ICSE Class 10 Maths MCQ Solutions Step-by-Step Guide
ICSE Class 10 Maths MCQ Solutions for Competency-Focused Practice
ICSE Class 10 Maths competency-focused MCQs test whether you can select the correct option by applying a formula, property or theorem quickly. This page explains the method behind the answer, using school-style working for multiple-choice practice from GST, recurring deposits, shares, inequalities, quadratic equations, ratio, mensuration, factor theorem and matrices.
For syllabus alignment, students should compare their school textbook and the CISCE official website. For related study material on this site, use the ICSE Class 10 solutions, ICSE Class 10 Maths solutions and ICSE Board study resources.
Method map for ICSE Class 10 Maths MCQs
In a competency-focused question, the option is usually hidden behind a small interpretation step. Before calculating, identify the chapter idea being tested.
| Question type | First step | What to check before choosing the option |
|---|---|---|
| GST | Separate GST into CGST and SGST for an intra-state sale. | Tax liability means output tax minus input tax, not the full GST collected. |
| Recurring deposit | Use the RD interest formula with n in months. | Do not treat RD as compound interest. |
| Shares and dividend | Find dividend on face value, then compare it with market value. | Rate of return is based on money invested, not face value. |
| Inequality | Isolate x carefully. | Reverse the inequality sign when dividing by a negative number. |
| Quadratic equation | Use substitution, factorisation or discriminant as required. | For equal roots, set b^2-4ac=0. |
| Factor theorem | Put the factor equal to zero and substitute the resulting value of x. | A factor gives remainder 0. |
| Matrix multiplication | Compare orders before multiplying. | AB and BA may both exist, but their orders can be different. |
Concept snapshot: one-minute elimination
Think of an MCQ as a locked door with four keys. The formula gives you the key shape, but the condition in the question tells you which lock to try. For example, in a GST question, the word liability points to the difference between output tax and input tax. In a factor theorem question, the phrase is a factor tells you that the remainder must be 0. This is why the first reading of the question matters as much as the calculation.
Formula and rule reference
| Topic | Formula or rule | Use in MCQs |
|---|---|---|
| GST split | \text{CGST}=\text{SGST}=\dfrac{\text{GST rate}}{2} | Use this for intra-state transactions. |
| GST liability | \text{Liability}=\text{Output tax}-\text{Input tax} | Use this when a trader buys and resells goods. |
| Recurring deposit interest | I=P\times \dfrac{n(n+1)}{2\times 12}\times \dfrac{r}{100} | P is the monthly instalment and n is the number of months. |
| Dividend per share | \text{Dividend}=\dfrac{\text{Dividend rate}}{100}\times \text{Face value} | Dividend is always calculated on face value. |
| Rate of return | \text{Return}=\dfrac{\text{Dividend}}{\text{Market value}}\times 100 | Compare the income with the actual investment. |
| Equal roots of a quadratic | b^2-4ac=0 | Use this when the roots are equal. |
| Remainder theorem | If f(x) is divided by x-a, the remainder is f(a). | For px+q, first solve px+q=0. |
| Area of right triangle | \text{Area}=\dfrac{1}{2}\times \text{base}\times \text{height} | Use perpendicular sides as base and height. |
| Curved surface area of cylinder | \text{CSA}=2\pi rh | Use radius, not diameter. |
Worked examples for competency-focused questions
Worked example 1: GST liability of a retailer
A retailer buys an article at a list price of \text{₹ }2500, marks it up by 20\%, and sells it in the same state. If GST is 12\%, find the retailer’s tax liability to the central government.
Step 1: In an intra-state sale, \text{CGST}=\dfrac{12\%}{2}=6\%.
Step 2: Find the CGST paid on purchase.
\text{Input CGST}=\frac{6}{100}\times 2500=\text{₹ }150
Step 3: Find the selling price after 20\% mark-up.
\text{Selling price}=2500+\frac{20}{100}\times 2500=2500+500=\text{₹ }3000
Step 4: Find the CGST collected on sale.
\text{Output CGST}=\frac{6}{100}\times 3000=\text{₹ }180
Step 5: Calculate tax liability.
\text{Tax liability}=180-150=\text{₹ }30
Final answer: \text{₹ }30.
Worked example 2: Recurring deposit with changed rate
A student deposits \text{₹ }1000 per month for 12 months at 5\% per annum. If the rate becomes 4\% per annum, find the monthly deposit required to earn the same interest.
Step 1: Use the RD formula.
I=P\times \frac{n(n+1)}{2\times 12}\times \frac{r}{100}
Step 2: Find the original interest.
I=1000\times \frac{12(13)}{2\times 12}\times \frac{5}{100}=1000\times \frac{13}{2}\times \frac{1}{20}=\text{₹ }325
Step 3: Let the new monthly deposit be x.
325=x\times \frac{12(13)}{2\times 12}\times \frac{4}{100}=x\times \frac{13}{2}\times \frac{1}{25}
Step 4: Solve for x.
x=\frac{325\times 2\times 25}{13}=\text{₹ }1250
Final answer: The required monthly deposit is \text{₹ }1250.
Worked example 3: Factor theorem in a polynomial MCQ
For the polynomial x^3+5x^2-kx-24, decide which listed linear expression is a factor when k=2.
Step 1: Put k=2. The polynomial becomes f(x)=x^3+5x^2-2x-24.
Step 2: Test x+4 by putting x=-4.
f(-4)=(-4)^3+5(-4)^2-2(-4)-24
=-64+80+8-24=0
Step 3: Since the remainder is 0, x+4 is a factor.
Final answer: x+4 is the required factor.
Step-by-step MCQ solutions
The following school-style solutions show the calculation behind each answer. The option number is stated only after the method is clear.
Question 1: GST liability to the central government
Step 1: GST is 12\%, so \text{CGST}=6\%.
Step 2: CGST paid on purchase is 6\% of 2500.
\text{Input CGST}=\frac{6}{100}\times 2500=\text{₹ }150
Step 3: The selling price after 20\% mark-up is 3000.
\text{SP}=2500+\frac{20}{100}\times 2500=\text{₹ }3000
Step 4: CGST collected on sale is 6\% of 3000.
\text{Output CGST}=\frac{6}{100}\times 3000=\text{₹ }180
Final answer: 180-150=\text{₹ }30, so option (c) is correct.
Question 2: SGST on a fan marked at \text{₹ }800
Step 1: If GST is 7\%, then \text{SGST}=\dfrac{7\%}{2}=3.5\%.
Step 2: Find 3.5\% of 800.
\text{SGST}=\frac{3.5}{100}\times 800=\text{₹ }28
Final answer: \text{₹ }28, so option (b) is correct.
Question 3: Nature of interest in a recurring deposit
Step 1: The recurring deposit formula used in ICSE Class 10 Maths is
I=P\times \frac{n(n+1)}{2\times 12}\times \frac{r}{100}.
Step 2: This formula is formed by adding simple interest on monthly instalments for their respective periods.
Step 3: It is not compound interest because interest is not added back to the principal each month.
Final answer: The calculation uses simple interest in the monthly-deposit form, so option (d) is the intended answer.
Question 4: Monthly deposit needed after a rate reduction
Step 1: Find the original interest for P=1000, n=12, r=5.
I=1000\times \frac{12(13)}{2\times 12}\times \frac{5}{100}
=1000\times \frac{13}{2}\times \frac{1}{20}=\text{₹ }325
Step 2: Let the new monthly deposit be x when r=4.
325=x\times \frac{12(13)}{2\times 12}\times \frac{4}{100}
325=x\times \frac{13}{2}\times \frac{1}{25}
Step 3: Solve the equation.
x=\frac{325\times 2\times 25}{13}=\text{₹ }1250
Final answer: \text{₹ }1250, so option (b) is correct.
Question 5: Comparing rate of return on shares
Step 1: For Mr. Das, dividend on one share is 12\% of the face value 100.
\text{Dividend}=\frac{12}{100}\times 100=\text{₹ }12
Step 2: His market value is 60, so
\text{Return}=\frac{12}{60}\times 100=20\%
Step 3: For Mr. Singh, dividend on one share is 16\% of 50.
\text{Dividend}=\frac{16}{100}\times 50=\text{₹ }8
Step 4: His market value is 40, so
\text{Return}=\frac{8}{40}\times 100=20\%
Final answer: Both have the same return of 20\%, so option (d) is correct.
Question 6: Money invested in shares
Step 1: Dividend on one share of face value 100 at 7.5\% is
\frac{7.5}{100}\times 100=\text{₹ }7.5
Step 2: Dividend on 10 shares is
10\times 7.5=\text{₹ }75
Step 3: Since the rate of return is 10\%,
\frac{75}{\text{Investment}}\times 100=10
Step 4: Solve for investment.
\text{Investment}=\frac{75\times 100}{10}=\text{₹ }750
Final answer: \text{₹ }750, so option (b) is correct.
Question 7: Solution set of -3\leq -4x+5, where x\in W
Step 1: Start with the inequality.
-3\leq -4x+5
Step 2: Subtract 5 from both sides.
-8\leq -4x
Step 3: Divide by -4 and reverse the inequality sign.
2\geq x
Step 4: Since x\in W, the possible values are whole numbers not greater than 2.
Final answer: \{0,1,2\}, so option (c) is correct.
Question 8: Solving -4x>8y
Step 1: Divide both sides by -4.
Step 2: Reverse the inequality sign because the divisor is negative.
-4x>8y \Rightarrow x<-2y
Final answer: x<-2y, so option (c) is correct.
Question 9: Equal roots of 2x^2-kx+k=0
Step 1: For equal roots, use b^2-4ac=0.
Step 2: Here a=2, b=-k, c=k.
(-k)^2-4(2)(k)=0
k^2-8k=0
Step 3: Factorise.
k(k-8)=0
Final answer: k=0 or k=8, so option (d) is correct.
Question 10: Finding a when x=-2 is a solution
Step 1: Substitute x=-2 in x^2+3a-x=0.
(-2)^2+3a-(-2)=0
4+3a+2=0
Step 2: Solve for a.
3a=-6 \Rightarrow a=-2
Final answer: a=-2, so option (b) is correct.
Question 11: Rounding 233.356 to two significant figures
Step 1: The first two significant digits in 233.356 are 2 and 3.
Step 2: The next digit is 3, which is less than 5, so the second significant digit remains 3.
233.356\approx 230 \quad \text{to two significant figures}
Final answer: 230, so option (d) is correct.
Question 12: Finding AC from area and side difference
Assumption: The adjoining figure is a right triangle with AB=x, BC=y, and AB\perp BC.
Step 1: Use the area condition.
30=\frac{1}{2}xy \Rightarrow xy=60
Step 2: Use the difference condition x-y=7, so x=y+7.
Step 3: Substitute in xy=60.
y(y+7)=60
y^2+7y-60=0
(y+12)(y-5)=0
Step 4: Reject y=-12 because a length cannot be negative. Hence y=5, and x=12.
Step 5: Use Pythagoras theorem.
AC^2=AB^2+BC^2=12^2+5^2=144+25=169
AC=13\text{ cm}
Final answer: 13\text{ cm}, so option (c) is correct.
Question 13: Continued proportion
Step 1: If p, q, r are in continued proportion, then
\frac{p}{q}=\frac{q}{r} \Rightarrow q^2=pr
Step 2: Test the relation p:r=p^2:q^2.
\frac{p}{r}=\frac{p^2}{q^2} \Rightarrow q^2=pr
Step 3: This matches the condition for continued proportion.
Final answer: p:r=p^2:q^2, so option (d) is correct.
Question 14: Curved surface area of a cylindrical glass
Step 1: Diameter to height ratio is 3:5, and the actual diameter is 6\text{ cm}.
\frac{6}{h}=\frac{3}{5} \Rightarrow h=10\text{ cm}
Step 2: Radius is half of diameter.
r=\frac{6}{2}=3\text{ cm}
Step 3: Use curved surface area of cylinder.
\text{CSA}=2\pi rh=2\pi(3)(10)=60\pi\text{ cm}^2
Final answer: 60\pi\text{ cm}^2, so option (b) is correct.
Question 15: Finding a from exact polynomial division
Step 1: If the quotient is x^2-1 and the divisor is 2x+a, then
2x^3+3x^2-2x-3=(2x+a)(x^2-1)
Step 2: Expand the right side.
(2x+a)(x^2-1)=2x^3-2x+ax^2-a
Step 3: Compare with 2x^3+3x^2-2x-3.
ax^2-a=3x^2-3
Final answer: a=3, so option (d) is correct.
Question 16: Choosing the factor when k=2
Step 1: Put k=2 in x^3+5x^2-kx-24.
f(x)=x^3+5x^2-2x-24
Step 2: Test x+4 by substituting x=-4.
f(-4)=(-4)^3+5(-4)^2-2(-4)-24
=-64+80+8-24=0
Final answer: x+4 is a factor, so option (c) is correct.
Question 17: Matrix products AB and BA
Step 1: Let A=\begin{bmatrix}a & b\end{bmatrix} and B=\begin{bmatrix}c\\ d\end{bmatrix}.
Step 2: The order of A is 1\times 2, and the order of B is 2\times 1.
Step 3: AB is possible because the inner orders are 2 and 2.
AB=\begin{bmatrix}a & b\end{bmatrix}\begin{bmatrix}c\\ d\end{bmatrix}=\begin{bmatrix}ac+bd\end{bmatrix}
Step 4: BA is possible because the inner orders are 1 and 1.
BA=\begin{bmatrix}c\\ d\end{bmatrix}\begin{bmatrix}a & b\end{bmatrix}=\begin{bmatrix}ca & cb\\ da & db\end{bmatrix}
Final answer: Both AB and BA are possible, but they are not equal in general. Option (c) is correct.
Quick answer index
| Question | Topic | Answer |
|---|---|---|
| 1 | GST | \text{₹ }30 |
| 2 | GST | \text{₹ }28 |
| 3 | Recurring deposit | Simple-interest form for monthly deposits |
| 4 | Recurring deposit | \text{₹ }1250 |
| 5 | Shares | Both returns are 20\% |
| 6 | Shares | \text{₹ }750 |
| 7 | Inequality | \{0,1,2\} |
| 8 | Inequality | x<-2y |
| 9 | Quadratic equation | k=0,8 |
| 10 | Substitution in equation | a=-2 |
| 11 | Significant figures | 230 |
| 12 | Right triangle | 13\text{ cm} |
| 13 | Continued proportion | p:r=p^2:q^2 |
| 14 | Cylinder | 60\pi\text{ cm}^2 |
| 15 | Polynomial division | a=3 |
| 16 | Factor theorem | x+4 |
| 17 | Matrices | Both AB and BA are possible |
Examiner’s mindset for ICSE Class 10 Maths MCQs
For objective questions, the final option receives the mark, but the rough work must still be dependable. A common marking habit in school tests is to award credit in longer competency questions for the formula, substitution and final option if working is asked. In GST, write input tax and output tax separately. In quadratic equations, state the discriminant condition before solving. In geometry and mensuration, show the unit in the final line.
Common mistakes students make
- GST mistake: Taking the retailer’s liability as the full CGST collected. The correction is \text{liability}=\text{output CGST}-\text{input CGST}.
- RD mistake: Using simple interest for 12 months on every instalment. The correction is to use \dfrac{n(n+1)}{2} because each instalment remains for a different period.
- Shares mistake: Calculating dividend on market value. Dividend is calculated on face value; return is calculated on market value.
- Inequality mistake: Forgetting to reverse the sign when dividing by a negative number, as in -4x>8y.
- Mensuration mistake: Using diameter in 2\pi rh. The formula needs radius r, so divide diameter by 2 first.
- Matrix mistake: Assuming AB=BA. Matrix multiplication depends on order, and AB and BA can have different orders.
How to practise these ICSE Maths solved competency focused questions
Use a three-pass method. In the first pass, solve only by identifying the topic and writing the required formula. In the second pass, calculate without looking at the options. In the third pass, compare your answer with the options and reject distractors that come from common mistakes, such as using GST instead of CGST or using diameter instead of radius.
For timed practice, keep a separate error log with columns for topic, wrong step and correction. This makes revision more useful than repeatedly reading the same solved page.
Frequently Asked Questions
How should I solve ICSE Class 10 Maths MCQs without guessing?
Start by identifying the chapter concept, write the formula, substitute the values, and only then look at the options. In ICSE Class 10 Maths, most competency-focused MCQs test a small interpretation step before the calculation.
Why is GST liability not equal to the full GST collected?
GST liability for a trader is the output tax collected on sale minus the input tax paid on purchase. For example, if output CGST is \text{₹ }180 and input CGST is \text{₹ }150, the liability is \text{₹ }30.
What is the key formula for recurring deposit MCQs?
The recurring deposit interest formula is I=P\times \dfrac{n(n+1)}{2\times 12}\times \dfrac{r}{100}, where P is the monthly instalment, n is the number of months and r is the annual rate.
How do I know when to reverse an inequality sign?
Reverse the inequality sign only when multiplying or dividing both sides by a negative number. For example, -4x>8y gives x<-2y after dividing by -4.
Are AB and BA always both possible in matrices?
No. AB is possible only when the number of columns of A equals the number of rows of B. BA must be checked separately. Even if both are possible, they need not be equal.