Nootan solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० आईसीएसई chapter • - Competency focused practice questions [Latest edition]

A retailer buys an article at its listed price from a wholesaler and sells it to a consumer in the same state after marking up the price by 20%. The list price of the article is 2500 and the rate of GST is 12%. What is the tax liability of the retailer to the central government?

Dev bought an electrical fan which has a marked price of ₹ 800. If the GST on the goods is 7%, then the SGST is ______.

₹ P is deposited for n number of months in a recurring deposit account which pays interest at the rate of r % per annum. The nature and time of interest calculated is ______.

Anwesha intended to open a Recurring Deposit account of ₹ 1000 per month for 1 year in a Bank, paying a 5% per annum rate of simple interest. The bank reduced the rate to 4% per annum. How much must Anwesha deposit monthly for 1 year so that her interest remains the same?

Mr. Das invests in ₹ 100, 12% shares of Company А available at ₹ 60 each. Mr. Singh invests in ₹ 50, 16% shares of Company B available at ₹ 40 each. Use this information to state which of the following statements is true?

Amit invested a certain sum of money in ₹ 100 shares, paying a 7.5% dividend. The rate of return on his investment is 10%. The money invested by Amit to purchase 10 shares is ______.

If – 3 ≤ – 4x + 5 and x ε W, then the solution set is ______.

If – 4x > 8y, then

The value/s of ‘k’ for which the quadratic equation 2x² – kx + k = 0 has equal roots is (are):

If x = –2 is one of the solutions of the quadratic equation x² + 3a – x = 0, then the value of ‘a’ is ______.

In solving a quadratic equation, one of the values of the variable x is 233.356. The solution rounded to two significant figures is ______.

In the adjoining diagram, AB = x cm, BC = у cm and x – y = 7 cm. Area of ΔABC = 30 cm². The length of AC is:

If p, q and r are in continued proportion, then:

The ratio of diameter to height of a Borosil cylindrical glass is 3 : 5. If the actual diameter of the glass is 6 cm, then the curved surface area of the glass is ______.

If the polynomial 2x³ + 3x² – 2x – 3 is completely divisible by (2x + a) and the quotient is equal to (x² – 1), then one of the values of a is ______.

If A = [a b] and B = `[(c), (d)]`, then:

Matrix A = `[(6, 9),(-4, k)]` such that A² = `[(0, 0),(0, 0)]`. Then k is ______.

If the sum of n terms of an arithmetic progression S_n = n² – n, then the third term of the series is ______.

Which of the following is NOT a geometric progression?

In the adjoining diagram, G is the centroid of ΔABC. A(3, –3), В(2, –6), C(x, y) and G(5, –5). The coordinates of point D are:

In the given diagram, O is the origin and P is the midpoint of AB. The equation of OP is:

In the given figure Line l₁ is a parallel to Line l₂. If line l₃ is perpendicular to Line l₁, then the slopes of lines l₂ and l₃ respectively are:

Which of the following lines cut the positive x-axis and positive y-axis at equal distances from the origin?

In the given diagram (not drawn to scale), railway stations A, B, C, P and Q are connected by straight tracks. Track PQ is parallel to BC. The time taken by a train travelling at 90km/hr to reach B from A by the shortest route is:

In the given diagram, ΔABC and ΔDEF (not drawn to scale) are such that ∠C = ∠F and `(AB)/(DE) = (BC)/(EF)`, then

In the adjoining diagram, ST is not parallel to PQ. The necessary and sufficient conditions for ΔPQR ~ ΔTSR is:

The scale factor of a picture and the actual height of Sonia is 20 cm: 1.6 m. If her height in the picture is 18 cm, then her actual height is ______.

In the adjoining figure, O is the centre of the circle and a semicircle is drawn on OA as the diameter. ∠APQ = 20°. The degree measure of ∠OAQ is:

In the given diagram, O is the centre of the circle and DE is a tangent at B. If ∠ABC = 50°, then values of x, y and z respectively are:

In the given figure, PT and QT are tangents to a circle such that ∠TPS = 45° and ∠TQS = 30°. Then, the value of x is:

A cylindrical metallic wire is stretched to double its length. Which of the following will NOT change for the wire after stretching?

A right circular cone has the radius of the base equal to the height of the cone. If the volume of the cone is 9702 cu. cm, then the diameter of the base of the cone is ______.

A solid sphere with a radius of 4 cm is cut into 4 identical pieces by two mutually perpendicular planes passing through its centre. Find the total surface area of one-quarter piece.

Two identical solid hemispheres are kept in contact to form a sphere. The ratio of the total surface areas of the two hemispheres to the surface area of the sphere formed is:

cosec²θ + sec²θ is equal to ______.

Given a = 3 sec² θ and b = 3 tan² θ – 2. The value of (a – b) is ______.

At a certain time of day, the ratio of the height of the pole to the length of its shadow is `1 : sqrt(3)`, then the angle of elevation of the sun at that time of the day is ______.

A man standing on a ship approaching the port towards the lighthouse is observing the top of the lighthouse. In 10 minutes, the angle of elevation of the top of the lighthouse changes from α to β. Then:

Assertion (A): The difference in class marks of the modal class and the median class of the following frequency distribution table is 0.

Reason (R): Modal class and median class are always the same for a given frequency distribution.

Assertion (A): For a collection of 11 arrayed data, the median is the middle number.

Reason (R): For the data 5, 9, 7, 13, 10, 11, 10, the median is 13.

Ankit had the option of investing in company A, where 7%, ₹ 100 shares are available at ₹ 120 or in company B, where 8%, ₹ 1000 shares are available at ₹ 1620.

Assertion (A): Investment in Company A is better than Company B.

Reason (R): The rate of income in Company A is better than in Company B.

Assertion (A): x³ + 2x² – x – 2 is a polynomial of degree 3.

Reason (R): x + 2 is a factor of the polynomial.

Assertion (A): The point (–2, 8) is invariant under reflection in line x = –2.

Reason (R): If a point has its x-coordinate 0, it is invariant under reflection in both axes.

When a die is cast with numbering on its faces, as shown, the ratio of the probability of getting a composite number to the probability of getting a prime number is ______.

The product of `A = [(1, -2),(-3, 4)]` and matrix M, AM = B where `B = [(2), (24)]`, then the order of matrix M is ______.

Given, a₁, а₂, а₃, ....... and b₁, b₂, b₃, ....... are real numbers such that a₁ – b₁ = a₂ – b₂ = a₃ – b₃ = .......... are all equal. a₁ – b₁, a₂ – b₂, a₃ – b₃ .......... forms a ______ progression.

Locus of a moving point is ______ if it moves such that it keeps a fixed distance from a fixed point.

The point of concurrence of the angle bisectors of a triangle is called the ______ of the triangle.

A shopkeeper marked a pressure cooker at ₹ 1800. The rate of GST on pressure cooker is 12%. The customer has only ₹ 1792 with him and he requests the shopkeeper to reduce the price so that he can buy the cooker in ₹ 1792. What percent discount must the shopkeeper give?

A man opened a recurring deposit account in a branch of PNB. The man deposits certain amount of money per month such that after 2 years, the interest accumulated is equal to his monthly deposits. Find the rate of interest per annum that the bank was paying for the recurring deposit account.

Akshay buys 350 shares of ₹ 50 par value of a company. The dividend declared by the company is 14%. If his return percent from the shares is 10%, find the market value of each share.

Solve for x, if `5/x + 4sqrt(3) = (2sqrt(3))/x^2, x = 0`

The marked price of a toy is same as the percentage of GST that is charged. The price of the toy is ₹ 24 including GST. Taking the marked price as x, form an equation and solve it to find x.

The mean proportion between two numbers is 6 and their third proportion is 48. Find the two numbers.

Pamela factorized the following polynomial:

2x³ + 3x² – 3x – 2

She found the result as (x + 2) (x – 1) (x – 2). Using remainder and factor theorem, verify whether her result is correct. If incorrect, give the correct result.

`A = [(-6, 0),(4, 2)]` and `B = [(1, 0),(1, 3)]`. Find matrix M, if `M = 1/2 A - 2B + 5l`, where l is the identity matrix.

(a) Write the nth term (T_n) of an Arithmetic Progression (A.P.) consisting of all whole numbers which are divisible by 3 and 7.

(b) How many of these are two-digit numbers? Write them.

(c) Find the sum of first 10 terms of this A.P.

Write the first five terms of the sequence given by `(sqrt(3))^n, n ∈ N`.

1. Is the sequence an A.P. or G.P?
2. If the sum of its first ten terms is `p(3 + sqrt(3))`, find the value of p.

ABC is a triangle as shown in the figure below.

1. Write down the coordinates of A, B and C on reflecting through the origin. 
2. Write down the coordinates of the point/s which remain invariant on reflecting the triangle ABC on the x-axis and y-axis respectively.

Determine the ratio in which the line y = 2 + 3x divides the line segment AB joining the points A(–3, 9) and B(4, 2).

Square ABCD lies in the third quadrant of a XY plane such that its vertex A is at (–3, –1) and the diagonal DB produced is equally inclined to both the axes. The diagonals AC and BD meets at P(–2, –2). Find the:

1. slope of BD
2. equation of AC

ABCD is a rectangle where side BC is twice side AB. If ΔACQ ~ ΔBAP, find area of ΔBAP : area of ΔACQ.

Given a triangle ABC and D is a point on BC such that BD = 4 cm and DC = x cm. If ∠BAD = ∠C and AB = 8 cm, then, 

1. prove that triangle ABD is similar to triangle CBA. 
2. find the value of ‘x’.

In the extract of Survey of India map G43S7, prepared on a scale of 2 cm to 1 km, a child finds the length of the cart track between two settlements is 7.6 cm. Find:

1. the actual length of the cart track on the ground. 
2. actual area of a grid square, if each has an area of 4 cm².

Construct a triangle ABC such that AB = 7 cm, BC = 6 cm and CA = 5 cm. (use ruler and compass to do so).

(a) Draw the locus of the points such that

(i) it is equidistant from BC and BA.

(ii) it is equidistant from points A and B. 

(b) Mark P where the loci (i) and (ii) meet, measure and write length of PA.

In the given figure O is the centre of the circle. ABCD is a quadrilateral where sides AB, BC, CD and DA touch the circle at E, F, G and H respectively. If AB = 15 cm, BC = 18 cm and AD = 24 cm, find the length of CD.

In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of:

1. ∠FAG 
2. ∠BСD 
3. ∠PDE

AB and CD intersect at the centre O of the circle given in the above diagram. If ∠EBA = 33° and ∠EAC = 82°, find.

1. ∠BAE
2. ∠BOC
3. ∠ODB

The ratio of the radius and the height of a solid metallic right circular cylinder is 7 : 27. This is melted and made into a cone of diameter 14 cm and slant height 25 cm. Find the height of the:

1. cone
2. cylinder

An inclined plane AC is prepared with its base AB which is `sqrt(3)` times its vertical height BC. The length of the inclined plane is 15 m. Find:

1. value of θ.
2. length of its base AB, in nearest metre.

Prove that tan²θ + cos²θ – 1 = tan²θ. sin²θ

The class mark and frequency of a data is given in the graph. From the graph, Find:

1. the table showing the class interval and frequency. 
2. the mean.

The mean of 5, 7, 8, 4 and m is n and the mean of 5, 7, 8, 4, m and n is m. Find the values of m and n.

The probability of selecting a blue marble and a red marble from a bag containing red, blue and green marbles is `1/3` and `1/5` respectively. If the bag contains 14 green marbles, then find:

1. number of red marbles. 
2. total number of marbles in the bag.

The following bill shows the GST rate and the marked price of items:

Find:

1. the value of x if the total GST on wheat flour and basmati rice is ₹ 45. 
2. the value of y, if CGST paid for detergent powder is ₹ 39.60. 
3. total amount to be paid (including GST) for the above bill.

Amit deposited ₹ 600 per month in a recurring deposit account. The bank pays a simple interest of 12% p.a. Calculate the:

1. number of monthly instalments Amit deposits to get a maturity amount of ₹ 11826? 
2. total interest paid by the bank. 
3. total amount deposited by him.

Aman has 500, ₹ 100 shares of a company quoted at ₹ 120, paying a 10% dividend. When the share price rises to ₹ 200 each, he sells all his shares. He invests half of the sale proceeds in ₹ 10, 12% shares at ₹ 25 and the remaining sale proceeds in ₹ 400, 9% shares at ₹ 500.

Find his:

1. sales proceeds.
2. investment in ₹ 10, 12% shares at ₹ 25.
3. original income.
4. change in income.

Solve the following inequation.

`(11 + 3x)/5 ≥ 3 - x > -3/2, x ∈ R`

1. Write the solution set. 
2. Represent the solution on the number line.

Determine whether the following quadratic equation has real roots.

5x² – 9x + 4 = 0

1. Give reasons for your answer.
2. If the equation has real roots, identify them.

The profit in rupees in a local restaurant and the number of customers who visited the restaurant are tabulated below for each week for one month.

Find:

1. if the number of customers and profit per week in continued proportion or not? Justify your answer.
2. the value of x and y.

Given, 9x² – 4 is a factor of 9x³ – mx² + nx + 8: 

1. find the value of m and n using the remainder and factor theorem.
2. factorise the given polynomial completely.

The marks scored by 100 students are given below:

A student in the class is selected at random. Find the probability that the student has scored:

1. less than 20. 
2. below 60 but 30 or more. 
3. more than or equal to 70. 
4. above 89.

Given, matrix `A = [(x, 1),(y, 2)]` and `B = [(x),(x - 2)]` such that AB is a null matrix. Find:

1. order of the null matrix. 
2. possible values of x and y.

The sum of a certain number of terms of the Arithmetic Progression (A.P.) 20, 17, 14, .... is 65. Find the:

1. number of terms.
2. last term.

1. Point P(2, –3) on reflection becomes P’(2, 3). Name the line of reflection (say L₁).
2. Point P’ is reflected to P” along the line (L₂), which is perpendicular to the line L₁ and passes through the point, which is invariant along both axes. Write the coordinates of Р”.
3. Name and write the coordinates of the point of intersection of the lines L₁ and L₂.
4. Point P is reflected to P” on reflection through the point named in the answer of part I of this question. Write the coordinates of P”. Comment on the location of the points P” and P”.

In the given figure, if the line segment AB is intercepted by the y-axis and x-axis at C and D, respectively, such that AC : AD = 1 : 4 and D is the midpoint of CB. Find the coordinates of D, C and В.

Find the equation of the straight line perpendicular to the line x + 2y = 4, which cuts an intercept of 2 units from the positive y-axis. Hence, find the intersection point of the two lines.

While preparing a Power Point presentation, ΔABC is enlarged along the side BC to ΔAB’C’, as shown in the diagram, such that BC : B’C’ is 3 : 5. Find:

1. AB : BB’
2. length AB, if BB’ = 4 cm.
3. Is ΔABC ~ ΔАВ’C’? Justify your answer.
4. ar (ΔABC) : ar (quad. BB’C’C).

In the adjoining diagram PQ, PR and ST are the tangents to the circle with centre O and radius 7 cm. Given OP = 25 cm.

Find:

1. length of ST
2. value of ∠OPQ, i.e., θ
3. ∠QUR, in nearest degree (use mathematical tables)

Use ruler and compass to answer this question. Construct a triangle ABC where AB = 5.5 cm, BC = 4.5 cm and angle ABC = 135°. Construct the circumcircle to the triangle ABC. Measure and write down the length of AC.

The curved surface area of a right circular cone is half of another right circular cone. If the ratio of their slant heights is 2 : 1 and that of their volumes is 3 : 1, find ratio of their:

1. radii
2. heights

A cylindrical drum is unloaded from a truck by rolling it down along a wooden plank. The length of the plank is 10 m and it is making an angle of 10° with the horizontal ground. Find the height from which the cylindrical drum was rolled down. Give your answer correct to 3 significant figures.

The data given below shows the marks of 12 students in a test, arranged in ascending order:

2, 3, 3, 3, 4, x, x + 2, 8, p, q, 8, 9

If the given value of the median and mode is 6 and 8 respectively, then find the values of x, p, q.

Solve the linear inequation, write down the solution set and represent it on the real number line:

5(2 – 4x) > 18 – 16x > 22 – 20x, x ∈ R

If a polynomial x³ + 2x² – ax + b leaves a remainder –6 when divided by x + 1 and the same polynomial has x – 2 as a factor, then find the values of a and b.

If `A = [(-1, 3),(2, 0)], B = [(1, -2),(0, 3)], C = [1 - 4]` and `D = [(4), (1)]`.

In the given figure(not drawn to scale), BC is parallel to EF, CD is parallel to FG, AE : EB = 2 : 3, ∠BAD = 70°, ∠ACB = 105°, ∠ADC = 40° and AC is bisector of ∠BAD.

1. Prove ΔAEF ~ ΔAGF 
2. Find: 

1. AG : AD 
2. area of ΔACB : area ΔACD 
3. area of quadrilateral ABCD : area of ΔАСВ.

In the given figure angle ABC = 700 and angle ACB = 500. Given, O is the centre of the circle and PT is the tangent to the circle. Then calculate the following angles.

1. ∠CBT
2. ∠BAT
3. ∠PBT
4. ∠APT

The daily wages of workers in a construction unit were recorded as follows:

Form a frequency distribution table with class intervals and find modal wage by plotting a histogram.

A bag contains 13 red cards, 13 black cards and 13 green cards. Each set of cards are numbered 1 to 13. From these cards, a card is drawn at random. What is the probability that the card drawn is a:

1. green card?
2. a card with an even number?
3. a red or black card with a number which is a multiple of three?

(For this question, use a graph paper. Scale: 1 cm = 1 unit along both x and y-axis.)

Plot points A(0, 3), B(4, 0), C(6, 2) and D(5, 0). Reflect the points as given below and write their coordinates:

Study the graph and answer the questions that follow:

1. Make a frequency table for the information provided in the graph.
2. The number of students whose height is less than 150 cm.
3. The total number of students.
4. The modal height.
5. The difference in the modal height and the mean height, if the average height of the students is 145.5 cm.

On seeing the above display board outside Pearl Stationery Shop, Chetan enters the shop to buy the following items:

The shopkeeper handed over the bill to Chetan saying that he has given further discount of 2% on total bill. Chetan became so happy hearing about the discount that he did not check the bill until he reached home. He later found out that though shopkeeper has given 2% discount as promised, he had also charged uniform 18% GST on all the items.

1. Calculate: 
   1. total selling price of all the items as per the offers displayed on the board.
   2. total amount to be paid by Chetan including GST with correct rates.
   3. actual amount charged by the shopkeeper.
2. Did the shopkeeper overcharge Chetan? Justify your answer.

Using remainder and factor theorem, show that (2x + 3) is a factor of the polynomial 2x² + 11x + 12. Hence, factorise it completely. What must be multiplied to the given polynomial so that x² + 3x – 4 is a factor of the resulting polynomial? Also, write the resulting polynomial.

The sequence 2, 9, 16, ...... is given.

1. Identify if the given sequence is an AP or a GP. Give reasons to support your answer.
2. Find the 20th term of the sequence.
3. Find the difference between the sum of its first 22 and 25 terms.
4. Is the term 102 belong to this sequencе?
5. If ‘k’ is added to each of the above terms, will the new sequence be in A.P. or G.P.?

Given the equations of two straight lines, L1 and L2 are x – y = 1 and x + y = 5 respectively. If L1 and L2 intersects at point Q(3, 2). Find: 

1. the equation of line L3 which is parallel to L1 and has y-intercept 3.
2. the value of k, if the line L3 meets the line L2 at a point P(k, 4).
3. the coordinate of R and the ratio PQ : QR, if line L2 meets x-axis at point R.

In the figure given above (not drawn to scale), AD || GE || BC, DE = 18 cm, EC = 3 cm, AD = 35 cm.

Find:

1. AF : FC 
2. length of EF 
3. area (trapezium ADEF) : area (ΔEFC)
4. BC : GF

(Use a ruler and a compass for this question.)

1. Construct the locus of a moving point which moves such that it keeps a fixed distance of 4.5 cm from a fixed-point O. 
2. Draw line segment AB of 6 cm where A and B are two points on the locus (a). 
3. Construct the locus of all points equidistant from A and B. Name the points of intersection of the loci (a) and (c) as P and Q respectively. 
4. Join PA. Find the locus of all points equidistant from AP and AB. 
5. Mark the point of intersection of the locus (a) and (d) as R. Measure and write down the length of AR.

(Use a ruler and a compass for this question.)

Construct a regular hexagon ABCDEF of side 4.3 cm and construct its circumscribed circle. Also, construct tangents to the circumscribed circle at points B and C which meets each other at point P. Measure and record ∠BPC.

A mathematics teacher uses certain amount of terracotta clay to form different shaped solids. First, she turned it into a sphere of radius 7 cm and then she made a right circular cone with base radius 14 cm. Find the height of the cone so formed. If the same clay is turned to make a right circular cylinder of height `7/3` cm, then find the radius of the cylinder so formed. Also, compare the total surface areas of sphere and cylinder so formed.

A tree (TS) of height 30 m stands in front of a tall building (AB). Two friends Rohit and Neha are standing at R and N respectively, along the same straight line joining the tree and the building (as shown in the diagram). Rohit, standing at a distance of 150 m from the foot of the building, observes the angle of elevation of the top of the building as 30°. Neha from her position observes that the top of the building and the tree has the same elevation of 60°. Find the:

1. height of the building 
2. distance between 
   1. Neha and the foot of the building 
   2. Rohit and Neha 
   3. Neha and the tree 
   4. building and the tree.

A life insurance agent found the following data of age distribution of 100 policy holders, where f is an unknown frequency.

1. If the mean age of the policy holders is 35.65 years, find the unknown frequency f. 
2. Find the median class of the distribution.