Can You Solve These 20 Tricky Math Problems for Classes IX and X CBSE ICSE

📚 20 Mathematical Challenges – CBSE, ICSE

Welcome! This slider shows two problems side by side on each page. Use Previous / Next buttons (or left/right arrow keys) to navigate through the 10 slides. Each problem has its own “Show hint” button – click to reveal a subtle nudge.

✨ Covers rational proofs, surd simplifications, Diophantine equations, and more. Suitable for grades 9–12 and competitive exams. Featuring 20 hand-picked problems covering everything from rational, irrational numbers to algebra and number theory. Whether you are prepping for a competitive exam or just love a good brain-teaser, this is for you. Suitable for students of IX and X of CBSE and ICSE. If these problems feel a bit intimidating, don't worry—they are designed to be! While standard high school mathematics focuses heavily on rote memorization and applying standard formulas to straightforward questions, this problem set bridges the gap between the regular classroom and competitive mathematics (like the AMC, math Olympiads, or advanced entrance exams). Instead of just asking you to "solve for x," these questions require creative algebraic manipulation, pattern recognition, and proof-based logical reasoning. They test whether you can combine multiple mathematical concepts to find an elegant solution, rather than just grinding through a standard algorithm. Key Mathematical Areas Covered This set of 20 problems targets several advanced domains: Advanced Algebra & Identities: Moving beyond basic factoring to use conditional identities (like the sum of cubes) and symmetric functions. Number Theory: Exploring the properties of prime numbers, divisibility rules, and solving basic Diophantine equations (finding integer solutions to polynomial equations). Complex Radicals & Surds: Tackling nested square roots, cube roots of binomial surds, and rationalizing multi-term denominators. Telescoping Series: Recognizing patterns in sequences that cancel each other out to reveal a clean, simple answer. Introductory Complex Numbers & Logarithms: Understanding the cyclical nature of imaginary numbers and manipulating logarithmic bases.

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✨ 20 problems, 10 slides, 2 per slide. Click "Show hint" for each problem individually. Left/right arrow keys navigate. Good luck! © Vinod Singh

Integral Calculus for JEE Main & Advanced: Practice Problems

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25 Multiple Choice Questions (MCQs) on Integral Calculus for students of class XI and XII preparing for board examinations or JEE Mains, IIT Advanced WBJEE or any other competitive entrance examination.

👨‍🏫 Author: Vinod Singh

📞 WA: +91-9038126497

Prime Maths

Advanced Integral Calculus - Definite and Indefinite Integration

Test your understanding of core concepts.Master Integral Calculus for Indian Statistical Institute (B. Math & B.Stat), JEE Main & Advanced. Practice hand-picked problems with step-by-step solutions, advanced shortcuts, and integration techniques.

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Question 1

Let \(f(x)=\frac{x}{(1+x^n)^{1/n}}\) for \(n \geq 2\) and \(g(x)=f \circ f \circ \dots \circ f\) (\(n\) times), then \(\int x^{n-2} g(x) dx \) is equal to

A\( \frac{1}{n(n+1)} (1+nx^n)^{1-\frac{1}{n}}+c\) B\( \frac{1}{n(n-1)} (1+nx^n)^{1-\frac{1}{n}}+c\) C\( \frac{1}{n(n-1)} (1+nx^n)^{1+\frac{1}{n}}+c \) D\( \frac{1}{n(n-1)} (1-nx^n)^{1-\frac{1}{n}}+c\)

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Question 2

\(\int_{\frac{1}{2026}}^{2026} \frac{tan^{-1} x}{x} dx \quad =\)

A \(\frac{\pi}{4} \ln 2026\) B \(\sqrt{2}\frac{\pi}{8} \ln 2026\) C \(\frac{\pi}{2} \ln 2026\) D \(\frac{\pi}{3} \ln 2026\)

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Question 3

\(\int \frac{dx}{\sqrt[3]{x}+\sqrt[5]{x}} \quad =\)

A \( = 15 \left( \frac{x^{2/3}}{10} - \frac{x^{8/15}}{8} + \frac{x^{2/5}}{6} - \frac{x^{4/15}}{4} + \frac{x^{2/15}}{2} - \frac{1}{2} \ln |x^{2/15} + 1| \right) + c \) B \( = 15 \left( \frac{x^{2/3}}{10} + \frac{x^{8/15}}{8} + \frac{x^{2/5}}{6} + \frac{x^{4/15}}{4} + \frac{x^{2/15}}{2} - \frac{1}{2} \ln |x^{2/15} + 1| \right) + c \) C \( = 15 \left( -\frac{x^{2/3}}{10} - \frac{x^{8/15}}{8} - \frac{x^{2/5}}{6} - \frac{x^{4/15}}{4} - \frac{x^{2/15}}{2} - \frac{1}{2} \ln |x^{2/15} + 1| \right) + c \) D \( = 17 \left( \frac{x^{2/3}}{10} - \frac{x^{8/15}}{8} + \frac{x^{2/5}}{6} - \frac{x^{4/15}}{4} + \frac{x^{2/15}}{2} - \frac{1}{2} \ln |x^{2/15} + 1| \right) + c \)

Put \(x=z^{15}\)

Question 4

\(\int (x^6+x^3) \sqrt[3]{x^3+2} \quad dx \quad =\)

A\(\frac{1}{2}(x^6+2x^3)^{\frac{3}{4}} +c\) B\(\frac{1}{8}(x^6+2x^3)^{\frac{5}{3}} +c\) C\(\frac{1}{8}(x^6+2x^3)^{\frac{4}{3}} +c\) D\(\frac{1}{8}(x^6+x^3)^{\frac{4}{3}} +c\)

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Question 5

\( \int_{\pi/2}^{5\pi/2} \frac{e^{tan^{-1}(\sin x)}}{e^{tan^{-1}(\sin x)}+e^{tan^{-1}(\cos x)}} dx \quad = \)

A\( 2\pi \) B\( \frac{\pi}{8}\) C\( \frac{\pi}{2} \) D\( \pi \)

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Question 6

\(\int \frac{dx}{\tan x + \sec x + \cot x + \csc x} dx \quad = \)

A\(\frac{1}{2}(\cos x - \sin x -x)+c\) B\(\frac{1}{2}(-\cos x + \sin x -x)+c\) C\(\frac{1}{2}(-\cos x + \sin x + x)+c\) D\(\frac{1}{2}(-\cos x - \sin x -x)+c\)

See that \(\frac{1}{\tan x + \sec x + \cot x + \csc x} = \frac{\sin x \cos x}{1+\sin x +\cos x}=\frac{1}{2}\frac{(\sin x +\cos x)^2-1^2}{1+\sin x +\cos x}\)

Question 7

If \( f \) is an even function and \(I= \int_{0}^{\pi/2}f(\cos 2x) \cos x \quad dx \), then

A \( I= \sqrt{2}\int_{0}^{\pi/4}f(\cos 2x) \cos x \quad dx \) B \( I= \sqrt{2}\int_{0}^{\pi/4}f(\cos 2x) \sin x \quad dx \) C \( I= \sqrt{2}\int_{0}^{\pi/4}f(\sin 2x) \cos x \quad dx \) D \( I= \sqrt{2}\int_{0}^{\pi/4}f(\sin 2x) \sin x \quad dx \)

Use King's Rule

Question 8

Let \( f \) be a polynomial function such that \( f(x^2+1) = x^4+5x^2+2\), for all \( x \in \mathbb{R}.\) Then \(\int_{0}^{3} f(x) dx \) is equal to

A \( \frac{41}{3} \) B \( \frac{33}{2} \) C \( \frac{27}{2} \) D \( \frac{5}{3} \)

See that \(f(x^2+1)=(x^2+1)^2+3(x^2+1)-2\)

Question 9

Evaluate \(\int_{-\pi/3}^{\pi/3} \frac{\pi+4x^3}{2-cos\big(|x|+\frac{\pi}{3}\big)} dx\)

A\(\frac{4\pi}{\sqrt{3}}tan^{-1}(\frac{1}{3})\) B\(\frac{4\pi}{\sqrt{2}}tan^{-1}(\frac{1}{2})\) C\(\frac{4\pi}{\sqrt{3}}tan^{-1}(\frac{1}{2})\) D\(\frac{4\pi}{\sqrt{2}}tan^{-1}(\frac{1}{3})\)

Express \(\frac{\pi+4x^3}{2-cos\big(|x|+\frac{\pi}{3}\big)}\) as a sum of an odd and an even function, then split

Question 10

Lrt \(f: (0,\infty) \rightarrow \mathbb{R}\) and \(F(x)=\int_{0}^{x} f(t) dt.\) If \(F(x^2)=x^2(1+x),\) then \(f(4)\) equals

A\(\frac{5}{4}\) B\(7\) C\(4\) D\(2\)

See that \(F'(x)=f(x)\) by fundamental theorem of integral calculus and \(f(x^2)=F'(x^2)= \frac{3x+2}{2}\)

Question 11

Evaluate \( \int sin^{-1} \big(\frac{2x+2}{\sqrt{4x^2+8x+13}} \big) dx.\)

A\((x+1)tan^{-1}\big(\frac{2x+2}{3}\big)+\frac{3}{4}ln|4x^2+8x+13|+c\) B\((x+1)tan^{-1}\big(\frac{2x+2}{3}\big)-\frac{1}{4}ln|4x^2+8x+13|+c\) C\((x+1)tan^{-1}\big(\frac{2x+2}{3}\big)-\frac{3}{2}ln|4x^2+8x+13|+c\) D\((x+1)tan^{-1}\big(\frac{2x+2}{3}\big)-\frac{3}{4}ln|4x^2+8x+13|+c\)

\(4x^2+8x+13=(2x+2)^2+3^2\) now put \( 2x+2 = t\) and then subsutitute \( t=3 \tan \theta \)

Question 12

Let \(T > 0\) be a fixed real number. Suppose \(f\) is a continuous function such that for all \(x \in \mathbb{R}, f(x+T)=f(x).\) If \(I=\int_{0}^{T} f(x) \\dx\) then the value of \(\int_{3}^{3+3T} f(2x) \\dx\) is

A\(3I\) B\(2I \) C\(\frac{3}{2}I\) D\(6I\)

\(\int_{3}^{3+3T} f(2x) \\dx\) = \(\frac{1}{2}\int_{6}^{6+6T} f(t) \\dt\) on substituting \(2x=t\) now split the integral form 6 to T, T to 2T, ...., 5T to 6T and 6T to 6+6T and think!

Question 13

The integral \(\int_{-1/a}^{1/a} \big( [x]+\ln (\frac{1+x}{1-x})\big) dx\) where \(a>1\) equals

A\(0\) B\(\frac{-1}{a}\) C\(1\) D\(a \ln \frac{1}{a}\)

Find out the odd function!

Question 14

For any natural number \(m\), evaluate \(\int (x^{3m}+x^{2m}+x^m)(2x^{2m}+3x^m+6)^{1/m} dx, x>0\)

A \( \frac{1}{6(m+1)} (2x^{3m}+3x^{2m}+6x^m)^{\frac{m-1}{m}}+c\) B \( \frac{1}{6(m-1)} (2x^{3m}+3x^{2m}+6x^m)^{\frac{m+1}{m}}+c\) C \( \frac{1}{(m+1)} (2x^{3m}+3x^{2m}+6x^m)^{\frac{m+1}{m}}+c\) D \( \frac{1}{6(m+1)} (2x^{3m}+3x^{2m}+6x^m)^{\frac{m+1}{m}}+c\)

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Question 15

Let \(f(x)= \frac{e^x}{1+e^x} \quad I_1 = \int_{f(-a)}^{f(a)}xg(x(1-x)) dx \) and \(I_2 = \int_{f(-a)}^{f(a)} g(x(1-x)) dx, \) then the value of \(\frac{I_2}{I_1}\) is

A\(3\) B\(4\) C\(2\) D\(-1\)

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Question 16

Evaluate \(\int_{0}^{2\pi} |1+2\sin x| dx \)

A\(\frac{2\pi}{3} (4+\sqrt{3})\) B\(\frac{4\pi}{3} (2+\sqrt{3})\) C\(\frac{\pi}{3} (4+\sqrt{3})\) D\(\frac{\pi}{3} (2+\sqrt{3})\)

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Question 17

The integral \(\int \frac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx \)

A\(\frac{-x^5}{(x^5+x^3+1)^{2}}+c\) B\(\frac{x^{10}}{2(x^5+x^3+1)^{2}}+c\) C\(\frac{x^5}{2(x^5+x^3+1)^{2}}+c\) D\(\frac{-x^{10}}{2(x^5+x^3+1)^{2}}+c\)

substitute \(\x=\frac{1}{t})

Question 18

If \(f\) and \(g\) be continuous functions on \([0,a]\) such that \(f(x)=f(a-x)\) and \(g(x)+g(a-x)=4,\) then \(\int_{0}^{a} f(x)g(x) dx \) is equal to

A\(4\int_{0}^{a} f(x)dx \) B\(4\int_{0}^{a} f(x)dx \) C\(2\int_{0}^{a} f(x)dx \) D\(-3\int_{0}^{a} f(x)dx \)

Apply King's Theorem

Question 19

The value of \(\int_{-\pi/2}^{\pi/2}\frac{x^2 \cos x}{1+e^x}dx\) is equal to

A\(\frac{\pi^2}{4}-2\) B\(\frac{\pi^2}{4}+2\) C\(\pi^2-e^{-\pi/2}\) D\(\pi^2-e^{-\pi/2}\)

Use \(\int_{a}^{b} f(x) dx = \) \(\int_{a}^{b} f(a+b-x) dx\) then add the resulting integral with the given one and then apply integration by parts

Question 20

The value of \(\int_{\sqrt{\log 2}}^{\sqrt{\log 3} }\frac{x \sin x^2}{\sin x^2+\sin (\log 6 -x^2)}dx\) is equal to

A\(\frac{1}{4} \log \frac{3}{2}\) B\(\frac{1}{2} \log \frac{3}{2}\) C\(\log \frac{3}{2}\) D\(\frac{1}{6} \log \frac{3}{2}\)

put \(\log 6 - x^2 = t\) and then apply \(\int_{a}^{b} f(x) dx = \int_{a}^{0b} f(a+b-x) dx\)

Question 21

The value of \(\int_{-2}^{0} (x^3+3x^2+3x+3+(x+1) \cos (x+1)) dx\) is equal to

A\( 0 \) B\( 3 \) C\( 4 \) D\( 1 \)

Use \((x+1)^3\) to reduce the expression then substitute \(x+1=z\)

Question 22

Let \( f(x) = 7\tan^8 x + 7 \tan^6 x -3 \tan^4 x -3 \tan^2 x \), for all \(x \in \big(\frac{-\pi}{2},\frac{\pi}{2} \big)\). Then, the correct expression is

A \(\int_{0}^{\pi/4} x f(x) dx = \frac{1}{12}\) B \(\int_{0}^{\pi/4} x f(x) dx = \frac{1}{6}\) C \(\int_{0}^{\pi/4} x f(x) dx = \frac{5}{12}\) D \(\int_{0}^{\pi/4} x f(x) dx = \frac{3}{4}\)

Easy!

Question 23

The value of \(\int_{0}^{1/2} \frac{1+\sqrt{3}}{((x+1)^2(1-x)^6)^{1/4}} dx\) is equal to

A\( 2 \) B\( \frac{1}{\sqrt{3}} \) C\( 3 \) D\( 4 \)

After simplifying the integrand substitute \(x= \cos \theta\)

Question 24

The value of \(\,(5050)\frac{\int_{0}^{1} (1 - x^{50})^{100} \, dx}{\int_{0}^{1} (1 - x^{50})^{101} \, dx}\) is

A\( 5050 \) B\( 5051 \) C\( 1 \) D\( 5500 \)

\(If I_n \int_{0}^{1} (1-x^{50})^n dx\) then by use of integration by parts it is easy to see thet \((1+50n)I_n=50nI_{n-1}\). Take \((1-x^{50})^n\) as the first function

Question 25

The value of \(\int_{-2}^{2} |1-x^2| dx\) is equal to

A\(2\) B\(3\) C\(\frac{3}{4}\) D\(4\)

\(\int_{-2}^{2} |1-x^2| dx=\)\(\int_{-2}^{-1} -(1-x^2) dx\) \(+\int_{-1}^{1} (1-x^2) dx\)\(+\int_{1}^{2} -(1-x^2) dx\)

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Circles for ICSE and CBSE | 25 Conceptual Problems

Prime Maths

Chapter Test Circles for CBSE, ICSE | Class X

👨‍🏫 Author: Singh

📞 WA: +91-9038126497

Prime Maths

Geometry - Circles

This quiz provides a thorough evaluation of key concepts and advanced skills in Circle Geometry, tailored for both CBSE and ICSE curriculums. It covers all core theorems and properties, including: Angle properties (Angle at the centre, Angles in the same segment). Cyclic Quadrilateral properties. Intersecting chord Theorem. Tangent-Secant theorems (Alternate Segment Theorem, Tangent-Radius properties). Perfect for exam preparation, this practice test is designed to challenge your understanding and refine your problem-solving techniques. CIRCLES | CBSE | ICSE . 25 multiple-choice questions designed to test both theoretical understanding and practical application.

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Question 1

From a point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle with centre at \(O\) and radius \(r\). If \(OP=2r\), then \(\triangle APB\) is:

Aan isosceles triangle Ba right angle triangle Can equilateral triangle Da scalene triangle

Question 2

In the given figure, \(AB\) is a chord of length \(16\) cm of a circle of radius \(10\) cm. The tangents at \(A\) and \(B\) interesct at point \(P\). The length of \(PA\) is:

A\(\frac{37}{7} \quad cm\) B\(\frac{37}{3} \quad cm\) C\(\frac{40}{7} \quad cm\) D\(\frac{40}{3} \quad cm\)

Question 3

The angle subtended by a tangent \(MN\) at the centre \(O\) of a circle, intercepted between two parallel tangets \(AM\) and \(BN\) of the same circle is:

A\(75^{\circ}\) B\(60^{\circ}\) C\(45^{\circ}\) D\(90^{\circ}\)

Question 4

In the figure \(AB\) is a common tangent of two circles intersecting at \(C\) and \(D\). The value of \(\angle ACB+ \angle ADB\) is:

A\(180^{\circ}\) B\(220^{\circ}\) C\(160^{\circ}\) D\(280^{\circ}\)

Question 5

In the given figure \(O\) is the centre of the circle. \(AB\) and \(AC\) are tangents drawn to the circle from point \(A\). If \(\angle BAC = 65^{\circ}\), then the measure of \(\angle BOC \) is:

A\(100^{\circ}\) B\(135^{\circ}\) C\(135^{\circ}\) D\(115^{\circ}\)

Question 6

\(AB\) is the diameter of the circle with centre \(O\). From a point \(P\) on the circle a perpendicular \(PN\) is drawn on \(AB\). Which of the following is true?

A\(PB^2 = AN.BN\) B\(PB^2 = AB.BN\) C\(PN^2 = AB.BN\) D\(PN^2 = AB.AP\)

Question 7

From an external point \(P\) of a circle with centre \(O\), two tangents \(PS\) and \(PT\) are drawn. \(QS\) is a chord of the circle parallel to \(PT\). If \( \angle SPT = 80^{\circ}\), then the value of \(\angle QST \) is

A\(55^{\circ}\) B\(45^{\circ}\) C\(50^{\circ}\) D\(60^{\circ}\)

Question 8

In the figure given below, \(PA\) is a tangent to the circle with centre \(O\) and \(PCB\) is a straight line. The measure on \(\angle OBC \) is

A\(20^{\circ}\) B\(30^{\circ}\) C\(40^{\circ}\) D\(35^{\circ}\)

Question 9

In the figure given below, a circle with centre \(O\) inscribed inside triangle \(LMN\). \(A\) and \(B\) are the points of tangency. The measure on \(\angle ANB \) is

A\(70^{\circ}\) B\(80^{\circ}\) C\(60^{\circ}\) D\(50^{\circ}\)

Question 10

In the diagram given below, \(\angle EDC = 90^{\circ}\). The tangent drawn to the circle at \(C\) makes an angle of \(50^{\circ}\) with \(AB\) produced.The measure on \(\angle ACB \) is

A\(60^{\circ}\) B\(50^{\circ}\) C\(30^{\circ}\) D\(40^{\circ}\)

Question 11

In the figure given below, \(CD\) is the diameter of the circle which meets the chord \(AB\) at \(P\) such that \(AP = BP = 12 \quad cm\). If \(DP = 8\) cm, find the radius of the circle is

A\(17 \) cm B\(11\) cm C\(7\) cm D\(13\) cm

Question 12

\(ABCD\) is a cyclic quadrilateral. If \(AD=AB\), \(\angle DAC = 60^{\circ}\), \(\angle BDC=50^{\circ}\) then the measure of \(\angle ACD\) is

A\(25^{\circ}\) B\(20^{\circ}\) C\(35^{\circ}\) D\(45^{\circ}\)

Question 13

In the figure given below, two circles touch each other at point \(A\). \(PQ\) is a direct common tangent, the point of contacts being \(P\) and \(Q\) respectively. The measure of \(\angle PAQ\) is

A\(90^{\circ}\) B\(45^{\circ}\) C\(120^{\circ}\) DCannot be determined

Question 14

In the diagram given below, \(O\) is centre of circle. The tangent \(PT\) meets the diameter \(RQ\) produced at \(P\). If \(PT = 6\) cm, \(QR = 9 \) cm, the length of \(PQ\) is ( Hint \(\triangle PQT \sim \triangle PTR\))

A\(1.5\) cm B\(2\) cm C\(3\) cm D\(3.5 \) cm

Question 15

In the given figure AC is the diameter of the circle with centre \(O\). \(CD\) is parallel to \(BE\). If \(\angle AOB = 80^{\circ}\) and \(\angle ACE = 20^{\circ}\), the sum of the angles \(\angle BEC\), \(\angle BCD\) and \(\angle CED\) is

A\(280^{\circ}\) B\(160^{\circ}\) C\(120^{\circ}\) D\(180^{\circ}\)

Question 16

In the given figure, \(O\) is the centre of the circle and \(AB\) is a tangent to the circle at \(B\).If \(\angle PQB =55^{\circ}\), sum of \(x,y\) and \(z\) is

A\(160^{\circ}\) B\(140^{\circ}\) C\(180^{\circ}\) D\(120^{\circ}\)

Question 17

In the figure given below, \(O\) is the centre of the circle and \(SP\) is a tangent.If \(\angle SRT = 65^{\circ}\), find the sum of \(x, y\) and \(z\).

A\(120^{\circ}\) B\(115^{\circ}\) C\(135^{\circ}\) D\(150^{\circ}\)

Question 18

The radii of two circles with center at \(A\) and \(B\) are \(11\) cm and \(6\) cm respectively. If \(PQ\) is the common tangent of the circles and \(AB = 13\) cm, length of PQ is

A13 cm B12 cm C17 cm D8.5 cm

Question 19

Suppose \(Q\) is a point on the circle with centre \(P\) and radius \(1\), as shown in the figure; \(R\) is a point outside thr circle such that \(QR = 1\) and \(\angle QRP = 2^{\circ}\). Let \(S\) be the point where the segment \(RP\) intersects the given circle. Then measure of \(\angle RQS\) equals.

A\(86^{\circ}\) B\(87^{\circ}\) C\(88^{\circ}\) D\(89^{\circ}\)

Question 20

The chords \(PQ\) and \(RS\) of a circle are extended to meet at the point \(Q\). If \(PQ = 6\) cm, \(OQ = 8\) cm, \(OS = 7\) cm, then length of \(RS\)

A\(12\) cm B\(9\) cm C\(10\) cm D\(16\) cm

Question 21

\(ABC\) is a triangle in which \(AB = 4\) cm, \(BC = 5\) cm and \(AC = 6\) cm. A circle is drawn to touch side \(BC\) at \(P\), side \(AB\) extended at \(Q\) and side \(AC\) extended at \(R\). Then, \(AQ\) equals:

A\(7\) cm B\(7.5\) cm C\(6.5\) cm D\(15\) cm

Question 22

A line from one vertex \(A\) of an equilateral \(\triangle ABC\) meets the opposite side \(BC\) in \(P\) and the circumcircle of \(\triangle ABC\) in \(Q\).If \(BQ = 4\) cm and \(CQ = 3\) cm, then \(PQ\) is equal to

A\(7 \) cm B\(\frac{4}{3}\) cm C\(\frac{12}{7}\) cm D\(2 \) cm

Question 23

In the figure, \(AB, AC\) and \(BC\) are three tangents touching the circle at \(D\), \(E\) and \(F\) respectively. If \(AC = 24\) cm, \(BC = 18\) cm and \( \angle ACB = 90^{\circ}\),the radius of the circle is

A\(3\) cm B\(4\) cm C\(5\) cm D\(6\) cm

Question 24

In the figure, \(ST\) is a tangents to the smaller circle, \(ABC\) is a straight line. If \(\angle TAD = 2x\) and \(\angle DPC = 3x\), find \(x\)

A\(30^{\circ}\) B\(36^{\circ}\) C\(40^{\circ}\) D\(42^{\circ}\)

Question 25

In the adjoining figure, \(ABC\) is a triangle in which, \(\angle B = 90^{\circ}\) and its incircle \(C_1\) has radius \(3\) units. A circle \(C_2\) of radius \(1\) unit touches sides \(AC\), \(BC\) and the circle \(C_1\). Then length \(AB\) is equal to

A\(3+6\sqrt{3}\) B\(10+3\sqrt{2}\) C\(10+2\sqrt{3}\) D\(9+3\sqrt{3}\)

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© 2026 Prime Maths | Created by Vinod Singh