Bihar Board Class 10 Mathematics Question Paper 2026 PDF with Solutions

The nth term of an A.P. is given by \( a_n = a + (n-1)d \).
For the 5th term:
\[ 11 = a + 4 \times 2 = a + 8 \Rightarrow a = 3 \]

Step 1: Analyze the options.

- (A) 1 — Incorrect value.

- (B) 2 — Does not satisfy the equation.

- (C) 3 — Correct value.

- (D) 4 — Too large.


Step 2: Conclusion.

Thus the first term is 3.



Final Answer: \[ \boxed{(C) \, 3} \] Quick Tip: General term of A.P.: \( a_n = a + (n-1)d \)

Given \( S_n = n^2 + 2n + 1 = (n+1)^2 \)
\[ t_n = S_n - S_{n-1} \]
\[ t_6 = 7^2 - 6^2 = 49 - 36 = 13 \]

Step 1: Analyze the options.

- (A) 29 — Incorrect.

- (B) 19 — Incorrect.

- (C) 15 — Incorrect.

- (D) None of these — Correct.


Step 2: Conclusion.

Actual 6th term is 13.



Final Answer: \[ \boxed{(D) \, None of these} \] Quick Tip: Term from sum: \( t_n = S_n - S_{n-1} \)

An A.P. has a constant difference between consecutive terms.

For option (C):
\[ 2x-x=x,\quad 3x-2x=x \]

Step 1: Analyze the options.

- (A) Differences not constant.

- (B) Differences increase.

- (C) Constant difference \(x\).

- (D) Differences not constant.


Step 2: Conclusion.

Thus option (C) is an A.P.



Final Answer: \[ \boxed{(C)} \] Quick Tip: Constant difference ⇒ Arithmetic Progression

Squares: 4, 16, 36, 64 …

Differences: 12, 20, 28 (not constant)

Step 1: Analyze the options.

- (A), (B), (C) — Constant differences.

- (D) — Not constant.


Step 2: Conclusion.

Hence (D) is not an A.P.



Final Answer: \[ \boxed{(D)} \] Quick Tip: Squares do not form an A.P.

\[ a=1,\ d=3,\ n=20 \]
\[ S_n = \frac{n}{2}[2a+(n-1)d] = 10[2+57] = 590 \]

Step 1: Analyze the options.

- Only option (C) matches the calculated sum.


Step 2: Conclusion.

Hence sum is 590.



Final Answer: \[ \boxed{(C) \, 590} \] Quick Tip: Use sum formula for A.P.

\[ \sin45^\circ=\cos45^\circ=\frac{1}{\sqrt2} \]
\[ \frac{1}{\sqrt2}+\frac{1}{\sqrt2}=1 \]

Step 1: Analyze the options.

- (A), (B), (C) do not equal 1.

- (D) equals 1.


Step 2: Conclusion.

Thus option (D) is correct.



Final Answer: \[ \boxed{(D)} \] Quick Tip: Memorize special angle values.

Using trigonometric identities:
\[ \tan^2 A = \frac{\sin^2 A}{\cos^2 A} \]
\[ \cos^2 A + \tan^2 A = \cos^2 A + \frac{\sin^2 A}{\cos^2 A} = \frac{\cos^4 A + \sin^2 A}{\cos^2 A} = \frac{1}{\sin^2 A} = \csc^2 A \]

Step 1: Analyze the options.

- (A) \( \sin^2 A \) — Incorrect

- (B) \( \csc^2 A \) — Correct

- (C) 1 — Incorrect

- (D) \( \tan^2 A \) — Incorrect


Step 2: Conclusion.

Therefore, the expression equals \( \csc^2 A \).



Final Answer: \[ \boxed{(B) \, \csc^2 A} \] Quick Tip: Convert all functions into sine and cosine to simplify identities.

From standard trigonometric values:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

Step 1: Analyze the options.

- (A) \( \sqrt{3} \) — Incorrect

- (B) \( \frac{\sqrt{3}}{2} \) — Incorrect

- (C) \( \frac{1}{\sqrt{3}} \) — Correct

- (D) 1 — Incorrect


Step 2: Conclusion.

Hence, the correct value is \( \frac{1}{\sqrt{3}} \).



Final Answer: \[ \boxed{(C) \, \frac{1}{\sqrt{3}}} \] Quick Tip: Memorize standard values for angles \(30^\circ, 45^\circ, 60^\circ\).

From standard trigonometric ratios:
\[ \cos 60^\circ = \frac{1}{2} \]

Step 1: Analyze the options.

- (A) \( \frac{1}{2} \) — Correct

- (B) \( \frac{\sqrt{3}}{2} \) — Incorrect

- (C) \( \frac{1}{\sqrt{2}} \) — Incorrect

- (D) 1 — Incorrect


Step 2: Conclusion.

Thus, the correct value is \( \frac{1}{2} \).



Final Answer: \[ \boxed{(A) \, \frac{1}{2}} \] Quick Tip: Cosine values decrease as angle increases from \(0^\circ\) to \(90^\circ\).

\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \Rightarrow \sin^2 60^\circ = \frac{3}{4} \]
\[ \tan 45^\circ = 1 \Rightarrow \tan^2 45^\circ = 1 \]
\[ \frac{3}{4} - 1 = 0 \]

Step 1: Analyze the options.

- (A) 1 — Incorrect

- (B) \( \frac{1}{2} \) — Incorrect

- (C) 2 — Incorrect

- (D) 0 — Correct


Step 2: Conclusion.

Hence, the result is zero.



Final Answer: \[ \boxed{(D) \, 0} \] Quick Tip: Square trigonometric values carefully before subtracting.

\[ 8\sin60^\circ = 4\sqrt{3}, \quad 8\cos60^\circ = 4 \]

Distance:
\[ d = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8 \]

Step 1: Analyze the options.

- (A) 8 — Correct

- (B) 25 — Incorrect

- (C) 64 — Incorrect

- (D) 1 — Incorrect


Step 2: Conclusion.

Therefore, the distance is 8.



Final Answer: \[ \boxed{(A) \, 8} \] Quick Tip: Use distance formula \( \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \).