviation and its coefficient from the following data by

- direct method
- short-cut method
- step-deviation

Marks |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |

No. of. Students |
7 |
12 |
24 |
10 |
7 |

7. Calculate the standard deviation and coefficient of variation:

Hight (cm) |
0-8 |
8-16 |
16-24 |
24-32 |
32-40 |

No.of plants |
6 |
7 |
10 |
8 |
9 |

8. Calculate the standard deviation and coefficient of variation from the following

x |
Less than 10 |
Less than 20 |
Less than 30 |
Less than 40 |
Less than 50 |

f |
12 |
19 |
24 |
33 |
40 |