CBSE PYQs & Competency Resources

Prove that $\sqrt{5}$ is an irrational number.

Find HCF and LCM of 72 and 120 using prime factorisation.

Find a quadratic polynomial whose zeroes are $3+\sqrt{2}$ and $3-\sqrt{2}$.

If $\alpha$ and $\beta$ are zeroes of $p(x) = x^2 - 5x + 6$, find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

Solve for $x$ and $y$: $2x + 3y = 11$ and $2x - 4y = -24$. Hence find $m$ for which $y = mx + 3$.

Find $k$ for which $(k-12)x^2 + 2(k-12)x + 2 = 0$ has real and equal roots.

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

If $a_n = 7 - 4n$, find $d$ and $S_{15}$.

The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms.

State and prove Basic Proportionality Theorem (Thales Theorem).

Find the ratio in which the $y$-axis divides the line segment joining $A(5, -6)$ and $B(-1, -4)$.

Prove that: $\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta} = \tan \theta$.

If $\tan(A+B) = \sqrt{3}$ and $\tan(A-B) = \frac{1}{\sqrt{3}}$ ($0^\circ < A+B \le 90^\circ, A > B$), find $A$ and $B$.

A 1.5 m boy observes a 30 m tower. Angle increases from $30^\circ$ to $60^\circ$. Find distance walked.

Prove that the lengths of tangents drawn from an external point to a circle are equal.

A quadrilateral $ABCD$ is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$.

Find the area of the shaded region in a square of side 14 cm with four identical semicircles inside.

A hemisphere surmounted by a cone ($r=2$ cm, $h=2$ cm). Find total volume. ($\pi = 3.14$)

Find the mode of the following data: 0-20 (6), 20-40 (8), 40-60 (10), 60-80 (12), 80-100 (6), 100-120 (5).

A box contains 90 discs numbered 1 to 90. Find probability of drawing: (i) a two-digit number, (ii) a perfect square.

Why is respiration exothermic? Write its balanced equation.

What is a redox reaction? Identify substance oxidized and reduced in: $\text{MnO}_2 + 4\text{HCl} \rightarrow \text{MnCl}_2 + \text{Cl}_2 + 2\text{H}_2\text{O}$.

What is chlor-alkali process? Write balanced chemical equation and uses of products.

Differentiate between roasting and calcination with balanced equations.

What is saponification? How does soap cleanse grease? Explain micelle formation with diagram.

Draw and explain the human heart. Trace the double circulation pathway.

Describe the role of bile juice, trypsin, and lipase in the digestion of food.

What is a reflex arc? Trace nerve pathway when touching a hot pan.

Draw a longitudinal section of a flower and label: Stigma, Style, Anther, Filament, Ovary.

How is the sex of a child determined genetically in human beings?

Object of 4 cm placed 25 cm from concave mirror ($f=15$ cm). Find image position, size, nature.

Why does the clear sky appear blue and the sun appear reddish at sunrise?

An electric heater ($R=20\,\Omega$) draws $5\,\text{A}$ for 30 seconds. Calculate heat produced.

Derive equivalent resistance for three resistors $R_1, R_2, R_3$ in parallel.

State Fleming's Left-Hand Rule. Explain principle of an electric motor.