ICSE Class 10 Maths Competency Questions Solutions
ICSE Class 10 Maths Short Answer Questions 2 Solutions
ICSE Class 10 Maths competency-focused Short Answer Questions 2 test whether you can choose the correct formula, substitute values and present the working clearly. This page gives step-by-step solutions for the available questions in this set: GST, recurring deposit, shares and dividends, inequations, quadratic equations, proportion, factor theorem, probability, matrices, arithmetic progression, reflection and coordinate geometry.
Method before solving
For each answer, identify the topic first. In commercial mathematics, write the taxable value, nominal value or monthly deposit before substituting. In algebra, write the theorem or formula before simplifying.
Concept snapshot: Treat a competency question like a word problem wearing a disguise. A bill points to GST, a bank deposit points to recurring deposits, a table points to proportion or probability, and a polynomial with a factor points to factor theorem.
Formula reference
| Topic | Formula / rule |
|---|---|
| GST | \text{GST}=\dfrac{\text{rate}}{100}\times\text{taxable value}; CGST is half of GST when equally split. |
| Recurring deposit | I=\dfrac{P n(n+1)r}{2\times 12\times 100}, maturity value =Pn+I. |
| Shares | Dividend =\text{number of shares}\times\dfrac{\text{rate}}{100}\times\text{nominal value}. |
| Quadratic equation | D=b^2-4ac. |
| Factor theorem | If x-a is a factor of f(x), then f(a)=0. |
| A.P. | S_n=\dfrac{n}{2}[2a+(n-1)d], l=a+(n-1)d. |
Worked examples
Example 1: GST
Find GST and CGST on ₹800 at 18\%.
Step 1: Apply the GST formula.
\text{GST}=\frac{18}{100}\times 800=144
Step 2: Find half for CGST.
\text{CGST}=\frac{144}{2}=72
Final answer: GST =₹144, CGST =₹72.
Example 2: Recurring deposit
Find interest when P=₹500, n=12, r=6\%.
Step 1: Substitute in the formula.
I=\frac{500\times 12\times 13\times 6}{2\times 12\times 100}=195
Final answer: Interest =₹195.
Example 3: Factor theorem
Check whether x-3 is a factor of x^2-5x+6.
Step 1: From x-3=0, x=3.
f(3)=3^2-5(3)+6=9-15+6=0
Final answer: Since the remainder is 0, x-3 is a factor.
Solved Short Answer Questions 2
Question 78: GST bill
Step 1: Taxable values: wheat flour =35\times 5=175, rice =180\times 5=900.
\frac{x}{100}\times 175+\frac{5}{100}\times 900=45
1.75x+45=45\Rightarrow x=0
Answer (a): x=0\%.
Step 2: For detergent, taxable value =220y. Since GST is 18\%, CGST is 9\%.
\frac{9}{100}\times 220y=39.60 \Rightarrow y=\frac{3960}{1980}=2
Answer (b): y=2 kg.
Step 3: Total amount including GST:
175+\left(900+\frac{5}{100}\times 900\right)+\left(440+\frac{18}{100}\times 440\right)
=175+945+519.20=1639.20
Final answer: Total bill =₹1639.20.
Question 79: Recurring deposit
Step 1: Let the number of instalments be n.
11826=600n+\frac{600\times n(n+1)\times 12}{2\times 12\times 100}
11826=600n+3n(n+1)
3n^2+603n-11826=0 \Rightarrow n^2+201n-3942=0
(n-18)(n+219)=0 \Rightarrow n=18
Answer (a): 18 instalments.
I=\frac{600\times 18\times 19\times 12}{2\times 12\times 100}=1026
Answer (b): Interest =₹1026.
\text{Total deposit}=600\times 18=10800
Answer (c): Total deposit =₹10800.
Question 80: Shares and dividend
\text{Sale proceeds}=500\times 200=100000
Answer (a): ₹100000.
\text{Investment in first new share}=\frac{100000}{2}=50000
Answer (b): ₹50000.
Step 1: Original dividend uses nominal value ₹100.
\text{Original income}=500\times\frac{10}{100}\times 100=5000
Answer (c): ₹5000.
Step 2: New income from ₹10, 12\% shares at ₹25:
\frac{50000}{25}\times\frac{12}{100}\times 10=2000\times 1.2=2400
Step 3: New income from ₹400, 9\% shares at ₹500:
\frac{50000}{500}\times\frac{9}{100}\times 400=100\times 36=3600
\text{Change}=2400+3600-5000=1000
Final answer (d): Income increases by ₹1000.
Question 81: Inequation
\frac{11+3x}{5}\ge 3-x \Rightarrow 11+3x\ge 15-5x \Rightarrow 8x\ge 4 \Rightarrow x\ge \frac{1}{2}
3-x>-\frac{3}{2}\Rightarrow -x>-\frac{9}{2}\Rightarrow x<\frac{9}{2}
Final answer: \left\{x:x\in\mathbb{R},\frac{1}{2}\le x<\frac{9}{2}\right\}. On the number line, use a closed dot at \frac{1}{2}, an open dot at \frac{9}{2}, and shade between them.
Question 82: Quadratic equation
Step 1: Here a=5, b=-9, c=4.
D=b^2-4ac=(-9)^2-4(5)(4)=81-80=1
Answer (a): Since D>0, the equation has two real roots.
5x^2-9x+4=5x^2-5x-4x+4
=5x(x-1)-4(x-1)=(5x-4)(x-1)
Answer (b): x=\frac{4}{5} or x=1.
Question 83: Proportion
\frac{1400}{28000}=\frac{1}{20},\qquad \frac{5600}{112000}=\frac{1}{20}
Answer (a): The data show direct proportion between customers and profit. It is not a continued proportion of three quantities.
\frac{x}{32140}=\frac{1}{20}\Rightarrow x=1607
\frac{3212}{y}=\frac{1}{20}\Rightarrow y=3212\times 20=64240
Final answer: x=1607, y=₹64240.
Question 84: Factor theorem
Step 1: 9x^2-4=(3x+2)(3x-2). Let f(x)=9x^3-mx^2-nx+8.
f\left(-\frac{2}{3}\right)=0\Rightarrow -\frac{8}{3}-\frac{4m}{9}+\frac{2n}{3}+8=0 \Rightarrow -2m+3n=-24
f\left(\frac{2}{3}\right)=0\Rightarrow \frac{8}{3}-\frac{4m}{9}-\frac{2n}{3}+8=0 \Rightarrow 2m+3n=48
6n=24\Rightarrow n=4,\qquad 2m+12=48\Rightarrow m=18
Answer (a): m=18, n=4.
9x^3-18x^2-4x+8=(9x^2-4)(x-2)
=(3x+2)(3x-2)(x-2)
Answer (b): Complete factorisation is (3x+2)(3x-2)(x-2).
Question 85: Probability
Step 1: Total students =100. The relevant frequencies are added from the grouped table.
P(\text{less than }20)=\frac{4+5}{100}=\frac{9}{100}
P(30\le \text{marks}<60)=\frac{7+13+12}{100}=\frac{32}{100}=\frac{8}{25}
P(\text{marks}\ge 70)=\frac{11+14+10}{100}=\frac{35}{100}=\frac{7}{20}
P(\text{above }89)=\frac{10}{100}=\frac{1}{10}
Final answers: (a) \frac{9}{100}, (b) \frac{8}{25}, (c) \frac{7}{20}, (d) \frac{1}{10}.
Question 86: Matrix product
Step 1: A is 2\times 2 and B is 2\times 1, so AB is 2\times 1.
Answer (a): Order of the null matrix is 2\times 1.
AB=\begin{bmatrix}x&1\\y&2\end{bmatrix}\begin{bmatrix}x\\x-2\end{bmatrix}=\begin{bmatrix}x^2+x-2\\xy+2x-4\end{bmatrix}
x^2+x-2=0\Rightarrow (x+2)(x-1)=0\Rightarrow x=1\text{ or }x=-2
x=1\Rightarrow y+2-4=0\Rightarrow y=2
x=-2\Rightarrow -2y-4-4=0\Rightarrow y=-4
Final answer (b): (x,y)=(1,2) or (x,y)=(-2,-4).
Question 87: Arithmetic progression
Step 1: a=20, d=-3, S_n=65.
65=\frac{n}{2}[2(20)+(n-1)(-3)]
65=\frac{n}{2}(43-3n)\Rightarrow 3n^2-43n+130=0
3n^2-30n-13n+130=0\Rightarrow (3n-13)(n-10)=0
n=\frac{13}{3}\text{ or }10
Answer (a): n=10, since the number of terms must be a whole number.
l=a+(n-1)d=20+9(-3)=-7
Answer (b): Last term =-7.
Question 88: Reflection
Step 1: P(2,-3)\mapsto P'(2,3); only the y-coordinate changes sign.
Answer (a): L_1 is the x-axis.
Step 2: The line perpendicular to the x-axis through the origin is the y-axis.
P'(2,3)\xrightarrow{\text{reflection in }y\text{-axis}}P''(-2,3)
Answer (b): P''=(-2,3).
Answer (c): L_1 and L_2 meet at (0,0).
P(2,-3)\xrightarrow{\text{reflection in origin}}P'''(-2,3)
Answer (d): P'' and P''' coincide.
Question 89: Coordinate-geometry intercept method
Assumption (edition may vary): Treat the figure as the standard case where AD meets the y-axis at C and the x-axis at D, AC:AD=1:4, and A=(p,q).
Step 1: Since AC:AD=1:4, AC:CD=1:3.
Step 2: Let D=(d,0). By section formula,
C=\left(\frac{3p+d}{4},\frac{3q}{4}\right)
Step 3: Since C lies on the y-axis, its x-coordinate is 0.
\frac{3p+d}{4}=0\Rightarrow d=-3p
Final answer: D=(-3p,0), C=\left(0,\frac{3q}{4}\right).
Quick answer index
| Question | Answer |
|---|---|
| 78 | x=0\%, y=2, total =₹1639.20 |
| 79 | 18 instalments, interest =₹1026, deposit =₹10800 |
| 80 | Sale proceeds =₹100000, original income =₹5000, increase =₹1000 |
| 81 | \frac{1}{2}\le x<\frac{9}{2} |
| 82 | Roots =\frac{4}{5},1 |
| 83 | x=1607, y=₹64240 |
| 84 | m=18, n=4, factors (3x+2)(3x-2)(x-2) |
| 85 | \frac{9}{100},\frac{8}{25},\frac{7}{20},\frac{1}{10} |
| 86 | Order 2\times 1; (x,y)=(1,2) or (-2,-4) |
| 87 | n=10, last term =-7 |
| 88 | L_1 is x-axis; P''=P'''=(-2,3) |
| 89 | D=(-3p,0), C=\left(0,\frac{3q}{4}\right) |
Examiner’s mindset
In ICSE-style Maths work, marks are commonly lost when students skip the formula, use the wrong value for substitution, or write only the final answer. Show the formula in GST, recurring deposit and A.P. questions; show the zero of each factor in factor theorem; and state the favourable cases in probability.
Common mistakes
- Dividend error: Use nominal value for dividend, not market value.
- Inequation sign error: Multiplying by -1 reverses the inequality sign.
- GST split error: If total GST is 18\%, then CGST is 9\%, not 18\%.
- A.P. root error: Reject n=\frac{13}{3} because the number of terms must be a whole number.
- Probability range error: For whole-number marks, above 89 belongs to the 90–100 group.
Related links
- ICSE Class 10 Maths resources
- Selina Maths Class 10 Solutions
- ML Aggarwal Class 10 Maths Solutions
- ICSE Class 10 Maths previous year papers
Sources used
Frequently Asked Questions
How should I write ICSE Class 10 Maths competency answers?
Write the formula, substitute the values, simplify line by line and end with a labelled final answer. This is the safest presentation for ICSE Class 10 Maths numerical questions.
Why is CGST taken as half of GST in Question 78?
When GST is split equally, CGST is half of the total GST rate. For 18\% GST, CGST is 9\%, so the equation is \frac{9}{100}\times 220y=39.60.
What is the recurring deposit formula used here?
For monthly deposit P, time n months and rate r\%, I=\frac{P n(n+1)r}{2\times 12\times 100} and maturity value =Pn+I.
How do I know if a quadratic equation has real roots?
Compute D=b^2-4ac. If D>0, the roots are real and distinct; if D=0, the roots are real and equal; if D<0, real roots do not exist.
What is the main mistake in shares and dividend questions?
Dividend is calculated on nominal value, not market value. Market value is used to find investment or sale proceeds.