1,050 practice questions for CBSE Class 10 Mathematics, organised by NCERT chapter (14 chapters). Every question includes a written explanation. Pick a chapter below, or try the samples first.
The ratio $\sec\theta$ is defined as the reciprocal of which trigonometric ratio?
A$\sin\theta$
B$\cos\theta$
C$\cot\theta$
D$\tan\theta$
Show answer and explanation
Correct answer: B - $\cos\theta$
By definition $\sec\theta = \frac{1}{\cos\theta}$. The reciprocal of $\sin\theta$ is $\mathrm{cosec}\,\theta$, and the reciprocal of $\tan\theta$ is $\cot\theta$. The reciprocal of $\cot\theta$ is $\tan\theta$ again. Only the cosine produces the secant.
Q2
medium
Tangents $PA$ and $PB$ drawn from an external point $P$ touch a circle with centre $O$ at $A$ and $B$. If $\angle OAB = 20^\circ$, find $\angle APB$.
A$70^\circ$
B$140^\circ$
C$20^\circ$
D$40^\circ$
Show answer and explanation
Correct answer: D - $40^\circ$
The radius $OA$ is perpendicular to the tangent $PA$, so $\angle PAB = 90^\circ - 20^\circ = 70^\circ$. Triangle $PAB$ is isosceles with $PA = PB$, so $\angle PBA = 70^\circ$ too and $\angle APB = 180^\circ - 140^\circ = 40^\circ$. The value $70^\circ$ stops at a base angle, $20^\circ$ copies the given angle, and $140^\circ$ is the sum of the two base angles.
Q3
medium
A circular pizza of radius $14\ \text{cm}$ is cut through the centre into $8$ equal slices, and three slices are eaten. Taking $\pi = \frac{22}{7}$, what is the area of the pizza left?
A$77\ \text{cm}^2$
B$231\ \text{cm}^2$
C$385\ \text{cm}^2$
D$462\ \text{cm}^2$
Show answer and explanation
Correct answer: C - $385\ \text{cm}^2$
The whole pizza has area $\frac{22}{7}\times 14\times 14 = 616\ \text{cm}^2$, so each of the $8$ equal slices is $77\ \text{cm}^2$. Five slices remain, giving $5\times 77 = 385\ \text{cm}^2$. $231\ \text{cm}^2$ is the part that was eaten, $77\ \text{cm}^2$ is one slice, and $462\ \text{cm}^2$ counts six slices instead of five.
Q4
easy
Two triangles are known to be congruent. Which statement about this pair is always correct?
AThey are similar only when they happen to be equilateral
BThey are also similar, with corresponding sides in the ratio $1:1$
CThey need not be similar, because congruence says nothing about their angles
DThey cannot be similar, because similar figures must differ in size
Show answer and explanation
Correct answer: B - They are also similar, with corresponding sides in the ratio $1:1$
Congruent triangles have equal corresponding angles and equal corresponding sides, so every side ratio is $1$ and the definition of similarity is met. Congruence is therefore the special case of similarity with scale factor $1$; it does not depend on the triangles being equilateral. Similar figures are allowed to be the same size, and congruence certainly does fix the angles.
Q5
medium
How many zeroes does the constant polynomial $p(x)=7$ have?
A$1$
BInfinitely many
C$0$
D$7$
Show answer and explanation
Correct answer: C - $0$
The value of $p(x)$ is $7$ for every $x$, so $p(x)$ is never $0$ and it has no zero at all; its graph is a horizontal line seven units above the x-axis. The answer $1$ confuses a constant with a linear polynomial. The answer $7$ quotes the value of the polynomial. Only the zero polynomial $p(x)=0$ has infinitely many zeroes.
Q6
easy
From the top of a vertical lighthouse $40\ \text{m}$ high, the angle of depression of a boat floating on the sea is $45^\circ$. Taking the sea as level with the foot of the lighthouse, how far is the boat from the foot?
A$40\ \text{m}$
B$\frac{40\sqrt{3}}{3}\ \text{m}$
C$40\sqrt{3}\ \text{m}$
D$20\ \text{m}$
Show answer and explanation
Correct answer: A - $40\ \text{m}$
The elevation of the top from the boat is also $45^\circ$, so distance $= \frac{40}{\tan 45^\circ} = 40\ \text{m}$. The value $40\sqrt{3}\ \text{m}$ belongs to a $30^\circ$ angle and $\frac{40\sqrt{3}}{3}\ \text{m}$ to a $60^\circ$ angle. The value $20\ \text{m}$ comes from halving the height, which no ratio requires.
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